{"id":783,"date":"2020-02-25T16:18:41","date_gmt":"2020-02-25T21:18:41","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/back-matter\/appendix-d-units-numbers-and-significant-figures\/"},"modified":"2020-02-25T16:18:41","modified_gmt":"2020-02-25T21:18:41","slug":"appendix-d-units-numbers-and-significant-figures","status":"publish","type":"back-matter","link":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/back-matter\/appendix-d-units-numbers-and-significant-figures\/","title":{"raw":"Appendix D Units, Numbers, and Significant Figures","rendered":"Appendix D Units, Numbers, and Significant Figures"},"content":{"raw":"<em>This content is originally from OpenStax College Chemistry 1st Canadian Edition.<\/em>\n\nThis appendix is broken into several sections:<em>\n<\/em>\n<ul>\n \t<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#intro-meas\">Introduction to Measurement<\/a><\/li>\n \t<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#exp-num\">Expressing Numbers<\/a><\/li>\n \t<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#sig-figs\">Significant Figures<\/a><\/li>\n \t<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#conv-units\">Converting Units<\/a><\/li>\n \t<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#oth-units\">Other Units: Temperature and Density<\/a><\/li>\n \t<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#exp-units\">Expressing Units<\/a><\/li>\n \t<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#add-exer\">Additional Exercises<\/a><\/li>\n<\/ul>\n<h1><a id=\"intro-meas\" href=\"\"><\/a>Introduction to Measurement<\/h1>\n<div id=\"ball-ch02_n01\" class=\"callout block\">\n<p id=\"ball-ch02_p01\" class=\"para\">Data suggest that a male child will weigh 50% of his adult weight at about 11 years of age. However, he will reach 50% of his adult height at only 2 years of age. It is obvious, then, that people eventually stop growing up but continue to grow out. Data also suggest that the average human height has been increasing over time. In industrialized countries, the average height of people increased 5.5 inches from 1810 to 1984. Most scientists attribute this simple, basic measurement of the human body to better health and nutrition.<\/p>\n\n\n[caption id=\"attachment_4607\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Stature-Percentile.png\"><img class=\"wp-image-742\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2018\/08\/Stature-Percentile-1.png\" alt=\"Stature Percentile\" width=\"400\" height=\"504\"><\/a> <strong>Figure 1.<\/strong> Stature-for-age percentiles: Boys, 2 to 20 years. Source: Chart courtesy of Centers for Disease Control and Prevention, http:\/\/www.cdc.gov\/nchs\/nhanes.htm#Set%201.[\/caption]\n\n<\/div>\n<p id=\"ball-ch02_p02\" class=\"para editable block\">In 1983, an Air Canada airplane had to make an emergency landing because it unexpectedly ran out of fuel; ground personnel had filled the fuel tanks with a certain number of pounds of fuel, not kilograms of fuel. In 1999, the Mars Climate Orbiter spacecraft was lost attempting to orbit Mars because the thrusters were programmed in terms of English units, even though the engineers built the spacecraft using metric units. In 1993, a nurse mistakenly administered 23 units of morphine to a patient rather than the \u201c2\u20133\u201d units prescribed. (The patient ultimately survived.) These incidents occurred because people weren\u2019t paying attention to quantities.<\/p>\n<p id=\"ball-ch02_p03\" class=\"para editable block\">Physics and chemistry, like all sciences, are quantitative. they deals with <em class=\"emphasis\">quantities<\/em>, things that have amounts and units. Dealing with quantities is very important in chemistry and physics, as is relating quantities to each other. In this chapter, we will discuss how we deal with numbers and units, including how they are combined and manipulated.<\/p>\n\n<h1><a id=\"exp-num\" href=\"\"><\/a>Expressing Numbers<\/h1>\n<div id=\"ball-ch02_s01\" class=\"section\" lang=\"en\">\n<p id=\"ball-ch02_s01_p01\" class=\"para editable block\">Quantities have two parts: the number and the unit. The number tells \u201chow many.\u201d It is important to be able to express numbers properly so that the quantities can be communicated properly.<\/p>\n<p id=\"ball-ch02_s01_p02\" class=\"para editable block\"><span class=\"margin_term\"><a class=\"glossterm\" href=\"\">Standard notation<\/a><\/span> is the straightforward expression of a number. Numbers such as 17, 101.5, and 0.00446 are expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as 306,000,000, or for very small numbers, such as 0.000000419, standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position.<\/p>\n<p id=\"ball-ch02_s01_p03\" class=\"para editable block\"><span class=\"margin_term\"><a class=\"glossterm\" href=\"\">Scientific notation<\/a><\/span> is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:<\/p>\n\n<div class=\"informaltable block\">\n<table style=\"border-spacing: 0px\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td>10<sup class=\"superscript\">0<\/sup><\/td>\n<td>= 1<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">1<\/sup><\/td>\n<td>= 10<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">2<\/sup><\/td>\n<td>= 100 = 10 \u00d7 10<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">3<\/sup><\/td>\n<td>= 1,000 = 10 \u00d7 10 \u00d7 10<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">4<\/sup><\/td>\n<td>= 10,000 = 10 \u00d7 10 \u00d7 10 \u00d7 10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"ball-ch02_s01_p04\" class=\"para editable block\">and so forth. The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">exponent<\/a><\/span>. (Scientific notation is sometimes called <em class=\"emphasis\">exponential notation<\/em>.) The exponent\u2019s value is equal to the number of zeros in the number expressed in standard notation.<\/p>\n<p id=\"ball-ch02_s01_p05\" class=\"para editable block\">Small numbers can also be expressed in scientific notation but with negative exponents:<\/p>\n\n<div class=\"informaltable block\">\n<table style=\"border-spacing: 0px\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td>10<sup class=\"superscript\">\u22121<\/sup><\/td>\n<td>= 0.1 = 1\/10<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">\u22122<\/sup><\/td>\n<td>= 0.01 = 1\/100<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">\u22123<\/sup><\/td>\n<td>= 0.001 = 1\/1,000<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">\u22124<\/sup><\/td>\n<td>= 0.0001 = 1\/10,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"ball-ch02_s01_p06\" class=\"para editable block\">and so forth. Again, the value of the exponent is equal to the number of zeros in the denominator of the associated fraction. A negative exponent implies a decimal number less than one.<\/p>\n<p id=\"ball-ch02_s01_p07\" class=\"para editable block\">A number is expressed in scientific notation by writing the first nonzero digit, then a decimal point, and then the rest of the digits. The part of a number in scientific notation that is multiplied by a power of 10 is called the <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">coefficient<\/a><\/span>. Then determine the power of 10 needed to make that number into the original number and multiply the written number by the proper power of 10. For example, to write 79,345 in scientific notation,<\/p>\n<span class=\"informalequation block\"><span class=\"mathphrase\">79,345 = 7.9345 \u00d7 10,000 = 7.9345 \u00d7 10<sup class=\"superscript\">4<\/sup><\/span><\/span>\n<p id=\"ball-ch02_s01_p08\" class=\"para editable block\">Thus, the number in scientific notation is 7.9345 \u00d7 10<sup class=\"superscript\">4<\/sup>. For small numbers, the same process is used, but the exponent for the power of 10 is negative:<\/p>\n<span class=\"informalequation block\"><span class=\"mathphrase\">0.000411 = 4.11 \u00d7 1\/10,000 = 4.11 \u00d7 10<sup class=\"superscript\">\u22124<\/sup><\/span><\/span>\n<p id=\"ball-ch02_s01_p09\" class=\"para editable block\">Typically, the extra zero digits at the end or the beginning of a number are not included.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 1<\/h3>\n<p id=\"ball-ch02_s01_p10\" class=\"para\">Express these numbers in scientific notation.<\/p>\n\n<ol id=\"ball-ch02_s01_l02\" class=\"orderedlist\">\n \t<li>306,000<\/li>\n \t<li>0.00884<\/li>\n \t<li>2,760,000<\/li>\n \t<li>0.000000559<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n\n<ol id=\"ball-ch02_s01_l03\" class=\"orderedlist\">\n \t<li>The number 306,000 is 3.06 times 100,000, or 3.06 times 10<sup class=\"superscript\">5<\/sup>. In scientific notation, the number is 3.06 \u00d7 10<sup class=\"superscript\">5<\/sup>.<\/li>\n \t<li>The number 0.00884 is 8.84 times 1\/1,000, which is 8.84 times 10<sup class=\"superscript\">\u22123<\/sup>. In scientific notation, the number is 8.84 \u00d7 10<sup class=\"superscript\">\u22123<\/sup>.<\/li>\n \t<li>The number 2,760,000 is 2.76 times 1,000,000, which is the same as 2.76 times 10<sup class=\"superscript\">6<\/sup>. In scientific notation, the number is written as 2.76 \u00d7 10<sup class=\"superscript\">6<\/sup>. Note that we omit the zeros at the end of the original number.<\/li>\n \t<li>The number 0.000000559 is 5.59 times 1\/10,000,000, which is 5.59 times 10<sup class=\"superscript\">\u22127<\/sup>. In scientific notation, the number is written as 5.59 \u00d7 10<sup class=\"superscript\">\u22127<\/sup>.<\/li>\n<\/ol>\n<p class=\"simpara\"><em class=\"emphasis bolditalic\">Test Yourself<\/em><\/p>\n<p id=\"ball-ch02_s01_p11\" class=\"para\">Express these numbers in scientific notation.<\/p>\n\n<ol id=\"ball-ch02_s01_l04\" class=\"orderedlist\">\n \t<li>23,070<\/li>\n \t<li>0.0009706<\/li>\n<\/ol>\n<p class=\"simpara\"><em class=\"emphasis\">Answers<\/em><\/p>\n\n<ol id=\"ball-ch02_s01_l05\" class=\"orderedlist\">\n \t<li>2.307 \u00d7 10<sup class=\"superscript\">4<\/sup><\/li>\n \t<li>9.706 \u00d7 10<sup class=\"superscript\">\u22124<\/sup><\/li>\n<\/ol>\n<\/div>\n<p id=\"ball-ch02_s01_p12\" class=\"para editable block\">Another way to determine the power of 10 in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between 1 and 10. The number of places equals the power of 10. This number is positive if you move the decimal point to the right and negative if you move the decimal point to the left.<\/p>\nMany quantities in chemistry are expressed in scientific notation. When performing calculations, you may have to enter a number in scientific notation into a calculator. Be sure you know how to correctly enter a number in scientific notation into your calculator. Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.\n<div id=\"ball-ch02_s01_f02\" class=\"figure large medium-height editable block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s01_l06\" class=\"itemizedlist\">\n \t<li>Standard notation expresses a number normally.<\/li>\n \t<li>Scientific notation expresses a number as a coefficient times a power of 10.<\/li>\n \t<li>The power of 10 is positive for numbers greater than 1 and negative for numbers between 0 and 1.<\/li>\n<\/ul>\n[caption id=\"attachment_3289\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/calc1.jpg\"><img class=\"wp-image-743\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/calc1-635x1024-1.jpg\" alt=\"This calculator shows only the coefficient and the power of 10 to represent the number in scientific notation. Thus, the number being displayed is 3.84951 \u00d7 1018, or 3,849,510,000,000,000,000. Source: \u201cCasio\u201dAsim Bijarani is licensed under Creative Commons Attribution 2.0 Generic\" width=\"400\" height=\"645\"><\/a> <strong>Figure 1.<\/strong> This calculator shows only the coefficient and the power of 10 to represent the number in scientific notation. Thus, the number being displayed is 3.84951 \u00d7 10<sup>18<\/sup>, or 3,849,510,000,000,000,000.<br>Source: \u201cCasio\u201dAsim Bijarani is licensed under Creative Commons Attribution 2.0 Generic[\/caption]\n\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol id=\"ball-ch02_s01_qs01_qd01\" class=\"qandadiv\">\n \t<li id=\"ball-ch02_s01_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p01\" class=\"para\">Express these numbers in scientific notation.<\/p>\n\n<\/div><\/li>\n<\/ol>\n(a) 56.9 (b) 563,100 (c) 0.0804 (d) 0.00000667\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p02\" class=\"para\">2. Express these numbers in scientific notation.<\/p>\n(a) \u2212890,000 (b) 602,000,000,000 (c) 0.0000004099 (d) 0.000000000000011\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p03\" class=\"para\">3. Express these numbers in scientific notation.<\/p>\n(a) 0.00656 (b) 65,600 (c) 4,567,000 (d) 0.000005507\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p04\" class=\"para\">4. Express these numbers in scientific notation.<\/p>\n(a) 65 (b) \u2212321.09 (c) 0.000077099 (d) 0.000000000218\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p05\" class=\"para\">5. Express these numbers in standard notation.<\/p>\n(a) 1.381 \u00d7 10<sup class=\"superscript\">5 <\/sup>(b) 5.22 \u00d7 10<sup class=\"superscript\">\u22127 <\/sup>(c) 9.998 \u00d7 10<sup class=\"superscript\">4<\/sup>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p06\" class=\"para\">6. Express these numbers in standard notation.<\/p>\n(a) 7.11 \u00d7 10<sup class=\"superscript\">\u22122 <\/sup>(b) 9.18 \u00d7 10<sup class=\"superscript\">2 <\/sup>(c) 3.09 \u00d7 10<sup class=\"superscript\">\u221210<\/sup>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p07\" class=\"para\">7. Express these numbers in standard notation.<\/p>\n(a) 8.09 \u00d7 10<sup class=\"superscript\">0 <\/sup>(b) 3.088 \u00d7 10<sup class=\"superscript\">\u22125 <\/sup>(c) \u22124.239 \u00d7 10<sup class=\"superscript\">2<\/sup>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p08\" class=\"para\">8. Express these numbers in standard notation.<\/p>\n(a) 2.87 \u00d7 10<sup class=\"superscript\">\u22128 <\/sup>(b) 1.78 \u00d7 10<sup class=\"superscript\">11 <\/sup>(c) 1.381 \u00d7 10<sup class=\"superscript\">\u221223<\/sup>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p09\" class=\"para\">9. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\n(a) 72.44 \u00d7 10<sup class=\"superscript\">3 <\/sup>(b) 9,943 \u00d7 10<sup class=\"superscript\">\u22125 <\/sup>(c) 588,399 \u00d7 10<sup class=\"superscript\">2<\/sup>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p10\" class=\"para\">10. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\n(a) 0.000077 \u00d7 10<sup class=\"superscript\">\u22127 <\/sup>(b) 0.000111 \u00d7 10<sup class=\"superscript\">8 <\/sup>(c) 602,000 \u00d7 10<sup class=\"superscript\">18<\/sup>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p11\" class=\"para\">11. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\n(a) 345.1 \u00d7 10<sup class=\"superscript\">2 <\/sup>(b) 0.234 \u00d7 10<sup class=\"superscript\">\u22123 <\/sup>(c) 1,800 \u00d7 10<sup class=\"superscript\">\u22122<\/sup>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p12\" class=\"para\">12. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\n(a) 8,099 \u00d7 10<sup class=\"superscript\">\u22128 <\/sup>(b) 34.5 \u00d7 10<sup class=\"superscript\">0 <\/sup>(c) 0.000332 \u00d7 10<sup class=\"superscript\">4<\/sup>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p13\" class=\"para\">13. Write these numbers in scientific notation by counting the number of places the decimal point is moved.<\/p>\n(a) 123,456.78 (b) 98,490 (c) 0.000000445\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p14\" class=\"para\">14. Write these numbers in scientific notation by counting the number of places the decimal point is moved.<\/p>\n(a) 0.000552 (b) 1,987 (c) 0.00000000887\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p15\" class=\"para\">15. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\n(a) 456 \u00d7 (7.4 \u00d7 10<sup class=\"superscript\">8<\/sup>) = ? (b) (3.02 \u00d7 10<sup class=\"superscript\">5<\/sup>) \u00f7 (9.04 \u00d7 10<sup class=\"superscript\">15<\/sup>) = ? (c) 0.0044 \u00d7 0.000833 = ?\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p16\" class=\"para\">16. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\n(a) 98,000 \u00d7 23,000 = ? (b) 98,000 \u00f7 23,000 = ? (c) (4.6 \u00d7 10<sup class=\"superscript\">\u22125<\/sup>) \u00d7 (2.09 \u00d7 10<sup class=\"superscript\">3<\/sup>) = ?\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p17\" class=\"para\">17. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\n(a) 45 \u00d7 132 \u00f7 882 = ? (b) [(6.37 \u00d7 10<sup class=\"superscript\">4<\/sup>) \u00d7 (8.44 \u00d7 10<sup class=\"superscript\">\u22124<\/sup>)] \u00f7 (3.2209 \u00d7 10<sup class=\"superscript\">15<\/sup>) = ?\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p18\" class=\"para\">18. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\n(a) (9.09 \u00d7 10<sup class=\"superscript\">8<\/sup>) \u00f7 [(6.33 \u00d7 10<sup class=\"superscript\">9<\/sup>) \u00d7 (4.066 \u00d7 10<sup class=\"superscript\">\u22127<\/sup>)] = ? (b) 9,345 \u00d7 34.866 \u00f7 0.00665 = ?\n\n<\/div>\n<\/div>\n<div class=\"layoutArea\">\n<div class=\"column\">\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<strong>Problems &amp; Exercises\n<\/strong>\n\n<strong>1.<\/strong> a) 5.69 \u00d7 10<sup>1 <\/sup>b) 5.631 \u00d7 10<sup>5 <\/sup>c) 8.04 \u00d7 10<sup>\u22122 <\/sup>d) 6.67 \u00d7 10<sup>\u22126<\/sup>\n\n<strong>3.<\/strong> a) 6.56 \u00d7 10<sup>\u22123 <\/sup>b) 6.56 \u00d7 10<sup>4 <\/sup>c) 4.567 \u00d7 10<sup>6 <\/sup>d) 5.507 \u00d7 10<sup>\u22126<\/sup>\n\n<strong>5.<\/strong> a) 138,100 b) 0.000000522 c) 99,980\n\n<strong>7.<\/strong> a) 8.09 b) 0.00003088 c) \u2212423.9\n\n<strong>9.<\/strong> a) 7.244 \u00d7 10<sup>4 <\/sup>b) 9.943 \u00d7 10<sup>\u22122 <\/sup>c) 5.88399 \u00d7 10<sup>7<\/sup>\n<strong>11.<\/strong> a) 3.451 \u00d7 10<sup>4 <\/sup>b) 2.34 \u00d7 10<sup>\u22124 <\/sup>c) 1.8 \u00d7 10<sup>1<\/sup>\n<div class=\"layoutArea\">\n<div class=\"column\">\n\n<strong>13.<\/strong> a) 1.2345678 \u00d7 10<sup>5 <\/sup>b) 9.849 \u00d7 10<sup>4 <\/sup>c) 4.45 \u00d7 10<sup>\u22127<\/sup>\n\n<strong>15.<\/strong> a) 3.3744 \u00d7 10<sup>11 <\/sup>b) 3.3407 \u00d7 10<sup>\u221211 <\/sup>c) 3.665 \u00d7 10<sup>\u22126<\/sup>\n\n<strong>17.<\/strong> a) 6.7346 \u00d7 10<sup>0 <\/sup>b) 1.6691 \u00d7 10<sup>\u221214<\/sup>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1><a id=\"sig-figs\" href=\"\"><\/a>Significant Figures<\/h1>\n<div id=\"ball-ch02_s03\" class=\"section\" lang=\"en\">\n<div id=\"ball-ch02_s03_n01\" class=\"learning_objectives editable block\">\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ol id=\"ball-ch02_s03_l01\">\n \t<li>Apply the concept of significant figures to limit a measurement to the proper number of digits.<\/li>\n \t<li>Recognize the number of significant figures in a given quantity.<\/li>\n \t<li>Limit mathematical results to the proper number of significant figures.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p id=\"ball-ch02_s03_p01\" class=\"para editable block\">If you use a calculator to evaluate the expression 337\/217, you will get the following:<\/p>\n<span class=\"informalequation block\">337217=1.55299539171...<\/span>\n<p id=\"ball-ch02_s03_p02\" class=\"para editable block\">and so on for many more digits. Although this answer is correct, it is somewhat presumptuous. You start with two values that each have three digits, and the answer has <em class=\"emphasis\">twelve<\/em> digits? That does not make much sense from a strict numerical point of view.<\/p>\n<p id=\"ball-ch02_s03_p03\" class=\"para editable block\">Consider using a ruler to measure the width of an object, as shown in <a class=\"xref\" href=\"#ball-ch02_s03_f01\">Figure 2.6 \"Expressing Width\"<\/a>. The object is definitely more than 1 cm long, so we know that the first digit in our measurement is 1. We see by counting the tick marks on the ruler that the object is at least three ticks after the 1. If each tick represents 0.1 cm, then we know the object is at least 1.3 cm wide. But our ruler does not have any more ticks between the 0.3 and the 0.4 marks, so we can\u2019t know exactly how much the next decimal place is. But with a practiced eye we can estimate it. Let us estimate it as about six-tenths of the way between the third and fourth tick marks, which estimates our hundredths place as 6, so we identify a measurement of 1.36 cm for the width of the object.<\/p>\n\n<div id=\"ball-ch02_s03_f01\" class=\"figure large medium-height editable block\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"482\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/eb7d58a9777dbb124396d7c8bcb75793-1.jpg\" alt=\"image\" width=\"482\" height=\"391\"> <strong>Figure 1.<\/strong> Expressing Width[\/caption]\n<p class=\"para\">What is the proper way to express the width of this object?<\/p>\n\n<\/div>\n<p id=\"ball-ch02_s03_p04\" class=\"para editable block\">Does it make any sense to try to report a thousandths place for the measurement? No, it doesn\u2019t; we are not exactly sure of the hundredths place (after all, it was an estimate only), so it would be fruitless to estimate a thousandths place. Our best measurement, then, stops at the hundredths place, and we report 1.36 cm as proper measurement.<\/p>\n<p id=\"ball-ch02_s03_p05\" class=\"para editable block\">This concept of reporting the proper number of digits in a measurement or a calculation is called <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">significant figures<\/a><\/span>. Significant figures (sometimes called significant digits) represent the limits of what values of a measurement or a calculation we are sure of. The convention for a measurement is that the quantity reported should be all known values and the first estimated value. The conventions for calculations are discussed as follows.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 3<\/h3>\n<p id=\"ball-ch02_s03_p06\" class=\"para\">Use each diagram to report a measurement to the proper number of significant figures.<\/p>\n\n<ol id=\"ball-ch02_s03_l02\" class=\"orderedlist\">\n \t<li>\n<div id=\"ball-ch02_s03_f02\" class=\"informalfigure small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"599\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/151d73b2e1318e4386f1be7579009b82-1.jpg\" alt=\"image\" width=\"599\" height=\"599\"> <strong>Figure 2.<\/strong> Pressure gauge in units of pounds per square inch[\/caption]\n\n<\/div><\/li>\n \t<li>\n<div id=\"ball-ch02_s03_f03\" class=\"informalfigure small\">\n\n[caption id=\"attachment_4613\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler.png\"><img class=\"wp-image-746\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-1.png\" alt=\"Ruler\" width=\"400\" height=\"255\"><\/a> <strong>Figure 3.<\/strong> A measuring ruler[\/caption]\n\n<\/div><\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n\n<ol id=\"ball-ch02_s03_l03\" class=\"orderedlist\">\n \t<li>The arrow is between 4.0 and 5.0, so the measurement is at least 4.0. The arrow is between the third and fourth small tick marks, so it\u2019s at least 0.3. We will have to estimate the last place. It looks like about one-third of the way across the space, so let us estimate the hundredths place as 3. Combining the digits, we have a measurement of 4.33 psi (psi stands for \u201cpounds per square inch\u201d and is a unit of pressure, like air in a tire). We say that the measurement is reported to three significant figures.<\/li>\n \t<li>The rectangle is at least 1.0 cm wide but certainly not 2.0 cm wide, so the first significant digit is 1. The rectangle\u2019s width is past the second tick mark but not the third; if each tick mark represents 0.1, then the rectangle is at least 0.2 in the next significant digit. We have to estimate the next place because there are no markings to guide us. It appears to be about halfway between 0.2 and 0.3, so we will estimate the next place to be a 5. Thus, the measured width of the rectangle is 1.25 cm. Again, the measurement is reported to three significant figures.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"section\" lang=\"en\"><\/div>\n<div class=\"section\" lang=\"en\">\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 1<\/h3>\n<p id=\"ball-ch02_s03_p07\" class=\"para\">What would be the reported width of this rectangle?<\/p>\n\n\n[caption id=\"attachment_4615\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Rectangle.png\"><img class=\"wp-image-747\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Rectangle-1.png\" alt=\"Rectangle\" width=\"400\" height=\"255\"><\/a> <strong>Figure 4.<\/strong> A measuring ruler[\/caption]\n\n<\/div>\n&nbsp;\n<p id=\"ball-ch02_s03_p09\" class=\"para editable block\">In many cases, you will be given a measurement. How can you tell by looking what digits are significant? For example, the reported population of the United States is 306,000,000. Does that mean that it is <em class=\"emphasis\">exactly<\/em> three hundred six million or is some estimation occurring?<\/p>\n<p id=\"ball-ch02_s03_p10\" class=\"para editable block\">The following conventions dictate which numbers in a reported measurement are significant and which are not significant:<\/p>\n\n<ol id=\"ball-ch02_s03_l04\" class=\"orderedlist editable block\">\n \t<li>Any nonzero digit is significant.<\/li>\n \t<li>Any zeros between nonzero digits (i.e., embedded zeros) are significant.<\/li>\n \t<li>Zeros at the end of a number without a decimal point (i.e., trailing zeros) are not significant; they serve only to put the significant digits in the correct positions. However, zeros at the end of any number with a decimal point are significant.<\/li>\n \t<li>Zeros at the beginning of a decimal number (i.e., leading zeros) are not significant; again, they serve only to put the significant digits in the correct positions.<\/li>\n<\/ol>\n<p id=\"ball-ch02_s03_p11\" class=\"para editable block\">So, by these rules, the population figure of the United States has only three significant figures: the 3, the 6, and the zero between them. The remaining six zeros simply put the 306 in the millions position.<\/p>\n\n<div id=\"ball-ch02_s03_f05\" class=\"figure large medium-height editable block\"><\/div>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 4<\/h3>\n<p id=\"ball-ch02_s03_p12\" class=\"para\">Give the number of significant figures in each measurement.<\/p>\n\n<ol id=\"ball-ch02_s03_l05\" class=\"orderedlist\">\n \t<li>36.7 m<\/li>\n \t<li>0.006606 s<\/li>\n \t<li>2,002 kg<\/li>\n \t<li>306,490,000 people<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n\n<ol id=\"ball-ch02_s03_l06\" class=\"orderedlist\">\n \t<li>By rule 1, all nonzero digits are significant, so this measurement has three significant figures.<\/li>\n \t<li>By rule 4, the first three zeros are not significant, but by rule 2 the zero between the sixes is; therefore, this number has four significant figures.<\/li>\n \t<li>By rule 2, the two zeros between the twos are significant, so this measurement has four significant figures.<\/li>\n \t<li>The four trailing zeros in the number are not significant, but the other five numbers are, so this number has five significant figures.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 2<\/h3>\n<p id=\"ball-ch02_s03_p13\" class=\"para\">Give the number of significant figures in each measurement.<\/p>\n\n<ol id=\"ball-ch02_s03_l07\" class=\"orderedlist\">\n \t<li>0.000601 m<\/li>\n \t<li>65.080 kg<\/li>\n<\/ol>\n<\/div>\n&nbsp;\n<div class=\"section\" lang=\"en\">\n<p id=\"ball-ch02_s03_p14\" class=\"para editable block\">How are significant figures handled in calculations? It depends on what type of calculation is being performed. If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column that all numbers have significant figures in common. For example, if you were to add 1.2 and 4.71, we note that the first number stops its significant figures in the tenths column, while the second number stops its significant figures in the hundredths column. We therefore limit our answer to the tenths column.<\/p>\n\n\n[caption id=\"attachment_4616\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Sig-Figs-1.png\"><img class=\"wp-image-748\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-1-1.png\" alt=\"Sig Figs 1\" width=\"400\" height=\"85\"><\/a> <strong>Figure 5.<\/strong> Math[\/caption]\n\n<div id=\"fwk-ball-eq02_001\" class=\"informalfigure large block\">\n<p id=\"ball-ch02_s03_p15\" class=\"para editable block\">We drop the last digit\u2014the 1\u2014because it is not significant to the final answer.<\/p>\n<p id=\"ball-ch02_s03_p16\" class=\"para editable block\">The dropping of positions in sums and differences brings up the topic of rounding. Although there are several conventions, in this text we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater and rounded down if the first dropped digit is less than 5.<\/p>\n\n\n[caption id=\"attachment_4617\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Sig-Figs-2.png\"><img class=\"wp-image-749\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-2-1.png\" alt=\"Sig Figs 2\" width=\"400\" height=\"85\"><\/a> <strong>Figure 6.<\/strong> More Math[\/caption]\n\n<div id=\"fwk-ball-eq02_002\" class=\"informalfigure large block\">\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 5<\/h3>\n<p id=\"ball-ch02_s03_p17\" class=\"para\">Express the final answer to the proper number of significant figures.<\/p>\n\n<ol id=\"ball-ch02_s03_l09\" class=\"orderedlist\">\n \t<li>101.2 + 18.702 = ?<\/li>\n \t<li>202.88 \u2212 1.013 = ?<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n\n<ol id=\"ball-ch02_s03_l10\" class=\"orderedlist\">\n \t<li>If we use a calculator to add these two numbers, we would get 119.902. However, most calculators do not understand significant figures, and we need to limit the final answer to the tenths place. Thus, we drop the 02 and report a final answer of 119.9 (rounding down).<\/li>\n \t<li>A calculator would answer 201.867. However, we have to limit our final answer to the hundredths place. Because the first number being dropped is 7, which is greater than 7, we round up and report a final answer of 201.87.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 3<\/h3>\n<p id=\"ball-ch02_s03_p18\" class=\"para\">Express the answer for 3.445 + 90.83 \u2212 72.4 to the proper number of significant figures.<\/p>\n\n<\/div>\n&nbsp;\n<p id=\"ball-ch02_s03_p20\" class=\"para editable block\">If the operations being performed are multiplication or division, the rule is as follows: limit the answer to the number of significant figures that the data value with the <em class=\"emphasis\">least<\/em> number of significant figures has. So if we are dividing 23 by 448, which have two and three significant figures each, we should limit the final reported answer to two significant figures (the lesser of two and three significant figures):<\/p>\n<span class=\"informalequation block\">23448=0.051339286...=0.051<\/span>\n<p id=\"ball-ch02_s03_p21\" class=\"para editable block\">The same rounding rules apply in multiplication and division as they do in addition and subtraction.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 6<\/h3>\n<p id=\"ball-ch02_s03_p22\" class=\"para\">Express the final answer to the proper number of significant figures.<\/p>\n\n<ol id=\"ball-ch02_s03_l11\" class=\"orderedlist\">\n \t<li>76.4 \u00d7 180.4 = ?<\/li>\n \t<li>934.9 \u00f7 0.00455 = ?<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n\n<ol id=\"ball-ch02_s03_l12\" class=\"orderedlist\">\n \t<li>The first number has three significant figures, while the second number has four significant figures. Therefore, we limit our final answer to three significant figures: 76.4 \u00d7 180.4 = 13,782.56 = 13,800.<\/li>\n \t<li>The first number has four significant figures, while the second number has three significant figures. Therefore we limit our final answer to three significant figures: 934.9 \u00f7 0.00455 = 205,472.5275\u2026 = 205,000.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 4<\/h3>\n<p id=\"ball-ch02_s03_p23\" class=\"para\">Express the final answer to the proper number of significant figures.<\/p>\n\n<ol id=\"ball-ch02_s03_l13\" class=\"orderedlist\">\n \t<li>22.4 \u00d7 8.314 = ?<\/li>\n \t<li>1.381 \u00f7 6.02 = ?<\/li>\n<\/ol>\n<\/div>\n&nbsp;\n<p id=\"ball-ch02_s03_p24\" class=\"para editable block\">As you have probably realized by now, the biggest issue in determining the number of significant figures in a value is the zero. Is the zero significant or not? One way to unambiguously determine whether a zero is significant or not is to write a number in scientific notation. Scientific notation will include zeros in the coefficient of the number <em class=\"emphasis\">only if they are significant<\/em>. Thus, the number 8.666 \u00d7 10<sup class=\"superscript\">6<\/sup> has four significant figures. However, the number 8.6660 \u00d7 10<sup class=\"superscript\">6<\/sup> has five significant figures. That last zero is significant; if it were not, it would not be written in the coefficient. So when in doubt about expressing the number of significant figures in a quantity, use scientific notation and include the number of zeros that are truly significant.<\/p>\n\n\n[caption id=\"attachment_3960\" align=\"aligncenter\" width=\"150\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/qrcode.23437479.png\"><img class=\"wp-image-74 size-thumbnail\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/qrcode.23437479-150x150-1.png\" alt=\"qrcode.23437479\" width=\"150\" height=\"150\"><\/a> <strong>Figure 7.<\/strong> Video source: Significant figures by keyj (https:\/\/viutube.viu.ca\/public\/media\/Significant+Figures\/0_0j38j93r)[\/caption]\n\n&nbsp;\n<div id=\"ball-ch02_s03_qs01\" class=\"qandaset block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s03_l15\" class=\"itemizedlist\">\n \t<li>Significant figures in a quantity indicate the number of known values plus one place that is estimated.<\/li>\n \t<li>There are rules for which numbers in a quantity are significant and which are not significant.<\/li>\n \t<li>In calculations involving addition and subtraction, limit significant figures based on the rightmost place that all values have in common.<\/li>\n \t<li>In calculations involving multiplication and division, limit significant figures to the least number of significant figures in all the data values.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol id=\"ball-ch02_s03_qs01_qd01\" class=\"qandadiv\">\n \t<li id=\"ball-ch02_s03_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p1\" class=\"para\">Express each measurement to the correct number of significant figures.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"599\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/f85b7d0b1d3f3563a5b973ef04349df3-1.jpg\" alt=\"image\" width=\"599\" height=\"599\"> <strong>Figure 8.<\/strong> Pressure gauge in units of pounds per square inch[\/caption]\n<p class=\"para\">a)<\/p>\n\n\n[caption id=\"attachment_751\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler-2.png\"><img class=\"wp-image-752\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-2-1.png\" alt=\"Ruler-2\" width=\"400\" height=\"255\"><\/a> <strong>Figure 9.<\/strong> A measuring ruler[\/caption]\n<p class=\"para\">b)<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s03_qs01_qd01_qa02\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p2\" class=\"para\">Express each measurement to the correct number of significant figures.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"599\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/06979677f6f13c11ea559e495ebcbf85-1.jpg\" alt=\"image\" width=\"599\" height=\"599\"> <strong>Figure 10.<\/strong> Pressure gauge in units of pounds per square inch[\/caption]\n<p class=\"para\">a)<\/p>\n\n\n[caption id=\"attachment_753\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler-3.png\"><img class=\"wp-image-754\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-3-1.png\" alt=\"Ruler-3\" width=\"400\" height=\"255\"><\/a> <strong>Figure 11.<\/strong> A measuring ruler[\/caption]\n<p class=\"para\">b)<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s03_qs01_qd01_qa03\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p3\" class=\"para\">How many significant figures do these numbers have?<\/p>\n\n<\/div><\/li>\n<\/ol>\n(a) 23 (b) 23.0 (c) 0.00023 (d) 0.0002302\n\n4. How many significant figures do these numbers have?\n\n(a) 5.44 \u00d7 10<sup class=\"superscript\">8 <\/sup>(b) 1.008 \u00d7 10<sup class=\"superscript\">\u22125 <\/sup>(c) 43.09 (d) 0.0000001381\n\n5. How many significant figures do these numbers have?\n\n(a) 765,890 (b) 765,890.0 (c) 1.2000 \u00d7 10<sup class=\"superscript\">5 <\/sup>(d) 0.0005060\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p6\" class=\"para\">6) How many significant figures do these numbers have?<\/p>\n(a) 0.009 (b) 0.0000009 (c) 65,444 (d) 65,040\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p7\" class=\"para\">7. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n(a) 56.0 + 3.44 = ? (b) 0.00665 + 1.004 = ? (c) 45.99 \u2212 32.8 = ? (d) 45.99 \u2212 32.8 + 75.02 = ?\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p8\" class=\"para\">8. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n(a) 1.005 + 17.88 = ? (b) 56,700 \u2212 324 = ? (c) 405,007 \u2212 123.3 = ? (d) 55.5 + 66.66 \u2212 77.777 = ?\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p9\" class=\"para\">9. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n(a) 56.7 \u00d7 66.99 = ? (b) 1.000 \u00f7 77 = ? (c) 1.000 \u00f7 77.0 = ? (d) 6.022 \u00d7 1.89 = ?\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p10\" class=\"para\">10. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n(a) 0.000440 \u00d7 17.22 = ? (b) 203,000 \u00f7 0.044 = ? (c) 67 \u00d7 85.0 \u00d7 0.0028 = ? (d) 999,999 \u00f7 3,310 = ?\n\n<\/div>\n<div class=\"question\">\n\n11. Write the number 87,449 in scientific notation with four significant figures.\n\n12. Write the number 0.000066600 in scientific notation with five significant figures.\n\n<\/div>\n<div class=\"question\">\n\n13. Write the number 306,000,000 in scientific notation to the proper number of significant figures.\n\n14. Write the number 0.0000558 in scientific notation with two significant figures.\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p13\" class=\"para\">15. Perform each calculation and limit each answer to three significant figures.<\/p>\n(a) 67,883 \u00d7 0.004321 = ? (b) (9.67 \u00d7 10<sup class=\"superscript\">3<\/sup>) \u00d7 0.0055087 = ?\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p14\" class=\"para\">16. Perform each calculation and limit each answer to four significant figures.<\/p>\n(a) 18,900 \u00d7 76.33 \u00f7 0.00336 = ? (b) 0.77604 \u00f7 76,003 \u00d7 8.888 = ?\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<strong>Check Your Understanding 1<\/strong>\n\n0.63 cm\n\n<strong>Check Your Understanding 2\n<\/strong>\n<ol id=\"ball-ch02_s03_l08\" class=\"orderedlist\">\n \t<li>three significant figures<\/li>\n \t<li>five significant figures<\/li>\n<\/ol>\n<strong>Check Your Understanding 3\n<\/strong>\n\n21.9\n\n<strong>Check Your Understanding 4<\/strong>\n<ol id=\"ball-ch02_s03_l14\" class=\"orderedlist\">\n \t<li>186<\/li>\n \t<li>0.229<\/li>\n<\/ol>\n<strong>Problems &amp; Exercises<\/strong>\n\n<strong>1.<\/strong> (a) 375 psi (b) 1.30 cm\n\n<strong>3.<\/strong> (a) two (b) three (c) two (d) four\n\n<strong>5.<\/strong> (a) five (b) seven (c) five (d) four\n\n<strong>7.<\/strong> (a) 59.4 (b) 1.011 (c) 13.2 (d) 88.2\n\n<strong>9.<\/strong> (a) 3.80 \u00d7 10<sup class=\"superscript\">3 <\/sup>(b) 0.013 (c) 0.0130 (d) 11.4\n\n<strong>11.<\/strong> (a) 8.745 \u00d7 10<sup class=\"superscript\">4 <\/sup>(b) 6.6600 \u00d7 10<sup class=\"superscript\">\u22125<\/sup>\n\n<strong>13.<\/strong> (a) 293 (b) 53.3\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1><a id=\"conv-units\" href=\"\"><\/a>Converting Units<\/h1>\n<p id=\"ball-ch02_s04_p01\" class=\"para editable block\">In <a class=\"xref\" href=\"\/douglasphys1107\/part\/appendix-c-units-numbers-and-significant-figures\/#exp-units\">\"Expressing Units\"<\/a>, we showed some examples of how to replace initial units with other units of the same type to get a numerical value that is easier to comprehend. In this section, we will formalize the process.<\/p>\n<p id=\"ball-ch02_s04_p02\" class=\"para editable block\">Consider a simple example: how many feet are there in 4 yards? Most people will almost automatically answer that there are 12 feet in 4 yards. How did you make this determination? Well, if there are 3 feet in 1 yard and there are 4 yards, then there are 4 \u00d7 3 = 12 feet in 4 yards.<\/p>\n<p id=\"ball-ch02_s04_p03\" class=\"para editable block\">This is correct, of course, but it is informal. Let us formalize it in a way that can be applied more generally. We know that 1 yard (yd) equals 3 feet (ft):<\/p>\n<span class=\"informalequation block\"><span class=\"mathphrase\">1 yd = 3 ft<\/span><\/span>\n<p id=\"ball-ch02_s04_p04\" class=\"para editable block\">In math, this expression is called an <em class=\"emphasis\">equality<\/em>. The rules of algebra say that you can change (i.e., multiply or divide or add or subtract) the equality (as long as you don\u2019t divide by zero) and the new expression will still be an equality. For example, if we divide both sides by 2, we get<\/p>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/Converting_Units_1.png\"><img class=\"wp-image-755 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Converting_Units_1.png\" alt=\"1\/2 yd = 3\/2 feet\" width=\"237\" height=\"107\"><\/a>\n<p id=\"ball-ch02_s04_p05\" class=\"para editable block\">We see that one-half of a yard equals 3\/2, or one and a half, feet\u2014something we also know to be true, so the above equation is still an equality. Going back to the original equality, suppose we divide both sides of the equation by 1 yard (number <em class=\"emphasis\">and<\/em> unit):<\/p>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_2.png\"><img class=\"wp-image-756 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_2.png\" alt=\"1\/1 yd = 3 ft\/ 1 yd\" width=\"232\" height=\"115\"><\/a>\n<p id=\"ball-ch02_s04_p06\" class=\"para editable block\">The expression is still an equality, by the rules of algebra. The left fraction equals 1. It has the same quantity in the numerator and the denominator, so it must equal 1. The quantities in the numerator and denominator cancel, both the number <em class=\"emphasis\">and<\/em> the unit:<\/p>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_3.png\"><img class=\"wp-image-757 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_3.png\" alt=\"1\/1 yd = 3 ft \/ 1 yd (cancelled units crossed out)\" width=\"215\" height=\"128\"><\/a>\n<p id=\"ball-ch02_s04_p07\" class=\"para editable block\">When everything cancels in a fraction, the fraction reduces to 1:<\/p>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_4.png\"><img class=\"wp-image-758 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_4.png\" alt=\"1 = 3 ft\/1 yd\" width=\"182\" height=\"97\"><\/a>\n<p id=\"ball-ch02_s04_p08\" class=\"para block\">We have an expression, <span class=\"inlineequation\">3 ft1 yd<\/span>, that equals 1. This is a strange way to write 1, but it makes sense: 3 ft equal 1 yd, so the quantities in the numerator and denominator are the same quantity, just expressed with different units. The expression <span class=\"inlineequation\">3 ft1 yd<\/span> is called a <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">conversion factor<\/a><\/span>, and it is used to formally change the unit of a quantity into another unit. (The process of converting units in such a formal fashion is sometimes called <em class=\"emphasis\">dimensional analysis<\/em> or the <em class=\"emphasis\">factor label method<\/em>.)<\/p>\n<p id=\"ball-ch02_s04_p09\" class=\"para editable block\">To see how this happens, let us start with the original quantity:<\/p>\n<span class=\"informalequation block\"><span class=\"mathphrase\">4 yd<\/span><\/span>\n<p id=\"ball-ch02_s04_p10\" class=\"para block\">Now let us multiply this quantity by 1. When you multiply anything by 1, you don\u2019t change the value of the quantity. Rather than multiplying by just 1, let us write 1 as <span class=\"inlineequation\">3 ft1 yd<\/span>:<\/p>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_5.png\"><img class=\"wp-image-759 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_5.png\" alt=\"4 yd x (3ft\/1yd)\" width=\"229\" height=\"113\"><\/a>\n<p id=\"ball-ch02_s04_p11\" class=\"para block\">The 4 yd term can be thought of as <span class=\"inlineequation\">4 yd\/1<\/span>; that is, it can be thought of as a fraction with 1 in the denominator. We are essentially multiplying fractions. If the same thing appears in the numerator and denominator of a fraction, they cancel. In this case, what cancels is the unit <em class=\"emphasis\">yard<\/em>:<\/p>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_6.png\"><img class=\"wp-image-760 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_6.png\" alt=\"4 yd x (3 ft\/ 1 yd) showing units cancel\" width=\"218\" height=\"115\"><\/a>\n\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_7.png\"><img class=\"wp-image-761 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_7.png\" alt=\"(4 x 3 ft)\/1 = 12 ft\/1 = 12 ft\" width=\"405\" height=\"107\"><\/a>\n<p id=\"ball-ch02_s04_p12\" class=\"para editable block\">That is all that we can cancel. Now, multiply and divide all the numbers to get the final answer:<\/p>\n<p id=\"ball-ch02_s04_p13\" class=\"para editable block\">Again, we get an answer of 12 ft, just as we did originally. But in this case, we used a more formal procedure that is applicable to a variety of problems.<\/p>\n<p id=\"ball-ch02_s04_p14\" class=\"para editable block\">How many millimeters are in 14.66 m? To answer this, we need to construct a conversion factor between millimeters and meters and apply it correctly to the original quantity. We start with the definition of a millimeter, which is<\/p>\n<span class=\"informalequation block\"><span class=\"mathphrase\">1 mm = 1\/1,000 m<\/span><\/span>\n<p id=\"ball-ch02_s04_p15\" class=\"para editable block\">The 1\/1,000 is what the prefix <em class=\"emphasis\">milli-<\/em> means. Most people are more comfortable working without fractions, so we will rewrite this equation by bringing the 1,000 into the numerator of the other side of the equation:<\/p>\n<span class=\"informalequation block\"><span class=\"mathphrase\">1,000 mm = 1 m<\/span><\/span>\n<p id=\"ball-ch02_s04_p16\" class=\"para editable block\">Now we construct a conversion factor by dividing one quantity into both sides. But now a question arises: which quantity do we divide by? It turns out that we have two choices, and the two choices will give us different conversion factors, both of which equal 1:<\/p>\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_8.png\"><img class=\"size-full wp-image-762 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_8.png\" alt=\"conversion facts equaling 1 m \/ 1000 mm\" width=\"672\" height=\"120\"><\/a><\/span>\n\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_23.png\"><img class=\"size-full wp-image-763 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_23.png\" alt=\"conversion factor 1m \/ 1000 mm\" width=\"512\" height=\"107\"><\/a>\n<p id=\"ball-ch02_s04_p17\" class=\"para editable block\">Which conversion factor do we use? The answer is based on <em class=\"emphasis\">what unit you want to get rid of in your initial quantity<\/em>. The original unit of our quantity is meters, which we want to convert to millimeters. Because the original unit is assumed to be in the numerator, to get rid of it, we want the meter unit in the <em class=\"emphasis\">denominator<\/em>; then they will cancel. Therefore, we will use the second conversion factor. Canceling units and performing the mathematics, we get<\/p>\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_10.png\"><img class=\"size-full wp-image-764 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_10.png\" alt=\"14.66 m x (1000 mm\/1 m) = 14660 mm\" width=\"498\" height=\"141\"><\/a><\/span>\n<p id=\"ball-ch02_s04_p18\" class=\"para editable block\">Note how m cancels, leaving mm, which is the unit of interest.<\/p>\n<p id=\"ball-ch02_s04_p19\" class=\"para editable block\">The ability to construct and apply proper conversion factors is a very powerful mathematical technique in chemistry. You need to master this technique if you are going to be successful in this and future courses.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 7<\/h3>\n<ol id=\"ball-ch02_s04_l02\" class=\"orderedlist\">\n \t<li>Convert 35.9 kL to liters.<\/li>\n \t<li>Convert 555 nm to meters.<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n\n<ol id=\"ball-ch02_s04_l03\" class=\"orderedlist\">\n \t<li>\n<p class=\"para\">We will use the fact that 1 kL = 1,000 L. Of the two conversion factors that can be defined, the one that will work is <span class=\"inlineequation\">1,000 L\/1 kL<\/span>. Applying this conversion factor, we get<\/p>\n<span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_11.png\"><img class=\"size-full wp-image-765 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_11.png\" alt=\"35.9 kL x (1000 L\/1 kL) = 35900 L\" width=\"417\" height=\"113\"><\/a><\/span><\/li>\n \t<li>\n<p class=\"para\">We will use the fact that 1 nm = 1\/1,000,000,000 m, which we will rewrite as 1,000,000,000 nm = 1 m, or 10<sup class=\"superscript\">9<\/sup> nm = 1 m. Of the two possible conversion factors, the appropriate one has the nm unit in the denominator: <span class=\"inlineequation\">1 m\/10<sup>9<\/sup> nm<\/span>. Applying this conversion factor, we get<\/p>\n<span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_12.png\"><img class=\"size-full wp-image-766 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_12.png\" alt=\"555 nm x (1 m\/ 10^9 nm) = 5.55 x 10^-7 m\" width=\"785\" height=\"129\"><\/a><\/span>\n<p id=\"ball-ch02_s04_p20\" class=\"para\">In the final step, we expressed the answer in scientific notation.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 1<\/h3>\n<ol id=\"ball-ch02_s04_l04\" class=\"orderedlist\">\n \t<li>Convert 67.08 \u03bcL to liters.<\/li>\n \t<li>Convert 56.8 m to kilometers.<\/li>\n<\/ol>\n<\/div>\n<p id=\"ball-ch02_s04_p21\" class=\"para editable block\">What if we have a derived unit that is the product of more than one unit, such as m<sup class=\"superscript\">2<\/sup>? Suppose we want to convert square meters to square centimeters? The key is to remember that m<sup class=\"superscript\">2<\/sup> means m \u00d7 m, which means we have <em class=\"emphasis\">two<\/em> meter units in our derived unit. That means we have to include <em class=\"emphasis\">two<\/em> conversion factors, one for each unit. For example, to convert 17.6 m<sup class=\"superscript\">2<\/sup> to square centimeters, we perform the conversion as follows:<\/p>\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_13.png\"><img class=\"size-full wp-image-767 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13.png\" alt=\"17.6 m^2 = 17.6 (mxm) x (100cm\/1m) x (100cm\/1m)=176000cm^2\" width=\"1188\" height=\"115\"><\/a><\/span>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 8<\/h3>\n<p id=\"ball-ch02_s04_p22\" class=\"para\">How many cubic centimeters are in 0.883 m<sup class=\"superscript\">3<\/sup>?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s04_p23\" class=\"para\">With an exponent of 3, we have three length units, so by extension we need to use three conversion factors between meters and centimeters. Thus, we have<\/p>\n<span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_14.png\"><img class=\"size-full wp-image-768 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14.png\" alt=\"0.883m^3 x (100cm\/1m) x (100cm\/1m) x (100cm\/1m) = 883000 cm^3\" width=\"1048\" height=\"99\"><\/a><\/span>\n<p id=\"ball-ch02_s04_p24\" class=\"para\">You should demonstrate to yourself that the three meter units do indeed cancel.<\/p>\n\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 2<\/h3>\nHow many cubic millimeters are present in 0.0923 m<sup class=\"superscript\">3<\/sup>?\n\n<\/div>\n<p id=\"ball-ch02_s04_p27\" class=\"para editable block\">Suppose the unit you want to convert is in the denominator of a derived unit; what then? Then, in the conversion factor, the unit you want to remove must be in the <em class=\"emphasis\">numerator<\/em>. This will cancel with the original unit in the denominator and introduce a new unit in the denominator. The following example illustrates this situation.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 9<\/h3>\n<p id=\"ball-ch02_s04_p28\" class=\"para\">Convert 88.4 m\/min to meters\/second.<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s04_p29\" class=\"para\">We want to change the unit in the denominator from minutes to seconds. Because there are 60 seconds in 1 minute (60 s = 1 min), we construct a conversion factor so that the unit we want to remove, minutes, is in the numerator: <span class=\"inlineequation\">1 min\/60 s<\/span>. Apply and perform the math:<\/p>\n<span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_15.png\"><img class=\"size-full wp-image-769 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_15.png\" alt=\"88.4m\/m x 1min\/60s = 1.47 m\/s\" width=\"411\" height=\"95\"><\/a><\/span>\n<p id=\"ball-ch02_s04_p30\" class=\"para\">Notice how the 88.4 automatically goes in the numerator. That\u2019s because any number can be thought of as being in the numerator of a fraction divided by 1.<\/p>\n\n<div id=\"ball-ch02_s04_f01\" class=\"figure small\">\n\n[caption id=\"attachment_3201\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/800px-Grapevinesnail_01.jpg\"><img class=\"wp-image-770\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Grapevinesnail_01-1.jpg\" alt=\"A common garden snail moves at a rate of about 0.2 m\/min, which is about 0.003 m\/s, which is 3 mm\/s! Source: \u201cGrapevine snail\u201dby J\u00fcrgen Schoneris licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.\" width=\"400\" height=\"236\"><\/a> <strong>Figure 1.<\/strong> How Fast Is Fast? A common garden snail moves at a rate of about 0.2 m\/min, which is about 0.003 m\/s, which is 3 mm\/s!<br>Source: \u201cGrapevine snail\u201dby J\u00fcrgen Schoneris licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.[\/caption]\n\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 3<\/h3>\n<p id=\"ball-ch02_s04_p31\" class=\"para\">Convert 0.203 m\/min to meters\/second.<\/p>\n\n<\/div>\n<p id=\"ball-ch02_s04_p33\" class=\"para editable block\">Sometimes there will be a need to convert from one unit with one numerical prefix to another unit with a different numerical prefix. How do we handle those conversions? Well, you could memorize the conversion factors that interrelate all numerical prefixes. Or you can go the easier route: first convert the quantity to the base unit, the unit with no numerical prefix, using the definition of the original prefix. Then convert the quantity in the base unit to the desired unit using the definition of the second prefix. You can do the conversion in two separate steps or as one long algebraic step. For example, to convert 2.77 kg to milligrams:<\/p>\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_16.png\"><img class=\"size-full wp-image-771 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_16.png\" alt=\"2.77 kg x 1000 g\/1kg = 2770 g (convert to the base unit of grams)\" width=\"918\" height=\"109\"><\/a><\/span>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_17.png\"><img class=\"wp-image-772 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17.png\" alt=\"2770 g x 1000 mg\/1g = 2770000 mg = 2.77x10^6 mg (convert to the desired unit)\" width=\"1139\" height=\"99\"><\/a>\n<p id=\"ball-ch02_s04_p34\" class=\"para editable block\">Alternatively, it can be done in a single multistep process:<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_18.png\"><img class=\"size-full wp-image-773 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_18.png\" alt=\"2.77kg x 1000g\/1kg x 1000 mg\/1g = 2770000 mg = 2.77 x 10^6 mg\" width=\"893\" height=\"109\"><\/a><\/span><\/p>\n<p id=\"ball-ch02_s04_p35\" class=\"para editable block\">You get the same answer either way.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 10<\/h3>\n<p id=\"ball-ch02_s04_p36\" class=\"para\">How many nanoseconds are in 368.09 \u03bcs?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s04_p37\" class=\"para\">You can either do this as a one-step conversion from microseconds to nanoseconds or convert to the base unit first and then to the final desired unit. We will use the second method here, showing the two steps in a single line. Using the definitions of the prefixes <em class=\"emphasis\">micro-<\/em> and <em class=\"emphasis\">nano-<\/em>,<\/p>\n<span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_21.png\"><img class=\"size-full wp-image-774 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_21.png\" alt=\"368.09 us x 1s\/10^6us x 10^9ns \/1s = 368090 ns = 3.608 x 10^5 ns\" width=\"871\" height=\"90\"><\/a><\/span>\n\n<\/div>\n<div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 4<\/h3>\n<p id=\"ball-ch02_s04_p38\" class=\"para\">How many milliliters are in 607.8 kL?<\/p>\n\n<\/div>\n&nbsp;\n<p id=\"ball-ch02_s04_p40\" class=\"para editable block\">When considering the significant figures of a final numerical answer in a conversion, there is one important case where a number does not impact the number of significant figures in a final answer\u2014the so-called <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">exact number<\/a><\/span>. An exact number is a number from a defined relationship, not a measured one. For example, the prefix <em class=\"emphasis\">kilo-<\/em> means 1,000\u2014<em class=\"emphasis\">exactly<\/em> 1,000, no more or no less. Thus, in constructing the conversion factor<\/p>\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_19.png\"><img class=\"size-full wp-image-775 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_19.png\" alt=\"1000 g\/1 kg\" width=\"127\" height=\"116\"><\/a><\/span>\n<p id=\"ball-ch02_s04_p41\" class=\"para editable block\">neither the 1,000 nor the 1 enter into our consideration of significant figures. The numbers in the numerator and denominator are defined exactly by what the prefix <em class=\"emphasis\">kilo-<\/em> means. Another way of thinking about it is that these numbers can be thought of as having an infinite number of significant figures, such as<\/p>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_untis_24.png\"><img class=\"size-full wp-image-776 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_untis_24.png\" alt=\"1000.0000000....g\/1.000000000... kg\" width=\"339\" height=\"109\"><\/a>\n<p id=\"ball-ch02_s04_p42\" class=\"para editable block\">The other numbers in the calculation will determine the number of significant figures in the final answer.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 11<\/h3>\n<p id=\"ball-ch02_s04_p43\" class=\"para\">A rectangular plot in a garden has the dimensions 36.7 cm by 128.8 cm. What is the area of the garden plot in square meters? Express your answer in the proper number of significant figures.<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s04_p44\" class=\"para\">Area is defined as the product of the two dimensions, which we then have to convert to square meters and express our final answer to the correct number of significant figures, which in this case will be three.<\/p>\n<span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_22.png\"><img class=\"size-full wp-image-777 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_22.png\" alt=\"36.7 cm x 128.8 cm x 1 m\/100cm x 1 m\/100 cm = 0.472696 m^2 = 0.473 m^2\" width=\"967\" height=\"101\"><\/a><\/span>\n<p id=\"ball-ch02_s04_p45\" class=\"para\">The 1 and 100 in the conversion factors do not affect the determination of significant figures because they are exact numbers, defined by the centi- prefix.<\/p>\n\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 5<\/h3>\n<p id=\"ball-ch02_s04_p46\" class=\"para\">What is the volume of a block in cubic meters whose dimensions are 2.1 cm \u00d7 34.0 cm \u00d7 118 cm?<\/p>\n\n<\/div>\n&nbsp;\n<div id=\"ball-ch02_s04_n07\" class=\"callout block\">\n<h3 class=\"title\">Chemistry (and physics and math...) is Everywhere: The Gimli Glider<\/h3>\n<p id=\"ball-ch02_s04_p48\" class=\"para\">On July 23, 1983, an Air Canada Boeing 767 jet had to glide to an emergency landing at Gimli Industrial Park Airport in Gimli, Manitoba, because it unexpectedly ran out of fuel during flight. There was no loss of life in the course of the emergency landing, only some minor injuries associated in part with the evacuation of the craft after landing. For the remainder of its operational life (the plane was retired in 2008), the aircraft was nicknamed \u201cthe Gimli Glider.\u201d<\/p>\n\n<div id=\"ball-ch02_s04_f02\" class=\"informalfigure large\">\n<div class=\"copyright\">\n\n[caption id=\"attachment_3203\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/800px-Aircanada.b767-300er.c-ggmx.arp_.jpg\"><img class=\"wp-image-778\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Aircanada.b767-300er.c-ggmx.arp_-1.jpg\" alt=\"The Gimli Glider is the Boeing 767 that ran out of fuel and glided to safety at Gimli Airport. The aircraft ran out of fuel because of confusion over the units used to express the amount of fuel. \u201cAircanada.b767\u2032\u2032 is in the the public domain.\" width=\"400\" height=\"293\"><\/a> <strong>Figure 2.<\/strong> The Gimli Glider is the Boeing 767 that ran out of fuel and glided to safety at Gimli Airport. The aircraft ran out of fuel because of confusion over the units used to express the amount of fuel.<br>\u201cAircanada.b767\u2032\u2032 is in the the public domain.[\/caption]\n<p class=\"para\"><\/p>\n\n<\/div>\n<\/div>\n<p id=\"ball-ch02_s04_p49\" class=\"para\">The 767 took off from Montreal on its way to Ottawa, ultimately heading for Edmonton, Canada. About halfway through the flight, all the engines on the plane began to shut down because of a lack of fuel. When the final engine cut off, all electricity (which was generated by the engines) was lost; the plane became, essentially, a powerless glider. Captain Robert Pearson was an experienced glider pilot, although he had never flown a glider the size of a 767. First Officer Maurice Quintal quickly determined that the aircraft would not be able make it to Winnipeg, the next large airport. He suggested his old Royal Air Force base at Gimli Station, one of whose runways was still being used as a community airport. Between the efforts of the pilots and the flight crew, they managed to get the airplane safely on the ground (although with buckled landing gear) and all passengers off safely.<\/p>\n<p id=\"ball-ch02_s04_p50\" class=\"para\">What happened? At the time, Canada was transitioning from the older English system to the metric system. The Boeing 767s were the first aircraft whose gauges were calibrated in the metric system of units (liters and kilograms) rather than the English system of units (gallons and pounds). Thus, when the fuel gauge read 22,300, the gauge meant kilograms, but the ground crew mistakenly fueled the plane with 22,300 <em class=\"emphasis\">pounds<\/em> of fuel. This ended up being just less than half of the fuel needed to make the trip, causing the engines to quit about halfway to Ottawa. Quick thinking and extraordinary skill saved the lives of 61 passengers and 8 crew members\u2014an incident that would not have occurred if people were watching their units.<\/p>\n\n<\/div>\n\n[caption id=\"attachment_3962\" align=\"aligncenter\" width=\"150\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/qrcode.23437561.png\"><img class=\"wp-image-779 size-thumbnail\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/qrcode.23437561-150x150-1.png\" alt=\"qrcode.23437561\" width=\"150\" height=\"150\"><\/a> <strong>Figure 3.<\/strong> Video source: Unit conversion by keyj (https:\/\/viutube.viu.ca\/public\/media\/Unit+Conversion\/0_h2w068q1)[\/caption]\n\n&nbsp;\n<div id=\"ball-ch02_s04_qs01\" class=\"qandaset block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s04_l06\" class=\"itemizedlist\">\n \t<li>Units can be converted to other units using the proper conversion factors.<\/li>\n \t<li>Conversion factors are constructed from equalities that relate two different units.<\/li>\n \t<li>Conversions can be a single step or multistep.<\/li>\n \t<li>Unit conversion is a powerful mathematical technique in chemistry that must be mastered.<\/li>\n \t<li>Exact numbers do not affect the determination of significant figures.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Exercises<\/h3>\n<ol id=\"ball-ch02_s04_qs01_qd01\" class=\"qandadiv\">\n \t<li id=\"ball-ch02_s04_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p1\" class=\"para\">Write the two conversion factors that exist between the two given units.<\/p>\n\n<\/div><\/li>\n<\/ol>\n(a) milliliters and liters (b) microseconds and seconds (c) kilometers and meters\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p2\" class=\"para\">2. Write the two conversion factors that exist between the two given units.<\/p>\n(a) kilograms and grams (b) milliseconds and seconds (c) centimeters and meters\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p3\" class=\"para\">3. Perform the following conversions.<\/p>\n(a) 5.4 km to meters (b) 0.665 m to millimeters (c) 0.665 m to kilometers\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p4\" class=\"para\">4. Perform the following conversions.<\/p>\n(a) 90.6 mL to liters (b) 0.00066 ML to liters (c) 750 L to kiloliters\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p5\" class=\"para\">5. Perform the following conversions.<\/p>\n(a) 17.8 \u03bcg to grams (b) 7.22 \u00d7 10<sup class=\"superscript\">2<\/sup> kg to grams (c) 0.00118 g to nanograms\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p6\" class=\"para\">6. Perform the following conversions.<\/p>\n(a) 833 ns to seconds (b) 5.809 s to milliseconds (c) 2.77 \u00d7 10<sup class=\"superscript\">6<\/sup> s to megaseconds\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p7\" class=\"para\">7. Perform the following conversions.<\/p>\n(a) 9.44 m<sup class=\"superscript\">2<\/sup> to square centimeters (b) 3.44 \u00d7 10<sup class=\"superscript\">8<\/sup> mm<sup class=\"superscript\">3<\/sup> to cubic meters\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p8\" class=\"para\">8. Perform the following conversions.<\/p>\n(a) 0.00444 cm<sup class=\"superscript\">3<\/sup> to cubic meters (b) 8.11 \u00d7 10<sup class=\"superscript\">2<\/sup> m<sup class=\"superscript\">2<\/sup> to square nanometers\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p9\" class=\"para\">9. Why would it be inappropriate to convert square centimeters to cubic meters?<\/p>\n<p class=\"para\">10. Why would it be inappropriate to convert from cubic meters to cubic seconds?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p13\" class=\"para\">11. Perform the following conversions.<\/p>\n(a) 45.0 m\/min to meters\/second (b) 0.000444 m\/s to micrometers\/second (c) 60.0 km\/h to kilometers\/second\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p14\" class=\"para\">12. Perform the following conversions.<\/p>\n(a) 3.4 \u00d7 10<sup class=\"superscript\">2<\/sup> cm\/s to centimeters\/minute (b) 26.6 mm\/s to millimeters\/hour (c) 13.7 kg\/L to kilograms\/milliliters\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p15\" class=\"para\">13. Perform the following conversions.<\/p>\n(a) 0.674 kL to milliliters (b) 2.81 \u00d7 10<sup class=\"superscript\">12<\/sup> mm to kilometers (c) 94.5 kg to milligrams\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p16\" class=\"para\">14. Perform the following conversions.<\/p>\n(a) 6.79 \u00d7 10<sup class=\"superscript\">\u22126<\/sup> kg to micrograms (b) 1.22 mL to kiloliters (c) 9.508 \u00d7 10<sup class=\"superscript\">\u22129<\/sup> ks to milliseconds\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p17\" class=\"para\">15. Perform the following conversions.<\/p>\n(a) 6.77 \u00d7 10<sup class=\"superscript\">14<\/sup> ms to kiloseconds (b) 34,550,000 cm to kilometers\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p18\" class=\"para\">16. Perform the following conversions.<\/p>\n(a) 4.701 \u00d7 10<sup class=\"superscript\">15<\/sup> mL to kiloliters (b) 8.022 \u00d7 10<sup class=\"superscript\">\u221211<\/sup> ks to microseconds\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p19\" class=\"para\">17. Perform the following conversions. Note that you will have to convert units in both the numerator and the denominator.<\/p>\n(a) 88 ft\/s to miles\/hour (Hint: use 5,280 ft = 1 mi.) (b) 0.00667 km\/h to meters\/second\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p20\" class=\"para\">18. Perform the following conversions. Note that you will have to convert units in both the numerator and the denominator.<\/p>\n(a) 3.88 \u00d7 10<sup class=\"superscript\">2<\/sup> mm\/s to kilometers\/hour (b) 1.004 kg\/L to grams\/milliliter\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p21\" class=\"para\">19. What is the area in square millimeters of a rectangle whose sides are 2.44 cm \u00d7 6.077 cm? Express the answer to the proper number of significant figures.<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p23\" class=\"para\">20. What is the volume in cubic centimeters of a cube with sides of 0.774 m? Express the answer to the proper number of significant figures.<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p25\" class=\"para\">21. The formula for the area of a triangle is 1\/2 \u00d7 base \u00d7 height. What is the area of a triangle in square centimeters if its base is 1.007 m and its height is 0.665 m? Express the answer to the proper number of significant figures.<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p27\" class=\"para\">22. The formula for the area of a triangle is 1\/2 \u00d7 base \u00d7 height. What is the area of a triangle in square meters if its base is 166 mm and its height is 930.0 mm? Express the answer to the proper number of significant figures.<\/p>\n\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<strong>Check Your Understanding 1<\/strong>\n<ol id=\"ball-ch02_s04_l05\" class=\"orderedlist\">\n \t<li>6.708 \u00d7 10<sup class=\"superscript\">\u22125<\/sup> L<\/li>\n \t<li>5.68 \u00d7 10<sup class=\"superscript\">\u22122<\/sup> km<\/li>\n<\/ol>\n<strong>Check Your Understanding 2<\/strong>\n\n9.23 \u00d7 10<sup class=\"superscript\">7<\/sup> mm<sup class=\"superscript\">3<\/sup>\n\n<strong>Check Your Understanding 3<\/strong>\n\n0.00338 m\/s or 3.38 \u00d7 10<sup class=\"superscript\">\u22123<\/sup> m\/s\n\n<strong>Check Your Understanding 4<\/strong>\n\n6.078 \u00d7 10<sup class=\"superscript\">8<\/sup> mL\n\n<strong>Check Your Understanding 5<\/strong>\n\n0.0084 m<sup class=\"superscript\">3<\/sup>\n\n<strong>Problems &amp; Exercises<\/strong>\n\n<span class=\"inlineequation\"><strong>1.<\/strong> (a) 1,000 mL\/1 L<\/span> and <span class=\"inlineequation\">1 L\/1,000 mL (b) 1,000,000 \u03bcs\/1 s<\/span> and <span class=\"inlineequation\">1 s\/1,000,000 \u03bcs (c) 1,000 m\/1 km<\/span> and <span class=\"inlineequation\">1 km1,000 m<\/span>\n\n<strong>3.<\/strong> (a) 5,400 m (b) 665 mm (c) 6.65 \u00d7 10<sup class=\"superscript\">\u22124<\/sup> km\n\n<strong>5.<\/strong> (a) 1.78 \u00d7 10<sup class=\"superscript\">\u22125<\/sup> g (b) 7.22 \u00d7 10<sup class=\"superscript\">5<\/sup> g (c) 1.18 \u00d7 10<sup class=\"superscript\">6<\/sup> ng\n\n<strong>7.<\/strong> (a) 94,400 cm<sup class=\"superscript\">2 <\/sup>(b) 0.344 m<sup class=\"superscript\">3<\/sup>\n\n<strong>9.<\/strong> One is a unit of area, and the other is a unit of volume.\n\n<strong>11.<\/strong> (a) 0.75 m\/s (b) 444 \u00b5m\/s (c) 1.666 \u00d7 10<sup class=\"superscript\">\u22122<\/sup> km\/s\n\n<strong>13.<\/strong> (a) 674,000 mL (b) 2.81 \u00d7 10<sup class=\"superscript\">6<\/sup> km (c) 9.45 \u00d7 10<sup class=\"superscript\">7<\/sup> mg\n\n<strong>15.<\/strong> (a) 6.77 \u00d7 10<sup class=\"superscript\">8<\/sup> ks (b) 345.5 km\n\n<strong>17.<\/strong> (a) 6.0 \u00d7 10<sup class=\"superscript\">1<\/sup> mi\/h (b) 0.00185 m\/s\n\n<strong>19.<\/strong> 1.48 \u00d7 10<sup class=\"superscript\">3<\/sup> mm<sup class=\"superscript\">2<\/sup>\n\n<strong>21.<\/strong> 3.35 \u00d7 10<sup class=\"superscript\">3<\/sup> cm<sup class=\"superscript\">2<\/sup>\n\n<\/div>\n<\/div>\n<\/div>\n<h1><a id=\"oth-units\" href=\"\"><\/a>Other Units: Temperature and Density<\/h1>\n<p id=\"ball-ch02_s05_p01\" class=\"para editable block\">There are other units in chemistry that are important, and we will cover others in the course of the entire book. One of the fundamental quantities in science is temperature. <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">Temperature<\/a><\/span> is a measure of the average amount of energy of motion, or <em class=\"emphasis\">kinetic energy<\/em>, a system contains. Temperatures are expressed using scales that use units called <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">degrees<\/a><\/span>, and there are several temperature scales in use. In the United States, the commonly used temperature scale is the <em class=\"emphasis\">Fahrenheit scale<\/em> (symbolized by \u00b0F and spoken as \u201cdegrees Fahrenheit\u201d). On this scale, the freezing point of liquid water (the temperature at which liquid water turns to solid ice) is 32 \u00b0F, and the boiling point of water (the temperature at which liquid water turns to steam) is 212 \u00b0F.<\/p>\n<p id=\"ball-ch02_s05_p02\" class=\"para editable block\">Science also uses other scales to express temperature. The Celsius scale (symbolized by \u00b0C and spoken as \u201cdegrees Celsius\u201d) is a temperature scale where 0 \u00b0C is the freezing point of water and 100 \u00b0C is the boiling point of water; the scale is divided into 100 divisions between these two landmarks and extended higher and lower. By comparing the Fahrenheit and Celsius scales, a conversion between the two scales can be determined:<\/p>\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_1.png\"><img class=\"size-full wp-image-47 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2017\/09\/other_units_1.png\" alt=\"oC = (oF-32) x 5\/9\" width=\"244\" height=\"90\"><\/a><\/span>\n\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_2.png\"><img class=\"size-full wp-image-48 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_2.png\" alt=\"oF = (oC x 9\/5) + 32\" width=\"265\" height=\"91\"><\/a><\/span>\n<p id=\"ball-ch02_s05_p03\" class=\"para editable block\">Using these formulas, we can convert from one temperature scale to another. The number 32 in the formulas is exact and does not count in significant figure determination.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 12<\/h3>\n<ol id=\"ball-ch02_s05_l02\" class=\"orderedlist\">\n \t<li>What is 98.6 \u00b0F in degrees Celsius?<\/li>\n \t<li>What is 25.0 \u00b0C in degrees Fahrenheit?<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n\n<ol id=\"ball-ch02_s05_l03\" class=\"orderedlist\">\n \t<li>\n<p class=\"para\">Using the first formula from above, we have<\/p>\n<span class=\"informalequation\">\u00b0C = (98.6 \u2013 32)\u2009\u00d7\u20095\/9 = 66.6\u2009\u00d7\u20095\/9 = 37.0 \u00b0C<\/span><\/li>\n \t<li>\n<p class=\"para\">Using the second formula from above, we have<\/p>\n<span class=\"informalequation\">\u00b0F = (25.0\u2009\u00d7\u20099\/5) + 3\/2 = 45.0 + 32 = 77.0 \u00b0F<\/span><\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 1<\/h3>\n<ol id=\"ball-ch02_s05_l04\" class=\"orderedlist\">\n \t<li>Convert 0 \u00b0F to degrees Celsius.<\/li>\n \t<li>Convert 212 \u00b0C to degrees Fahrenheit.<\/li>\n<\/ol>\n<\/div>\n&nbsp;\n<p id=\"ball-ch02_s05_p04\" class=\"para editable block\">The fundamental unit of temperature (another fundamental unit of science, bringing us to four) in SI is the <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">kelvin<\/a><\/span> (K). The Kelvin temperature scale (note that the name of the scale capitalizes the word <em class=\"emphasis\">Kelvin<\/em>, but the unit itself is lowercase) uses degrees that are the same size as the Celsius degree, but the numerical scale is shifted up by 273.15 units. That is, the conversion between the Kelvin and Celsius scales is as follows:<\/p>\n<span class=\"informalequation block\"><span class=\"mathphrase\">K = \u00b0C + 273.15<\/span><\/span>\n<span class=\"informalequation block\"><span class=\"mathphrase\">\u00b0C = K \u2212 273.15<\/span><\/span>\n<p id=\"ball-ch02_s05_p05\" class=\"para editable block\">For most purposes, it is acceptable to use 273 instead of 273.15. Note that the Kelvin scale does not use the word <em class=\"emphasis\">degrees<\/em>; a temperature of 295 K is spoken of as \u201ctwo hundred ninety-five kelvins\u201d and not \u201ctwo hundred ninety-five degrees Kelvin.\u201d<\/p>\n<p id=\"ball-ch02_s05_p06\" class=\"para editable block\">The reason that the Kelvin scale is defined this way is because there exists a minimum possible temperature called <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">absolute zero<\/a><\/span>. The Kelvin temperature scale is set so that 0 K is absolute zero, and temperature is counted upward from there. Normal room temperature is about 295 K, as seen in the following example.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 13<\/h3>\n<p id=\"ball-ch02_s05_p07\" class=\"para\">If normal room temperature is 72.0 \u00b0F, what is room temperature in degrees Celsius and kelvins?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s05_p08\" class=\"para\">First, we use the formula to determine the temperature in degrees Celsius:<\/p>\n<span class=\"informalequation\">\u00b0C = (72.0 \u2013 32)\u2009\u00d7\u20095\/9 = 40.0\u2009\u00d7\u20095\/9 = 22.2 \u00b0C<\/span>\n<p id=\"ball-ch02_s05_p09\" class=\"para\">Then we use the appropriate formula above to determine the temperature in the Kelvin scale:<\/p>\n<span class=\"informalequation\"><span class=\"mathphrase\">K = 22.2 \u00b0C + 273.15 = 295.4 K<\/span><\/span>\n<p id=\"ball-ch02_s05_p10\" class=\"para\">So, room temperature is about 295 K.<\/p>\n\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 2<\/h3>\n<p id=\"ball-ch02_s05_p11\" class=\"para\">What is 98.6 \u00b0F on the Kelvin scale?<\/p>\n\n<\/div>\n&nbsp;\n<p id=\"ball-ch02_s05_p13\" class=\"para editable block\"><a class=\"xref\" href=\"#ball-ch02_s05_f01\">Figure 2.9 \"Fahrenheit, Celsius, and Kelvin Temperatures\"<\/a> compares the three temperature scales. Note that science uses the Celsius and Kelvin scales almost exclusively; virtually no practicing chemist expresses laboratory-measured temperatures with the Fahrenheit scale. (In fact, the United States is one of the few countries in the world that still uses the Fahrenheit scale on a daily basis. The other two countries are Liberia and Myanmar [formerly Burma].<\/p>\n\n<div id=\"ball-ch02_s05_f01\" class=\"figure large editable block\">\n<p class=\"title\"><span class=\"title-prefix\">Figure 2.9<\/span> Fahrenheit, Celsius, and Kelvin Temperatures<\/p>\n\n\n[caption id=\"attachment_4622\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Temperatures.png\"><img class=\"wp-image-49\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Temperatures-1.png\" alt=\"Temperatures\" width=\"400\" height=\"355\"><\/a> <strong>Figure 1.<\/strong> Fahrenheit, Celsius, and Kelvin Temperatures[\/caption]\n<p class=\"para\">A comparison of the three temperature scales.<\/p>\n\n<\/div>\n<p id=\"ball-ch02_s05_p14\" class=\"para editable block\"><span class=\"margin_term\"><a class=\"glossterm\" href=\"\">Density <\/a><\/span>is a physical property that is defined as a substance\u2019s mass divided by its volume:<\/p>\n<span class=\"informalequation block\">density = mass\/volume or d = m\/V<\/span>\n<p id=\"ball-ch02_s05_p15\" class=\"para editable block\">Density is usually a measured property of a substance, so its numerical value affects the significant figures in a calculation. Notice that density is defined in terms of two dissimilar units, mass and volume. That means that density overall has derived units, just like velocity. Common units for density include g\/mL, g\/cm<sup class=\"superscript\">3<\/sup>, g\/L, kg\/L, and even kg\/m<sup class=\"superscript\">3<\/sup>. Densities for some common substances are listed in <a class=\"xref\" href=\"#ball-ch02_s05_t01\">Table 2.2 \"Densities of Some Common Substances\"<\/a>.<\/p>\n\n<div id=\"ball-ch02_s05_t01\" class=\"table block\">\n<p class=\"title\"><span class=\"title-prefix\">Table 2.2<\/span> Densities of Some Common Substances<\/p>\n\n<table style=\"border-spacing: 0px;width: 689px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th style=\"width: 203.517px\">Substance<\/th>\n<th style=\"width: 461.483px\">Density (g\/mL or g\/cm<sup class=\"superscript\">3<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 203.517px\">water<\/td>\n<td style=\"width: 461.483px\">1.0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">gold<\/td>\n<td style=\"width: 461.483px\">19.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">mercury<\/td>\n<td style=\"width: 461.483px\">13.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">air<\/td>\n<td style=\"width: 461.483px\">0.0012<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">cork<\/td>\n<td style=\"width: 461.483px\">0.22\u20130.26<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">aluminum<\/td>\n<td style=\"width: 461.483px\">2.7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">iron<\/td>\n<td style=\"width: 461.483px\">7.87<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"ball-ch02_s05_p16\" class=\"para editable block\">Because of how it is defined, density can act as a conversion factor for switching between units of mass and volume. For example, suppose you have a sample of aluminum that has a volume of 7.88 cm<sup class=\"superscript\">3<\/sup>. How can you determine what mass of aluminum you have without measuring it? You can use the volume to calculate it. If you multiply the given volume by the known density (from <a class=\"xref\" href=\"#ball-ch02_s05_t01\">Table 2.2 \"Densities of Some Common Substances\"<\/a>), the volume units will cancel and leave you with mass units, telling you the mass of the sample:<\/p>\n<span class=\"informalequation block\"> 7.88 cm<sup>3<\/sup>\u2009\u00d7\u20092.7 g\/cm<sup>3 <\/sup>= 21 g of aluminum<\/span>\n<p id=\"ball-ch02_s05_p17\" class=\"para editable block\">where we have limited our answer to two significant figures.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 14<\/h3>\n<p id=\"ball-ch02_s05_p18\" class=\"para\">What is the mass of 44.6 mL of mercury?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s05_p19\" class=\"para\">Use the density from <a class=\"xref\" href=\"#ball-ch02_s05_t01\">Table 2.2 \"Densities of Some Common Substances\"<\/a> as a conversion factor to go from volume to mass:<\/p>\n<span class=\"informalequation\">44.6 mL\u2009\u00d7\u200913.6 g\/mL = 607 g<\/span>\n<p id=\"ball-ch02_s05_p20\" class=\"para\">The mass of the mercury is 607 g.<\/p>\n\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 3<\/h3>\n<p id=\"ball-ch02_s05_p21\" class=\"emphasis bolditalic\">What is the mass of 25.0 cm<sup class=\"superscript\">3<\/sup> of iron?<\/p>\n\n<\/div>\n&nbsp;\n<p id=\"ball-ch02_s05_p23\" class=\"para editable block\">Density can also be used as a conversion factor to convert mass to volume\u2014but care must be taken. We have already demonstrated that the number that goes with density normally goes in the numerator when density is written as a fraction. Take the density of gold, for example:<\/p>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_3.png\"><img class=\"size-full wp-image-50 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_3.png\" alt=\"d = 19.3 g\/1 mL\" width=\"312\" height=\"115\"><\/a>\n\nAlthough this was not previously pointed out, it can be assumed that there is a 1 in the denominator:\n\nThat is, the density value tells us that we have 19.3 grams for every 1 milliliter of volume, and the 1 is an exact number. When we want to use density to convert from mass to volume, the numerator and denominator of density need to be switched\u2014that is, we must take the <em class=\"emphasis\">reciprocal<\/em> of the density. In so doing, we move not only the units but also the numbers:\n\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_4.png\"><img class=\"size-full wp-image-51 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_4.png\" alt=\"1\/d = 1mL\/19.3g\" width=\"203\" height=\"106\"><\/a><\/span>\n<p id=\"ball-ch02_s05_p26\" class=\"para editable block\">This reciprocal density is still a useful conversion factor, but now the mass unit will cancel and the volume unit will be introduced. Thus, if we want to know the volume of 45.9 g of gold, we would set up the conversion as follows:<\/p>\n<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_5.png\"><img class=\"size-full wp-image-52 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_5.png\" alt=\"45.9 g x 1mL\/19.3g = 2.38 mL\" width=\"376\" height=\"96\"><\/a><\/span>\n<p id=\"ball-ch02_s05_p27\" class=\"para editable block\">Note how the mass units cancel, leaving the volume unit, which is what we\u2019re looking for.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 15<\/h3>\n<p id=\"ball-ch02_s05_p28\" class=\"para\">A cork stopper from a bottle of wine has a mass of 3.78 g. If the density of cork is 0.22 g\/cm<sup class=\"superscript\">3<\/sup>, what is the volume of the cork?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s05_p29\" class=\"para\">To use density as a conversion factor, we need to take the reciprocal so that the mass unit of density is in the denominator. Taking the reciprocal, we find<\/p>\n<span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_6.png\"><img class=\"size-full wp-image-53 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_6.png\" alt=\"1\/d = 1cm^3\/0.22g\" width=\"189\" height=\"111\"><\/a><\/span>\n<p id=\"ball-ch02_s05_p30\" class=\"para\">We can use this expression as the conversion factor. So<\/p>\n<span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_7.png\"><img class=\"size-full wp-image-54 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_7.png\" alt=\"3.78 g x 1 cm^3\/0.22g = 17.2 cm^3\" width=\"381\" height=\"101\"><\/a><\/span>\n\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 4<\/h3>\n<p id=\"ball-ch02_s05_p31\" class=\"para\">What is the volume of 3.78 g of gold?<\/p>\n\n<\/div>\n&nbsp;\n<p id=\"ball-ch02_s05_p33\" class=\"para editable block\">Care must be used with density as a conversion factor. Make sure the mass units are the same, or the volume units are the same, before using density to convert to a different unit. Often, the unit of the given quantity must be first converted to the appropriate unit before applying density as a conversion factor.<\/p>\n\n<div id=\"ball-ch02_s05_n06\" class=\"callout block\">\n<h3 class=\"title\">Food and Drink App: Cooking Temperatures<\/h3>\n<p id=\"ball-ch02_s05_p78\" class=\"para\">Because degrees Fahrenheit is the common temperature scale in the United States, kitchen appliances, such as ovens, are calibrated in that scale. A cool oven may be only 150\u00b0F, while a cake may be baked at 350\u00b0F and a chicken roasted at 400\u00b0F. The broil setting on many ovens is 500\u00b0F, which is typically the highest temperature setting on a household oven.<\/p>\n<p id=\"ball-ch02_s05_p79\" class=\"para\">People who live at high altitudes, typically 2,000 ft above sea level or higher, are sometimes urged to use slightly different cooking instructions on some products, such as cakes and bread, because water boils at a lower temperature the higher in altitude you go, meaning that foods cook slower. For example, in Cleveland water typically boils at 212\u00b0F (100\u00b0C), but in Denver, the Mile-High City, water boils at about 200\u00b0F (93.3\u00b0C), which can significantly lengthen cooking times. Good cooks need to be aware of this.<\/p>\n<p id=\"ball-ch02_s05_p80\" class=\"para\">At the other end is pressure cooking. A pressure cooker is a closed vessel that allows steam to build up additional pressure, which increases the temperature at which water boils. A good pressure cooker can get to temperatures as high as 252\u00b0F (122\u00b0C); at these temperatures, food cooks much faster than it normally would. Great care must be used with pressure cookers because of the high pressure and high temperature. (When a pressure cooker is used to sterilize medical instruments, it is called an <em class=\"emphasis\">autoclave<\/em>.)<\/p>\n<p id=\"ball-ch02_s05_p81\" class=\"para\">Other countries use the Celsius scale for everyday purposes. Therefore, oven dials in their kitchens are marked in degrees Celsius. It can be confusing for US cooks to use ovens abroad\u2014a 425\u00b0F oven in the United States is equivalent to a 220\u00b0C oven in other countries. These days, many oven thermometers are marked with both temperature scales.<\/p>\n\n<div id=\"ball-ch02_s05_qs01\" class=\"qandaset block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s05_l06\" class=\"itemizedlist\">\n \t<li>Chemistry uses the Celsius and Kelvin scales to express temperatures.<\/li>\n \t<li>A temperature on the Kelvin scale is the Celsius temperature plus 273.15.<\/li>\n \t<li>The minimum possible temperature is absolute zero and is assigned 0 K on the Kelvin scale.<\/li>\n \t<li>Density relates a substance\u2019s mass and volume.<\/li>\n \t<li>Density can be used to calculate volume from a given mass or mass from a given volume.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol id=\"ball-ch02_s05_qs01_qd01\" class=\"qandadiv\">\n \t<li id=\"ball-ch02_s05_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p1\" class=\"para\">Perform the following conversions.<\/p>\n\n<\/div><\/li>\n<\/ol>\n(a) 255\u00b0F to degrees Celsius (b) \u2212255\u00b0F to degrees Celsius (c) 50.0\u00b0C to degrees Fahrenheit (d) \u221250.0\u00b0C to degrees Fahrenheit\n\n2. Perform the following conversions.\n\n(a) 1,065\u00b0C to degrees Fahrenheit (b) \u2212222\u00b0C to degrees Fahrenheit (c) 400.0\u00b0F to degrees Celsius (d) 200.0\u00b0F to degrees Celsius\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p3\" class=\"para\">3. Perform the following conversions.<\/p>\n(a) 100.0\u00b0C to kelvins (b) \u2212100.0\u00b0C to kelvins (c) 100 K to degrees Celsius (d) 300 K to degrees Celsius\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p4\" class=\"para\">4. Perform the following conversions.<\/p>\n(a) 1,000.0 K to degrees Celsius (b) 50.0 K to degrees Celsius (c) 37.0\u00b0C to kelvins (d) \u221237.0\u00b0C to kelvins\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p5\" class=\"para\">5. Convert 0 K to degrees Celsius. What is the significance of the temperature in degrees Celsius?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p7\" class=\"para\">6. Convert 0 K to degrees Fahrenheit. What is the significance of the temperature in degrees Fahrenheit?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p9\" class=\"para\">7. The hottest temperature ever recorded on the surface of the earth was 136\u00b0F in Libya in 1922. What is the temperature in degrees Celsius and in kelvins?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p11\" class=\"para\">8. The coldest temperature ever recorded on the surface of the earth was \u2212128.6\u00b0F in Vostok, Antarctica, in 1983. What is the temperature in degrees Celsius and in kelvins?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p13\" class=\"para\">9. Give at least three possible units for density.<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p15\" class=\"para\">10. What are the units when density is inverted? Give three examples.<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p17\" class=\"para\">11. A sample of iron has a volume of 48.2 cm<sup class=\"superscript\">3<\/sup>. What is its mass?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p19\" class=\"para\">12. A sample of air has a volume of 1,015 mL. What is its mass?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p21\" class=\"para\">13. The volume of hydrogen used by the <em class=\"emphasis\">Hindenburg<\/em>, the German airship that exploded in New Jersey in 1937, was 2.000 \u00d7 10<sup class=\"superscript\">8<\/sup> L. If hydrogen gas has a density of 0.0899 g\/L, what mass of hydrogen was used by the airship?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p23\" class=\"para\">14. The volume of an Olympic-sized swimming pool is 2.50 \u00d7 10<sup class=\"superscript\">9<\/sup> cm<sup class=\"superscript\">3<\/sup>. If the pool is filled with alcohol (<em class=\"emphasis\">d<\/em> = 0.789 g\/cm<sup class=\"superscript\">3<\/sup>), what mass of alcohol is in the pool?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p25\" class=\"para\">15. A typical engagement ring has 0.77 cm<sup class=\"superscript\">3<\/sup> of gold. What mass of gold is present?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p27\" class=\"para\">16. A typical mercury thermometer has 0.039 mL of mercury in it. What mass of mercury is in the thermometer?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p29\" class=\"para\">17. What is the volume of 100.0 g of lead if lead has a density of 11.34 g\/cm<sup class=\"superscript\">3<\/sup>?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p31\" class=\"para\">18. What is the volume of 255.0 g of uranium if uranium has a density of 19.05 g\/cm<sup class=\"superscript\">3<\/sup>?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p33\" class=\"para\">19. What is the volume in liters of 222 g of neon if neon has a density of 0.900 g\/L?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p35\" class=\"para\">20. What is the volume in liters of 20.5 g of sulfur hexafluoride if sulfur hexafluoride has a density of 6.164 g\/L?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p37\" class=\"para\">21. Which has the greater volume, 100.0 g of iron (<em class=\"emphasis\">d<\/em> = 7.87 g\/cm<sup class=\"superscript\">3<\/sup>) or 75.0 g of gold (<em class=\"emphasis\">d<\/em> = 19.3 g\/cm<sup class=\"superscript\">3<\/sup>)?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p39\" class=\"para\">22. Which has the greater volume, 100.0 g of hydrogen gas (<em class=\"emphasis\">d<\/em> = 0.0000899 g\/cm<sup class=\"superscript\">3<\/sup>) or 25.0 g of argon gas (<em class=\"emphasis\">d<\/em> = 0.00178 g\/cm<sup class=\"superscript\">3<\/sup>)?<\/p>\n\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<strong>Check Your Understanding 1\n<\/strong>\n<ol id=\"ball-ch02_s05_l05\" class=\"orderedlist\">\n \t<li>\u221217.8 \u00b0C<\/li>\n \t<li>414 \u00b0F<\/li>\n<\/ol>\n<strong>Check Your Understanding 2\n<\/strong>\n\n310.2 K\n\n<strong>Check Your Understanding 3\n<\/strong>\n\n197 g\n\n<strong>Check Your Understanding 4\n<\/strong>\n\n0.196 cm<sup class=\"superscript\">3<\/sup>\n\n<strong>Problems &amp; Exercises<\/strong>\n\n<strong>1.<\/strong> (a) 124\u00b0C (b) \u2212159\u00b0C (c) 122\u00b0F (d) \u221258\u00b0F\n\n<strong>3.<\/strong> (a) 373 K (b) 173 K (c) \u2212173\u00b0C (d) 27\u00b0C\n\n<strong>5.<\/strong> \u2212273\u00b0C. This is the lowest possible temperature in degrees Celsius.\n\n<strong>7.<\/strong> 57.8\u00b0C; 331 K\n\n<strong>9.<\/strong> g\/mL, g\/L, and kg\/L (answers will vary)\n\n<strong>11.<\/strong> 379 g\n\n<strong>13.<\/strong> 1.80 \u00d7 10<sup class=\"superscript\">7<\/sup> g\n\n<strong>15.<\/strong> 15 g\n\n<strong>17.<\/strong> 8.818 cm<sup class=\"superscript\">3<\/sup>\n\n<strong>19.<\/strong> 247 L\n\n<strong>21.<\/strong> The 100.0 g of iron has the greater volume.\n\n<\/div>\n&nbsp;\n\n<\/div>\n<\/div>\n<h1><a id=\"exp-units\" href=\"\"><\/a>Expressing Units<\/h1>\n<p id=\"ball-ch02_s02_p01\" class=\"para editable block\">A number indicates \u201chow much,\u201d but the unit indicates \u201cof what.\u201d The \u201cof what\u201d is important when communicating a quantity. For example, if you were to ask a friend how close you are to Lake Erie and your friend says \u201csix,\u201d then your friend isn\u2019t giving you complete information. Six <em class=\"emphasis\">what<\/em>? Six miles? Six inches? Six city blocks? The actual distance to the lake depends on what units you use.<\/p>\n<p id=\"ball-ch02_s02_p02\" class=\"para editable block\">Chemistry, like most sciences, uses the International System of Units, or SI for short. (The letters <em class=\"emphasis\">SI<\/em> stand for the French \u201cle Syst\u00e8me International d\u2019unit\u00e9s.\u201d) SI specifies certain units for various types of quantities, based on seven <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">fundamental units<\/a><\/span> for various quantities. We will use most of the fundamental units in chemistry. Initially, we will deal with three fundamental units. The meter (m) is the SI unit of length. It is a little longer than a yard (see <a class=\"xref\" href=\"#ball-ch02_s02_f01\">Figure 2.3 \"The Meter\"<\/a>). The SI unit of mass is the kilogram (kg), which is about 2.2 pounds (lb). The SI unit of time is the second (s).<\/p>\n\n<div id=\"ball-ch02_s02_f01\" class=\"figure large medium-height editable block\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"380\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/67d75b028a8e56c36f0622ce6b20547e-1.jpg\" alt=\"image\" width=\"380\" height=\"375\"> <strong>Figure 1.<\/strong> The Meter[\/caption]\n<p class=\"para\">The SI standard unit of length, the meter, is a little longer than a yard.<\/p>\n\n<\/div>\n<p id=\"ball-ch02_s02_p03\" class=\"para editable block\">To express a quantity, you need to combine a number with a unit. If you have a length that is 2.4 m, then you express that length as simply 2.4 m. A time of 15,000 s can be expressed as 1.5 \u00d7 10<sup class=\"superscript\">4<\/sup> s in scientific notation.<\/p>\n<p id=\"ball-ch02_s02_p04\" class=\"para editable block\">Sometimes, a given unit is not an appropriate size to easily express a quantity. For example, the width of a human hair is very small, and it doesn\u2019t make much sense to express it in meters. SI also defines a series of <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">numerical prefixes<\/a><\/span> that refer to multiples or fractions of a fundamental unit to make a unit more conveniently sized for a specific quantity. <a class=\"xref\" href=\"#ball-ch02_s02_t01\">Table 2.1 \"Multiplicative Prefixes for SI Units\"<\/a> lists the prefixes, their abbreviations, and their multiplicative factors. Some of the prefixes, such as kilo-, mega-, and giga-, represent more than one of the fundamental unit, while other prefixes, such as centi-, milli-, and micro-, represent fractions of the original unit. Note, too, that once again we are using powers of 10. Each prefix is a multiple of or fraction of a power of 10.<\/p>\n\n<div id=\"ball-ch02_s02_t01\" class=\"table block\">\n<p class=\"title\"><span class=\"title-prefix\">Table 2.1<\/span> Multiplicative Prefixes for SI Units<\/p>\n\n<table style=\"border-spacing: 0px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th>Prefix<\/th>\n<th align=\"center\">Abbreviation<\/th>\n<th align=\"center\">Multiplicative Amount<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>giga-<\/td>\n<td align=\"center\">G<\/td>\n<td align=\"center\">1,000,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>mega-<\/td>\n<td align=\"center\">M<\/td>\n<td align=\"center\">1,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>kilo-<\/td>\n<td align=\"center\">k<\/td>\n<td align=\"center\">1,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>deci-<\/td>\n<td align=\"center\">d<\/td>\n<td align=\"center\">1\/10 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>centi-<\/td>\n<td align=\"center\">c<\/td>\n<td align=\"center\">1\/100 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>milli-<\/td>\n<td align=\"center\">m<\/td>\n<td align=\"center\">1\/1,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>micro-<\/td>\n<td align=\"center\">\u03bc*<\/td>\n<td align=\"center\">1\/1,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>nano-<\/td>\n<td align=\"center\">n<\/td>\n<td align=\"center\">1\/1,000,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>pico-<\/td>\n<td align=\"center\">p<\/td>\n<td align=\"center\">1\/1,000,000,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<th colspan=\"3\">* The letter <em class=\"emphasis\">\u03bc<\/em> is the Greek letter lowercase equivalent to an m and is called \u201cmu\u201d (pronounced \u201cmyoo\u201d).<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"ball-ch02_s02_p05\" class=\"para editable block\">To use the fractions to generate new units, simply combine the prefix with the unit itself; the abbreviation for the new unit is the combination of the abbreviation for the prefix and the abbreviation of the unit. For example, the kilometer (km) is 1,000 \u00d7 meter, or 1,000 m. Thus, 5 kilometers (5 km) is equal to 5,000 m. Similarly, a millisecond (ms) is 1\/1,000 \u00d7 second, or one-thousandth of a second. Thus, 25 ms is 25 thousandths of a second. You will need to become proficient in combining prefixes and units. (You may recognize that one of our fundamental units, the kilogram, automatically has a prefix-unit combination, the kilogram. The word <em class=\"emphasis\">kilogram<\/em> means 1,000 g.)<\/p>\n<p id=\"ball-ch02_s02_p06\" class=\"para editable block\">In addition to the fundamental units, SI also allows for <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">derived units<\/a><\/span> based on a fundamental unit or units. There are many derived units used in science. For example, the derived unit for area comes from the idea that area is defined as width times height. Because both width and height are lengths, they both have the fundamental unit of meter, so the unit of area is meter \u00d7 meter, or meter<sup class=\"superscript\">2<\/sup> (m<sup class=\"superscript\">2<\/sup>). This is sometimes spoken as \u201csquare meters.\u201d A unit with a prefix can also be used to derive a unit for area, so we can also have cm<sup class=\"superscript\">2<\/sup>, mm<sup class=\"superscript\">2<\/sup>, or km<sup class=\"superscript\">2<\/sup> as acceptable units for area.<\/p>\n<p id=\"ball-ch02_s02_p07\" class=\"para editable block\">Volume is defined as length times width times height, so it has units of meter \u00d7 meter \u00d7 meter or meter<sup class=\"superscript\">3<\/sup> (m<sup class=\"superscript\">3<\/sup>), sometimes spoken as \u201ccubic meters.\u201d The cubic meter is a rather large unit, however, so another unit is defined that is somewhat more manageable: the liter (L). A liter is 1\/1,000th of a cubic meter and is a little more than 1 quart in volume (see <a class=\"xref\" href=\"#ball-ch02_s02_f02\">Figure 2.4 \"The Liter\"<\/a>). Prefixes can also be used with the liter unit, so we can speak of milliliters (1\/1,000th of a liter; mL) and kiloliters (1,000 L; kL).<\/p>\n\n<div id=\"ball-ch02_s02_f02\" class=\"figure small editable block\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"380\"]<img src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/6866a70a4b52e0fd95f9dd3e1f3426a2-1.jpg\" alt=\"image\" width=\"380\" height=\"375\"> <strong>Figure 2.<\/strong> The Liter[\/caption]\n<p class=\"para\">The SI unit of volume, the liter, is slightly larger than 1 quart.<\/p>\n\n<\/div>\n<p id=\"ball-ch02_s02_p08\" class=\"para editable block\">Another definition of a liter is one-tenth of a meter cubed. Because one-tenth of a meter is 10 cm, then a liter is equal to 1,000 cm<sup class=\"superscript\">3<\/sup> (<a class=\"xref\" href=\"#ball-ch02_s02_f03\">Figure 2.5 \"The Size of 1 Liter\"<\/a>). Because 1 L equals 1,000 mL, we conclude that 1 mL equals 1 cm<sup class=\"superscript\">3<\/sup>; thus, these units are interchangeable.<\/p>\n\n<div id=\"ball-ch02_s02_f03\" class=\"figure large editable block\">\n\n[caption id=\"attachment_4611\" align=\"aligncenter\" width=\"400\"]<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Size-of-a-Liter.png\"><img class=\"wp-image-782\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Size-of-a-Liter-1.png\" alt=\"Size of a Liter\" width=\"400\" height=\"281\"><\/a> <strong>Figure 3.<\/strong> The Size of 1 Liter[\/caption]\n<p class=\"para\">One liter equals 1,000 cm<sup class=\"superscript\">3<\/sup>, so 1 cm<sup class=\"superscript\">3<\/sup> is the same as 1 mL.<\/p>\n\n<\/div>\n<p id=\"ball-ch02_s02_p09\" class=\"para editable block\">Units not only are multiplied together but also can be divided. For example, if you are traveling at one meter for every second of time elapsed, your velocity is 1 meter per second, or 1 m\/s. The word <em class=\"emphasis\">per<\/em> implies division, so velocity is determined by dividing a distance quantity by a time quantity. Other units for velocity include kilometers per hour (km\/h) or even micrometers per nanosecond (\u03bcm\/ns). Later, we will see other derived units that can be expressed as fractions.<\/p>\n\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 2<\/h3>\n<ol id=\"ball-ch02_s02_l02\" class=\"orderedlist\">\n \t<li>A human hair has a diameter of about 6.0 \u00d7 10<sup class=\"superscript\">\u22125<\/sup> m. Suggest an appropriate unit for this measurement and write the diameter of a human hair in terms of that unit.<\/li>\n \t<li>What is the velocity of a car if it goes 25 m in 5.0 s?<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n\n<ol id=\"ball-ch02_s02_l03\" class=\"orderedlist\">\n \t<li>The scientific notation 10<sup class=\"superscript\">\u22125<\/sup> is close to 10<sup class=\"superscript\">\u22126<\/sup>, which defines the micro- prefix. Let us use micrometers as the unit for hair diameter. The number 6.0 \u00d7 10<sup class=\"superscript\">\u22125<\/sup> can be written as 60 \u00d7 10<sup class=\"superscript\">\u22126<\/sup>, and a micrometer is 10<sup class=\"superscript\">\u22126<\/sup> m, so the diameter of a human hair is about 60 \u03bcm.<\/li>\n \t<li>If velocity is defined as a distance quantity divided by a time quantity, then velocity is 25 meters\/5.0 seconds. Dividing the numbers gives us 25\/5.0 = 5.0, and dividing the units gives us meters\/second, or m\/s. The velocity is 5.0 m\/s.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 1<\/h3>\n<ol id=\"ball-ch02_s02_l04\" class=\"orderedlist\">\n \t<li>Express the volume of an Olympic-sized swimming pool, 2,500,000 L, in more appropriate units.<\/li>\n \t<li>A common garden snail moves about 6.1 m in 30 min. What is its velocity in meters per minute (m\/min)?<\/li>\n<\/ol>\n<\/div>\n&nbsp;\n<div id=\"ball-ch02_s02_n03\" class=\"key_takeaways editable block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s02_l06\" class=\"itemizedlist\">\n \t<li>Numbers tell \u201chow much,\u201d and units tell \u201cof what.\u201d<\/li>\n \t<li>Chemistry uses a set of fundamental units and derived units from SI units.<\/li>\n \t<li>Chemistry uses a set of prefixes that represent multiples or fractions of units.<\/li>\n \t<li>Units can be multiplied and divided to generate new units for quantities.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol id=\"ball-ch02_s02_qs01_qd01\" class=\"qandadiv\">\n \t<li id=\"ball-ch02_s02_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p1\" class=\"para\">Identify the unit in each quantity.<\/p>\n\n<\/div><\/li>\n<\/ol>\n(a) 2 boxes of crayons (b) 3.5 grams of gold\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p2\" class=\"para\">2. Identify the unit in each quantity.<\/p>\n(a) 32 oz of cheddar cheese (b) 0.045 cm<sup class=\"superscript\">3<\/sup> of water\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p3\" class=\"para\">3. Identify the unit in each quantity.<\/p>\n(a) 9.58 s (the current world record in the 100 m dash) (b) 6.14 m (the current world record in the pole vault)\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p4\" class=\"para\">4. Identify the unit in each quantity.<\/p>\n(a) 2 dozen eggs (b) 2.4 km\/s (the escape velocity of the moon, which is the velocity you need at the surface to escape the moon\u2019s gravity)\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p5\" class=\"para\">5. Indicate what multiplier each prefix represents.<\/p>\n(a) k (b) m (c) M\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p6\" class=\"para\">6. Indicate what multiplier each prefix represents.<\/p>\n(a) c (b) G (c) \u03bc\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p7\" class=\"para\">7. Give the prefix that represents each multiplier.<\/p>\n(a) 1\/1,000th \u00d7 (b) 1,000 \u00d7 (c) 1,000,000,000 \u00d7\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p8\" class=\"para\">8. Give the prefix that represents each multiplier.<\/p>\n(a) 1\/1,000,000,000th \u00d7 (b) 1\/100th \u00d7 (c) 1,000,000 \u00d7\n\n9. Complete the following table with the missing information.\n\n<\/div>\n<div class=\"question\">\n<div class=\"informaltable\">\n<table style=\"border-color: #000000;border-spacing: 0px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th>Unit<\/th>\n<th align=\"center\">Abbreviation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>kilosecond<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">mL<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">Mg<\/td>\n<\/tr>\n<tr>\n<td>centimeter<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p10\" class=\"para\">10.Complete the following table with the missing information.<\/p>\n\n<div class=\"informaltable\">\n<table style=\"border-color: #000000;border-spacing: 0px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th>Unit<\/th>\n<th align=\"center\">Abbreviation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>kilometer per second<\/td>\n<\/tr>\n<tr>\n<td>second<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">cm<sup class=\"superscript\">3<\/sup><\/td>\n<\/tr>\n<tr>\n<td align=\"center\">\u03bcL<\/td>\n<\/tr>\n<tr>\n<td>nanosecond<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n11. Express each quantity in a more appropriate unit. There may be more than one acceptable answer.\n\n<\/div>\n<\/div>\n(a) 3.44 \u00d7 10<sup class=\"superscript\">\u22126<\/sup> s (b) 3,500 L(c) 0.045 m\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p12\" class=\"para\">12. Express each quantity in a more appropriate unit. There may be more than one acceptable answer.<\/p>\n(a) 0.000066 m\/s (Hint: you need consider only the unit in the numerator.) (b) 4.66 \u00d7 10<sup class=\"superscript\">6<\/sup> s (c) 7,654 L\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p13\" class=\"para\">13. Express each quantity in a more appropriate unit. There may be more than one acceptable answer.<\/p>\n(a) 43,600 mL (b) 0.0000044 m (c) 1,438 ms\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p14\" class=\"para\">14. Express each quantity in a more appropriate unit. There may be more than one acceptable answer.<\/p>\n(a) 0.000000345 m<sup class=\"superscript\">3 <\/sup>(b) 47,000,000 mm<sup class=\"superscript\">3 <\/sup>(c) 0.00665 L\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p15\" class=\"para\">15. Multiplicative prefixes are used for other units as well, such as computer memory. The basic unit of computer memory is the byte (b). What is the unit for one million bytes?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p17\" class=\"para\">16. You may have heard the terms <em class=\"emphasis\">microscale<\/em> or <em class=\"emphasis\">nanoscale<\/em> to represent the sizes of small objects. What units of length do you think are useful at these scales? What fractions of the fundamental unit of length are these units?<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p19\" class=\"para\">17. Acceleration is defined as a change in velocity per time. Propose a unit for acceleration in terms of the fundamental SI units.<\/p>\n\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p21\" class=\"para\">18. Density is defined as the mass of an object divided by its volume. Propose a unit of density in terms of the fundamental SI units.<\/p>\n\n<\/div>\n&nbsp;\n\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<strong>Check Your Understanding 1<\/strong>\n<ol id=\"ball-ch02_s02_l05\" class=\"orderedlist\">\n \t<li>2.5 ML<\/li>\n \t<li>0.203 m\/min<\/li>\n<\/ol>\n<strong>Problems &amp; Exercises<\/strong>\n\n<strong>1.<\/strong> (a) boxes of crayons (b) grams of gold\n\n<strong>3.<\/strong> (a) seconds (b) meters\n\n<strong>5.<\/strong> (a) 1,000 \u00d7 (b) 1\/1,000 \u00d7 (c) 1,000,000 \u00d7\n\n<strong>7.<\/strong> (a) milli- (b) kilo- (c) giga-\n\n<strong>9.<\/strong>\n<div class=\"informaltable\">\n<table style=\"border-color: #000000;border-spacing: 0px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th>Unit<\/th>\n<th align=\"center\">Abbreviation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>kilosecond<\/td>\n<td align=\"center\">ks<\/td>\n<\/tr>\n<tr>\n<td>milliliter<\/td>\n<td align=\"center\">mL<\/td>\n<\/tr>\n<tr>\n<td>megagram<\/td>\n<td align=\"center\">Mg<\/td>\n<\/tr>\n<tr>\n<td>centimeter<\/td>\n<td align=\"center\">cm<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<strong>11<\/strong>. (a) 3.44 \u03bcs (b) 3.5 kL (c) 4.5 cm\n\n<strong>13.<\/strong> (a) 43.6 L ( b) 4.4 \u00b5m (c) 1.438 s\n\n<strong>15.<\/strong> megabytes (Mb)\n\n<strong>17.<\/strong> meters\/second<sup class=\"superscript\">2<\/sup>\n\n<\/div>\n&nbsp;\n\n<\/div>\n<h1><a id=\"add-exer\" href=\"\"><\/a>Additional Exercises<\/h1>\n<div class=\"bcc-box bcc-info\">\n<h3 class=\"title\">Additional Exercises<\/h3>\n<ol>\n \t<li>Evaluate 0.00000000552 \u00d7 0.0000000006188 and express the answer in scientific notation. You may have to rewrite the original numbers in scientific notation first.<\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa02\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p3\" class=\"para\">Evaluate 333,999,500,000 \u00f7 0.00000000003396 and express the answer in scientific notation. You may need to rewrite the original numbers in scientific notation first.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa03\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p5\" class=\"para\">Express the number 6.022 \u00d7 10<sup class=\"superscript\">23<\/sup> in standard notation.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa04\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p7\" class=\"para\">Express the number 6.626 \u00d7 10<sup class=\"superscript\">\u221234<\/sup> in standard notation.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa05\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p9\" class=\"para\">When powers of 10 are multiplied together, the powers are added together. For example, 10<sup class=\"superscript\">2<\/sup> \u00d7 10<sup class=\"superscript\">3<\/sup> = 10<sup class=\"superscript\">2+3<\/sup> = 10<sup class=\"superscript\">5<\/sup>. With this in mind, can you evaluate (4.506 \u00d7 10<sup class=\"superscript\">4<\/sup>) \u00d7 (1.003 \u00d7 10<sup class=\"superscript\">2<\/sup>) without entering scientific notation into your calculator?<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa06\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p11\" class=\"para\">When powers of 10 are divided into each other, the bottom exponent is subtracted from the top exponent. For example, 10<sup class=\"superscript\">5<\/sup>\/10<sup class=\"superscript\">3<\/sup> = 10<sup class=\"superscript\">5\u22123<\/sup> = 10<sup class=\"superscript\">2<\/sup>. With this in mind, can you evaluate (8.552 \u00d7 10<sup class=\"superscript\">6<\/sup>) \u00f7 (3.129 \u00d7 10<sup class=\"superscript\">3<\/sup>) without entering scientific notation into your calculator?<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa07\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p13\" class=\"para\">Consider the quantity two dozen eggs. Is the number in this quantity \u201ctwo\u201d or \u201ctwo dozen\u201d? Justify your choice.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa08\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p15\" class=\"para\">Consider the quantity two dozen eggs. Is the unit in this quantity \u201ceggs\u201d or \u201cdozen eggs\u201d? Justify your choice.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa09\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p17\" class=\"para\">Fill in the blank: 1 km = ______________ \u03bcm.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa10\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p19\" class=\"para\">Fill in the blank: 1 Ms = ______________ ns.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa11\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p21\" class=\"para\">Fill in the blank: 1 cL = ______________ ML.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa12\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p23\" class=\"para\">Fill in the blank: 1 mg = ______________ kg.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa13\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p25\" class=\"para\">Express 67.3 km\/h in meters\/second.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa14\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p27\" class=\"para\">Express 0.00444 m\/s in kilometers\/hour.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa15\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p29\" class=\"para\">Using the idea that 1.602 km = 1.000 mi, convert a speed of 60.0 mi\/h into kilometers\/hour.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa16\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p31\" class=\"para\">Using the idea that 1.602 km = 1.000 mi, convert a speed of 60.0 km\/h into miles\/hour.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa17\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p33\" class=\"para\">Convert 52.09 km\/h into meters\/second.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa18\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p35\" class=\"para\">Convert 2.155 m\/s into kilometers\/hour.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa19\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p37\" class=\"para\">Use the formulas for converting degrees Fahrenheit into degrees Celsius to determine the relative size of the Fahrenheit degree over the Celsius degree.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa20\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p39\" class=\"para\">Use the formulas for converting degrees Celsius into kelvins to determine the relative size of the Celsius degree over kelvins.<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa21\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p41\" class=\"para\">What is the mass of 12.67 L of mercury?<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa22\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p43\" class=\"para\">What is the mass of 0.663 m<sup class=\"superscript\">3<\/sup> of air?<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa23\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p45\" class=\"para\">What is the volume of 2.884 kg of gold?<\/p>\n\n<\/div><\/li>\n \t<li id=\"ball-ch02_s06_qs01_qd01_qa24\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p47\" class=\"para\">What is the volume of 40.99 kg of cork? Assume a density of 0.22 g\/cm<sup class=\"superscript\">3<\/sup>.<\/p>\n\n<\/div><\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Answers<\/h3>\n<strong>1<\/strong>. 3.42 \u00d7 10<sup>\u221218 <\/sup>\n\n<strong>3. <\/strong>602,200,000,000,000,000,000,000\n\n<strong>5. <\/strong>4.520 \u00d7 10<sup>6 <\/sup>\n\n<strong>7 . <\/strong>The quantity is two; dozen is the unit.\n<div>\n\n<strong>9. <\/strong>1,000,000,000\n\n<strong>11. <\/strong>1\/100,000,000\n\n<strong>13. <\/strong>18.7 m\/s\n\n<strong>15. <\/strong>96.1 km\/h\n\n<strong>17. <\/strong>14.47 m\/s\n\n<strong>19. <\/strong>One Fahrenheit degree is nine-fifths the size of a Celsius degree.\n\n<strong>21. <\/strong>1.72 \u00d7 10<sup>5<\/sup> g\n\n<strong>23. <\/strong>149 mL\n\n<\/div>\n<\/div>","rendered":"<p><em>This content is originally from OpenStax College Chemistry 1st Canadian Edition.<\/em><\/p>\n<p>This appendix is broken into several sections:<em><br \/>\n<\/em><\/p>\n<ul>\n<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#intro-meas\">Introduction to Measurement<\/a><\/li>\n<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#exp-num\">Expressing Numbers<\/a><\/li>\n<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#sig-figs\">Significant Figures<\/a><\/li>\n<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#conv-units\">Converting Units<\/a><\/li>\n<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#oth-units\">Other Units: Temperature and Density<\/a><\/li>\n<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#exp-units\">Expressing Units<\/a><\/li>\n<li><a href=\"\/douglasphys1107\/back-matter\/appendix-c-units-numbers-and-significant-figures\/#add-exer\">Additional Exercises<\/a><\/li>\n<\/ul>\n<h1><a id=\"intro-meas\" href=\"\"><\/a>Introduction to Measurement<\/h1>\n<div id=\"ball-ch02_n01\" class=\"callout block\">\n<p id=\"ball-ch02_p01\" class=\"para\">Data suggest that a male child will weigh 50% of his adult weight at about 11 years of age. However, he will reach 50% of his adult height at only 2 years of age. It is obvious, then, that people eventually stop growing up but continue to grow out. Data also suggest that the average human height has been increasing over time. In industrialized countries, the average height of people increased 5.5 inches from 1810 to 1984. Most scientists attribute this simple, basic measurement of the human body to better health and nutrition.<\/p>\n<figure id=\"attachment_4607\" aria-describedby=\"caption-attachment-4607\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Stature-Percentile.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-742\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2018\/08\/Stature-Percentile-1.png\" alt=\"Stature Percentile\" width=\"400\" height=\"504\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2018\/08\/Stature-Percentile-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2018\/08\/Stature-Percentile-1-238x300.png 238w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2018\/08\/Stature-Percentile-1-65x82.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2018\/08\/Stature-Percentile-1-225x284.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2018\/08\/Stature-Percentile-1-350x441.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-4607\" class=\"wp-caption-text\"><strong>Figure 1.<\/strong> Stature-for-age percentiles: Boys, 2 to 20 years. Source: Chart courtesy of Centers for Disease Control and Prevention, http:\/\/www.cdc.gov\/nchs\/nhanes.htm#Set%201.<\/figcaption><\/figure>\n<\/div>\n<p id=\"ball-ch02_p02\" class=\"para editable block\">In 1983, an Air Canada airplane had to make an emergency landing because it unexpectedly ran out of fuel; ground personnel had filled the fuel tanks with a certain number of pounds of fuel, not kilograms of fuel. In 1999, the Mars Climate Orbiter spacecraft was lost attempting to orbit Mars because the thrusters were programmed in terms of English units, even though the engineers built the spacecraft using metric units. In 1993, a nurse mistakenly administered 23 units of morphine to a patient rather than the \u201c2\u20133\u201d units prescribed. (The patient ultimately survived.) These incidents occurred because people weren\u2019t paying attention to quantities.<\/p>\n<p id=\"ball-ch02_p03\" class=\"para editable block\">Physics and chemistry, like all sciences, are quantitative. they deals with <em class=\"emphasis\">quantities<\/em>, things that have amounts and units. Dealing with quantities is very important in chemistry and physics, as is relating quantities to each other. In this chapter, we will discuss how we deal with numbers and units, including how they are combined and manipulated.<\/p>\n<h1><a id=\"exp-num\" href=\"\"><\/a>Expressing Numbers<\/h1>\n<div id=\"ball-ch02_s01\" class=\"section\" lang=\"en\">\n<p id=\"ball-ch02_s01_p01\" class=\"para editable block\">Quantities have two parts: the number and the unit. The number tells \u201chow many.\u201d It is important to be able to express numbers properly so that the quantities can be communicated properly.<\/p>\n<p id=\"ball-ch02_s01_p02\" class=\"para editable block\"><span class=\"margin_term\"><a class=\"glossterm\" href=\"\">Standard notation<\/a><\/span> is the straightforward expression of a number. Numbers such as 17, 101.5, and 0.00446 are expressed in standard notation. For relatively small numbers, standard notation is fine. However, for very large numbers, such as 306,000,000, or for very small numbers, such as 0.000000419, standard notation can be cumbersome because of the number of zeros needed to place nonzero numbers in the proper position.<\/p>\n<p id=\"ball-ch02_s01_p03\" class=\"para editable block\"><span class=\"margin_term\"><a class=\"glossterm\" href=\"\">Scientific notation<\/a><\/span> is an expression of a number using powers of 10. Powers of 10 are used to express numbers that have many zeros:<\/p>\n<div class=\"informaltable block\">\n<table style=\"border-spacing: 0px\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td>10<sup class=\"superscript\">0<\/sup><\/td>\n<td>= 1<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">1<\/sup><\/td>\n<td>= 10<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">2<\/sup><\/td>\n<td>= 100 = 10 \u00d7 10<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">3<\/sup><\/td>\n<td>= 1,000 = 10 \u00d7 10 \u00d7 10<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">4<\/sup><\/td>\n<td>= 10,000 = 10 \u00d7 10 \u00d7 10 \u00d7 10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"ball-ch02_s01_p04\" class=\"para editable block\">and so forth. The raised number to the right of the 10 indicating the number of factors of 10 in the original number is the <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">exponent<\/a><\/span>. (Scientific notation is sometimes called <em class=\"emphasis\">exponential notation<\/em>.) The exponent\u2019s value is equal to the number of zeros in the number expressed in standard notation.<\/p>\n<p id=\"ball-ch02_s01_p05\" class=\"para editable block\">Small numbers can also be expressed in scientific notation but with negative exponents:<\/p>\n<div class=\"informaltable block\">\n<table style=\"border-spacing: 0px\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td>10<sup class=\"superscript\">\u22121<\/sup><\/td>\n<td>= 0.1 = 1\/10<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">\u22122<\/sup><\/td>\n<td>= 0.01 = 1\/100<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">\u22123<\/sup><\/td>\n<td>= 0.001 = 1\/1,000<\/td>\n<\/tr>\n<tr>\n<td>10<sup class=\"superscript\">\u22124<\/sup><\/td>\n<td>= 0.0001 = 1\/10,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"ball-ch02_s01_p06\" class=\"para editable block\">and so forth. Again, the value of the exponent is equal to the number of zeros in the denominator of the associated fraction. A negative exponent implies a decimal number less than one.<\/p>\n<p id=\"ball-ch02_s01_p07\" class=\"para editable block\">A number is expressed in scientific notation by writing the first nonzero digit, then a decimal point, and then the rest of the digits. The part of a number in scientific notation that is multiplied by a power of 10 is called the <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">coefficient<\/a><\/span>. Then determine the power of 10 needed to make that number into the original number and multiply the written number by the proper power of 10. For example, to write 79,345 in scientific notation,<\/p>\n<p><span class=\"informalequation block\"><span class=\"mathphrase\">79,345 = 7.9345 \u00d7 10,000 = 7.9345 \u00d7 10<sup class=\"superscript\">4<\/sup><\/span><\/span><\/p>\n<p id=\"ball-ch02_s01_p08\" class=\"para editable block\">Thus, the number in scientific notation is 7.9345 \u00d7 10<sup class=\"superscript\">4<\/sup>. For small numbers, the same process is used, but the exponent for the power of 10 is negative:<\/p>\n<p><span class=\"informalequation block\"><span class=\"mathphrase\">0.000411 = 4.11 \u00d7 1\/10,000 = 4.11 \u00d7 10<sup class=\"superscript\">\u22124<\/sup><\/span><\/span><\/p>\n<p id=\"ball-ch02_s01_p09\" class=\"para editable block\">Typically, the extra zero digits at the end or the beginning of a number are not included.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 1<\/h3>\n<p id=\"ball-ch02_s01_p10\" class=\"para\">Express these numbers in scientific notation.<\/p>\n<ol id=\"ball-ch02_s01_l02\" class=\"orderedlist\">\n<li>306,000<\/li>\n<li>0.00884<\/li>\n<li>2,760,000<\/li>\n<li>0.000000559<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n<ol id=\"ball-ch02_s01_l03\" class=\"orderedlist\">\n<li>The number 306,000 is 3.06 times 100,000, or 3.06 times 10<sup class=\"superscript\">5<\/sup>. In scientific notation, the number is 3.06 \u00d7 10<sup class=\"superscript\">5<\/sup>.<\/li>\n<li>The number 0.00884 is 8.84 times 1\/1,000, which is 8.84 times 10<sup class=\"superscript\">\u22123<\/sup>. In scientific notation, the number is 8.84 \u00d7 10<sup class=\"superscript\">\u22123<\/sup>.<\/li>\n<li>The number 2,760,000 is 2.76 times 1,000,000, which is the same as 2.76 times 10<sup class=\"superscript\">6<\/sup>. In scientific notation, the number is written as 2.76 \u00d7 10<sup class=\"superscript\">6<\/sup>. Note that we omit the zeros at the end of the original number.<\/li>\n<li>The number 0.000000559 is 5.59 times 1\/10,000,000, which is 5.59 times 10<sup class=\"superscript\">\u22127<\/sup>. In scientific notation, the number is written as 5.59 \u00d7 10<sup class=\"superscript\">\u22127<\/sup>.<\/li>\n<\/ol>\n<p class=\"simpara\"><em class=\"emphasis bolditalic\">Test Yourself<\/em><\/p>\n<p id=\"ball-ch02_s01_p11\" class=\"para\">Express these numbers in scientific notation.<\/p>\n<ol id=\"ball-ch02_s01_l04\" class=\"orderedlist\">\n<li>23,070<\/li>\n<li>0.0009706<\/li>\n<\/ol>\n<p class=\"simpara\"><em class=\"emphasis\">Answers<\/em><\/p>\n<ol id=\"ball-ch02_s01_l05\" class=\"orderedlist\">\n<li>2.307 \u00d7 10<sup class=\"superscript\">4<\/sup><\/li>\n<li>9.706 \u00d7 10<sup class=\"superscript\">\u22124<\/sup><\/li>\n<\/ol>\n<\/div>\n<p id=\"ball-ch02_s01_p12\" class=\"para editable block\">Another way to determine the power of 10 in scientific notation is to count the number of places you need to move the decimal point to get a numerical value between 1 and 10. The number of places equals the power of 10. This number is positive if you move the decimal point to the right and negative if you move the decimal point to the left.<\/p>\n<p>Many quantities in chemistry are expressed in scientific notation. When performing calculations, you may have to enter a number in scientific notation into a calculator. Be sure you know how to correctly enter a number in scientific notation into your calculator. Different models of calculators require different actions for properly entering scientific notation. If in doubt, consult your instructor immediately.<\/p>\n<div id=\"ball-ch02_s01_f02\" class=\"figure large medium-height editable block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s01_l06\" class=\"itemizedlist\">\n<li>Standard notation expresses a number normally.<\/li>\n<li>Scientific notation expresses a number as a coefficient times a power of 10.<\/li>\n<li>The power of 10 is positive for numbers greater than 1 and negative for numbers between 0 and 1.<\/li>\n<\/ul>\n<figure id=\"attachment_3289\" aria-describedby=\"caption-attachment-3289\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/calc1.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-743\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/calc1-635x1024-1.jpg\" alt=\"This calculator shows only the coefficient and the power of 10 to represent the number in scientific notation. Thus, the number being displayed is 3.84951 \u00d7 1018, or 3,849,510,000,000,000,000. Source: \u201cCasio\u201dAsim Bijarani is licensed under Creative Commons Attribution 2.0 Generic\" width=\"400\" height=\"645\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/calc1-635x1024-1.jpg 635w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/calc1-635x1024-1-186x300.jpg 186w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/calc1-635x1024-1-65x105.jpg 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/calc1-635x1024-1-225x363.jpg 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/calc1-635x1024-1-350x564.jpg 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-3289\" class=\"wp-caption-text\"><strong>Figure 1.<\/strong> This calculator shows only the coefficient and the power of 10 to represent the number in scientific notation. Thus, the number being displayed is 3.84951 \u00d7 10<sup>18<\/sup>, or 3,849,510,000,000,000,000.<br \/>Source: \u201cCasio\u201dAsim Bijarani is licensed under Creative Commons Attribution 2.0 Generic<\/figcaption><\/figure>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol id=\"ball-ch02_s01_qs01_qd01\" class=\"qandadiv\">\n<li id=\"ball-ch02_s01_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p01\" class=\"para\">Express these numbers in scientific notation.<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<p>(a) 56.9 (b) 563,100 (c) 0.0804 (d) 0.00000667<\/p>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p02\" class=\"para\">2. Express these numbers in scientific notation.<\/p>\n<p>(a) \u2212890,000 (b) 602,000,000,000 (c) 0.0000004099 (d) 0.000000000000011<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p03\" class=\"para\">3. Express these numbers in scientific notation.<\/p>\n<p>(a) 0.00656 (b) 65,600 (c) 4,567,000 (d) 0.000005507<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p04\" class=\"para\">4. Express these numbers in scientific notation.<\/p>\n<p>(a) 65 (b) \u2212321.09 (c) 0.000077099 (d) 0.000000000218<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p05\" class=\"para\">5. Express these numbers in standard notation.<\/p>\n<p>(a) 1.381 \u00d7 10<sup class=\"superscript\">5 <\/sup>(b) 5.22 \u00d7 10<sup class=\"superscript\">\u22127 <\/sup>(c) 9.998 \u00d7 10<sup class=\"superscript\">4<\/sup><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p06\" class=\"para\">6. Express these numbers in standard notation.<\/p>\n<p>(a) 7.11 \u00d7 10<sup class=\"superscript\">\u22122 <\/sup>(b) 9.18 \u00d7 10<sup class=\"superscript\">2 <\/sup>(c) 3.09 \u00d7 10<sup class=\"superscript\">\u221210<\/sup><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p07\" class=\"para\">7. Express these numbers in standard notation.<\/p>\n<p>(a) 8.09 \u00d7 10<sup class=\"superscript\">0 <\/sup>(b) 3.088 \u00d7 10<sup class=\"superscript\">\u22125 <\/sup>(c) \u22124.239 \u00d7 10<sup class=\"superscript\">2<\/sup><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p08\" class=\"para\">8. Express these numbers in standard notation.<\/p>\n<p>(a) 2.87 \u00d7 10<sup class=\"superscript\">\u22128 <\/sup>(b) 1.78 \u00d7 10<sup class=\"superscript\">11 <\/sup>(c) 1.381 \u00d7 10<sup class=\"superscript\">\u221223<\/sup><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p09\" class=\"para\">9. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\n<p>(a) 72.44 \u00d7 10<sup class=\"superscript\">3 <\/sup>(b) 9,943 \u00d7 10<sup class=\"superscript\">\u22125 <\/sup>(c) 588,399 \u00d7 10<sup class=\"superscript\">2<\/sup><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p10\" class=\"para\">10. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\n<p>(a) 0.000077 \u00d7 10<sup class=\"superscript\">\u22127 <\/sup>(b) 0.000111 \u00d7 10<sup class=\"superscript\">8 <\/sup>(c) 602,000 \u00d7 10<sup class=\"superscript\">18<\/sup><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p11\" class=\"para\">11. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\n<p>(a) 345.1 \u00d7 10<sup class=\"superscript\">2 <\/sup>(b) 0.234 \u00d7 10<sup class=\"superscript\">\u22123 <\/sup>(c) 1,800 \u00d7 10<sup class=\"superscript\">\u22122<\/sup><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p12\" class=\"para\">12. These numbers are not written in proper scientific notation. Rewrite them so that they are in proper scientific notation.<\/p>\n<p>(a) 8,099 \u00d7 10<sup class=\"superscript\">\u22128 <\/sup>(b) 34.5 \u00d7 10<sup class=\"superscript\">0 <\/sup>(c) 0.000332 \u00d7 10<sup class=\"superscript\">4<\/sup><\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p13\" class=\"para\">13. Write these numbers in scientific notation by counting the number of places the decimal point is moved.<\/p>\n<p>(a) 123,456.78 (b) 98,490 (c) 0.000000445<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p14\" class=\"para\">14. Write these numbers in scientific notation by counting the number of places the decimal point is moved.<\/p>\n<p>(a) 0.000552 (b) 1,987 (c) 0.00000000887<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p15\" class=\"para\">15. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\n<p>(a) 456 \u00d7 (7.4 \u00d7 10<sup class=\"superscript\">8<\/sup>) = ? (b) (3.02 \u00d7 10<sup class=\"superscript\">5<\/sup>) \u00f7 (9.04 \u00d7 10<sup class=\"superscript\">15<\/sup>) = ? (c) 0.0044 \u00d7 0.000833 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p16\" class=\"para\">16. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\n<p>(a) 98,000 \u00d7 23,000 = ? (b) 98,000 \u00f7 23,000 = ? (c) (4.6 \u00d7 10<sup class=\"superscript\">\u22125<\/sup>) \u00d7 (2.09 \u00d7 10<sup class=\"superscript\">3<\/sup>) = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p17\" class=\"para\">17. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\n<p>(a) 45 \u00d7 132 \u00f7 882 = ? (b) [(6.37 \u00d7 10<sup class=\"superscript\">4<\/sup>) \u00d7 (8.44 \u00d7 10<sup class=\"superscript\">\u22124<\/sup>)] \u00f7 (3.2209 \u00d7 10<sup class=\"superscript\">15<\/sup>) = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s01_qs01_p18\" class=\"para\">18. Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation.<\/p>\n<p>(a) (9.09 \u00d7 10<sup class=\"superscript\">8<\/sup>) \u00f7 [(6.33 \u00d7 10<sup class=\"superscript\">9<\/sup>) \u00d7 (4.066 \u00d7 10<sup class=\"superscript\">\u22127<\/sup>)] = ? (b) 9,345 \u00d7 34.866 \u00f7 0.00665 = ?<\/p>\n<\/div>\n<\/div>\n<div class=\"layoutArea\">\n<div class=\"column\">\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Problems &amp; Exercises<br \/>\n<\/strong><\/p>\n<p><strong>1.<\/strong> a) 5.69 \u00d7 10<sup>1 <\/sup>b) 5.631 \u00d7 10<sup>5 <\/sup>c) 8.04 \u00d7 10<sup>\u22122 <\/sup>d) 6.67 \u00d7 10<sup>\u22126<\/sup><\/p>\n<p><strong>3.<\/strong> a) 6.56 \u00d7 10<sup>\u22123 <\/sup>b) 6.56 \u00d7 10<sup>4 <\/sup>c) 4.567 \u00d7 10<sup>6 <\/sup>d) 5.507 \u00d7 10<sup>\u22126<\/sup><\/p>\n<p><strong>5.<\/strong> a) 138,100 b) 0.000000522 c) 99,980<\/p>\n<p><strong>7.<\/strong> a) 8.09 b) 0.00003088 c) \u2212423.9<\/p>\n<p><strong>9.<\/strong> a) 7.244 \u00d7 10<sup>4 <\/sup>b) 9.943 \u00d7 10<sup>\u22122 <\/sup>c) 5.88399 \u00d7 10<sup>7<\/sup><br \/>\n<strong>11.<\/strong> a) 3.451 \u00d7 10<sup>4 <\/sup>b) 2.34 \u00d7 10<sup>\u22124 <\/sup>c) 1.8 \u00d7 10<sup>1<\/sup><\/p>\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p><strong>13.<\/strong> a) 1.2345678 \u00d7 10<sup>5 <\/sup>b) 9.849 \u00d7 10<sup>4 <\/sup>c) 4.45 \u00d7 10<sup>\u22127<\/sup><\/p>\n<p><strong>15.<\/strong> a) 3.3744 \u00d7 10<sup>11 <\/sup>b) 3.3407 \u00d7 10<sup>\u221211 <\/sup>c) 3.665 \u00d7 10<sup>\u22126<\/sup><\/p>\n<p><strong>17.<\/strong> a) 6.7346 \u00d7 10<sup>0 <\/sup>b) 1.6691 \u00d7 10<sup>\u221214<\/sup><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1><a id=\"sig-figs\" href=\"\"><\/a>Significant Figures<\/h1>\n<div id=\"ball-ch02_s03\" class=\"section\" lang=\"en\">\n<div id=\"ball-ch02_s03_n01\" class=\"learning_objectives editable block\">\n<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ol id=\"ball-ch02_s03_l01\">\n<li>Apply the concept of significant figures to limit a measurement to the proper number of digits.<\/li>\n<li>Recognize the number of significant figures in a given quantity.<\/li>\n<li>Limit mathematical results to the proper number of significant figures.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p id=\"ball-ch02_s03_p01\" class=\"para editable block\">If you use a calculator to evaluate the expression 337\/217, you will get the following:<\/p>\n<p><span class=\"informalequation block\">337217=1.55299539171&#8230;<\/span><\/p>\n<p id=\"ball-ch02_s03_p02\" class=\"para editable block\">and so on for many more digits. Although this answer is correct, it is somewhat presumptuous. You start with two values that each have three digits, and the answer has <em class=\"emphasis\">twelve<\/em> digits? That does not make much sense from a strict numerical point of view.<\/p>\n<p id=\"ball-ch02_s03_p03\" class=\"para editable block\">Consider using a ruler to measure the width of an object, as shown in <a class=\"xref\" href=\"#ball-ch02_s03_f01\">Figure 2.6 &#8220;Expressing Width&#8221;<\/a>. The object is definitely more than 1 cm long, so we know that the first digit in our measurement is 1. We see by counting the tick marks on the ruler that the object is at least three ticks after the 1. If each tick represents 0.1 cm, then we know the object is at least 1.3 cm wide. But our ruler does not have any more ticks between the 0.3 and the 0.4 marks, so we can\u2019t know exactly how much the next decimal place is. But with a practiced eye we can estimate it. Let us estimate it as about six-tenths of the way between the third and fourth tick marks, which estimates our hundredths place as 6, so we identify a measurement of 1.36 cm for the width of the object.<\/p>\n<div id=\"ball-ch02_s03_f01\" class=\"figure large medium-height editable block\">\n<figure style=\"width: 482px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/eb7d58a9777dbb124396d7c8bcb75793-1.jpg\" alt=\"image\" width=\"482\" height=\"391\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> Expressing Width<\/figcaption><\/figure>\n<p class=\"para\">What is the proper way to express the width of this object?<\/p>\n<\/div>\n<p id=\"ball-ch02_s03_p04\" class=\"para editable block\">Does it make any sense to try to report a thousandths place for the measurement? No, it doesn\u2019t; we are not exactly sure of the hundredths place (after all, it was an estimate only), so it would be fruitless to estimate a thousandths place. Our best measurement, then, stops at the hundredths place, and we report 1.36 cm as proper measurement.<\/p>\n<p id=\"ball-ch02_s03_p05\" class=\"para editable block\">This concept of reporting the proper number of digits in a measurement or a calculation is called <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">significant figures<\/a><\/span>. Significant figures (sometimes called significant digits) represent the limits of what values of a measurement or a calculation we are sure of. The convention for a measurement is that the quantity reported should be all known values and the first estimated value. The conventions for calculations are discussed as follows.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 3<\/h3>\n<p id=\"ball-ch02_s03_p06\" class=\"para\">Use each diagram to report a measurement to the proper number of significant figures.<\/p>\n<ol id=\"ball-ch02_s03_l02\" class=\"orderedlist\">\n<li>\n<div id=\"ball-ch02_s03_f02\" class=\"informalfigure small\">\n<figure style=\"width: 599px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/151d73b2e1318e4386f1be7579009b82-1.jpg\" alt=\"image\" width=\"599\" height=\"599\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong> Pressure gauge in units of pounds per square inch<\/figcaption><\/figure>\n<\/div>\n<\/li>\n<li>\n<div id=\"ball-ch02_s03_f03\" class=\"informalfigure small\">\n<figure id=\"attachment_4613\" aria-describedby=\"caption-attachment-4613\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-746\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-1.png\" alt=\"Ruler\" width=\"400\" height=\"255\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-1-300x192.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-1-65x41.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-1-225x144.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-1-350x223.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-4613\" class=\"wp-caption-text\"><strong>Figure 3.<\/strong> A measuring ruler<\/figcaption><\/figure>\n<\/div>\n<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n<ol id=\"ball-ch02_s03_l03\" class=\"orderedlist\">\n<li>The arrow is between 4.0 and 5.0, so the measurement is at least 4.0. The arrow is between the third and fourth small tick marks, so it\u2019s at least 0.3. We will have to estimate the last place. It looks like about one-third of the way across the space, so let us estimate the hundredths place as 3. Combining the digits, we have a measurement of 4.33 psi (psi stands for \u201cpounds per square inch\u201d and is a unit of pressure, like air in a tire). We say that the measurement is reported to three significant figures.<\/li>\n<li>The rectangle is at least 1.0 cm wide but certainly not 2.0 cm wide, so the first significant digit is 1. The rectangle\u2019s width is past the second tick mark but not the third; if each tick mark represents 0.1, then the rectangle is at least 0.2 in the next significant digit. We have to estimate the next place because there are no markings to guide us. It appears to be about halfway between 0.2 and 0.3, so we will estimate the next place to be a 5. Thus, the measured width of the rectangle is 1.25 cm. Again, the measurement is reported to three significant figures.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"section\" lang=\"en\"><\/div>\n<div class=\"section\" lang=\"en\">\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 1<\/h3>\n<p id=\"ball-ch02_s03_p07\" class=\"para\">What would be the reported width of this rectangle?<\/p>\n<figure id=\"attachment_4615\" aria-describedby=\"caption-attachment-4615\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Rectangle.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-747\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Rectangle-1.png\" alt=\"Rectangle\" width=\"400\" height=\"255\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Rectangle-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Rectangle-1-300x192.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Rectangle-1-65x41.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Rectangle-1-225x144.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Rectangle-1-350x223.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-4615\" class=\"wp-caption-text\"><strong>Figure 4.<\/strong> A measuring ruler<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"ball-ch02_s03_p09\" class=\"para editable block\">In many cases, you will be given a measurement. How can you tell by looking what digits are significant? For example, the reported population of the United States is 306,000,000. Does that mean that it is <em class=\"emphasis\">exactly<\/em> three hundred six million or is some estimation occurring?<\/p>\n<p id=\"ball-ch02_s03_p10\" class=\"para editable block\">The following conventions dictate which numbers in a reported measurement are significant and which are not significant:<\/p>\n<ol id=\"ball-ch02_s03_l04\" class=\"orderedlist editable block\">\n<li>Any nonzero digit is significant.<\/li>\n<li>Any zeros between nonzero digits (i.e., embedded zeros) are significant.<\/li>\n<li>Zeros at the end of a number without a decimal point (i.e., trailing zeros) are not significant; they serve only to put the significant digits in the correct positions. However, zeros at the end of any number with a decimal point are significant.<\/li>\n<li>Zeros at the beginning of a decimal number (i.e., leading zeros) are not significant; again, they serve only to put the significant digits in the correct positions.<\/li>\n<\/ol>\n<p id=\"ball-ch02_s03_p11\" class=\"para editable block\">So, by these rules, the population figure of the United States has only three significant figures: the 3, the 6, and the zero between them. The remaining six zeros simply put the 306 in the millions position.<\/p>\n<div id=\"ball-ch02_s03_f05\" class=\"figure large medium-height editable block\"><\/div>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 4<\/h3>\n<p id=\"ball-ch02_s03_p12\" class=\"para\">Give the number of significant figures in each measurement.<\/p>\n<ol id=\"ball-ch02_s03_l05\" class=\"orderedlist\">\n<li>36.7 m<\/li>\n<li>0.006606 s<\/li>\n<li>2,002 kg<\/li>\n<li>306,490,000 people<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n<ol id=\"ball-ch02_s03_l06\" class=\"orderedlist\">\n<li>By rule 1, all nonzero digits are significant, so this measurement has three significant figures.<\/li>\n<li>By rule 4, the first three zeros are not significant, but by rule 2 the zero between the sixes is; therefore, this number has four significant figures.<\/li>\n<li>By rule 2, the two zeros between the twos are significant, so this measurement has four significant figures.<\/li>\n<li>The four trailing zeros in the number are not significant, but the other five numbers are, so this number has five significant figures.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 2<\/h3>\n<p id=\"ball-ch02_s03_p13\" class=\"para\">Give the number of significant figures in each measurement.<\/p>\n<ol id=\"ball-ch02_s03_l07\" class=\"orderedlist\">\n<li>0.000601 m<\/li>\n<li>65.080 kg<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"section\" lang=\"en\">\n<p id=\"ball-ch02_s03_p14\" class=\"para editable block\">How are significant figures handled in calculations? It depends on what type of calculation is being performed. If the calculation is an addition or a subtraction, the rule is as follows: limit the reported answer to the rightmost column that all numbers have significant figures in common. For example, if you were to add 1.2 and 4.71, we note that the first number stops its significant figures in the tenths column, while the second number stops its significant figures in the hundredths column. We therefore limit our answer to the tenths column.<\/p>\n<figure id=\"attachment_4616\" aria-describedby=\"caption-attachment-4616\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Sig-Figs-1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-748\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-1-1.png\" alt=\"Sig Figs 1\" width=\"400\" height=\"85\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-1-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-1-1-300x64.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-1-1-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-1-1-225x48.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-1-1-350x74.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-4616\" class=\"wp-caption-text\"><strong>Figure 5.<\/strong> Math<\/figcaption><\/figure>\n<div id=\"fwk-ball-eq02_001\" class=\"informalfigure large block\">\n<p id=\"ball-ch02_s03_p15\" class=\"para editable block\">We drop the last digit\u2014the 1\u2014because it is not significant to the final answer.<\/p>\n<p id=\"ball-ch02_s03_p16\" class=\"para editable block\">The dropping of positions in sums and differences brings up the topic of rounding. Although there are several conventions, in this text we will adopt the following rule: the final answer should be rounded up if the first dropped digit is 5 or greater and rounded down if the first dropped digit is less than 5.<\/p>\n<figure id=\"attachment_4617\" aria-describedby=\"caption-attachment-4617\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Sig-Figs-2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-749\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-2-1.png\" alt=\"Sig Figs 2\" width=\"400\" height=\"85\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-2-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-2-1-300x64.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-2-1-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-2-1-225x48.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Sig-Figs-2-1-350x74.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-4617\" class=\"wp-caption-text\"><strong>Figure 6.<\/strong> More Math<\/figcaption><\/figure>\n<div id=\"fwk-ball-eq02_002\" class=\"informalfigure large block\">\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 5<\/h3>\n<p id=\"ball-ch02_s03_p17\" class=\"para\">Express the final answer to the proper number of significant figures.<\/p>\n<ol id=\"ball-ch02_s03_l09\" class=\"orderedlist\">\n<li>101.2 + 18.702 = ?<\/li>\n<li>202.88 \u2212 1.013 = ?<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n<ol id=\"ball-ch02_s03_l10\" class=\"orderedlist\">\n<li>If we use a calculator to add these two numbers, we would get 119.902. However, most calculators do not understand significant figures, and we need to limit the final answer to the tenths place. Thus, we drop the 02 and report a final answer of 119.9 (rounding down).<\/li>\n<li>A calculator would answer 201.867. However, we have to limit our final answer to the hundredths place. Because the first number being dropped is 7, which is greater than 7, we round up and report a final answer of 201.87.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 3<\/h3>\n<p id=\"ball-ch02_s03_p18\" class=\"para\">Express the answer for 3.445 + 90.83 \u2212 72.4 to the proper number of significant figures.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"ball-ch02_s03_p20\" class=\"para editable block\">If the operations being performed are multiplication or division, the rule is as follows: limit the answer to the number of significant figures that the data value with the <em class=\"emphasis\">least<\/em> number of significant figures has. So if we are dividing 23 by 448, which have two and three significant figures each, we should limit the final reported answer to two significant figures (the lesser of two and three significant figures):<\/p>\n<p><span class=\"informalequation block\">23448=0.051339286&#8230;=0.051<\/span><\/p>\n<p id=\"ball-ch02_s03_p21\" class=\"para editable block\">The same rounding rules apply in multiplication and division as they do in addition and subtraction.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 6<\/h3>\n<p id=\"ball-ch02_s03_p22\" class=\"para\">Express the final answer to the proper number of significant figures.<\/p>\n<ol id=\"ball-ch02_s03_l11\" class=\"orderedlist\">\n<li>76.4 \u00d7 180.4 = ?<\/li>\n<li>934.9 \u00f7 0.00455 = ?<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n<ol id=\"ball-ch02_s03_l12\" class=\"orderedlist\">\n<li>The first number has three significant figures, while the second number has four significant figures. Therefore, we limit our final answer to three significant figures: 76.4 \u00d7 180.4 = 13,782.56 = 13,800.<\/li>\n<li>The first number has four significant figures, while the second number has three significant figures. Therefore we limit our final answer to three significant figures: 934.9 \u00f7 0.00455 = 205,472.5275\u2026 = 205,000.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 4<\/h3>\n<p id=\"ball-ch02_s03_p23\" class=\"para\">Express the final answer to the proper number of significant figures.<\/p>\n<ol id=\"ball-ch02_s03_l13\" class=\"orderedlist\">\n<li>22.4 \u00d7 8.314 = ?<\/li>\n<li>1.381 \u00f7 6.02 = ?<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"ball-ch02_s03_p24\" class=\"para editable block\">As you have probably realized by now, the biggest issue in determining the number of significant figures in a value is the zero. Is the zero significant or not? One way to unambiguously determine whether a zero is significant or not is to write a number in scientific notation. Scientific notation will include zeros in the coefficient of the number <em class=\"emphasis\">only if they are significant<\/em>. Thus, the number 8.666 \u00d7 10<sup class=\"superscript\">6<\/sup> has four significant figures. However, the number 8.6660 \u00d7 10<sup class=\"superscript\">6<\/sup> has five significant figures. That last zero is significant; if it were not, it would not be written in the coefficient. So when in doubt about expressing the number of significant figures in a quantity, use scientific notation and include the number of zeros that are truly significant.<\/p>\n<figure id=\"attachment_3960\" aria-describedby=\"caption-attachment-3960\" style=\"width: 150px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/qrcode.23437479.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-74 size-thumbnail\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/qrcode.23437479-150x150-1.png\" alt=\"qrcode.23437479\" width=\"150\" height=\"150\" \/><\/a><figcaption id=\"caption-attachment-3960\" class=\"wp-caption-text\"><strong>Figure 7.<\/strong> Video source: Significant figures by keyj (https:\/\/viutube.viu.ca\/public\/media\/Significant+Figures\/0_0j38j93r)<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<div id=\"ball-ch02_s03_qs01\" class=\"qandaset block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s03_l15\" class=\"itemizedlist\">\n<li>Significant figures in a quantity indicate the number of known values plus one place that is estimated.<\/li>\n<li>There are rules for which numbers in a quantity are significant and which are not significant.<\/li>\n<li>In calculations involving addition and subtraction, limit significant figures based on the rightmost place that all values have in common.<\/li>\n<li>In calculations involving multiplication and division, limit significant figures to the least number of significant figures in all the data values.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol id=\"ball-ch02_s03_qs01_qd01\" class=\"qandadiv\">\n<li id=\"ball-ch02_s03_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p1\" class=\"para\">Express each measurement to the correct number of significant figures.<\/p>\n<figure style=\"width: 599px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/f85b7d0b1d3f3563a5b973ef04349df3-1.jpg\" alt=\"image\" width=\"599\" height=\"599\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong> Pressure gauge in units of pounds per square inch<\/figcaption><\/figure>\n<p class=\"para\">a)<\/p>\n<figure id=\"attachment_751\" aria-describedby=\"caption-attachment-751\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler-2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-752\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-2-1.png\" alt=\"Ruler-2\" width=\"400\" height=\"255\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-2-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-2-1-300x192.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-2-1-65x41.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-2-1-225x144.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-2-1-350x223.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-751\" class=\"wp-caption-text\"><strong>Figure 9.<\/strong> A measuring ruler<\/figcaption><\/figure>\n<p class=\"para\">b)<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s03_qs01_qd01_qa02\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p2\" class=\"para\">Express each measurement to the correct number of significant figures.<\/p>\n<figure style=\"width: 599px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/06979677f6f13c11ea559e495ebcbf85-1.jpg\" alt=\"image\" width=\"599\" height=\"599\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10.<\/strong> Pressure gauge in units of pounds per square inch<\/figcaption><\/figure>\n<p class=\"para\">a)<\/p>\n<figure id=\"attachment_753\" aria-describedby=\"caption-attachment-753\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Ruler-3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-754\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-3-1.png\" alt=\"Ruler-3\" width=\"400\" height=\"255\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-3-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-3-1-300x192.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-3-1-65x41.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-3-1-225x144.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Ruler-3-1-350x223.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-753\" class=\"wp-caption-text\"><strong>Figure 11.<\/strong> A measuring ruler<\/figcaption><\/figure>\n<p class=\"para\">b)<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s03_qs01_qd01_qa03\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p3\" class=\"para\">How many significant figures do these numbers have?<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<p>(a) 23 (b) 23.0 (c) 0.00023 (d) 0.0002302<\/p>\n<p>4. How many significant figures do these numbers have?<\/p>\n<p>(a) 5.44 \u00d7 10<sup class=\"superscript\">8 <\/sup>(b) 1.008 \u00d7 10<sup class=\"superscript\">\u22125 <\/sup>(c) 43.09 (d) 0.0000001381<\/p>\n<p>5. How many significant figures do these numbers have?<\/p>\n<p>(a) 765,890 (b) 765,890.0 (c) 1.2000 \u00d7 10<sup class=\"superscript\">5 <\/sup>(d) 0.0005060<\/p>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p6\" class=\"para\">6) How many significant figures do these numbers have?<\/p>\n<p>(a) 0.009 (b) 0.0000009 (c) 65,444 (d) 65,040<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p7\" class=\"para\">7. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p>(a) 56.0 + 3.44 = ? (b) 0.00665 + 1.004 = ? (c) 45.99 \u2212 32.8 = ? (d) 45.99 \u2212 32.8 + 75.02 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p8\" class=\"para\">8. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p>(a) 1.005 + 17.88 = ? (b) 56,700 \u2212 324 = ? (c) 405,007 \u2212 123.3 = ? (d) 55.5 + 66.66 \u2212 77.777 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p9\" class=\"para\">9. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p>(a) 56.7 \u00d7 66.99 = ? (b) 1.000 \u00f7 77 = ? (c) 1.000 \u00f7 77.0 = ? (d) 6.022 \u00d7 1.89 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p10\" class=\"para\">10. Compute and express each answer with the proper number of significant figures, rounding as necessary.<\/p>\n<p>(a) 0.000440 \u00d7 17.22 = ? (b) 203,000 \u00f7 0.044 = ? (c) 67 \u00d7 85.0 \u00d7 0.0028 = ? (d) 999,999 \u00f7 3,310 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p>11. Write the number 87,449 in scientific notation with four significant figures.<\/p>\n<p>12. Write the number 0.000066600 in scientific notation with five significant figures.<\/p>\n<\/div>\n<div class=\"question\">\n<p>13. Write the number 306,000,000 in scientific notation to the proper number of significant figures.<\/p>\n<p>14. Write the number 0.0000558 in scientific notation with two significant figures.<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p13\" class=\"para\">15. Perform each calculation and limit each answer to three significant figures.<\/p>\n<p>(a) 67,883 \u00d7 0.004321 = ? (b) (9.67 \u00d7 10<sup class=\"superscript\">3<\/sup>) \u00d7 0.0055087 = ?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s03_qs01_p14\" class=\"para\">16. Perform each calculation and limit each answer to four significant figures.<\/p>\n<p>(a) 18,900 \u00d7 76.33 \u00f7 0.00336 = ? (b) 0.77604 \u00f7 76,003 \u00d7 8.888 = ?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Check Your Understanding 1<\/strong><\/p>\n<p>0.63 cm<\/p>\n<p><strong>Check Your Understanding 2<br \/>\n<\/strong><\/p>\n<ol id=\"ball-ch02_s03_l08\" class=\"orderedlist\">\n<li>three significant figures<\/li>\n<li>five significant figures<\/li>\n<\/ol>\n<p><strong>Check Your Understanding 3<br \/>\n<\/strong><\/p>\n<p>21.9<\/p>\n<p><strong>Check Your Understanding 4<\/strong><\/p>\n<ol id=\"ball-ch02_s03_l14\" class=\"orderedlist\">\n<li>186<\/li>\n<li>0.229<\/li>\n<\/ol>\n<p><strong>Problems &amp; Exercises<\/strong><\/p>\n<p><strong>1.<\/strong> (a) 375 psi (b) 1.30 cm<\/p>\n<p><strong>3.<\/strong> (a) two (b) three (c) two (d) four<\/p>\n<p><strong>5.<\/strong> (a) five (b) seven (c) five (d) four<\/p>\n<p><strong>7.<\/strong> (a) 59.4 (b) 1.011 (c) 13.2 (d) 88.2<\/p>\n<p><strong>9.<\/strong> (a) 3.80 \u00d7 10<sup class=\"superscript\">3 <\/sup>(b) 0.013 (c) 0.0130 (d) 11.4<\/p>\n<p><strong>11.<\/strong> (a) 8.745 \u00d7 10<sup class=\"superscript\">4 <\/sup>(b) 6.6600 \u00d7 10<sup class=\"superscript\">\u22125<\/sup><\/p>\n<p><strong>13.<\/strong> (a) 293 (b) 53.3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1><a id=\"conv-units\" href=\"\"><\/a>Converting Units<\/h1>\n<p id=\"ball-ch02_s04_p01\" class=\"para editable block\">In <a class=\"xref\" href=\"\/douglasphys1107\/part\/appendix-c-units-numbers-and-significant-figures\/#exp-units\">&#8220;Expressing Units&#8221;<\/a>, we showed some examples of how to replace initial units with other units of the same type to get a numerical value that is easier to comprehend. In this section, we will formalize the process.<\/p>\n<p id=\"ball-ch02_s04_p02\" class=\"para editable block\">Consider a simple example: how many feet are there in 4 yards? Most people will almost automatically answer that there are 12 feet in 4 yards. How did you make this determination? Well, if there are 3 feet in 1 yard and there are 4 yards, then there are 4 \u00d7 3 = 12 feet in 4 yards.<\/p>\n<p id=\"ball-ch02_s04_p03\" class=\"para editable block\">This is correct, of course, but it is informal. Let us formalize it in a way that can be applied more generally. We know that 1 yard (yd) equals 3 feet (ft):<\/p>\n<p><span class=\"informalequation block\"><span class=\"mathphrase\">1 yd = 3 ft<\/span><\/span><\/p>\n<p id=\"ball-ch02_s04_p04\" class=\"para editable block\">In math, this expression is called an <em class=\"emphasis\">equality<\/em>. The rules of algebra say that you can change (i.e., multiply or divide or add or subtract) the equality (as long as you don\u2019t divide by zero) and the new expression will still be an equality. For example, if we divide both sides by 2, we get<\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/Converting_Units_1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-755 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Converting_Units_1.png\" alt=\"1\/2 yd = 3\/2 feet\" width=\"237\" height=\"107\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Converting_Units_1.png 237w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Converting_Units_1-65x29.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Converting_Units_1-225x102.png 225w\" sizes=\"auto, (max-width: 237px) 100vw, 237px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p05\" class=\"para editable block\">We see that one-half of a yard equals 3\/2, or one and a half, feet\u2014something we also know to be true, so the above equation is still an equality. Going back to the original equality, suppose we divide both sides of the equation by 1 yard (number <em class=\"emphasis\">and<\/em> unit):<\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-756 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_2.png\" alt=\"1\/1 yd = 3 ft\/ 1 yd\" width=\"232\" height=\"115\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_2.png 232w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_2-65x32.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_2-225x112.png 225w\" sizes=\"auto, (max-width: 232px) 100vw, 232px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p06\" class=\"para editable block\">The expression is still an equality, by the rules of algebra. The left fraction equals 1. It has the same quantity in the numerator and the denominator, so it must equal 1. The quantities in the numerator and denominator cancel, both the number <em class=\"emphasis\">and<\/em> the unit:<\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-757 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_3.png\" alt=\"1\/1 yd = 3 ft \/ 1 yd (cancelled units crossed out)\" width=\"215\" height=\"128\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_3.png 215w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_3-65x39.png 65w\" sizes=\"auto, (max-width: 215px) 100vw, 215px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p07\" class=\"para editable block\">When everything cancels in a fraction, the fraction reduces to 1:<\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-758 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_4.png\" alt=\"1 = 3 ft\/1 yd\" width=\"182\" height=\"97\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_4.png 182w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_4-65x35.png 65w\" sizes=\"auto, (max-width: 182px) 100vw, 182px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p08\" class=\"para block\">We have an expression, <span class=\"inlineequation\">3 ft1 yd<\/span>, that equals 1. This is a strange way to write 1, but it makes sense: 3 ft equal 1 yd, so the quantities in the numerator and denominator are the same quantity, just expressed with different units. The expression <span class=\"inlineequation\">3 ft1 yd<\/span> is called a <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">conversion factor<\/a><\/span>, and it is used to formally change the unit of a quantity into another unit. (The process of converting units in such a formal fashion is sometimes called <em class=\"emphasis\">dimensional analysis<\/em> or the <em class=\"emphasis\">factor label method<\/em>.)<\/p>\n<p id=\"ball-ch02_s04_p09\" class=\"para editable block\">To see how this happens, let us start with the original quantity:<\/p>\n<p><span class=\"informalequation block\"><span class=\"mathphrase\">4 yd<\/span><\/span><\/p>\n<p id=\"ball-ch02_s04_p10\" class=\"para block\">Now let us multiply this quantity by 1. When you multiply anything by 1, you don\u2019t change the value of the quantity. Rather than multiplying by just 1, let us write 1 as <span class=\"inlineequation\">3 ft1 yd<\/span>:<\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-759 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_5.png\" alt=\"4 yd x (3ft\/1yd)\" width=\"229\" height=\"113\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_5.png 229w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_5-65x32.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_5-225x111.png 225w\" sizes=\"auto, (max-width: 229px) 100vw, 229px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p11\" class=\"para block\">The 4 yd term can be thought of as <span class=\"inlineequation\">4 yd\/1<\/span>; that is, it can be thought of as a fraction with 1 in the denominator. We are essentially multiplying fractions. If the same thing appears in the numerator and denominator of a fraction, they cancel. In this case, what cancels is the unit <em class=\"emphasis\">yard<\/em>:<\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_6.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-760 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_6.png\" alt=\"4 yd x (3 ft\/ 1 yd) showing units cancel\" width=\"218\" height=\"115\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_6.png 218w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_6-65x34.png 65w\" sizes=\"auto, (max-width: 218px) 100vw, 218px\" \/><\/a><\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_7.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-761 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_7.png\" alt=\"(4 x 3 ft)\/1 = 12 ft\/1 = 12 ft\" width=\"405\" height=\"107\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_7.png 405w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_7-300x79.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_7-65x17.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_7-225x59.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_7-350x92.png 350w\" sizes=\"auto, (max-width: 405px) 100vw, 405px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p12\" class=\"para editable block\">That is all that we can cancel. Now, multiply and divide all the numbers to get the final answer:<\/p>\n<p id=\"ball-ch02_s04_p13\" class=\"para editable block\">Again, we get an answer of 12 ft, just as we did originally. But in this case, we used a more formal procedure that is applicable to a variety of problems.<\/p>\n<p id=\"ball-ch02_s04_p14\" class=\"para editable block\">How many millimeters are in 14.66 m? To answer this, we need to construct a conversion factor between millimeters and meters and apply it correctly to the original quantity. We start with the definition of a millimeter, which is<\/p>\n<p><span class=\"informalequation block\"><span class=\"mathphrase\">1 mm = 1\/1,000 m<\/span><\/span><\/p>\n<p id=\"ball-ch02_s04_p15\" class=\"para editable block\">The 1\/1,000 is what the prefix <em class=\"emphasis\">milli-<\/em> means. Most people are more comfortable working without fractions, so we will rewrite this equation by bringing the 1,000 into the numerator of the other side of the equation:<\/p>\n<p><span class=\"informalequation block\"><span class=\"mathphrase\">1,000 mm = 1 m<\/span><\/span><\/p>\n<p id=\"ball-ch02_s04_p16\" class=\"para editable block\">Now we construct a conversion factor by dividing one quantity into both sides. But now a question arises: which quantity do we divide by? It turns out that we have two choices, and the two choices will give us different conversion factors, both of which equal 1:<\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_8.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-762 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_8.png\" alt=\"conversion facts equaling 1 m \/ 1000 mm\" width=\"672\" height=\"120\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_8.png 672w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_8-300x54.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_8-65x12.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_8-225x40.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_8-350x63.png 350w\" sizes=\"auto, (max-width: 672px) 100vw, 672px\" \/><\/a><\/span><\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_23.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-763 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_23.png\" alt=\"conversion factor 1m \/ 1000 mm\" width=\"512\" height=\"107\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_23.png 512w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_23-300x63.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_23-65x14.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_23-225x47.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_23-350x73.png 350w\" sizes=\"auto, (max-width: 512px) 100vw, 512px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p17\" class=\"para editable block\">Which conversion factor do we use? The answer is based on <em class=\"emphasis\">what unit you want to get rid of in your initial quantity<\/em>. The original unit of our quantity is meters, which we want to convert to millimeters. Because the original unit is assumed to be in the numerator, to get rid of it, we want the meter unit in the <em class=\"emphasis\">denominator<\/em>; then they will cancel. Therefore, we will use the second conversion factor. Canceling units and performing the mathematics, we get<\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_10.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-764 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_10.png\" alt=\"14.66 m x (1000 mm\/1 m) = 14660 mm\" width=\"498\" height=\"141\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_10.png 498w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_10-300x85.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_10-65x18.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_10-225x64.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_10-350x99.png 350w\" sizes=\"auto, (max-width: 498px) 100vw, 498px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s04_p18\" class=\"para editable block\">Note how m cancels, leaving mm, which is the unit of interest.<\/p>\n<p id=\"ball-ch02_s04_p19\" class=\"para editable block\">The ability to construct and apply proper conversion factors is a very powerful mathematical technique in chemistry. You need to master this technique if you are going to be successful in this and future courses.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 7<\/h3>\n<ol id=\"ball-ch02_s04_l02\" class=\"orderedlist\">\n<li>Convert 35.9 kL to liters.<\/li>\n<li>Convert 555 nm to meters.<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n<ol id=\"ball-ch02_s04_l03\" class=\"orderedlist\">\n<li>\n<p class=\"para\">We will use the fact that 1 kL = 1,000 L. Of the two conversion factors that can be defined, the one that will work is <span class=\"inlineequation\">1,000 L\/1 kL<\/span>. Applying this conversion factor, we get<\/p>\n<p><span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_11.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-765 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_11.png\" alt=\"35.9 kL x (1000 L\/1 kL) = 35900 L\" width=\"417\" height=\"113\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_11.png 417w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_11-300x81.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_11-65x18.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_11-225x61.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_11-350x95.png 350w\" sizes=\"auto, (max-width: 417px) 100vw, 417px\" \/><\/a><\/span><\/li>\n<li>\n<p class=\"para\">We will use the fact that 1 nm = 1\/1,000,000,000 m, which we will rewrite as 1,000,000,000 nm = 1 m, or 10<sup class=\"superscript\">9<\/sup> nm = 1 m. Of the two possible conversion factors, the appropriate one has the nm unit in the denominator: <span class=\"inlineequation\">1 m\/10<sup>9<\/sup> nm<\/span>. Applying this conversion factor, we get<\/p>\n<p><span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_12.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-766 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_12.png\" alt=\"555 nm x (1 m\/ 10^9 nm) = 5.55 x 10^-7 m\" width=\"785\" height=\"129\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_12.png 785w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_12-300x49.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_12-768x126.png 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_12-65x11.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_12-225x37.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_12-350x58.png 350w\" sizes=\"auto, (max-width: 785px) 100vw, 785px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s04_p20\" class=\"para\">In the final step, we expressed the answer in scientific notation.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 1<\/h3>\n<ol id=\"ball-ch02_s04_l04\" class=\"orderedlist\">\n<li>Convert 67.08 \u03bcL to liters.<\/li>\n<li>Convert 56.8 m to kilometers.<\/li>\n<\/ol>\n<\/div>\n<p id=\"ball-ch02_s04_p21\" class=\"para editable block\">What if we have a derived unit that is the product of more than one unit, such as m<sup class=\"superscript\">2<\/sup>? Suppose we want to convert square meters to square centimeters? The key is to remember that m<sup class=\"superscript\">2<\/sup> means m \u00d7 m, which means we have <em class=\"emphasis\">two<\/em> meter units in our derived unit. That means we have to include <em class=\"emphasis\">two<\/em> conversion factors, one for each unit. For example, to convert 17.6 m<sup class=\"superscript\">2<\/sup> to square centimeters, we perform the conversion as follows:<\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_13.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-767 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13.png\" alt=\"17.6 m^2 = 17.6 (mxm) x (100cm\/1m) x (100cm\/1m)=176000cm^2\" width=\"1188\" height=\"115\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13.png 1188w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13-300x29.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13-1024x99.png 1024w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13-768x74.png 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13-65x6.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13-225x22.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_13-350x34.png 350w\" sizes=\"auto, (max-width: 1188px) 100vw, 1188px\" \/><\/a><\/span><\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 8<\/h3>\n<p id=\"ball-ch02_s04_p22\" class=\"para\">How many cubic centimeters are in 0.883 m<sup class=\"superscript\">3<\/sup>?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s04_p23\" class=\"para\">With an exponent of 3, we have three length units, so by extension we need to use three conversion factors between meters and centimeters. Thus, we have<\/p>\n<p><span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_14.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-768 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14.png\" alt=\"0.883m^3 x (100cm\/1m) x (100cm\/1m) x (100cm\/1m) = 883000 cm^3\" width=\"1048\" height=\"99\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14.png 1048w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14-300x28.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14-1024x97.png 1024w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14-768x73.png 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14-65x6.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14-225x21.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_14-350x33.png 350w\" sizes=\"auto, (max-width: 1048px) 100vw, 1048px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s04_p24\" class=\"para\">You should demonstrate to yourself that the three meter units do indeed cancel.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 2<\/h3>\n<p>How many cubic millimeters are present in 0.0923 m<sup class=\"superscript\">3<\/sup>?<\/p>\n<\/div>\n<p id=\"ball-ch02_s04_p27\" class=\"para editable block\">Suppose the unit you want to convert is in the denominator of a derived unit; what then? Then, in the conversion factor, the unit you want to remove must be in the <em class=\"emphasis\">numerator<\/em>. This will cancel with the original unit in the denominator and introduce a new unit in the denominator. The following example illustrates this situation.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 9<\/h3>\n<p id=\"ball-ch02_s04_p28\" class=\"para\">Convert 88.4 m\/min to meters\/second.<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s04_p29\" class=\"para\">We want to change the unit in the denominator from minutes to seconds. Because there are 60 seconds in 1 minute (60 s = 1 min), we construct a conversion factor so that the unit we want to remove, minutes, is in the numerator: <span class=\"inlineequation\">1 min\/60 s<\/span>. Apply and perform the math:<\/p>\n<p><span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_15.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-769 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_15.png\" alt=\"88.4m\/m x 1min\/60s = 1.47 m\/s\" width=\"411\" height=\"95\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_15.png 411w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_15-300x69.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_15-65x15.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_15-225x52.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_15-350x81.png 350w\" sizes=\"auto, (max-width: 411px) 100vw, 411px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s04_p30\" class=\"para\">Notice how the 88.4 automatically goes in the numerator. That\u2019s because any number can be thought of as being in the numerator of a fraction divided by 1.<\/p>\n<div id=\"ball-ch02_s04_f01\" class=\"figure small\">\n<figure id=\"attachment_3201\" aria-describedby=\"caption-attachment-3201\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/800px-Grapevinesnail_01.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-770\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Grapevinesnail_01-1.jpg\" alt=\"A common garden snail moves at a rate of about 0.2 m\/min, which is about 0.003 m\/s, which is 3 mm\/s! Source: \u201cGrapevine snail\u201dby J\u00fcrgen Schoneris licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.\" width=\"400\" height=\"236\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Grapevinesnail_01-1.jpg 800w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Grapevinesnail_01-1-300x177.jpg 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Grapevinesnail_01-1-768x453.jpg 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Grapevinesnail_01-1-65x38.jpg 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Grapevinesnail_01-1-225x133.jpg 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Grapevinesnail_01-1-350x207.jpg 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-3201\" class=\"wp-caption-text\"><strong>Figure 1.<\/strong> How Fast Is Fast? A common garden snail moves at a rate of about 0.2 m\/min, which is about 0.003 m\/s, which is 3 mm\/s!<br \/>Source: \u201cGrapevine snail\u201dby J\u00fcrgen Schoneris licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 3<\/h3>\n<p id=\"ball-ch02_s04_p31\" class=\"para\">Convert 0.203 m\/min to meters\/second.<\/p>\n<\/div>\n<p id=\"ball-ch02_s04_p33\" class=\"para editable block\">Sometimes there will be a need to convert from one unit with one numerical prefix to another unit with a different numerical prefix. How do we handle those conversions? Well, you could memorize the conversion factors that interrelate all numerical prefixes. Or you can go the easier route: first convert the quantity to the base unit, the unit with no numerical prefix, using the definition of the original prefix. Then convert the quantity in the base unit to the desired unit using the definition of the second prefix. You can do the conversion in two separate steps or as one long algebraic step. For example, to convert 2.77 kg to milligrams:<\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_16.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-771 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_16.png\" alt=\"2.77 kg x 1000 g\/1kg = 2770 g (convert to the base unit of grams)\" width=\"918\" height=\"109\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_16.png 918w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_16-300x36.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_16-768x91.png 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_16-65x8.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_16-225x27.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_16-350x42.png 350w\" sizes=\"auto, (max-width: 918px) 100vw, 918px\" \/><\/a><\/span><br \/>\n<a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_17.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-772 size-full aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17.png\" alt=\"2770 g x 1000 mg\/1g = 2770000 mg = 2.77x10^6 mg (convert to the desired unit)\" width=\"1139\" height=\"99\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17.png 1139w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17-300x26.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17-1024x89.png 1024w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17-768x67.png 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17-65x6.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17-225x20.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_17-350x30.png 350w\" sizes=\"auto, (max-width: 1139px) 100vw, 1139px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p34\" class=\"para editable block\">Alternatively, it can be done in a single multistep process:<span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_18.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-773 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_18.png\" alt=\"2.77kg x 1000g\/1kg x 1000 mg\/1g = 2770000 mg = 2.77 x 10^6 mg\" width=\"893\" height=\"109\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_18.png 893w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_18-300x37.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_18-768x94.png 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_18-65x8.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_18-225x27.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_18-350x43.png 350w\" sizes=\"auto, (max-width: 893px) 100vw, 893px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s04_p35\" class=\"para editable block\">You get the same answer either way.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 10<\/h3>\n<p id=\"ball-ch02_s04_p36\" class=\"para\">How many nanoseconds are in 368.09 \u03bcs?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s04_p37\" class=\"para\">You can either do this as a one-step conversion from microseconds to nanoseconds or convert to the base unit first and then to the final desired unit. We will use the second method here, showing the two steps in a single line. Using the definitions of the prefixes <em class=\"emphasis\">micro-<\/em> and <em class=\"emphasis\">nano-<\/em>,<\/p>\n<p><span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_21.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-774 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_21.png\" alt=\"368.09 us x 1s\/10^6us x 10^9ns \/1s = 368090 ns = 3.608 x 10^5 ns\" width=\"871\" height=\"90\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_21.png 871w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_21-300x31.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_21-768x79.png 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_21-65x7.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_21-225x23.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_21-350x36.png 350w\" sizes=\"auto, (max-width: 871px) 100vw, 871px\" \/><\/a><\/span><\/p>\n<\/div>\n<div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 4<\/h3>\n<p id=\"ball-ch02_s04_p38\" class=\"para\">How many milliliters are in 607.8 kL?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"ball-ch02_s04_p40\" class=\"para editable block\">When considering the significant figures of a final numerical answer in a conversion, there is one important case where a number does not impact the number of significant figures in a final answer\u2014the so-called <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">exact number<\/a><\/span>. An exact number is a number from a defined relationship, not a measured one. For example, the prefix <em class=\"emphasis\">kilo-<\/em> means 1,000\u2014<em class=\"emphasis\">exactly<\/em> 1,000, no more or no less. Thus, in constructing the conversion factor<\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_19.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-775 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_19.png\" alt=\"1000 g\/1 kg\" width=\"127\" height=\"116\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_19.png 127w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_19-65x59.png 65w\" sizes=\"auto, (max-width: 127px) 100vw, 127px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s04_p41\" class=\"para editable block\">neither the 1,000 nor the 1 enter into our consideration of significant figures. The numbers in the numerator and denominator are defined exactly by what the prefix <em class=\"emphasis\">kilo-<\/em> means. Another way of thinking about it is that these numbers can be thought of as having an infinite number of significant figures, such as<\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_untis_24.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-776 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_untis_24.png\" alt=\"1000.0000000....g\/1.000000000... kg\" width=\"339\" height=\"109\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_untis_24.png 339w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_untis_24-300x96.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_untis_24-65x21.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_untis_24-225x72.png 225w\" sizes=\"auto, (max-width: 339px) 100vw, 339px\" \/><\/a><\/p>\n<p id=\"ball-ch02_s04_p42\" class=\"para editable block\">The other numbers in the calculation will determine the number of significant figures in the final answer.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 11<\/h3>\n<p id=\"ball-ch02_s04_p43\" class=\"para\">A rectangular plot in a garden has the dimensions 36.7 cm by 128.8 cm. What is the area of the garden plot in square meters? Express your answer in the proper number of significant figures.<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s04_p44\" class=\"para\">Area is defined as the product of the two dimensions, which we then have to convert to square meters and express our final answer to the correct number of significant figures, which in this case will be three.<\/p>\n<p><span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/converting_units_22.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-777 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_22.png\" alt=\"36.7 cm x 128.8 cm x 1 m\/100cm x 1 m\/100 cm = 0.472696 m^2 = 0.473 m^2\" width=\"967\" height=\"101\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_22.png 967w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_22-300x31.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_22-768x80.png 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_22-65x7.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_22-225x24.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/converting_units_22-350x37.png 350w\" sizes=\"auto, (max-width: 967px) 100vw, 967px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s04_p45\" class=\"para\">The 1 and 100 in the conversion factors do not affect the determination of significant figures because they are exact numbers, defined by the centi- prefix.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 5<\/h3>\n<p id=\"ball-ch02_s04_p46\" class=\"para\">What is the volume of a block in cubic meters whose dimensions are 2.1 cm \u00d7 34.0 cm \u00d7 118 cm?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"ball-ch02_s04_n07\" class=\"callout block\">\n<h3 class=\"title\">Chemistry (and physics and math&#8230;) is Everywhere: The Gimli Glider<\/h3>\n<p id=\"ball-ch02_s04_p48\" class=\"para\">On July 23, 1983, an Air Canada Boeing 767 jet had to glide to an emergency landing at Gimli Industrial Park Airport in Gimli, Manitoba, because it unexpectedly ran out of fuel during flight. There was no loss of life in the course of the emergency landing, only some minor injuries associated in part with the evacuation of the craft after landing. For the remainder of its operational life (the plane was retired in 2008), the aircraft was nicknamed \u201cthe Gimli Glider.\u201d<\/p>\n<div id=\"ball-ch02_s04_f02\" class=\"informalfigure large\">\n<div class=\"copyright\">\n<figure id=\"attachment_3203\" aria-describedby=\"caption-attachment-3203\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/800px-Aircanada.b767-300er.c-ggmx.arp_.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-778\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Aircanada.b767-300er.c-ggmx.arp_-1.jpg\" alt=\"The Gimli Glider is the Boeing 767 that ran out of fuel and glided to safety at Gimli Airport. The aircraft ran out of fuel because of confusion over the units used to express the amount of fuel. \u201cAircanada.b767\u2032\u2032 is in the the public domain.\" width=\"400\" height=\"293\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Aircanada.b767-300er.c-ggmx.arp_-1.jpg 800w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Aircanada.b767-300er.c-ggmx.arp_-1-300x219.jpg 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Aircanada.b767-300er.c-ggmx.arp_-1-768x562.jpg 768w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Aircanada.b767-300er.c-ggmx.arp_-1-65x48.jpg 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Aircanada.b767-300er.c-ggmx.arp_-1-225x165.jpg 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/800px-Aircanada.b767-300er.c-ggmx.arp_-1-350x256.jpg 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-3203\" class=\"wp-caption-text\"><strong>Figure 2.<\/strong> The Gimli Glider is the Boeing 767 that ran out of fuel and glided to safety at Gimli Airport. The aircraft ran out of fuel because of confusion over the units used to express the amount of fuel.<br \/>\u201cAircanada.b767\u2032\u2032 is in the the public domain.<\/figcaption><\/figure>\n<p class=\"para\">\n<\/div>\n<\/div>\n<p id=\"ball-ch02_s04_p49\" class=\"para\">The 767 took off from Montreal on its way to Ottawa, ultimately heading for Edmonton, Canada. About halfway through the flight, all the engines on the plane began to shut down because of a lack of fuel. When the final engine cut off, all electricity (which was generated by the engines) was lost; the plane became, essentially, a powerless glider. Captain Robert Pearson was an experienced glider pilot, although he had never flown a glider the size of a 767. First Officer Maurice Quintal quickly determined that the aircraft would not be able make it to Winnipeg, the next large airport. He suggested his old Royal Air Force base at Gimli Station, one of whose runways was still being used as a community airport. Between the efforts of the pilots and the flight crew, they managed to get the airplane safely on the ground (although with buckled landing gear) and all passengers off safely.<\/p>\n<p id=\"ball-ch02_s04_p50\" class=\"para\">What happened? At the time, Canada was transitioning from the older English system to the metric system. The Boeing 767s were the first aircraft whose gauges were calibrated in the metric system of units (liters and kilograms) rather than the English system of units (gallons and pounds). Thus, when the fuel gauge read 22,300, the gauge meant kilograms, but the ground crew mistakenly fueled the plane with 22,300 <em class=\"emphasis\">pounds<\/em> of fuel. This ended up being just less than half of the fuel needed to make the trip, causing the engines to quit about halfway to Ottawa. Quick thinking and extraordinary skill saved the lives of 61 passengers and 8 crew members\u2014an incident that would not have occurred if people were watching their units.<\/p>\n<\/div>\n<figure id=\"attachment_3962\" aria-describedby=\"caption-attachment-3962\" style=\"width: 150px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/07\/qrcode.23437561.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-779 size-thumbnail\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/qrcode.23437561-150x150-1.png\" alt=\"qrcode.23437561\" width=\"150\" height=\"150\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/qrcode.23437561-150x150-1.png 150w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/qrcode.23437561-150x150-1-65x65.png 65w\" sizes=\"auto, (max-width: 150px) 100vw, 150px\" \/><\/a><figcaption id=\"caption-attachment-3962\" class=\"wp-caption-text\"><strong>Figure 3.<\/strong> Video source: Unit conversion by keyj (https:\/\/viutube.viu.ca\/public\/media\/Unit+Conversion\/0_h2w068q1)<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<div id=\"ball-ch02_s04_qs01\" class=\"qandaset block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s04_l06\" class=\"itemizedlist\">\n<li>Units can be converted to other units using the proper conversion factors.<\/li>\n<li>Conversion factors are constructed from equalities that relate two different units.<\/li>\n<li>Conversions can be a single step or multistep.<\/li>\n<li>Unit conversion is a powerful mathematical technique in chemistry that must be mastered.<\/li>\n<li>Exact numbers do not affect the determination of significant figures.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Exercises<\/h3>\n<ol id=\"ball-ch02_s04_qs01_qd01\" class=\"qandadiv\">\n<li id=\"ball-ch02_s04_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p1\" class=\"para\">Write the two conversion factors that exist between the two given units.<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<p>(a) milliliters and liters (b) microseconds and seconds (c) kilometers and meters<\/p>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p2\" class=\"para\">2. Write the two conversion factors that exist between the two given units.<\/p>\n<p>(a) kilograms and grams (b) milliseconds and seconds (c) centimeters and meters<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p3\" class=\"para\">3. Perform the following conversions.<\/p>\n<p>(a) 5.4 km to meters (b) 0.665 m to millimeters (c) 0.665 m to kilometers<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p4\" class=\"para\">4. Perform the following conversions.<\/p>\n<p>(a) 90.6 mL to liters (b) 0.00066 ML to liters (c) 750 L to kiloliters<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p5\" class=\"para\">5. Perform the following conversions.<\/p>\n<p>(a) 17.8 \u03bcg to grams (b) 7.22 \u00d7 10<sup class=\"superscript\">2<\/sup> kg to grams (c) 0.00118 g to nanograms<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p6\" class=\"para\">6. Perform the following conversions.<\/p>\n<p>(a) 833 ns to seconds (b) 5.809 s to milliseconds (c) 2.77 \u00d7 10<sup class=\"superscript\">6<\/sup> s to megaseconds<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p7\" class=\"para\">7. Perform the following conversions.<\/p>\n<p>(a) 9.44 m<sup class=\"superscript\">2<\/sup> to square centimeters (b) 3.44 \u00d7 10<sup class=\"superscript\">8<\/sup> mm<sup class=\"superscript\">3<\/sup> to cubic meters<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p8\" class=\"para\">8. Perform the following conversions.<\/p>\n<p>(a) 0.00444 cm<sup class=\"superscript\">3<\/sup> to cubic meters (b) 8.11 \u00d7 10<sup class=\"superscript\">2<\/sup> m<sup class=\"superscript\">2<\/sup> to square nanometers<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p9\" class=\"para\">9. Why would it be inappropriate to convert square centimeters to cubic meters?<\/p>\n<p class=\"para\">10. Why would it be inappropriate to convert from cubic meters to cubic seconds?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p13\" class=\"para\">11. Perform the following conversions.<\/p>\n<p>(a) 45.0 m\/min to meters\/second (b) 0.000444 m\/s to micrometers\/second (c) 60.0 km\/h to kilometers\/second<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p14\" class=\"para\">12. Perform the following conversions.<\/p>\n<p>(a) 3.4 \u00d7 10<sup class=\"superscript\">2<\/sup> cm\/s to centimeters\/minute (b) 26.6 mm\/s to millimeters\/hour (c) 13.7 kg\/L to kilograms\/milliliters<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p15\" class=\"para\">13. Perform the following conversions.<\/p>\n<p>(a) 0.674 kL to milliliters (b) 2.81 \u00d7 10<sup class=\"superscript\">12<\/sup> mm to kilometers (c) 94.5 kg to milligrams<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p16\" class=\"para\">14. Perform the following conversions.<\/p>\n<p>(a) 6.79 \u00d7 10<sup class=\"superscript\">\u22126<\/sup> kg to micrograms (b) 1.22 mL to kiloliters (c) 9.508 \u00d7 10<sup class=\"superscript\">\u22129<\/sup> ks to milliseconds<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p17\" class=\"para\">15. Perform the following conversions.<\/p>\n<p>(a) 6.77 \u00d7 10<sup class=\"superscript\">14<\/sup> ms to kiloseconds (b) 34,550,000 cm to kilometers<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p18\" class=\"para\">16. Perform the following conversions.<\/p>\n<p>(a) 4.701 \u00d7 10<sup class=\"superscript\">15<\/sup> mL to kiloliters (b) 8.022 \u00d7 10<sup class=\"superscript\">\u221211<\/sup> ks to microseconds<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p19\" class=\"para\">17. Perform the following conversions. Note that you will have to convert units in both the numerator and the denominator.<\/p>\n<p>(a) 88 ft\/s to miles\/hour (Hint: use 5,280 ft = 1 mi.) (b) 0.00667 km\/h to meters\/second<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p20\" class=\"para\">18. Perform the following conversions. Note that you will have to convert units in both the numerator and the denominator.<\/p>\n<p>(a) 3.88 \u00d7 10<sup class=\"superscript\">2<\/sup> mm\/s to kilometers\/hour (b) 1.004 kg\/L to grams\/milliliter<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p21\" class=\"para\">19. What is the area in square millimeters of a rectangle whose sides are 2.44 cm \u00d7 6.077 cm? Express the answer to the proper number of significant figures.<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p23\" class=\"para\">20. What is the volume in cubic centimeters of a cube with sides of 0.774 m? Express the answer to the proper number of significant figures.<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p25\" class=\"para\">21. The formula for the area of a triangle is 1\/2 \u00d7 base \u00d7 height. What is the area of a triangle in square centimeters if its base is 1.007 m and its height is 0.665 m? Express the answer to the proper number of significant figures.<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s04_qs01_p27\" class=\"para\">22. The formula for the area of a triangle is 1\/2 \u00d7 base \u00d7 height. What is the area of a triangle in square meters if its base is 166 mm and its height is 930.0 mm? Express the answer to the proper number of significant figures.<\/p>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Check Your Understanding 1<\/strong><\/p>\n<ol id=\"ball-ch02_s04_l05\" class=\"orderedlist\">\n<li>6.708 \u00d7 10<sup class=\"superscript\">\u22125<\/sup> L<\/li>\n<li>5.68 \u00d7 10<sup class=\"superscript\">\u22122<\/sup> km<\/li>\n<\/ol>\n<p><strong>Check Your Understanding 2<\/strong><\/p>\n<p>9.23 \u00d7 10<sup class=\"superscript\">7<\/sup> mm<sup class=\"superscript\">3<\/sup><\/p>\n<p><strong>Check Your Understanding 3<\/strong><\/p>\n<p>0.00338 m\/s or 3.38 \u00d7 10<sup class=\"superscript\">\u22123<\/sup> m\/s<\/p>\n<p><strong>Check Your Understanding 4<\/strong><\/p>\n<p>6.078 \u00d7 10<sup class=\"superscript\">8<\/sup> mL<\/p>\n<p><strong>Check Your Understanding 5<\/strong><\/p>\n<p>0.0084 m<sup class=\"superscript\">3<\/sup><\/p>\n<p><strong>Problems &amp; Exercises<\/strong><\/p>\n<p><span class=\"inlineequation\"><strong>1.<\/strong> (a) 1,000 mL\/1 L<\/span> and <span class=\"inlineequation\">1 L\/1,000 mL (b) 1,000,000 \u03bcs\/1 s<\/span> and <span class=\"inlineequation\">1 s\/1,000,000 \u03bcs (c) 1,000 m\/1 km<\/span> and <span class=\"inlineequation\">1 km1,000 m<\/span><\/p>\n<p><strong>3.<\/strong> (a) 5,400 m (b) 665 mm (c) 6.65 \u00d7 10<sup class=\"superscript\">\u22124<\/sup> km<\/p>\n<p><strong>5.<\/strong> (a) 1.78 \u00d7 10<sup class=\"superscript\">\u22125<\/sup> g (b) 7.22 \u00d7 10<sup class=\"superscript\">5<\/sup> g (c) 1.18 \u00d7 10<sup class=\"superscript\">6<\/sup> ng<\/p>\n<p><strong>7.<\/strong> (a) 94,400 cm<sup class=\"superscript\">2 <\/sup>(b) 0.344 m<sup class=\"superscript\">3<\/sup><\/p>\n<p><strong>9.<\/strong> One is a unit of area, and the other is a unit of volume.<\/p>\n<p><strong>11.<\/strong> (a) 0.75 m\/s (b) 444 \u00b5m\/s (c) 1.666 \u00d7 10<sup class=\"superscript\">\u22122<\/sup> km\/s<\/p>\n<p><strong>13.<\/strong> (a) 674,000 mL (b) 2.81 \u00d7 10<sup class=\"superscript\">6<\/sup> km (c) 9.45 \u00d7 10<sup class=\"superscript\">7<\/sup> mg<\/p>\n<p><strong>15.<\/strong> (a) 6.77 \u00d7 10<sup class=\"superscript\">8<\/sup> ks (b) 345.5 km<\/p>\n<p><strong>17.<\/strong> (a) 6.0 \u00d7 10<sup class=\"superscript\">1<\/sup> mi\/h (b) 0.00185 m\/s<\/p>\n<p><strong>19.<\/strong> 1.48 \u00d7 10<sup class=\"superscript\">3<\/sup> mm<sup class=\"superscript\">2<\/sup><\/p>\n<p><strong>21.<\/strong> 3.35 \u00d7 10<sup class=\"superscript\">3<\/sup> cm<sup class=\"superscript\">2<\/sup><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h1><a id=\"oth-units\" href=\"\"><\/a>Other Units: Temperature and Density<\/h1>\n<p id=\"ball-ch02_s05_p01\" class=\"para editable block\">There are other units in chemistry that are important, and we will cover others in the course of the entire book. One of the fundamental quantities in science is temperature. <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">Temperature<\/a><\/span> is a measure of the average amount of energy of motion, or <em class=\"emphasis\">kinetic energy<\/em>, a system contains. Temperatures are expressed using scales that use units called <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">degrees<\/a><\/span>, and there are several temperature scales in use. In the United States, the commonly used temperature scale is the <em class=\"emphasis\">Fahrenheit scale<\/em> (symbolized by \u00b0F and spoken as \u201cdegrees Fahrenheit\u201d). On this scale, the freezing point of liquid water (the temperature at which liquid water turns to solid ice) is 32 \u00b0F, and the boiling point of water (the temperature at which liquid water turns to steam) is 212 \u00b0F.<\/p>\n<p id=\"ball-ch02_s05_p02\" class=\"para editable block\">Science also uses other scales to express temperature. The Celsius scale (symbolized by \u00b0C and spoken as \u201cdegrees Celsius\u201d) is a temperature scale where 0 \u00b0C is the freezing point of water and 100 \u00b0C is the boiling point of water; the scale is divided into 100 divisions between these two landmarks and extended higher and lower. By comparing the Fahrenheit and Celsius scales, a conversion between the two scales can be determined:<\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_1.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-47 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2017\/09\/other_units_1.png\" alt=\"oC = (oF-32) x 5\/9\" width=\"244\" height=\"90\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2017\/09\/other_units_1.png 244w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2017\/09\/other_units_1-65x24.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2017\/09\/other_units_1-225x83.png 225w\" sizes=\"auto, (max-width: 244px) 100vw, 244px\" \/><\/a><\/span><\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_2.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-48 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_2.png\" alt=\"oF = (oC x 9\/5) + 32\" width=\"265\" height=\"91\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_2.png 265w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_2-65x22.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_2-225x77.png 225w\" sizes=\"auto, (max-width: 265px) 100vw, 265px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s05_p03\" class=\"para editable block\">Using these formulas, we can convert from one temperature scale to another. The number 32 in the formulas is exact and does not count in significant figure determination.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 12<\/h3>\n<ol id=\"ball-ch02_s05_l02\" class=\"orderedlist\">\n<li>What is 98.6 \u00b0F in degrees Celsius?<\/li>\n<li>What is 25.0 \u00b0C in degrees Fahrenheit?<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n<ol id=\"ball-ch02_s05_l03\" class=\"orderedlist\">\n<li>\n<p class=\"para\">Using the first formula from above, we have<\/p>\n<p><span class=\"informalequation\">\u00b0C = (98.6 \u2013 32)\u2009\u00d7\u20095\/9 = 66.6\u2009\u00d7\u20095\/9 = 37.0 \u00b0C<\/span><\/li>\n<li>\n<p class=\"para\">Using the second formula from above, we have<\/p>\n<p><span class=\"informalequation\">\u00b0F = (25.0\u2009\u00d7\u20099\/5) + 3\/2 = 45.0 + 32 = 77.0 \u00b0F<\/span><\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 1<\/h3>\n<ol id=\"ball-ch02_s05_l04\" class=\"orderedlist\">\n<li>Convert 0 \u00b0F to degrees Celsius.<\/li>\n<li>Convert 212 \u00b0C to degrees Fahrenheit.<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"ball-ch02_s05_p04\" class=\"para editable block\">The fundamental unit of temperature (another fundamental unit of science, bringing us to four) in SI is the <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">kelvin<\/a><\/span> (K). The Kelvin temperature scale (note that the name of the scale capitalizes the word <em class=\"emphasis\">Kelvin<\/em>, but the unit itself is lowercase) uses degrees that are the same size as the Celsius degree, but the numerical scale is shifted up by 273.15 units. That is, the conversion between the Kelvin and Celsius scales is as follows:<\/p>\n<p><span class=\"informalequation block\"><span class=\"mathphrase\">K = \u00b0C + 273.15<\/span><\/span><br \/>\n<span class=\"informalequation block\"><span class=\"mathphrase\">\u00b0C = K \u2212 273.15<\/span><\/span><\/p>\n<p id=\"ball-ch02_s05_p05\" class=\"para editable block\">For most purposes, it is acceptable to use 273 instead of 273.15. Note that the Kelvin scale does not use the word <em class=\"emphasis\">degrees<\/em>; a temperature of 295 K is spoken of as \u201ctwo hundred ninety-five kelvins\u201d and not \u201ctwo hundred ninety-five degrees Kelvin.\u201d<\/p>\n<p id=\"ball-ch02_s05_p06\" class=\"para editable block\">The reason that the Kelvin scale is defined this way is because there exists a minimum possible temperature called <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">absolute zero<\/a><\/span>. The Kelvin temperature scale is set so that 0 K is absolute zero, and temperature is counted upward from there. Normal room temperature is about 295 K, as seen in the following example.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 13<\/h3>\n<p id=\"ball-ch02_s05_p07\" class=\"para\">If normal room temperature is 72.0 \u00b0F, what is room temperature in degrees Celsius and kelvins?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s05_p08\" class=\"para\">First, we use the formula to determine the temperature in degrees Celsius:<\/p>\n<p><span class=\"informalequation\">\u00b0C = (72.0 \u2013 32)\u2009\u00d7\u20095\/9 = 40.0\u2009\u00d7\u20095\/9 = 22.2 \u00b0C<\/span><\/p>\n<p id=\"ball-ch02_s05_p09\" class=\"para\">Then we use the appropriate formula above to determine the temperature in the Kelvin scale:<\/p>\n<p><span class=\"informalequation\"><span class=\"mathphrase\">K = 22.2 \u00b0C + 273.15 = 295.4 K<\/span><\/span><\/p>\n<p id=\"ball-ch02_s05_p10\" class=\"para\">So, room temperature is about 295 K.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 2<\/h3>\n<p id=\"ball-ch02_s05_p11\" class=\"para\">What is 98.6 \u00b0F on the Kelvin scale?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"ball-ch02_s05_p13\" class=\"para editable block\"><a class=\"xref\" href=\"#ball-ch02_s05_f01\">Figure 2.9 &#8220;Fahrenheit, Celsius, and Kelvin Temperatures&#8221;<\/a> compares the three temperature scales. Note that science uses the Celsius and Kelvin scales almost exclusively; virtually no practicing chemist expresses laboratory-measured temperatures with the Fahrenheit scale. (In fact, the United States is one of the few countries in the world that still uses the Fahrenheit scale on a daily basis. The other two countries are Liberia and Myanmar [formerly Burma].<\/p>\n<div id=\"ball-ch02_s05_f01\" class=\"figure large editable block\">\n<p class=\"title\"><span class=\"title-prefix\">Figure 2.9<\/span> Fahrenheit, Celsius, and Kelvin Temperatures<\/p>\n<figure id=\"attachment_4622\" aria-describedby=\"caption-attachment-4622\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Temperatures.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-49\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Temperatures-1.png\" alt=\"Temperatures\" width=\"400\" height=\"355\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Temperatures-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Temperatures-1-300x266.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Temperatures-1-65x58.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Temperatures-1-225x200.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Temperatures-1-350x310.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-4622\" class=\"wp-caption-text\"><strong>Figure 1.<\/strong> Fahrenheit, Celsius, and Kelvin Temperatures<\/figcaption><\/figure>\n<p class=\"para\">A comparison of the three temperature scales.<\/p>\n<\/div>\n<p id=\"ball-ch02_s05_p14\" class=\"para editable block\"><span class=\"margin_term\"><a class=\"glossterm\" href=\"\">Density <\/a><\/span>is a physical property that is defined as a substance\u2019s mass divided by its volume:<\/p>\n<p><span class=\"informalequation block\">density = mass\/volume or d = m\/V<\/span><\/p>\n<p id=\"ball-ch02_s05_p15\" class=\"para editable block\">Density is usually a measured property of a substance, so its numerical value affects the significant figures in a calculation. Notice that density is defined in terms of two dissimilar units, mass and volume. That means that density overall has derived units, just like velocity. Common units for density include g\/mL, g\/cm<sup class=\"superscript\">3<\/sup>, g\/L, kg\/L, and even kg\/m<sup class=\"superscript\">3<\/sup>. Densities for some common substances are listed in <a class=\"xref\" href=\"#ball-ch02_s05_t01\">Table 2.2 &#8220;Densities of Some Common Substances&#8221;<\/a>.<\/p>\n<div id=\"ball-ch02_s05_t01\" class=\"table block\">\n<p class=\"title\"><span class=\"title-prefix\">Table 2.2<\/span> Densities of Some Common Substances<\/p>\n<table style=\"border-spacing: 0px;width: 689px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th style=\"width: 203.517px\">Substance<\/th>\n<th style=\"width: 461.483px\">Density (g\/mL or g\/cm<sup class=\"superscript\">3<\/sup>)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 203.517px\">water<\/td>\n<td style=\"width: 461.483px\">1.0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">gold<\/td>\n<td style=\"width: 461.483px\">19.3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">mercury<\/td>\n<td style=\"width: 461.483px\">13.6<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">air<\/td>\n<td style=\"width: 461.483px\">0.0012<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">cork<\/td>\n<td style=\"width: 461.483px\">0.22\u20130.26<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">aluminum<\/td>\n<td style=\"width: 461.483px\">2.7<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 203.517px\">iron<\/td>\n<td style=\"width: 461.483px\">7.87<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"ball-ch02_s05_p16\" class=\"para editable block\">Because of how it is defined, density can act as a conversion factor for switching between units of mass and volume. For example, suppose you have a sample of aluminum that has a volume of 7.88 cm<sup class=\"superscript\">3<\/sup>. How can you determine what mass of aluminum you have without measuring it? You can use the volume to calculate it. If you multiply the given volume by the known density (from <a class=\"xref\" href=\"#ball-ch02_s05_t01\">Table 2.2 &#8220;Densities of Some Common Substances&#8221;<\/a>), the volume units will cancel and leave you with mass units, telling you the mass of the sample:<\/p>\n<p><span class=\"informalequation block\"> 7.88 cm<sup>3<\/sup>\u2009\u00d7\u20092.7 g\/cm<sup>3 <\/sup>= 21 g of aluminum<\/span><\/p>\n<p id=\"ball-ch02_s05_p17\" class=\"para editable block\">where we have limited our answer to two significant figures.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 14<\/h3>\n<p id=\"ball-ch02_s05_p18\" class=\"para\">What is the mass of 44.6 mL of mercury?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s05_p19\" class=\"para\">Use the density from <a class=\"xref\" href=\"#ball-ch02_s05_t01\">Table 2.2 &#8220;Densities of Some Common Substances&#8221;<\/a> as a conversion factor to go from volume to mass:<\/p>\n<p><span class=\"informalequation\">44.6 mL\u2009\u00d7\u200913.6 g\/mL = 607 g<\/span><\/p>\n<p id=\"ball-ch02_s05_p20\" class=\"para\">The mass of the mercury is 607 g.<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 3<\/h3>\n<p id=\"ball-ch02_s05_p21\" class=\"emphasis bolditalic\">What is the mass of 25.0 cm<sup class=\"superscript\">3<\/sup> of iron?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"ball-ch02_s05_p23\" class=\"para editable block\">Density can also be used as a conversion factor to convert mass to volume\u2014but care must be taken. We have already demonstrated that the number that goes with density normally goes in the numerator when density is written as a fraction. Take the density of gold, for example:<\/p>\n<p><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-50 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_3.png\" alt=\"d = 19.3 g\/1 mL\" width=\"312\" height=\"115\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_3.png 312w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_3-300x111.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_3-65x24.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_3-225x83.png 225w\" sizes=\"auto, (max-width: 312px) 100vw, 312px\" \/><\/a><\/p>\n<p>Although this was not previously pointed out, it can be assumed that there is a 1 in the denominator:<\/p>\n<p>That is, the density value tells us that we have 19.3 grams for every 1 milliliter of volume, and the 1 is an exact number. When we want to use density to convert from mass to volume, the numerator and denominator of density need to be switched\u2014that is, we must take the <em class=\"emphasis\">reciprocal<\/em> of the density. In so doing, we move not only the units but also the numbers:<\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_4.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-51 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_4.png\" alt=\"1\/d = 1mL\/19.3g\" width=\"203\" height=\"106\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_4.png 203w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_4-65x34.png 65w\" sizes=\"auto, (max-width: 203px) 100vw, 203px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s05_p26\" class=\"para editable block\">This reciprocal density is still a useful conversion factor, but now the mass unit will cancel and the volume unit will be introduced. Thus, if we want to know the volume of 45.9 g of gold, we would set up the conversion as follows:<\/p>\n<p><span class=\"informalequation block\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_5.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-52 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_5.png\" alt=\"45.9 g x 1mL\/19.3g = 2.38 mL\" width=\"376\" height=\"96\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_5.png 376w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_5-300x77.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_5-65x17.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_5-225x57.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_5-350x89.png 350w\" sizes=\"auto, (max-width: 376px) 100vw, 376px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s05_p27\" class=\"para editable block\">Note how the mass units cancel, leaving the volume unit, which is what we\u2019re looking for.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 15<\/h3>\n<p id=\"ball-ch02_s05_p28\" class=\"para\">A cork stopper from a bottle of wine has a mass of 3.78 g. If the density of cork is 0.22 g\/cm<sup class=\"superscript\">3<\/sup>, what is the volume of the cork?<\/p>\n<p class=\"simpara\">Solution<\/p>\n<p id=\"ball-ch02_s05_p29\" class=\"para\">To use density as a conversion factor, we need to take the reciprocal so that the mass unit of density is in the denominator. Taking the reciprocal, we find<\/p>\n<p><span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_6.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-53 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_6.png\" alt=\"1\/d = 1cm^3\/0.22g\" width=\"189\" height=\"111\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_6.png 189w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_6-65x38.png 65w\" sizes=\"auto, (max-width: 189px) 100vw, 189px\" \/><\/a><\/span><\/p>\n<p id=\"ball-ch02_s05_p30\" class=\"para\">We can use this expression as the conversion factor. So<\/p>\n<p><span class=\"informalequation\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2015\/11\/other_units_7.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-54 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_7.png\" alt=\"3.78 g x 1 cm^3\/0.22g = 17.2 cm^3\" width=\"381\" height=\"101\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_7.png 381w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_7-300x80.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_7-65x17.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_7-225x60.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/other_units_7-350x93.png 350w\" sizes=\"auto, (max-width: 381px) 100vw, 381px\" \/><\/a><\/span><\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 4<\/h3>\n<p id=\"ball-ch02_s05_p31\" class=\"para\">What is the volume of 3.78 g of gold?<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"ball-ch02_s05_p33\" class=\"para editable block\">Care must be used with density as a conversion factor. Make sure the mass units are the same, or the volume units are the same, before using density to convert to a different unit. Often, the unit of the given quantity must be first converted to the appropriate unit before applying density as a conversion factor.<\/p>\n<div id=\"ball-ch02_s05_n06\" class=\"callout block\">\n<h3 class=\"title\">Food and Drink App: Cooking Temperatures<\/h3>\n<p id=\"ball-ch02_s05_p78\" class=\"para\">Because degrees Fahrenheit is the common temperature scale in the United States, kitchen appliances, such as ovens, are calibrated in that scale. A cool oven may be only 150\u00b0F, while a cake may be baked at 350\u00b0F and a chicken roasted at 400\u00b0F. The broil setting on many ovens is 500\u00b0F, which is typically the highest temperature setting on a household oven.<\/p>\n<p id=\"ball-ch02_s05_p79\" class=\"para\">People who live at high altitudes, typically 2,000 ft above sea level or higher, are sometimes urged to use slightly different cooking instructions on some products, such as cakes and bread, because water boils at a lower temperature the higher in altitude you go, meaning that foods cook slower. For example, in Cleveland water typically boils at 212\u00b0F (100\u00b0C), but in Denver, the Mile-High City, water boils at about 200\u00b0F (93.3\u00b0C), which can significantly lengthen cooking times. Good cooks need to be aware of this.<\/p>\n<p id=\"ball-ch02_s05_p80\" class=\"para\">At the other end is pressure cooking. A pressure cooker is a closed vessel that allows steam to build up additional pressure, which increases the temperature at which water boils. A good pressure cooker can get to temperatures as high as 252\u00b0F (122\u00b0C); at these temperatures, food cooks much faster than it normally would. Great care must be used with pressure cookers because of the high pressure and high temperature. (When a pressure cooker is used to sterilize medical instruments, it is called an <em class=\"emphasis\">autoclave<\/em>.)<\/p>\n<p id=\"ball-ch02_s05_p81\" class=\"para\">Other countries use the Celsius scale for everyday purposes. Therefore, oven dials in their kitchens are marked in degrees Celsius. It can be confusing for US cooks to use ovens abroad\u2014a 425\u00b0F oven in the United States is equivalent to a 220\u00b0C oven in other countries. These days, many oven thermometers are marked with both temperature scales.<\/p>\n<div id=\"ball-ch02_s05_qs01\" class=\"qandaset block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s05_l06\" class=\"itemizedlist\">\n<li>Chemistry uses the Celsius and Kelvin scales to express temperatures.<\/li>\n<li>A temperature on the Kelvin scale is the Celsius temperature plus 273.15.<\/li>\n<li>The minimum possible temperature is absolute zero and is assigned 0 K on the Kelvin scale.<\/li>\n<li>Density relates a substance\u2019s mass and volume.<\/li>\n<li>Density can be used to calculate volume from a given mass or mass from a given volume.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol id=\"ball-ch02_s05_qs01_qd01\" class=\"qandadiv\">\n<li id=\"ball-ch02_s05_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p1\" class=\"para\">Perform the following conversions.<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<p>(a) 255\u00b0F to degrees Celsius (b) \u2212255\u00b0F to degrees Celsius (c) 50.0\u00b0C to degrees Fahrenheit (d) \u221250.0\u00b0C to degrees Fahrenheit<\/p>\n<p>2. Perform the following conversions.<\/p>\n<p>(a) 1,065\u00b0C to degrees Fahrenheit (b) \u2212222\u00b0C to degrees Fahrenheit (c) 400.0\u00b0F to degrees Celsius (d) 200.0\u00b0F to degrees Celsius<\/p>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p3\" class=\"para\">3. Perform the following conversions.<\/p>\n<p>(a) 100.0\u00b0C to kelvins (b) \u2212100.0\u00b0C to kelvins (c) 100 K to degrees Celsius (d) 300 K to degrees Celsius<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p4\" class=\"para\">4. Perform the following conversions.<\/p>\n<p>(a) 1,000.0 K to degrees Celsius (b) 50.0 K to degrees Celsius (c) 37.0\u00b0C to kelvins (d) \u221237.0\u00b0C to kelvins<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p5\" class=\"para\">5. Convert 0 K to degrees Celsius. What is the significance of the temperature in degrees Celsius?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p7\" class=\"para\">6. Convert 0 K to degrees Fahrenheit. What is the significance of the temperature in degrees Fahrenheit?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p9\" class=\"para\">7. The hottest temperature ever recorded on the surface of the earth was 136\u00b0F in Libya in 1922. What is the temperature in degrees Celsius and in kelvins?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p11\" class=\"para\">8. The coldest temperature ever recorded on the surface of the earth was \u2212128.6\u00b0F in Vostok, Antarctica, in 1983. What is the temperature in degrees Celsius and in kelvins?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p13\" class=\"para\">9. Give at least three possible units for density.<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p15\" class=\"para\">10. What are the units when density is inverted? Give three examples.<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p17\" class=\"para\">11. A sample of iron has a volume of 48.2 cm<sup class=\"superscript\">3<\/sup>. What is its mass?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p19\" class=\"para\">12. A sample of air has a volume of 1,015 mL. What is its mass?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p21\" class=\"para\">13. The volume of hydrogen used by the <em class=\"emphasis\">Hindenburg<\/em>, the German airship that exploded in New Jersey in 1937, was 2.000 \u00d7 10<sup class=\"superscript\">8<\/sup> L. If hydrogen gas has a density of 0.0899 g\/L, what mass of hydrogen was used by the airship?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p23\" class=\"para\">14. The volume of an Olympic-sized swimming pool is 2.50 \u00d7 10<sup class=\"superscript\">9<\/sup> cm<sup class=\"superscript\">3<\/sup>. If the pool is filled with alcohol (<em class=\"emphasis\">d<\/em> = 0.789 g\/cm<sup class=\"superscript\">3<\/sup>), what mass of alcohol is in the pool?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p25\" class=\"para\">15. A typical engagement ring has 0.77 cm<sup class=\"superscript\">3<\/sup> of gold. What mass of gold is present?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p27\" class=\"para\">16. A typical mercury thermometer has 0.039 mL of mercury in it. What mass of mercury is in the thermometer?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p29\" class=\"para\">17. What is the volume of 100.0 g of lead if lead has a density of 11.34 g\/cm<sup class=\"superscript\">3<\/sup>?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p31\" class=\"para\">18. What is the volume of 255.0 g of uranium if uranium has a density of 19.05 g\/cm<sup class=\"superscript\">3<\/sup>?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p33\" class=\"para\">19. What is the volume in liters of 222 g of neon if neon has a density of 0.900 g\/L?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p35\" class=\"para\">20. What is the volume in liters of 20.5 g of sulfur hexafluoride if sulfur hexafluoride has a density of 6.164 g\/L?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p37\" class=\"para\">21. Which has the greater volume, 100.0 g of iron (<em class=\"emphasis\">d<\/em> = 7.87 g\/cm<sup class=\"superscript\">3<\/sup>) or 75.0 g of gold (<em class=\"emphasis\">d<\/em> = 19.3 g\/cm<sup class=\"superscript\">3<\/sup>)?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s05_qs01_p39\" class=\"para\">22. Which has the greater volume, 100.0 g of hydrogen gas (<em class=\"emphasis\">d<\/em> = 0.0000899 g\/cm<sup class=\"superscript\">3<\/sup>) or 25.0 g of argon gas (<em class=\"emphasis\">d<\/em> = 0.00178 g\/cm<sup class=\"superscript\">3<\/sup>)?<\/p>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Check Your Understanding 1<br \/>\n<\/strong><\/p>\n<ol id=\"ball-ch02_s05_l05\" class=\"orderedlist\">\n<li>\u221217.8 \u00b0C<\/li>\n<li>414 \u00b0F<\/li>\n<\/ol>\n<p><strong>Check Your Understanding 2<br \/>\n<\/strong><\/p>\n<p>310.2 K<\/p>\n<p><strong>Check Your Understanding 3<br \/>\n<\/strong><\/p>\n<p>197 g<\/p>\n<p><strong>Check Your Understanding 4<br \/>\n<\/strong><\/p>\n<p>0.196 cm<sup class=\"superscript\">3<\/sup><\/p>\n<p><strong>Problems &amp; Exercises<\/strong><\/p>\n<p><strong>1.<\/strong> (a) 124\u00b0C (b) \u2212159\u00b0C (c) 122\u00b0F (d) \u221258\u00b0F<\/p>\n<p><strong>3.<\/strong> (a) 373 K (b) 173 K (c) \u2212173\u00b0C (d) 27\u00b0C<\/p>\n<p><strong>5.<\/strong> \u2212273\u00b0C. This is the lowest possible temperature in degrees Celsius.<\/p>\n<p><strong>7.<\/strong> 57.8\u00b0C; 331 K<\/p>\n<p><strong>9.<\/strong> g\/mL, g\/L, and kg\/L (answers will vary)<\/p>\n<p><strong>11.<\/strong> 379 g<\/p>\n<p><strong>13.<\/strong> 1.80 \u00d7 10<sup class=\"superscript\">7<\/sup> g<\/p>\n<p><strong>15.<\/strong> 15 g<\/p>\n<p><strong>17.<\/strong> 8.818 cm<sup class=\"superscript\">3<\/sup><\/p>\n<p><strong>19.<\/strong> 247 L<\/p>\n<p><strong>21.<\/strong> The 100.0 g of iron has the greater volume.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<h1><a id=\"exp-units\" href=\"\"><\/a>Expressing Units<\/h1>\n<p id=\"ball-ch02_s02_p01\" class=\"para editable block\">A number indicates \u201chow much,\u201d but the unit indicates \u201cof what.\u201d The \u201cof what\u201d is important when communicating a quantity. For example, if you were to ask a friend how close you are to Lake Erie and your friend says \u201csix,\u201d then your friend isn\u2019t giving you complete information. Six <em class=\"emphasis\">what<\/em>? Six miles? Six inches? Six city blocks? The actual distance to the lake depends on what units you use.<\/p>\n<p id=\"ball-ch02_s02_p02\" class=\"para editable block\">Chemistry, like most sciences, uses the International System of Units, or SI for short. (The letters <em class=\"emphasis\">SI<\/em> stand for the French \u201cle Syst\u00e8me International d\u2019unit\u00e9s.\u201d) SI specifies certain units for various types of quantities, based on seven <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">fundamental units<\/a><\/span> for various quantities. We will use most of the fundamental units in chemistry. Initially, we will deal with three fundamental units. The meter (m) is the SI unit of length. It is a little longer than a yard (see <a class=\"xref\" href=\"#ball-ch02_s02_f01\">Figure 2.3 &#8220;The Meter&#8221;<\/a>). The SI unit of mass is the kilogram (kg), which is about 2.2 pounds (lb). The SI unit of time is the second (s).<\/p>\n<div id=\"ball-ch02_s02_f01\" class=\"figure large medium-height editable block\">\n<figure style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/67d75b028a8e56c36f0622ce6b20547e-1.jpg\" alt=\"image\" width=\"380\" height=\"375\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> The Meter<\/figcaption><\/figure>\n<p class=\"para\">The SI standard unit of length, the meter, is a little longer than a yard.<\/p>\n<\/div>\n<p id=\"ball-ch02_s02_p03\" class=\"para editable block\">To express a quantity, you need to combine a number with a unit. If you have a length that is 2.4 m, then you express that length as simply 2.4 m. A time of 15,000 s can be expressed as 1.5 \u00d7 10<sup class=\"superscript\">4<\/sup> s in scientific notation.<\/p>\n<p id=\"ball-ch02_s02_p04\" class=\"para editable block\">Sometimes, a given unit is not an appropriate size to easily express a quantity. For example, the width of a human hair is very small, and it doesn\u2019t make much sense to express it in meters. SI also defines a series of <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">numerical prefixes<\/a><\/span> that refer to multiples or fractions of a fundamental unit to make a unit more conveniently sized for a specific quantity. <a class=\"xref\" href=\"#ball-ch02_s02_t01\">Table 2.1 &#8220;Multiplicative Prefixes for SI Units&#8221;<\/a> lists the prefixes, their abbreviations, and their multiplicative factors. Some of the prefixes, such as kilo-, mega-, and giga-, represent more than one of the fundamental unit, while other prefixes, such as centi-, milli-, and micro-, represent fractions of the original unit. Note, too, that once again we are using powers of 10. Each prefix is a multiple of or fraction of a power of 10.<\/p>\n<div id=\"ball-ch02_s02_t01\" class=\"table block\">\n<p class=\"title\"><span class=\"title-prefix\">Table 2.1<\/span> Multiplicative Prefixes for SI Units<\/p>\n<table style=\"border-spacing: 0px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th>Prefix<\/th>\n<th align=\"center\">Abbreviation<\/th>\n<th align=\"center\">Multiplicative Amount<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>giga-<\/td>\n<td align=\"center\">G<\/td>\n<td align=\"center\">1,000,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>mega-<\/td>\n<td align=\"center\">M<\/td>\n<td align=\"center\">1,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>kilo-<\/td>\n<td align=\"center\">k<\/td>\n<td align=\"center\">1,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>deci-<\/td>\n<td align=\"center\">d<\/td>\n<td align=\"center\">1\/10 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>centi-<\/td>\n<td align=\"center\">c<\/td>\n<td align=\"center\">1\/100 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>milli-<\/td>\n<td align=\"center\">m<\/td>\n<td align=\"center\">1\/1,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>micro-<\/td>\n<td align=\"center\">\u03bc*<\/td>\n<td align=\"center\">1\/1,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>nano-<\/td>\n<td align=\"center\">n<\/td>\n<td align=\"center\">1\/1,000,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<td>pico-<\/td>\n<td align=\"center\">p<\/td>\n<td align=\"center\">1\/1,000,000,000,000 \u00d7<\/td>\n<\/tr>\n<tr>\n<th colspan=\"3\">* The letter <em class=\"emphasis\">\u03bc<\/em> is the Greek letter lowercase equivalent to an m and is called \u201cmu\u201d (pronounced \u201cmyoo\u201d).<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p id=\"ball-ch02_s02_p05\" class=\"para editable block\">To use the fractions to generate new units, simply combine the prefix with the unit itself; the abbreviation for the new unit is the combination of the abbreviation for the prefix and the abbreviation of the unit. For example, the kilometer (km) is 1,000 \u00d7 meter, or 1,000 m. Thus, 5 kilometers (5 km) is equal to 5,000 m. Similarly, a millisecond (ms) is 1\/1,000 \u00d7 second, or one-thousandth of a second. Thus, 25 ms is 25 thousandths of a second. You will need to become proficient in combining prefixes and units. (You may recognize that one of our fundamental units, the kilogram, automatically has a prefix-unit combination, the kilogram. The word <em class=\"emphasis\">kilogram<\/em> means 1,000 g.)<\/p>\n<p id=\"ball-ch02_s02_p06\" class=\"para editable block\">In addition to the fundamental units, SI also allows for <span class=\"margin_term\"><a class=\"glossterm\" href=\"\">derived units<\/a><\/span> based on a fundamental unit or units. There are many derived units used in science. For example, the derived unit for area comes from the idea that area is defined as width times height. Because both width and height are lengths, they both have the fundamental unit of meter, so the unit of area is meter \u00d7 meter, or meter<sup class=\"superscript\">2<\/sup> (m<sup class=\"superscript\">2<\/sup>). This is sometimes spoken as \u201csquare meters.\u201d A unit with a prefix can also be used to derive a unit for area, so we can also have cm<sup class=\"superscript\">2<\/sup>, mm<sup class=\"superscript\">2<\/sup>, or km<sup class=\"superscript\">2<\/sup> as acceptable units for area.<\/p>\n<p id=\"ball-ch02_s02_p07\" class=\"para editable block\">Volume is defined as length times width times height, so it has units of meter \u00d7 meter \u00d7 meter or meter<sup class=\"superscript\">3<\/sup> (m<sup class=\"superscript\">3<\/sup>), sometimes spoken as \u201ccubic meters.\u201d The cubic meter is a rather large unit, however, so another unit is defined that is somewhat more manageable: the liter (L). A liter is 1\/1,000th of a cubic meter and is a little more than 1 quart in volume (see <a class=\"xref\" href=\"#ball-ch02_s02_f02\">Figure 2.4 &#8220;The Liter&#8221;<\/a>). Prefixes can also be used with the liter unit, so we can speak of milliliters (1\/1,000th of a liter; mL) and kiloliters (1,000 L; kL).<\/p>\n<div id=\"ball-ch02_s02_f02\" class=\"figure small editable block\">\n<figure style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/6866a70a4b52e0fd95f9dd3e1f3426a2-1.jpg\" alt=\"image\" width=\"380\" height=\"375\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong> The Liter<\/figcaption><\/figure>\n<p class=\"para\">The SI unit of volume, the liter, is slightly larger than 1 quart.<\/p>\n<\/div>\n<p id=\"ball-ch02_s02_p08\" class=\"para editable block\">Another definition of a liter is one-tenth of a meter cubed. Because one-tenth of a meter is 10 cm, then a liter is equal to 1,000 cm<sup class=\"superscript\">3<\/sup> (<a class=\"xref\" href=\"#ball-ch02_s02_f03\">Figure 2.5 &#8220;The Size of 1 Liter&#8221;<\/a>). Because 1 L equals 1,000 mL, we conclude that 1 mL equals 1 cm<sup class=\"superscript\">3<\/sup>; thus, these units are interchangeable.<\/p>\n<div id=\"ball-ch02_s02_f03\" class=\"figure large editable block\">\n<figure id=\"attachment_4611\" aria-describedby=\"caption-attachment-4611\" style=\"width: 400px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/opentextbc.ca\/introductorychemistry\/wp-content\/uploads\/sites\/17\/2014\/09\/Size-of-a-Liter.png\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-782\" src=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104\/wp-content\/uploads\/sites\/1393\/2021\/05\/Size-of-a-Liter-1.png\" alt=\"Size of a Liter\" width=\"400\" height=\"281\" srcset=\"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Size-of-a-Liter-1.png 600w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Size-of-a-Liter-1-300x211.png 300w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Size-of-a-Liter-1-65x46.png 65w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Size-of-a-Liter-1-225x158.png 225w, https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-content\/uploads\/sites\/1393\/2021\/05\/Size-of-a-Liter-1-350x246.png 350w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/a><figcaption id=\"caption-attachment-4611\" class=\"wp-caption-text\"><strong>Figure 3.<\/strong> The Size of 1 Liter<\/figcaption><\/figure>\n<p class=\"para\">One liter equals 1,000 cm<sup class=\"superscript\">3<\/sup>, so 1 cm<sup class=\"superscript\">3<\/sup> is the same as 1 mL.<\/p>\n<\/div>\n<p id=\"ball-ch02_s02_p09\" class=\"para editable block\">Units not only are multiplied together but also can be divided. For example, if you are traveling at one meter for every second of time elapsed, your velocity is 1 meter per second, or 1 m\/s. The word <em class=\"emphasis\">per<\/em> implies division, so velocity is determined by dividing a distance quantity by a time quantity. Other units for velocity include kilometers per hour (km\/h) or even micrometers per nanosecond (\u03bcm\/ns). Later, we will see other derived units that can be expressed as fractions.<\/p>\n<div class=\"textbox shaded\">\n<h3 class=\"title\">Example 2<\/h3>\n<ol id=\"ball-ch02_s02_l02\" class=\"orderedlist\">\n<li>A human hair has a diameter of about 6.0 \u00d7 10<sup class=\"superscript\">\u22125<\/sup> m. Suggest an appropriate unit for this measurement and write the diameter of a human hair in terms of that unit.<\/li>\n<li>What is the velocity of a car if it goes 25 m in 5.0 s?<\/li>\n<\/ol>\n<p class=\"simpara\">Solution<\/p>\n<ol id=\"ball-ch02_s02_l03\" class=\"orderedlist\">\n<li>The scientific notation 10<sup class=\"superscript\">\u22125<\/sup> is close to 10<sup class=\"superscript\">\u22126<\/sup>, which defines the micro- prefix. Let us use micrometers as the unit for hair diameter. The number 6.0 \u00d7 10<sup class=\"superscript\">\u22125<\/sup> can be written as 60 \u00d7 10<sup class=\"superscript\">\u22126<\/sup>, and a micrometer is 10<sup class=\"superscript\">\u22126<\/sup> m, so the diameter of a human hair is about 60 \u03bcm.<\/li>\n<li>If velocity is defined as a distance quantity divided by a time quantity, then velocity is 25 meters\/5.0 seconds. Dividing the numbers gives us 25\/5.0 = 5.0, and dividing the units gives us meters\/second, or m\/s. The velocity is 5.0 m\/s.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Check Your Understanding 1<\/h3>\n<ol id=\"ball-ch02_s02_l04\" class=\"orderedlist\">\n<li>Express the volume of an Olympic-sized swimming pool, 2,500,000 L, in more appropriate units.<\/li>\n<li>A common garden snail moves about 6.1 m in 30 min. What is its velocity in meters per minute (m\/min)?<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"ball-ch02_s02_n03\" class=\"key_takeaways editable block\">\n<div class=\"bcc-box bcc-success\">\n<h3>Key Takeaways<\/h3>\n<ul id=\"ball-ch02_s02_l06\" class=\"itemizedlist\">\n<li>Numbers tell \u201chow much,\u201d and units tell \u201cof what.\u201d<\/li>\n<li>Chemistry uses a set of fundamental units and derived units from SI units.<\/li>\n<li>Chemistry uses a set of prefixes that represent multiples or fractions of units.<\/li>\n<li>Units can be multiplied and divided to generate new units for quantities.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Problems &amp; Exercises<\/h3>\n<ol id=\"ball-ch02_s02_qs01_qd01\" class=\"qandadiv\">\n<li id=\"ball-ch02_s02_qs01_qd01_qa01\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p1\" class=\"para\">Identify the unit in each quantity.<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<p>(a) 2 boxes of crayons (b) 3.5 grams of gold<\/p>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p2\" class=\"para\">2. Identify the unit in each quantity.<\/p>\n<p>(a) 32 oz of cheddar cheese (b) 0.045 cm<sup class=\"superscript\">3<\/sup> of water<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p3\" class=\"para\">3. Identify the unit in each quantity.<\/p>\n<p>(a) 9.58 s (the current world record in the 100 m dash) (b) 6.14 m (the current world record in the pole vault)<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p4\" class=\"para\">4. Identify the unit in each quantity.<\/p>\n<p>(a) 2 dozen eggs (b) 2.4 km\/s (the escape velocity of the moon, which is the velocity you need at the surface to escape the moon\u2019s gravity)<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p5\" class=\"para\">5. Indicate what multiplier each prefix represents.<\/p>\n<p>(a) k (b) m (c) M<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p6\" class=\"para\">6. Indicate what multiplier each prefix represents.<\/p>\n<p>(a) c (b) G (c) \u03bc<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p7\" class=\"para\">7. Give the prefix that represents each multiplier.<\/p>\n<p>(a) 1\/1,000th \u00d7 (b) 1,000 \u00d7 (c) 1,000,000,000 \u00d7<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p8\" class=\"para\">8. Give the prefix that represents each multiplier.<\/p>\n<p>(a) 1\/1,000,000,000th \u00d7 (b) 1\/100th \u00d7 (c) 1,000,000 \u00d7<\/p>\n<p>9. Complete the following table with the missing information.<\/p>\n<\/div>\n<div class=\"question\">\n<div class=\"informaltable\">\n<table style=\"border-color: #000000;border-spacing: 0px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th>Unit<\/th>\n<th align=\"center\">Abbreviation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>kilosecond<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">mL<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">Mg<\/td>\n<\/tr>\n<tr>\n<td>centimeter<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p10\" class=\"para\">10.Complete the following table with the missing information.<\/p>\n<div class=\"informaltable\">\n<table style=\"border-color: #000000;border-spacing: 0px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th>Unit<\/th>\n<th align=\"center\">Abbreviation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>kilometer per second<\/td>\n<\/tr>\n<tr>\n<td>second<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">cm<sup class=\"superscript\">3<\/sup><\/td>\n<\/tr>\n<tr>\n<td align=\"center\">\u03bcL<\/td>\n<\/tr>\n<tr>\n<td>nanosecond<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>11. Express each quantity in a more appropriate unit. There may be more than one acceptable answer.<\/p>\n<\/div>\n<\/div>\n<p>(a) 3.44 \u00d7 10<sup class=\"superscript\">\u22126<\/sup> s (b) 3,500 L(c) 0.045 m<\/p>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p12\" class=\"para\">12. Express each quantity in a more appropriate unit. There may be more than one acceptable answer.<\/p>\n<p>(a) 0.000066 m\/s (Hint: you need consider only the unit in the numerator.) (b) 4.66 \u00d7 10<sup class=\"superscript\">6<\/sup> s (c) 7,654 L<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p13\" class=\"para\">13. Express each quantity in a more appropriate unit. There may be more than one acceptable answer.<\/p>\n<p>(a) 43,600 mL (b) 0.0000044 m (c) 1,438 ms<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p14\" class=\"para\">14. Express each quantity in a more appropriate unit. There may be more than one acceptable answer.<\/p>\n<p>(a) 0.000000345 m<sup class=\"superscript\">3 <\/sup>(b) 47,000,000 mm<sup class=\"superscript\">3 <\/sup>(c) 0.00665 L<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p15\" class=\"para\">15. Multiplicative prefixes are used for other units as well, such as computer memory. The basic unit of computer memory is the byte (b). What is the unit for one million bytes?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p17\" class=\"para\">16. You may have heard the terms <em class=\"emphasis\">microscale<\/em> or <em class=\"emphasis\">nanoscale<\/em> to represent the sizes of small objects. What units of length do you think are useful at these scales? What fractions of the fundamental unit of length are these units?<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p19\" class=\"para\">17. Acceleration is defined as a change in velocity per time. Propose a unit for acceleration in terms of the fundamental SI units.<\/p>\n<\/div>\n<div class=\"question\">\n<p id=\"ball-ch02_s02_qs01_p21\" class=\"para\">18. Density is defined as the mass of an object divided by its volume. Propose a unit of density in terms of the fundamental SI units.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Solutions<\/h3>\n<p><strong>Check Your Understanding 1<\/strong><\/p>\n<ol id=\"ball-ch02_s02_l05\" class=\"orderedlist\">\n<li>2.5 ML<\/li>\n<li>0.203 m\/min<\/li>\n<\/ol>\n<p><strong>Problems &amp; Exercises<\/strong><\/p>\n<p><strong>1.<\/strong> (a) boxes of crayons (b) grams of gold<\/p>\n<p><strong>3.<\/strong> (a) seconds (b) meters<\/p>\n<p><strong>5.<\/strong> (a) 1,000 \u00d7 (b) 1\/1,000 \u00d7 (c) 1,000,000 \u00d7<\/p>\n<p><strong>7.<\/strong> (a) milli- (b) kilo- (c) giga-<\/p>\n<p><strong>9.<\/strong><\/p>\n<div class=\"informaltable\">\n<table style=\"border-color: #000000;border-spacing: 0px\" cellpadding=\"0\">\n<thead>\n<tr>\n<th>Unit<\/th>\n<th align=\"center\">Abbreviation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>kilosecond<\/td>\n<td align=\"center\">ks<\/td>\n<\/tr>\n<tr>\n<td>milliliter<\/td>\n<td align=\"center\">mL<\/td>\n<\/tr>\n<tr>\n<td>megagram<\/td>\n<td align=\"center\">Mg<\/td>\n<\/tr>\n<tr>\n<td>centimeter<\/td>\n<td align=\"center\">cm<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p><strong>11<\/strong>. (a) 3.44 \u03bcs (b) 3.5 kL (c) 4.5 cm<\/p>\n<p><strong>13.<\/strong> (a) 43.6 L ( b) 4.4 \u00b5m (c) 1.438 s<\/p>\n<p><strong>15.<\/strong> megabytes (Mb)<\/p>\n<p><strong>17.<\/strong> meters\/second<sup class=\"superscript\">2<\/sup><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<h1><a id=\"add-exer\" href=\"\"><\/a>Additional Exercises<\/h1>\n<div class=\"bcc-box bcc-info\">\n<h3 class=\"title\">Additional Exercises<\/h3>\n<ol>\n<li>Evaluate 0.00000000552 \u00d7 0.0000000006188 and express the answer in scientific notation. You may have to rewrite the original numbers in scientific notation first.<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa02\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p3\" class=\"para\">Evaluate 333,999,500,000 \u00f7 0.00000000003396 and express the answer in scientific notation. You may need to rewrite the original numbers in scientific notation first.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa03\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p5\" class=\"para\">Express the number 6.022 \u00d7 10<sup class=\"superscript\">23<\/sup> in standard notation.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa04\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p7\" class=\"para\">Express the number 6.626 \u00d7 10<sup class=\"superscript\">\u221234<\/sup> in standard notation.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa05\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p9\" class=\"para\">When powers of 10 are multiplied together, the powers are added together. For example, 10<sup class=\"superscript\">2<\/sup> \u00d7 10<sup class=\"superscript\">3<\/sup> = 10<sup class=\"superscript\">2+3<\/sup> = 10<sup class=\"superscript\">5<\/sup>. With this in mind, can you evaluate (4.506 \u00d7 10<sup class=\"superscript\">4<\/sup>) \u00d7 (1.003 \u00d7 10<sup class=\"superscript\">2<\/sup>) without entering scientific notation into your calculator?<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa06\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p11\" class=\"para\">When powers of 10 are divided into each other, the bottom exponent is subtracted from the top exponent. For example, 10<sup class=\"superscript\">5<\/sup>\/10<sup class=\"superscript\">3<\/sup> = 10<sup class=\"superscript\">5\u22123<\/sup> = 10<sup class=\"superscript\">2<\/sup>. With this in mind, can you evaluate (8.552 \u00d7 10<sup class=\"superscript\">6<\/sup>) \u00f7 (3.129 \u00d7 10<sup class=\"superscript\">3<\/sup>) without entering scientific notation into your calculator?<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa07\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p13\" class=\"para\">Consider the quantity two dozen eggs. Is the number in this quantity \u201ctwo\u201d or \u201ctwo dozen\u201d? Justify your choice.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa08\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p15\" class=\"para\">Consider the quantity two dozen eggs. Is the unit in this quantity \u201ceggs\u201d or \u201cdozen eggs\u201d? Justify your choice.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa09\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p17\" class=\"para\">Fill in the blank: 1 km = ______________ \u03bcm.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa10\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p19\" class=\"para\">Fill in the blank: 1 Ms = ______________ ns.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa11\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p21\" class=\"para\">Fill in the blank: 1 cL = ______________ ML.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa12\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p23\" class=\"para\">Fill in the blank: 1 mg = ______________ kg.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa13\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p25\" class=\"para\">Express 67.3 km\/h in meters\/second.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa14\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p27\" class=\"para\">Express 0.00444 m\/s in kilometers\/hour.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa15\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p29\" class=\"para\">Using the idea that 1.602 km = 1.000 mi, convert a speed of 60.0 mi\/h into kilometers\/hour.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa16\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p31\" class=\"para\">Using the idea that 1.602 km = 1.000 mi, convert a speed of 60.0 km\/h into miles\/hour.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa17\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p33\" class=\"para\">Convert 52.09 km\/h into meters\/second.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa18\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p35\" class=\"para\">Convert 2.155 m\/s into kilometers\/hour.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa19\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p37\" class=\"para\">Use the formulas for converting degrees Fahrenheit into degrees Celsius to determine the relative size of the Fahrenheit degree over the Celsius degree.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa20\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p39\" class=\"para\">Use the formulas for converting degrees Celsius into kelvins to determine the relative size of the Celsius degree over kelvins.<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa21\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p41\" class=\"para\">What is the mass of 12.67 L of mercury?<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa22\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p43\" class=\"para\">What is the mass of 0.663 m<sup class=\"superscript\">3<\/sup> of air?<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa23\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p45\" class=\"para\">What is the volume of 2.884 kg of gold?<\/p>\n<\/div>\n<\/li>\n<li id=\"ball-ch02_s06_qs01_qd01_qa24\" class=\"qandaentry\">\n<div class=\"question\">\n<p id=\"ball-ch02_s06_qs01_p47\" class=\"para\">What is the volume of 40.99 kg of cork? Assume a density of 0.22 g\/cm<sup class=\"superscript\">3<\/sup>.<\/p>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Answers<\/h3>\n<p><strong>1<\/strong>. 3.42 \u00d7 10<sup>\u221218 <\/sup><\/p>\n<p><strong>3. <\/strong>602,200,000,000,000,000,000,000<\/p>\n<p><strong>5. <\/strong>4.520 \u00d7 10<sup>6 <\/sup><\/p>\n<p><strong>7 . <\/strong>The quantity is two; dozen is the unit.<\/p>\n<div>\n<p><strong>9. <\/strong>1,000,000,000<\/p>\n<p><strong>11. <\/strong>1\/100,000,000<\/p>\n<p><strong>13. <\/strong>18.7 m\/s<\/p>\n<p><strong>15. <\/strong>96.1 km\/h<\/p>\n<p><strong>17. <\/strong>14.47 m\/s<\/p>\n<p><strong>19. <\/strong>One Fahrenheit degree is nine-fifths the size of a Celsius degree.<\/p>\n<p><strong>21. <\/strong>1.72 \u00d7 10<sup>5<\/sup> g<\/p>\n<p><strong>23. <\/strong>149 mL<\/p>\n<\/div>\n<\/div>\n","protected":false},"author":9,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"class_list":["post-783","back-matter","type-back-matter","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/back-matter\/783","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/comments?post=783"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/back-matter\/783\/revisions"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/back-matter\/783\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/media?parent=783"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/pressbooks\/v2\/back-matter-type?post=783"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/contributor?post=783"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/douglasphys1104summer2021\/wp-json\/wp\/v2\/license?post=783"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}