{"id":73,"date":"2021-07-23T09:18:56","date_gmt":"2021-07-23T13:18:56","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/aperrott\/chapter\/mathematical-treatment-of-measurement-results\/"},"modified":"2022-06-22T09:37:50","modified_gmt":"2022-06-22T13:37:50","slug":"mathematical-treatment-of-measurement-results","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/aperrott\/chapter\/mathematical-treatment-of-measurement-results\/","title":{"raw":"1.6 Mathematical Treatment of Measurement Results","rendered":"1.6 Mathematical Treatment of Measurement Results"},"content":{"raw":"<strong><span style=\"font-family: 'Cormorant Garamond', serif;font-size: 1.602em;background-color: #cbd4b6;color: #000000\">Learning Objectives<\/span><\/strong>\r\n<div class=\"textbox textbox--learning-objectives\">\r\n\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities<\/li>\r\n \t<li>Use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-idm319461744\">It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the <em data-effect=\"italics\">time<\/em> required for the athlete to run from the starting line to the finish line, and the <em data-effect=\"italics\">distance<\/em> between these two lines, and then computing <em data-effect=\"italics\">speed<\/em> from the equation that relates these three properties:<\/p>\r\n<img class=\"wp-image-1003 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6h-300x114.png\" alt=\"\" width=\"158\" height=\"60\" \/>\r\n\r\nAn Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of\r\n\r\n<img class=\"wp-image-1004 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6i-300x96.png\" alt=\"\" width=\"159\" height=\"51\" \/>\r\n<p id=\"fs-idm308822992\">Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100\/10 = 10) <em data-effect=\"italics\">and likewise<\/em> dividing the units of each measured quantity to yield the unit of the computed quantity (m\/s = m\/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m. The same relation among the three properties is used, but in this case, the two quantities provided are a speed (10 m\/s) and a distance (25 m). To yield the sought property, time, the equation must be rearranged appropriately:<\/p>\r\n<img class=\"wp-image-1005 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6j-300x122.png\" alt=\"\" width=\"148\" height=\"60\" \/>\r\n<p id=\"fs-idm24572080\">The time can then be computed as:<\/p>\r\n<img class=\"wp-image-1006 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6k-300x107.png\" alt=\"\" width=\"129\" height=\"46\" \/>\r\n<p id=\"fs-idm316688784\">Again, arithmetic on the numbers (25\/10 = 2.5) was accompanied by the same arithmetic on the units (m\/(m\/s) = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m\/m), the result is \u201c1\u201d\u2014or, as commonly phrased, the units \u201ccancel.\u201d<\/p>\r\n<p id=\"fs-idp44099792\">These calculations are examples of a versatile mathematical approach known as<strong> dimensional analysis<\/strong> (or the<strong> factor-label method<\/strong>). Dimensional analysis is based on this premise: <em data-effect=\"italics\">the units of quantities must be subjected to the same mathematical operations as their associated numbers<\/em>. This method can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.<\/p>\r\n\r\n<div id=\"fs-idm285086480\" class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\"><strong>Conversion Factors and Dimensional Analysis<\/strong><\/h3>\r\n<p id=\"fs-idm273312256\">A ratio of two equivalent quantities expressed with different measurement units can be used as a <span data-type=\"term\">unit conversion factor<\/span>. For example, the lengths of 2.54 cm and 1 in. are equivalent (by definition), and so a unit conversion factor may be derived from the ratio,<\/p>\r\n&nbsp;\r\n<div id=\"fs-idm256748160\" data-type=\"equation\"><img class=\"size-medium wp-image-996 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6a-300x59.png\" alt=\"\" width=\"300\" height=\"59\" \/><\/div>\r\n<p id=\"fs-idm205801120\">Several other commonly used conversion factors are given in <a class=\"autogenerated-content\" href=\"#fs-idm222237232\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"fs-idm222237232\" class=\"top-titled\" summary=\"This table is divided into 3 columns. They are titled length, volume, and mass. The following units are under the length column: 1 meter is equal to 1.0936 yards, 1 inch is equal to 2.54 cm 1 kilometer is equal to 0.62137 miles, 1 mile is equal to 1609.3 meters. The following units are under the volume column: 1 liter is equal to 1.0567 quarts, 1 quart is equal to 0.94635 meters, one cubic foot is equal to 28.317 liters, 1 tablespoon is equal to 14.787 milliliters. The following units are under the mass column: 1 kilogram is equal to 2.2046 pounds, 1 pound is equal to 453.59 grams, 1 avoirdupois ounce is equal to 28.349 grams, 1 troy ounce is equal to 31.103 grams.\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"3\" data-align=\"center\">Common Conversion Factors<\/th>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<th data-align=\"left\">Length<\/th>\r\n<th data-align=\"left\">Volume<\/th>\r\n<th data-align=\"left\">Mass<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">1 m = 1.0936 yd<\/td>\r\n<td data-align=\"left\">1 L = 1.0567 qt<\/td>\r\n<td data-align=\"left\">1 kg = 2.2046 lb<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">1 in. = 2.54 cm (exact)<\/td>\r\n<td data-align=\"left\">1 qt = 0.94635 L<\/td>\r\n<td data-align=\"left\">1 lb = 453.59 g<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">1 km = 0.62137 mi<\/td>\r\n<td data-align=\"left\">1 ft<sup>3<\/sup> = 28.317 L<\/td>\r\n<td data-align=\"left\">1 (avoirdupois) oz = 28.349 g<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-align=\"left\">1 mi = 1609.3 m<\/td>\r\n<td data-align=\"left\">1 tbsp = 14.787 mL<\/td>\r\n<td data-align=\"left\">1 (troy) oz = 31.103 g<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-idm221472400\">When a quantity (such as distance in inches) is multiplied by an appropriate unit conversion factor, the quantity is converted to an equivalent value with different units (such as distance in centimeters). For example, a basketball player\u2019s vertical jump of 34 inches can be converted to centimeters by:<\/p>\r\n\r\n<div id=\"fs-idm153590912\" data-type=\"equation\"><img class=\"wp-image-997 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6b-300x79.png\" alt=\"\" width=\"235\" height=\"62\" \/><\/div>\r\n<p id=\"fs-idm291646272\">Since this simple arithmetic involves <em data-effect=\"italics\">quantities<\/em>, the premise of dimensional analysis requires that we multiply both <em data-effect=\"italics\">numbers and units<\/em>. The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield <img class=\"alignnone wp-image-998\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6f.png\" alt=\"\" width=\"40\" height=\"21\" \/>. Just as for numbers, a ratio of identical units is also numerically equal to one, <img class=\"alignnone wp-image-999\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6g.png\" alt=\"\" width=\"40\" height=\"20\" \/> and the unit product thus simplifies to <em data-effect=\"italics\">cm<\/em>. (When identical units divide to yield a factor of 1, they are said to \u201ccancel.\u201d) Dimensional analysis may be used to confirm the proper application of unit conversion factors as demonstrated in the following example.<\/p>\r\n\r\n<div id=\"fs-idm150235328\" class=\"textbox textbox--examples\" data-type=\"example\">\r\n<p id=\"fs-idp22709840\"><strong>Using a Unit Conversion Factor:<\/strong><\/p>\r\nThe mass of a competition frisbee is 125 g. Convert its mass to ounces using the unit conversion factor derived from the relationship 1 oz = 28.349 g (<a class=\"autogenerated-content\" href=\"#fs-idm222237232\">(Figure)<\/a>).\r\n<p id=\"fs-idm290807904\"><strong>Solution:<\/strong><\/p>\r\nGiven the conversion factor, the mass in ounces may be derived using an equation similar to the one used for converting length from inches to centimeters.\r\n\r\n<img class=\"size-medium wp-image-1001 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6c-300x34.png\" alt=\"\" width=\"300\" height=\"34\" \/>\r\n<div id=\"fs-idm138056288\" data-type=\"equation\"><\/div>\r\n<p id=\"fs-idm300877280\">The unit conversion factor may be represented as:<\/p>\r\n\r\n<div id=\"fs-idm233281680\" data-type=\"equation\"><img class=\"wp-image-1002 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6d-300x85.png\" alt=\"\" width=\"190\" height=\"54\" \/><\/div>\r\n<p id=\"fs-idm369973600\">The correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.<\/p>\r\n\r\n<div id=\"fs-idm222314304\" data-type=\"equation\"><\/div>\r\n<div data-type=\"equation\"><img class=\"wp-image-1000 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6e-300x83.png\" alt=\"\" width=\"336\" height=\"93\" \/><\/div>\r\n<p id=\"fs-idm224226288\"><strong>Check Your Learning:<\/strong><\/p>\r\nConvert a volume of 9.345 qt to liters.\r\n\r\n&nbsp;\r\n<div id=\"fs-idm209829536\" data-type=\"note\">\r\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\r\n<p id=\"fs-idm130267872\">8.844 L<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-idm262838208\">Beyond simple unit conversions, the factor-label method can be used to solve more complex problems involving computations. Regardless of the details, the basic approach is the same\u2014all the <em data-effect=\"italics\">factors<\/em> involved in the calculation must be appropriately oriented to ensure that their <em data-effect=\"italics\">labels<\/em> (units) will appropriately cancel and\/or combine to yield the desired unit in the result. As your study of chemistry continues, you will encounter many opportunities to apply this approach.<\/p>\r\n\r\n<div id=\"fs-idm305814320\" class=\"textbox textbox--examples\" data-type=\"example\">\r\n<p id=\"fs-idm280005808\"><strong>Computing Quantities from Measurement Results and Known Mathematical Relations:<\/strong><\/p>\r\nWhat is the density of common antifreeze in units of g\/mL? A 4.00-qt sample of the antifreeze weighs 9.26 lb.\r\n<p id=\"fs-idp14199296\"><strong>Solution:<\/strong><\/p>\r\nSince <img class=\"alignnone wp-image-1009\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6l-300x71.png\" alt=\"\" width=\"131\" height=\"31\" \/>, we need to divide the mass in grams by the volume in milliliters. In general: the number of units of B = the number of units of A \u00d7 unit conversion factor. The necessary conversion factors are given in <a class=\"autogenerated-content\" href=\"#fs-idm222237232\">(Figure)<\/a>: 1 lb = 453.59 g; 1 L = 1.0567 qt; 1 L = 1,000 mL. Mass may be converted from pounds to grams as follows:\r\n<div id=\"fs-idm336821696\" data-type=\"equation\"><img class=\"size-medium wp-image-1010 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6m-300x55.png\" alt=\"\" width=\"300\" height=\"55\" \/><\/div>\r\n<p id=\"fs-idm244171104\">Volume may be converted from quarts to millimeters via two steps:<\/p>\r\n<img class=\"wp-image-1011 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-300x161.png\" alt=\"\" width=\"496\" height=\"266\" \/>\r\n<p id=\"fs-idm219351440\"><strong>Check Your Learning:<\/strong><\/p>\r\nWhat is the volume in liters of 1.000 oz, given that 1 L = 1.0567 qt and 1 qt = 32 oz (exactly)?\r\n\r\n&nbsp;\r\n<div id=\"fs-idm207792224\" data-type=\"note\">\r\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\r\n<p id=\"fs-idm292695264\">2.956 \u00d7 10<sup>\u22122<\/sup> L<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-idm306560960\" class=\"textbox textbox--examples\" data-type=\"example\">\r\n<p id=\"fs-idm338996800\"><strong>Computing Quantities from Measurement Results and Known Mathematical Relations:<\/strong><\/p>\r\nWhile being driven from Philadelphia to Atlanta, a distance of about 1250 km, a 2014 Lamborghini Aventador Roadster uses 213 L gasoline.\r\n<p id=\"fs-idm127359616\">(a) What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?<\/p>\r\n<p id=\"fs-idm315428528\">(b) If gasoline costs $3.80 per gallon, what was the fuel cost for this trip?<\/p>\r\n<p id=\"fs-idm327742336\"><strong>Solution:<\/strong><\/p>\r\n<img class=\"wp-image-1012 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-300x214.png\" alt=\"\" width=\"519\" height=\"370\" \/>\r\n<p id=\"fs-idm125734208\"><strong>Check Your Learning:<\/strong><\/p>\r\nA Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits).\r\n<p id=\"fs-idm247721312\">(a) What (average) fuel economy, in miles per gallon, did the Prius get during this trip?<\/p>\r\n<p id=\"fs-idm292238912\">(b) If gasoline costs $3.90 per gallon, what was the fuel cost for this trip?<\/p>\r\n&nbsp;\r\n<div id=\"fs-idp13830304\" data-type=\"note\">\r\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\r\n<p id=\"fs-idm210440528\">(a) 51 mi\/gal; (b) $62<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-idm206910464\" class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\"><strong>Conversion of Temperature Units<\/strong><\/h3>\r\n<p id=\"fs-idm262720192\">We use the word <strong>temperature <\/strong>to refer to the hotness or coldness of a substance. One way we measure a change in temperature is to use the fact that most substances expand when their temperature increases and contract when their temperature decreases. The mercury or alcohol in a common glass thermometer changes its volume as the temperature changes, and the position of the trapped liquid along a printed scale may be used as a measure of temperature.<\/p>\r\n<p id=\"fs-idm308860096\">Temperature scales are defined relative to selected reference temperatures: Two of the most commonly used are the freezing and boiling temperatures of water at a specified atmospheric pressure. On the Celsius scale, 0 \u00b0C is defined as the freezing temperature of water and 100 \u00b0C as the boiling temperature of water. The space between the two temperatures is divided into 100 equal intervals, which we call degrees.<\/p>\r\n<p id=\"fs-idm131802496\">As mentioned earlier in this chapter, the SI unit of temperature is the kelvin (K). Unlike the Celsius scale, the kelvin scale is an absolute temperature scale in which 0 (zero) K corresponds to the lowest temperature that can theoretically be achieved. Since the kelvin temperature scale is absolute, a degree symbol is not included in the unit abbreviation, K. The early 19th-century discovery of the relationship between a gas\u2019s volume and temperature suggested that the volume of a gas would be zero at \u2212273.15 \u00b0C. In 1848, British physicist William Thompson, who later adopted the title of Lord Kelvin, proposed an absolute temperature scale based on this concept (further treatment of this topic is provided in this text\u2019s chapter on gases).<\/p>\r\n<p id=\"fs-idm309875456\">The freezing temperature of water on this scale is 273.15 K and its boiling temperature is 373.15 K. Notice the numerical difference in these two reference temperatures is 100, the same as for the Celsius scale, and so a temperature <strong>change<\/strong> in K is the same as a temperature <strong>change<\/strong> in <sup>o<\/sup>C.<\/p>\r\n<img class=\"wp-image-1013 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6p-300x158.png\" alt=\"\" width=\"142\" height=\"75\" \/>\r\n<p id=\"fs-idm288599552\">The 273.15 in these equations has been determined experimentally, so it is not exact. <a class=\"autogenerated-content\" href=\"#CNX_Chem_01_06_TempScales\">(Figure)<\/a> shows the relationship the Celsius and Kelvin temperature scales.<\/p>\r\n\r\n<div id=\"CNX_Chem_01_06_TempScales\" class=\"bc-figure figure\">\r\n<div class=\"bc-figcaption figcaption\"><img class=\"alignnone size-medium wp-image-1014\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q-234x300.png\" alt=\"\" width=\"234\" height=\"300\" \/><\/div>\r\n<\/div>\r\n<p id=\"fs-idm296725200\">Although the kelvin (absolute) temperature scale is the official SI temperature scale, Celsius is commonly used in many scientific contexts and is the scale of choice for nonscience contexts in almost all areas of the world.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-idm292051392\" class=\"summary\" data-depth=\"1\">\r\n<h3 data-type=\"title\"><strong>Key Concepts and Summary<\/strong><\/h3>\r\n<p id=\"fs-idm126307616\">Measurements are made using a variety of units. It is often useful or necessary to convert a measured quantity from one unit into another. These conversions are accomplished using unit conversion factors, which are derived by simple applications of a mathematical approach called the factor-label method or dimensional analysis. This strategy is also employed to calculate sought quantities using measured quantities and appropriate mathematical relations.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-idm321708320\" class=\"exercises\" data-depth=\"1\"><\/div>\r\n<div class=\"textbox shaded\" data-type=\"glossary\">\r\n<h3 data-type=\"glossary-title\"><strong>Glossary<\/strong><\/h3>\r\n<dl id=\"fs-idm216300016\">\r\n \t<dt>dimensional analysis<\/dt>\r\n \t<dd id=\"fs-idp11307184\">(also, factor-label method) versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-idm327357936\">\r\n \t<dt>temperature<\/dt>\r\n \t<dd id=\"fs-idm327390944\">intensive property representing the hotness or coldness of matter<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-idm307214528\">\r\n \t<dt>unit conversion factor<\/dt>\r\n \t<dd id=\"fs-idm126354208\">ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<p><strong><span style=\"font-family: 'Cormorant Garamond', serif;font-size: 1.602em;background-color: #cbd4b6;color: #000000\">Learning Objectives<\/span><\/strong><\/p>\n<div class=\"textbox textbox--learning-objectives\">\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities<\/li>\n<li>Use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-idm319461744\">It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the <em data-effect=\"italics\">time<\/em> required for the athlete to run from the starting line to the finish line, and the <em data-effect=\"italics\">distance<\/em> between these two lines, and then computing <em data-effect=\"italics\">speed<\/em> from the equation that relates these three properties:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1003 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6h-300x114.png\" alt=\"\" width=\"158\" height=\"60\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6h-300x114.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6h-65x25.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6h-225x85.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6h-350x133.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6h.png 406w\" sizes=\"auto, (max-width: 158px) 100vw, 158px\" \/><\/p>\n<p>An Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1004 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6i-300x96.png\" alt=\"\" width=\"159\" height=\"51\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6i-300x96.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6i-65x21.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6i-225x72.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6i-350x112.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6i.png 393w\" sizes=\"auto, (max-width: 159px) 100vw, 159px\" \/><\/p>\n<p id=\"fs-idm308822992\">Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100\/10 = 10) <em data-effect=\"italics\">and likewise<\/em> dividing the units of each measured quantity to yield the unit of the computed quantity (m\/s = m\/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m. The same relation among the three properties is used, but in this case, the two quantities provided are a speed (10 m\/s) and a distance (25 m). To yield the sought property, time, the equation must be rearranged appropriately:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1005 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6j-300x122.png\" alt=\"\" width=\"148\" height=\"60\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6j-300x122.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6j-65x26.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6j-225x91.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6j-350x142.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6j.png 394w\" sizes=\"auto, (max-width: 148px) 100vw, 148px\" \/><\/p>\n<p id=\"fs-idm24572080\">The time can then be computed as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1006 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6k-300x107.png\" alt=\"\" width=\"129\" height=\"46\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6k-300x107.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6k-65x23.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6k-225x80.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6k.png 350w\" sizes=\"auto, (max-width: 129px) 100vw, 129px\" \/><\/p>\n<p id=\"fs-idm316688784\">Again, arithmetic on the numbers (25\/10 = 2.5) was accompanied by the same arithmetic on the units (m\/(m\/s) = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m\/m), the result is \u201c1\u201d\u2014or, as commonly phrased, the units \u201ccancel.\u201d<\/p>\n<p id=\"fs-idp44099792\">These calculations are examples of a versatile mathematical approach known as<strong> dimensional analysis<\/strong> (or the<strong> factor-label method<\/strong>). Dimensional analysis is based on this premise: <em data-effect=\"italics\">the units of quantities must be subjected to the same mathematical operations as their associated numbers<\/em>. This method can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.<\/p>\n<div id=\"fs-idm285086480\" class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>Conversion Factors and Dimensional Analysis<\/strong><\/h3>\n<p id=\"fs-idm273312256\">A ratio of two equivalent quantities expressed with different measurement units can be used as a <span data-type=\"term\">unit conversion factor<\/span>. For example, the lengths of 2.54 cm and 1 in. are equivalent (by definition), and so a unit conversion factor may be derived from the ratio,<\/p>\n<p>&nbsp;<\/p>\n<div id=\"fs-idm256748160\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-996 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6a-300x59.png\" alt=\"\" width=\"300\" height=\"59\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6a-300x59.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6a-768x151.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6a-65x13.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6a-225x44.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6a-350x69.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6a.png 836w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/div>\n<p id=\"fs-idm205801120\">Several other commonly used conversion factors are given in <a class=\"autogenerated-content\" href=\"#fs-idm222237232\">(Figure)<\/a>.<\/p>\n<table id=\"fs-idm222237232\" class=\"top-titled\" summary=\"This table is divided into 3 columns. They are titled length, volume, and mass. The following units are under the length column: 1 meter is equal to 1.0936 yards, 1 inch is equal to 2.54 cm 1 kilometer is equal to 0.62137 miles, 1 mile is equal to 1609.3 meters. The following units are under the volume column: 1 liter is equal to 1.0567 quarts, 1 quart is equal to 0.94635 meters, one cubic foot is equal to 28.317 liters, 1 tablespoon is equal to 14.787 milliliters. The following units are under the mass column: 1 kilogram is equal to 2.2046 pounds, 1 pound is equal to 453.59 grams, 1 avoirdupois ounce is equal to 28.349 grams, 1 troy ounce is equal to 31.103 grams.\">\n<thead>\n<tr>\n<th colspan=\"3\" data-align=\"center\">Common Conversion Factors<\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-align=\"left\">Length<\/th>\n<th data-align=\"left\">Volume<\/th>\n<th data-align=\"left\">Mass<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-align=\"left\">1 m = 1.0936 yd<\/td>\n<td data-align=\"left\">1 L = 1.0567 qt<\/td>\n<td data-align=\"left\">1 kg = 2.2046 lb<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">1 in. = 2.54 cm (exact)<\/td>\n<td data-align=\"left\">1 qt = 0.94635 L<\/td>\n<td data-align=\"left\">1 lb = 453.59 g<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">1 km = 0.62137 mi<\/td>\n<td data-align=\"left\">1 ft<sup>3<\/sup> = 28.317 L<\/td>\n<td data-align=\"left\">1 (avoirdupois) oz = 28.349 g<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-align=\"left\">1 mi = 1609.3 m<\/td>\n<td data-align=\"left\">1 tbsp = 14.787 mL<\/td>\n<td data-align=\"left\">1 (troy) oz = 31.103 g<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-idm221472400\">When a quantity (such as distance in inches) is multiplied by an appropriate unit conversion factor, the quantity is converted to an equivalent value with different units (such as distance in centimeters). For example, a basketball player\u2019s vertical jump of 34 inches can be converted to centimeters by:<\/p>\n<div id=\"fs-idm153590912\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-997 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6b-300x79.png\" alt=\"\" width=\"235\" height=\"62\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6b-300x79.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6b-65x17.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6b-225x59.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6b-350x93.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6b.png 643w\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" \/><\/div>\n<p id=\"fs-idm291646272\">Since this simple arithmetic involves <em data-effect=\"italics\">quantities<\/em>, the premise of dimensional analysis requires that we multiply both <em data-effect=\"italics\">numbers and units<\/em>. The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-998\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6f.png\" alt=\"\" width=\"40\" height=\"21\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6f.png 166w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6f-65x35.png 65w\" sizes=\"auto, (max-width: 40px) 100vw, 40px\" \/>. Just as for numbers, a ratio of identical units is also numerically equal to one, <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-999\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6g.png\" alt=\"\" width=\"40\" height=\"20\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6g.png 164w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6g-65x32.png 65w\" sizes=\"auto, (max-width: 40px) 100vw, 40px\" \/> and the unit product thus simplifies to <em data-effect=\"italics\">cm<\/em>. (When identical units divide to yield a factor of 1, they are said to \u201ccancel.\u201d) Dimensional analysis may be used to confirm the proper application of unit conversion factors as demonstrated in the following example.<\/p>\n<div id=\"fs-idm150235328\" class=\"textbox textbox--examples\" data-type=\"example\">\n<p id=\"fs-idp22709840\"><strong>Using a Unit Conversion Factor:<\/strong><\/p>\n<p>The mass of a competition frisbee is 125 g. Convert its mass to ounces using the unit conversion factor derived from the relationship 1 oz = 28.349 g (<a class=\"autogenerated-content\" href=\"#fs-idm222237232\">(Figure)<\/a>).<\/p>\n<p id=\"fs-idm290807904\"><strong>Solution:<\/strong><\/p>\n<p>Given the conversion factor, the mass in ounces may be derived using an equation similar to the one used for converting length from inches to centimeters.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1001 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6c-300x34.png\" alt=\"\" width=\"300\" height=\"34\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6c-300x34.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6c-768x86.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6c-65x7.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6c-225x25.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6c-350x39.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6c.png 890w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<div id=\"fs-idm138056288\" data-type=\"equation\"><\/div>\n<p id=\"fs-idm300877280\">The unit conversion factor may be represented as:<\/p>\n<div id=\"fs-idm233281680\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1002 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6d-300x85.png\" alt=\"\" width=\"190\" height=\"54\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6d-300x85.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6d-65x18.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6d-225x64.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6d-350x99.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6d.png 543w\" sizes=\"auto, (max-width: 190px) 100vw, 190px\" \/><\/div>\n<p id=\"fs-idm369973600\">The correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.<\/p>\n<div id=\"fs-idm222314304\" data-type=\"equation\"><\/div>\n<div data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1000 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6e-300x83.png\" alt=\"\" width=\"336\" height=\"93\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6e-300x83.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6e-768x213.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6e-65x18.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6e-225x62.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6e-350x97.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6e.png 947w\" sizes=\"auto, (max-width: 336px) 100vw, 336px\" \/><\/div>\n<p id=\"fs-idm224226288\"><strong>Check Your Learning:<\/strong><\/p>\n<p>Convert a volume of 9.345 qt to liters.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"fs-idm209829536\" data-type=\"note\">\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\n<p id=\"fs-idm130267872\">8.844 L<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-idm262838208\">Beyond simple unit conversions, the factor-label method can be used to solve more complex problems involving computations. Regardless of the details, the basic approach is the same\u2014all the <em data-effect=\"italics\">factors<\/em> involved in the calculation must be appropriately oriented to ensure that their <em data-effect=\"italics\">labels<\/em> (units) will appropriately cancel and\/or combine to yield the desired unit in the result. As your study of chemistry continues, you will encounter many opportunities to apply this approach.<\/p>\n<div id=\"fs-idm305814320\" class=\"textbox textbox--examples\" data-type=\"example\">\n<p id=\"fs-idm280005808\"><strong>Computing Quantities from Measurement Results and Known Mathematical Relations:<\/strong><\/p>\n<p>What is the density of common antifreeze in units of g\/mL? A 4.00-qt sample of the antifreeze weighs 9.26 lb.<\/p>\n<p id=\"fs-idp14199296\"><strong>Solution:<\/strong><\/p>\n<p>Since <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1009\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6l-300x71.png\" alt=\"\" width=\"131\" height=\"31\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6l-300x71.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6l-65x15.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6l-225x53.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6l-350x82.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6l.png 374w\" sizes=\"auto, (max-width: 131px) 100vw, 131px\" \/>, we need to divide the mass in grams by the volume in milliliters. In general: the number of units of B = the number of units of A \u00d7 unit conversion factor. The necessary conversion factors are given in <a class=\"autogenerated-content\" href=\"#fs-idm222237232\">(Figure)<\/a>: 1 lb = 453.59 g; 1 L = 1.0567 qt; 1 L = 1,000 mL. Mass may be converted from pounds to grams as follows:<\/p>\n<div id=\"fs-idm336821696\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-1010 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6m-300x55.png\" alt=\"\" width=\"300\" height=\"55\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6m-300x55.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6m-768x142.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6m-65x12.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6m-225x41.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6m-350x65.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6m.png 830w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/div>\n<p id=\"fs-idm244171104\">Volume may be converted from quarts to millimeters via two steps:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1011 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-300x161.png\" alt=\"\" width=\"496\" height=\"266\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-300x161.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-1024x550.png 1024w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-768x412.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-1536x824.png 1536w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-2048x1099.png 2048w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-65x35.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-225x121.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6n-350x188.png 350w\" sizes=\"auto, (max-width: 496px) 100vw, 496px\" \/><\/p>\n<p id=\"fs-idm219351440\"><strong>Check Your Learning:<\/strong><\/p>\n<p>What is the volume in liters of 1.000 oz, given that 1 L = 1.0567 qt and 1 qt = 32 oz (exactly)?<\/p>\n<p>&nbsp;<\/p>\n<div id=\"fs-idm207792224\" data-type=\"note\">\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\n<p id=\"fs-idm292695264\">2.956 \u00d7 10<sup>\u22122<\/sup> L<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-idm306560960\" class=\"textbox textbox--examples\" data-type=\"example\">\n<p id=\"fs-idm338996800\"><strong>Computing Quantities from Measurement Results and Known Mathematical Relations:<\/strong><\/p>\n<p>While being driven from Philadelphia to Atlanta, a distance of about 1250 km, a 2014 Lamborghini Aventador Roadster uses 213 L gasoline.<\/p>\n<p id=\"fs-idm127359616\">(a) What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?<\/p>\n<p id=\"fs-idm315428528\">(b) If gasoline costs $3.80 per gallon, what was the fuel cost for this trip?<\/p>\n<p id=\"fs-idm327742336\"><strong>Solution:<\/strong><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1012 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-300x214.png\" alt=\"\" width=\"519\" height=\"370\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-300x214.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-1024x731.png 1024w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-768x548.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-1536x1096.png 1536w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-65x46.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-225x161.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o-350x250.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6o.png 1571w\" sizes=\"auto, (max-width: 519px) 100vw, 519px\" \/><\/p>\n<p id=\"fs-idm125734208\"><strong>Check Your Learning:<\/strong><\/p>\n<p>A Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits).<\/p>\n<p id=\"fs-idm247721312\">(a) What (average) fuel economy, in miles per gallon, did the Prius get during this trip?<\/p>\n<p id=\"fs-idm292238912\">(b) If gasoline costs $3.90 per gallon, what was the fuel cost for this trip?<\/p>\n<p>&nbsp;<\/p>\n<div id=\"fs-idp13830304\" data-type=\"note\">\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\n<p id=\"fs-idm210440528\">(a) 51 mi\/gal; (b) $62<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-idm206910464\" class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>Conversion of Temperature Units<\/strong><\/h3>\n<p id=\"fs-idm262720192\">We use the word <strong>temperature <\/strong>to refer to the hotness or coldness of a substance. One way we measure a change in temperature is to use the fact that most substances expand when their temperature increases and contract when their temperature decreases. The mercury or alcohol in a common glass thermometer changes its volume as the temperature changes, and the position of the trapped liquid along a printed scale may be used as a measure of temperature.<\/p>\n<p id=\"fs-idm308860096\">Temperature scales are defined relative to selected reference temperatures: Two of the most commonly used are the freezing and boiling temperatures of water at a specified atmospheric pressure. On the Celsius scale, 0 \u00b0C is defined as the freezing temperature of water and 100 \u00b0C as the boiling temperature of water. The space between the two temperatures is divided into 100 equal intervals, which we call degrees.<\/p>\n<p id=\"fs-idm131802496\">As mentioned earlier in this chapter, the SI unit of temperature is the kelvin (K). Unlike the Celsius scale, the kelvin scale is an absolute temperature scale in which 0 (zero) K corresponds to the lowest temperature that can theoretically be achieved. Since the kelvin temperature scale is absolute, a degree symbol is not included in the unit abbreviation, K. The early 19th-century discovery of the relationship between a gas\u2019s volume and temperature suggested that the volume of a gas would be zero at \u2212273.15 \u00b0C. In 1848, British physicist William Thompson, who later adopted the title of Lord Kelvin, proposed an absolute temperature scale based on this concept (further treatment of this topic is provided in this text\u2019s chapter on gases).<\/p>\n<p id=\"fs-idm309875456\">The freezing temperature of water on this scale is 273.15 K and its boiling temperature is 373.15 K. Notice the numerical difference in these two reference temperatures is 100, the same as for the Celsius scale, and so a temperature <strong>change<\/strong> in K is the same as a temperature <strong>change<\/strong> in <sup>o<\/sup>C.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1013 aligncenter\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6p-300x158.png\" alt=\"\" width=\"142\" height=\"75\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6p-300x158.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6p-65x34.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6p-225x119.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6p-350x185.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6p.png 447w\" sizes=\"auto, (max-width: 142px) 100vw, 142px\" \/><\/p>\n<p id=\"fs-idm288599552\">The 273.15 in these equations has been determined experimentally, so it is not exact. <a class=\"autogenerated-content\" href=\"#CNX_Chem_01_06_TempScales\">(Figure)<\/a> shows the relationship the Celsius and Kelvin temperature scales.<\/p>\n<div id=\"CNX_Chem_01_06_TempScales\" class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-1014\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q-234x300.png\" alt=\"\" width=\"234\" height=\"300\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q-234x300.png 234w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q-799x1024.png 799w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q-768x984.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q-65x83.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q-225x288.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q-350x448.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/1.6q.png 978w\" sizes=\"auto, (max-width: 234px) 100vw, 234px\" \/><\/div>\n<\/div>\n<p id=\"fs-idm296725200\">Although the kelvin (absolute) temperature scale is the official SI temperature scale, Celsius is commonly used in many scientific contexts and is the scale of choice for nonscience contexts in almost all areas of the world.<\/p>\n<\/div>\n<div id=\"fs-idm292051392\" class=\"summary\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>Key Concepts and Summary<\/strong><\/h3>\n<p id=\"fs-idm126307616\">Measurements are made using a variety of units. It is often useful or necessary to convert a measured quantity from one unit into another. These conversions are accomplished using unit conversion factors, which are derived by simple applications of a mathematical approach called the factor-label method or dimensional analysis. This strategy is also employed to calculate sought quantities using measured quantities and appropriate mathematical relations.<\/p>\n<\/div>\n<div id=\"fs-idm321708320\" class=\"exercises\" data-depth=\"1\"><\/div>\n<div class=\"textbox shaded\" data-type=\"glossary\">\n<h3 data-type=\"glossary-title\"><strong>Glossary<\/strong><\/h3>\n<dl id=\"fs-idm216300016\">\n<dt>dimensional analysis<\/dt>\n<dd id=\"fs-idp11307184\">(also, factor-label method) versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities<\/dd>\n<\/dl>\n<dl id=\"fs-idm327357936\">\n<dt>temperature<\/dt>\n<dd id=\"fs-idm327390944\">intensive property representing the hotness or coldness of matter<\/dd>\n<\/dl>\n<dl id=\"fs-idm307214528\">\n<dt>unit conversion factor<\/dt>\n<dd id=\"fs-idm126354208\">ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":1392,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[48],"contributor":[],"license":[],"class_list":["post-73","chapter","type-chapter","status-publish","hentry","chapter-type-numberless"],"part":30,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters\/73","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/users\/1392"}],"version-history":[{"count":10,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters\/73\/revisions"}],"predecessor-version":[{"id":2104,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters\/73\/revisions\/2104"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/parts\/30"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters\/73\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/media?parent=73"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapter-type?post=73"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/contributor?post=73"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/license?post=73"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}