{"id":1399,"date":"2018-04-11T22:51:38","date_gmt":"2018-04-12T02:51:38","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/chapter\/3-3-molarity\/"},"modified":"2018-06-22T23:58:55","modified_gmt":"2018-06-23T03:58:55","slug":"3-3-molarity","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/chapter\/3-3-molarity\/","title":{"raw":"7.3 Molarity","rendered":"7.3 Molarity"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Describe the fundamental properties of solutions<\/li>\r\n \t<li>Calculate solution concentrations using molarity<\/li>\r\n \t<li>Perform dilution calculations using the dilution equation<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn preceding sections, we focused on the composition of substances: samples of matter that contain only one type of element or compound. However, mixtures\u2014samples of matter containing two or more substances physically\u00a0combined\u2014are more commonly encountered in nature than are pure substances.\r\n\r\n<section id=\"fs-idm10227280\">S<span style=\"font-family: Tinos, Georgia, serif;font-size: 16px\">imilar to a pure substance, the relative composition of a mixture plays an important role in determining its properties. The relative amount of oxygen in a planet\u2019s atmosphere determines its ability to sustain aerobic life. The relative amounts of iron, carbon, nickel, and other elements in steel (a mixture known as an \u201calloy\u201d) determine its physical strength and resistance to corrosion. The relative amount of the active ingredient in a medicine determines its effectiveness in achieving the desired pharmacological effect. The relative amount of sugar in a beverage determines its sweetness (see <\/span><a href=\"#CNX_Chem_03_03_espresso\" class=\"autogenerated-content\" style=\"font-family: Tinos, Georgia, serif;font-size: 16px\">Figure 1<\/a><span style=\"font-family: Tinos, Georgia, serif;font-size: 16px\">). In this section, we will describe one of the most common ways in which the relative compositions of mixtures may be quantified.<\/span><\/section>\r\n<figure id=\"CNX_Chem_03_03_espresso\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"281\"]<a href=\"https:\/\/opentextbc.ca\/chemistry\/wp-content\/uploads\/sites\/150\/2016\/05\/CNX_Chem_03_03_espresso.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_03_03_espresso-2.jpg\" alt=\"A picture is shown of sugar being poured from a spoon into a cup.\" width=\"281\" height=\"246\" class=\"\" \/><\/a> <strong>Figure 1.<\/strong> Sugar is one of many components in the complex mixture known as coffee. The amount of sugar in a given amount of coffee is an important determinant of the beverage\u2019s sweetness. (credit: Jane Whitney)[\/caption]<\/figure>\r\n<section id=\"fs-idm10227280\">\r\n<h2>\u00a0Solutions<\/h2>\r\n<p id=\"fs-idm60561504\">We have previously defined solutions as homogeneous mixtures, meaning that the composition of the mixture (and therefore its properties) is uniform throughout its entire volume. Solutions occur frequently in nature and have also been implemented in many forms of manmade technology. We will explore a more thorough treatment of solution properties in the chapter on solutions and colloids, but here we will introduce some of the basic properties of solutions.<\/p>\r\n<p id=\"fs-idm635664\">The relative amount of a given solution component is known as its <strong>concentration<\/strong>. Often, though not always, a solution contains one component with a concentration that is significantly greater than that of all other components. This component is called the <strong>solvent<\/strong> and may be viewed as the medium in which the other components are dispersed, or <strong>dissolved<\/strong>. Solutions in which water is the solvent are, of course, very common on our planet. A solution in which water is the solvent is called an <strong>aqueous solution<\/strong>.<\/p>\r\n<p id=\"fs-idm64279024\">A <strong>solute<\/strong> is a component of a solution that is typically present at a much lower concentration than the solvent. Solute concentrations are often described with qualitative terms such as <strong>dilute<\/strong> (of relatively low concentration) and <strong>concentrated<\/strong> (of relatively high concentration).<\/p>\r\n<p id=\"fs-idm62212352\">Concentrations may be quantitatively assessed using a wide variety of measurement units, each convenient for particular applications. <strong>Molarity (<em>M<\/em>)<\/strong> is a useful concentration unit for many applications in chemistry. Molarity is defined as the number of moles of solute in exactly 1 liter (1 L) of the solution:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm57520096\" style=\"text-align: center\">$latex M = \\frac{\\text{mol solute}}{\\text{L solution}}$<\/div>\r\n<div class=\"textbox shaded\" id=\"fs-idm98982768\">\r\n<h3>Example 1<\/h3>\r\n<p id=\"fs-idm10181424\">A 355-mL soft drink sample contains 0.133 mol of sucrose (table sugar). What is the molar concentration of sucrose in the beverage?<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm60197152\"><strong>Solution<\/strong>\r\nSince the molar amount of solute and the volume of solution are both given, the molarity can be calculated using the definition of molarity. Per this definition, the solution volume must be converted from mL to L:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm77939616\" style=\"text-align: center\">$latex M = \\frac{\\text{mol solute}}{\\text{L solution}} = \\frac{0.133 \\;\\text{mol}}{355 \\;\\text{mL} \\times \\frac{1 \\;\\text{L}}{1000 \\;\\text{mL}}} = 0.375 \\; M$<\/div>\r\n&nbsp;\r\n<p id=\"fs-idm883648\"><em><strong>Test Yourself<\/strong><\/em>\r\nA teaspoon of table sugar contains about 0.01 mol sucrose. What is the molarity of sucrose if a teaspoon of sugar has been dissolved in a cup of tea with a volume of 200 mL?<\/p>\r\n&nbsp;\r\n\r\n<em><strong>Answer<\/strong><\/em>\r\n\r\n0.05 M\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\" id=\"fs-idm64107376\">\r\n<h3>Example 2<\/h3>\r\n<p id=\"fs-idm58170960\">How much sugar (mol) is contained in a modest sip (~10 mL) of the soft drink from <a href=\"#fs-idm98982768\" class=\"autogenerated-content\">Example 1<\/a>?<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm768\"><strong>Solution<\/strong>\r\nIn this case, we can rearrange the definition of molarity to isolate the quantity sought, moles of sugar. We then substitute the value for molarity that we derived in <a href=\"#fs-idm98982768\" class=\"autogenerated-content\">Example 1<\/a>, 0.375 <em>M<\/em>:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm72962928\">\r\n<p style=\"text-align: center\">$latex M = \\frac{\\text{mol solute}}{\\text{L solution}}$\r\n$latex \\text{mol solute} = M \\times \\text{L solution}$<\/p>\r\n<p style=\"text-align: center\">$latex \\text{mol solute} = 0.375 \\;\\frac{\\text{mol sugar}}{\\text{L}} \\times (10 \\;\\text{mL} \\times \\frac{1 \\text{L}}{1000 \\;\\text{mL}}) = 0.004 \\;\\text{mol sugar}$<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<p id=\"fs-idm97768960\"><em><strong>Test Yourself<\/strong><\/em>\r\nWhat volume (mL) of the sweetened tea described in <a href=\"#fs-idm98982768\" class=\"autogenerated-content\">Example 1<\/a> contains the same amount of sugar (mol) as 10 mL of the soft drink in this example?<\/p>\r\n&nbsp;\r\n\r\n<em><strong>Answer<\/strong><\/em>\r\n\r\n80 mL\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\" id=\"fs-idm81897840\">\r\n<h3>Example 3<\/h3>\r\n<p id=\"fs-idm98918048\">Distilled white vinegar (<a href=\"#CNX_Chem_03_04_vinegar\" class=\"autogenerated-content\">Figure 2<\/a>) is a solution of acetic acid, CH<sub>3<\/sub>CO<sub>2<\/sub>H, in water. A 0.500-L vinegar solution contains 25.2 g of acetic acid. What is the concentration of the acetic acid solution in units of molarity?<\/p>\r\n\r\n\r\n[caption id=\"attachment_1726\" align=\"aligncenter\" width=\"300\"]<a href=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2-300x286.jpg\" alt=\"\" width=\"300\" height=\"286\" class=\"size-medium wp-image-1726\" \/><\/a> <strong>Figure 2.<\/strong> Distilled white vinegar is a solution of acetic acid in water.[\/caption]\r\n<figure id=\"CNX_Chem_03_04_vinegar\"><\/figure>\r\n&nbsp;\r\n<p id=\"fs-idm82387696\"><strong>Solution<\/strong>\r\nAs in previous textbox shaded, the definition of molarity is the primary equation used to calculate the quantity sought. In this case, the mass of solute is provided instead of its molar amount, so we must use the solute\u2019s molar mass to obtain the amount of solute in moles:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm81742128\" style=\"text-align: center\">$latex M = \\frac{\\text{mol solute}}{\\text{L solution}} = \\frac{25.2 \\;\\text{g CH}_3\\text{CO}_2\\text{H} \\times \\frac{1 \\;\\text{mol CH}_3\\text{CO}_2\\text{H}}{60.052 \\;\\text{g CH}_3\\text{CO}_2\\text{H}}}{0.500 \\;\\text{L solution}} = 0.839 \\; M$<\/div>\r\n&nbsp;\r\n<p id=\"fs-idm61725600\"><em><strong>Test Yourself<\/strong><\/em>\r\nCalculate the molarity of 6.52 g of CoCl<sub>2<\/sub> (128.9 g\/mol) dissolved in an aqueous solution with a total volume of 75.0 mL.<\/p>\r\n&nbsp;\r\n\r\n<em><strong>Answer<\/strong><\/em>\r\n\r\n0.674 M\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\" id=\"fs-idm104693104\">\r\n<h3>Example 4<\/h3>\r\n<p id=\"fs-idm2448752\">How many grams of NaCl are contained in 0.250 L of a 5.30-<em>M<\/em> solution?<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm122143856\"><strong>Solution<\/strong>\r\nThe volume and molarity of the solution are specified, so the amount (mol) of solute is easily computed as demonstrated in <a href=\"#fs-idm64107376\" class=\"autogenerated-content\">Example 2<\/a>:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm85599968\" style=\"text-align: center\">\r\n\r\n$latex M = \\;\\frac{\\text{mol solute}}{\\text{L solution}}$\r\n$latex \\text{mol solute} = M \\times \\text{L solution}$\r\n$latex \\text{mol solute} = 5.30 \\;\\frac{\\text{mol NaCl}}{\\text{L}} \\times 0.250 \\;\\text{L} = \\underline{1.32}50 \\;\\text{mol NaCl with 3 sig figs}$\r\n<p id=\"fs-idm108465360\" style=\"text-align: left\">Finally, this molar amount is used to derive the mass of NaCl:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm1818176\">$latex \\underline{1.32}50 \\;\\text{mol NaCl} \\times \\frac{58.4425 \\;\\text{g NaCl}}{\\text{mol NaCl}} = 77.4 \\;\\text{g NaCl}$<\/div>\r\n&nbsp;\r\n<p id=\"fs-idm122705296\" style=\"text-align: left\"><em><strong>Test Yourself<\/strong><\/em>\r\nHow many grams of CaCl<sub>2<\/sub> (110.98 g\/mol) are contained in 250.0 mL of a 0.200-<em>M<\/em> solution of calcium chloride?<\/p>\r\n&nbsp;\r\n<p style=\"text-align: left\"><em><strong>Answer<\/strong><\/em><\/p>\r\n<p style=\"text-align: left\">5.55 g CaCl<sub>2<\/sub><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-idm75726960\">When performing calculations stepwise, as in <a href=\"#fs-idm104693104\" class=\"autogenerated-content\">Example 4<\/a>, it is important to refrain from rounding any intermediate calculation results, which can lead to rounding errors in the final result. In <a href=\"#fs-idm104693104\" class=\"autogenerated-content\">Example 4<\/a>, the molar amount of NaCl computed in the first step, 1.325 mol, would be properly rounded to 1.32 mol if it were to be reported; however, although the last digit (5) is not significant, it must be retained as a guard digit in the intermediate calculation. If we had not retained this guard digit, the final calculation for the mass of NaCl would have been 77.1 g, a difference of 0.3 g.<\/p>\r\n<p id=\"fs-idm105202592\">In addition to retaining a guard digit for intermediate calculations, we can also avoid rounding errors by performing computations in a single step (see <a href=\"#fs-idm88061984\" class=\"autogenerated-content\">Example 5<\/a>). This eliminates intermediate steps so that only the final result is rounded.<\/p>\r\n\r\n<div class=\"textbox shaded\" id=\"fs-idm88061984\">\r\n<h3>Example 5<\/h3>\r\n<p id=\"fs-idm111967328\">In <a href=\"#fs-idm81897840\" class=\"autogenerated-content\">Example 3<\/a>, we found the typical concentration of vinegar to be 0.839 <em>M<\/em>. What volume of vinegar contains 75.6 g of acetic acid?<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm108289568\"><strong>Solution<\/strong>\r\nFirst, use the molar mass to calculate moles of acetic acid from the given mass:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm112801328\" style=\"text-align: center\">$latex \\text{g solute} \\times \\frac{\\text{mol solute}}{\\text{g solute}} = \\text{mol solute}$<\/div>\r\n<p id=\"fs-idm67563520\">Then, use the molarity of the solution to calculate the volume of solution containing this molar amount of solute:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm79868224\" style=\"text-align: center\">$latex \\text{mol solute} \\times \\frac{\\text{L solution}}{\\text{mol solute}} = \\text{L solution}$<\/div>\r\n<p id=\"fs-idm141610240\">Combining these two steps into one yields:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm104886656\">\r\n<p style=\"text-align: center\">$latex \\text{g solute} \\times \\frac{\\text{mol solute}}{\\text{g solute}} \\times \\frac{\\text{L solution}}{\\text{mol solute}} = \\text{L solution}$$latex 75.6 \\;\\text{g CH}_3\\text{CO}_2\\text{H} (\\frac{\\text{mol CH}_3\\text{CO}_2\\text{H}}{60.053 \\;\\text{g}}) (\\frac{\\text{L solution}}{0.839 \\;\\text{mol CH}_3\\text{CO}_2\\text{H}}) = 1.50 \\;\\text{L solution}$<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n<p id=\"fs-idm59437344\"><em><strong>Test Yourself<\/strong><\/em>\r\nWhat volume of a 1.50-<em>M<\/em> KBr solution contains 66.0 g KBr?<\/p>\r\n&nbsp;\r\n\r\n<em><strong>Answer<\/strong><\/em>\r\n\r\n0.370 L\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-idm97949648\">\r\n<h2>Dilution of Solutions<\/h2>\r\n<p id=\"fs-idm134700400\"><strong>Dilution<\/strong> is the process whereby the concentration of a solution is lessened by the addition of solvent. For example, we might say that a glass of iced tea becomes increasingly diluted as the ice melts. The water from the melting ice increases the volume of the solvent (water) and the overall volume of the solution (iced tea), thereby reducing the relative concentrations of the solutes that give the beverage its taste (<a href=\"#CNX_Chem_03_04_dilution\" class=\"autogenerated-content\">Figure 3<\/a>).<\/p>\r\n\r\n<figure id=\"CNX_Chem_03_04_dilution\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"422\"]<a href=\"https:\/\/opentextbc.ca\/chemistry\/wp-content\/uploads\/sites\/150\/2016\/05\/CNX_Chem_03_04_dilution.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_03_04_dilution-2.jpg\" alt=\"This figure shows two graduated cylinders side-by-side. The first has about half as much blue liquid as the second. The blue liquid is darker in the first cylinder than in the second.\" width=\"422\" height=\"250\" class=\"\" \/><\/a> <strong>Figure 3.<\/strong> Both solutions contain the same mass of copper nitrate. The solution on the right is more dilute because the copper nitrate is dissolved in more solvent. (credit: Mark Ott)[\/caption]<\/figure>\r\n<p id=\"fs-idm77995840\">Dilution is also a common means of preparing solutions of a desired concentration. By adding solvent to a measured portion of a more concentrated <em>stock solution<\/em>, we can achieve a particular concentration. For example, commercial pesticides are typically sold as solutions in which the active ingredients are far more concentrated than is appropriate for their application. Before they can be used on crops, the pesticides must be diluted. This is also a very common practice for the preparation of a number of common laboratory reagents (<a href=\"#CNX_Chem_03_04_solution\" class=\"autogenerated-content\">Figure 4<\/a>).<\/p>\r\n&nbsp;\r\n<figure id=\"CNX_Chem_03_04_solution\"><figcaption>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<a href=\"https:\/\/opentextbc.ca\/chemistry\/wp-content\/uploads\/sites\/150\/2016\/05\/CNX_Chem_03_04_solution.jpg\"><img src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_03_04_solution-2.jpg\" alt=\"This figure shows two photos. In the first, there is an empty glass container, 4.75 g of K M n O subscript 4 powder on a white circle, and a bottle of distilled water. In the second photo the powder and about half the water have been added to the glass container. The liquid in the glass container is almost black in color.\" width=\"975\" height=\"405\" \/><\/a> <strong>Figure 4.<\/strong> A solution of KMnO<sub>4<\/sub> is prepared by mixing water with 4.74 g of KMnO<sub>4<\/sub> in a flask. (credit: modification of work by Mark Ott)[\/caption]\r\n\r\n<\/figcaption><\/figure>\r\n<p id=\"fs-idm77996832\">A simple mathematical relationship can be used to relate the volumes and concentrations of a solution before and after the dilution process. According to the definition of molarity, the molar amount of solute in a solution is equal to the product of the solution\u2019s molarity and its volume in liters:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm32193760\" style=\"text-align: center\">$latex n = ML $<\/div>\r\n<p id=\"fs-idm104760288\">Expressions like these may be written for a solution before and after it is diluted:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm18261728\" style=\"text-align: center\">$latex n_1 = M_1L_1$<\/div>\r\n<div class=\"equation\" id=\"fs-idm61402048\" style=\"text-align: center\">$latex n_2 = M_2L_2$<\/div>\r\n<p id=\"fs-idm123215808\">where the subscripts \u201c1\u201d and \u201c2\u201d refer to the solution before and after the dilution, respectively. Since the dilution process <em>does not change the amount of solute in the solution,<\/em><em>n<\/em><sub>1<\/sub> = <em>n<\/em><sub>2<\/sub>. Thus, these two equations may be set equal to one another:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idp188032944\" style=\"text-align: center\">$latex M_1L_1 = M_2L_2$<\/div>\r\n<p id=\"fs-idm103311152\">This relation is commonly referred to as the dilution equation. Although we derived this equation using molarity as the unit of concentration and liters as the unit of volume, other units of concentration and volume may be used, so long as the units properly cancel per the factor-label method. Reflecting this versatility, the dilution equation is often written in the more general form:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm69146864\" style=\"text-align: center\">$latex C_1V_1 = C_2V_2$<\/div>\r\n<p id=\"fs-idm81143040\">where <em>C<\/em> and <em>V<\/em> are concentration and volume, respectively.<\/p>\r\n\r\n<div id=\"fs-idm98324208\" class=\"textbox shaded\">\r\n\r\n<img src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/OSC_Interactive_200-7-2.png\" alt=\"\u00a0\" width=\"138\" height=\"86\" class=\"alignleft\" \/>\r\n\r\n&nbsp;\r\n<p id=\"fs-idm105278864\">Use the <a href=\"http:\/\/openstaxcollege.org\/l\/16Phetsolvents\">simulation<\/a> to explore the relations between solute amount, solution volume, and concentration and to confirm the dilution equation.<\/p>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\" id=\"fs-idm81737600\">\r\n<h3>Example 6<\/h3>\r\n<p id=\"fs-idm65542512\">If 0.850 L of a 5.00-<em>M<\/em> solution of copper nitrate, Cu(NO<sub>3<\/sub>)<sub>2<\/sub>, is diluted to a volume of 1.80 L by the addition of water, what is the molarity of the diluted solution?<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm63250768\"><strong>Solution<\/strong>\r\nWe are given the volume and concentration of a stock solution, <em>V<\/em><sub>1<\/sub> and <em>C<\/em><sub>1<\/sub>, and the volume of the resultant diluted solution, <em>V<\/em><sub>2<\/sub>. We need to find the concentration of the diluted solution, <em>C<\/em><sub>2<\/sub>. We thus rearrange the dilution equation in order to isolate <em>C<\/em><sub>2<\/sub>:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm82973104\" style=\"text-align: center\">$latex C_1V_1 = C_2V_2$\r\n$latex C_2 = \\frac{C_1V_1}{V_2}$<\/div>\r\n<p id=\"fs-idm81759792\">Since the stock solution is being diluted by more than two-fold (volume is increased from 0.85 L to 1.80 L), we would expect the diluted solution\u2019s concentration to be less than one-half 5 <em>M<\/em>. We will compare this ballpark estimate to the calculated result to check for any gross errors in computation (for example, such as an improper substitution of the given quantities). Substituting the given values for the terms on the right side of this equation yields:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm61572480\" style=\"text-align: center\">$latex C_2 = \\frac{0.850 \\;\\text{L} \\times 5.00 \\frac{\\text{mol}}{\\text{L}}}{1.80 \\;\\text{L}} = 2.36 \\;M$<\/div>\r\n<p id=\"fs-idm125598048\">This result compares well to our ballpark estimate (it\u2019s a bit less than one-half the stock concentration, 5 <em>M<\/em>).<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm105016944\"><em><strong>Test Yourself<\/strong><\/em>\r\nWhat is the concentration of the solution that results from diluting 25.0 mL of a 2.04-<em>M<\/em> solution of CH<sub>3<\/sub>OH to 500.0 mL?<\/p>\r\n&nbsp;\r\n\r\n<em><strong>Answer<\/strong><\/em>\r\n\r\n0.102 <em>M<\/em> CH<sub>3<\/sub>OH\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\" id=\"fs-idm81033056\">\r\n<h3>Example 7<\/h3>\r\n<p id=\"fs-idm90576336\">What volume of 0.12 <em>M<\/em> HBr can be prepared from 11 mL (0.011 L) of 0.45 <em>M<\/em> HBr?<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm75791728\"><strong>Solution<\/strong>\r\nWe are given the volume and concentration of a stock solution, <em>V<\/em><sub>1<\/sub> and <em>C<\/em><sub>1<\/sub>, and the concentration of the resultant diluted solution, <em>C<\/em><sub>2<\/sub>. We need to find the volume of the diluted solution, <em>V<\/em><sub>2<\/sub>. We thus rearrange the dilution equation in order to isolate <em>V<\/em><sub>2<\/sub>:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm80850240\" style=\"text-align: center\">$latex C_1V_1 = C_2V_2$\r\n$latex V_2 = \\frac{C_1V_1}{C_2}$<\/div>\r\n<p id=\"fs-idm61209504\">Since the diluted concentration (0.12 <em>M<\/em>) is slightly more than one-fourth the original concentration (0.45 <em>M<\/em>), we would expect the volume of the diluted solution to be roughly four times the original volume, or around 44 mL. Substituting the given values and solving for the unknown volume yields:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm82787888\" style=\"text-align: center\">$latex V_2 = \\frac{(0.45\\;M)(0.011 \\;\\text{L})}{0.12 \\; M}$\r\n$latex V_2 = 0.041 \\;\\text{L}$<\/div>\r\n<p id=\"fs-idm80652096\">The volume of the 0.12-<em>M<\/em> solution is 0.041 L (41 mL). The result is reasonable and compares well with our rough estimate.<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm78684320\"><em><strong>Test Yourself<\/strong><\/em>\r\nA laboratory experiment calls for 0.125 <em>M<\/em> HNO<sub>3<\/sub>. What volume of 0.125 <em>M<\/em> HNO<sub>3<\/sub> can be prepared from 0.250 L of 1.88 <em>M<\/em> HNO<sub>3<\/sub>?<\/p>\r\n&nbsp;\r\n\r\n<em><strong>Answer<\/strong><\/em>\r\n\r\n3.76 L\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\" id=\"fs-idm60422464\">\r\n<h3>Example 8<\/h3>\r\n<p id=\"fs-idm58713072\">What volume of 1.59 <em>M<\/em> KOH is required to prepare 5.00 L of 0.100 <em>M<\/em> KOH?<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm72581456\"><strong>Solution<\/strong>\r\nWe are given the concentration of a stock solution, <em>C<\/em><sub>1<\/sub>, and the volume and concentration of the resultant diluted solution, <em>V<\/em><sub>2<\/sub> and <em>C<\/em><sub>2<\/sub>. We need to find the volume of the stock solution, <em>V<\/em><sub>1<\/sub>. We thus rearrange the dilution equation in order to isolate <em>V<\/em><sub>1<\/sub>:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm108330752\" style=\"text-align: center\">$latex C_1V_1 = C_2V_2$\r\n$latex V_2 = \\frac{C_2V_2}{C_2}$<\/div>\r\n<p id=\"fs-idm108418736\">Since the concentration of the diluted solution 0.100 <em>M<\/em> is roughly one-sixteenth that of the stock solution (1.59 <em>M<\/em>), we would expect the volume of the stock solution to be about one-sixteenth that of the diluted solution, or around 0.3 liters. Substituting the given values and solving for the unknown volume yields:<\/p>\r\n\r\n<div class=\"equation\" id=\"fs-idm129889216\" style=\"text-align: center\">$latex V_1 = \\frac{(0.100\\;M)(5.00 \\;\\text{L})}{1.59 \\; M}$\r\n$latex V_1 = 0.314 \\;\\text{L}$<\/div>\r\n<p id=\"fs-idm85673088\">Thus, we would need 0.314 L of the 1.59-<em>M<\/em> solution to prepare the desired solution. This result is consistent with our rough estimate.<\/p>\r\n&nbsp;\r\n<p id=\"fs-idm61586320\"><em><strong>Test Yourself<\/strong><\/em>\r\nWhat volume of a 0.575-<em>M<\/em> solution of glucose, C<sub>6<\/sub>H<sub>12<\/sub>O<sub>6<\/sub>, can be prepared from 50.00 mL of a 3.00-<em>M<\/em> glucose solution?<\/p>\r\n&nbsp;\r\n\r\n<em><strong>Answer<\/strong><\/em>\r\n\r\n0.261 L\r\n\r\n<\/div>\r\n<\/section><section id=\"fs-idm102255792\" class=\"summary\">\r\n<h2>Key Concepts and Summary<\/h2>\r\n<p id=\"fs-idm67554336\">Solutions are homogeneous mixtures. Many solutions contain one component, called the solvent, in which other components, called solutes, are dissolved. An aqueous solution is one for which the solvent is water. The concentration of a solution is a measure of the relative amount of solute in a given amount of solution. Concentrations may be measured using various units, with one very useful unit being molarity, defined as the number of moles of solute per liter of solution. The solute concentration of a solution may be decreased by adding solvent, a process referred to as dilution. The dilution equation is a simple relation between concentrations and volumes of a solution before and after dilution.<\/p>\r\n\r\n<\/section><section id=\"fs-idm26459312\" class=\"key-equations\">\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-idm26458432\">\r\n \t<li>$latex M = \\frac{\\text{mol solute}}{\\text{L solution}}$<\/li>\r\n \t<li><em>C<\/em><sub>1<\/sub><em>V<\/em><sub>1<\/sub> = <em>C<\/em><sub>2<\/sub><em>V<\/em><sub>2<\/sub><\/li>\r\n<\/ul>\r\n<\/section><section id=\"fs-idm92499152\" class=\"exercises\">\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Exercises<\/h3>\r\n1. What information do we need to calculate the molarity of a sulfuric acid solution?\r\n\r\n2. Determine the molarity for each of the following solutions:\r\n<p id=\"fs-idm72574192\">a) 0.444 mol of CoCl<sub>2<\/sub> in 0.654 L of solution<\/p>\r\n<p id=\"fs-idm1278048\">b) 98.0 g of phosphoric acid, H<sub>3<\/sub>PO<sub>4<\/sub>, in 1.00 L of solution<\/p>\r\n<p id=\"fs-idm76759104\">c) 0.2074 g of calcium hydroxide, Ca(OH)<sub>2<\/sub>, in 40.00 mL of solution<\/p>\r\n<p id=\"fs-idm62727632\">d) 10.5 kg of Na<sub>2<\/sub>SO<sub>4<\/sub>\u00b710H<sub>2<\/sub>O in 18.60 L of solution<\/p>\r\n<p id=\"fs-idm62190816\">e) 7.0 \u00d7 10<sup>\u22123<\/sup> mol of I<sub>2<\/sub> in 100.0 mL of solution<\/p>\r\n<p id=\"fs-idm76892960\">f) 1.8 \u00d7 10<sup>4<\/sup> mg of HCl in 0.075 L of solution<\/p>\r\n3. Consider this question: What is the mass of the solute in 0.500 L of 0.30 <em>M<\/em> glucose, C<sub>6<\/sub>H<sub>12<\/sub>O<sub>6<\/sub>, used for intravenous injection?\r\n<p id=\"fs-idm75876656\">a) Outline the steps necessary to answer the question.<\/p>\r\n<p id=\"fs-idp41676320\">b) Answer the question.<\/p>\r\n4. Calculate the number of moles and the mass of the solute in each of the following solutions:\r\n<p id=\"fs-idm102145824\">a) 2.00 L of 18.5 <em>M<\/em> H<sub>2<\/sub>SO<sub>4<\/sub>, concentrated sulfuric acid<\/p>\r\n<p id=\"fs-idm94203104\">b) 100.0 mL of 3.8 \u00d7 10<sup>\u22125<\/sup><em>M<\/em> NaCN, the minimum lethal concentration of sodium cyanide in blood serum<\/p>\r\n<p id=\"fs-idm60008496\">c) 5.50 L of 13.3 <em>M<\/em> H<sub>2<\/sub>CO, the formaldehyde used to \u201cfix\u201d tissue samples<\/p>\r\n<p id=\"fs-idm91658464\">d) 325 mL of 1.8 \u00d7 10<sup>\u22126<\/sup><em>M<\/em> FeSO<sub>4<\/sub>, the minimum concentration of iron sulfate detectable by taste in drinking water<\/p>\r\n5. Consider this question: What is the molarity of KMnO<sub>4<\/sub> in a solution of 0.0908 g of KMnO<sub>4<\/sub> in 0.500 L of solution?\r\n<p id=\"fs-idm39482304\">a) Outline the steps necessary to answer the question.<\/p>\r\n<p id=\"fs-idm44707952\">b) Answer the question.<\/p>\r\n6. Calculate the molarity of each of the following solutions:\r\n<p id=\"fs-idm26577024\">a) 0.195 g of cholesterol, C<sub>27<\/sub>H<sub>46<\/sub>O, in 0.100 L of serum, the average concentration of cholesterol in human serum<\/p>\r\n<p id=\"fs-idm26575872\">b) 4.25 g of NH<sub>3<\/sub> in 0.500 L of solution, the concentration of NH<sub>3<\/sub> in household ammonia<\/p>\r\n<p id=\"fs-idm26574720\">c) 1.49 kg of isopropyl alcohol, C<sub>3<\/sub>H<sub>7<\/sub>OH, in 2.50 L of solution, the concentration of isopropyl alcohol in rubbing alcohol<\/p>\r\n<p id=\"fs-idm26573568\">d) 0.029 g of I<sub>2<\/sub> in 0.100 L of solution, the solubility of I<sub>2<\/sub> in water at 20 \u00b0C<\/p>\r\n7. There is about 1.0 g of calcium, as Ca<sup>2+<\/sup>, in 1.0 L of milk. What is the molarity of Ca<sup>2+<\/sup> in milk?\r\n\r\n8. If 0.1718 L of a 0.3556-<em>M<\/em> C<sub>3<\/sub>H<sub>7<\/sub>OH solution is diluted to a concentration of 0.1222 <em>M<\/em>, what is the volume of the resulting solution?\r\n\r\n9. What volume of a 0.33-<em>M<\/em> C<sub>12<\/sub>H<sub>22<\/sub>O<sub>11<\/sub> solution can be diluted to prepare 25 mL of a solution with a concentration of 0.025 <em>M<\/em>?\r\n\r\n10. What is the molarity of the diluted solution when each of the following solutions is diluted to the given final volume?\r\n<p id=\"fs-idm27671600\">a) 1.00 L of a 0.250-<em>M<\/em> solution of Fe(NO<sub>3<\/sub>)<sub>3<\/sub> is diluted to a final volume of 2.00 L<\/p>\r\n<p id=\"fs-idm26632768\">b) 0.5000 L of a 0.1222-<em>M<\/em> solution of C<sub>3<\/sub>H<sub>7<\/sub>OH is diluted to a final volume of 1.250 L<\/p>\r\n<p id=\"fs-idm26631104\">c) 2.35 L of a 0.350-<em>M<\/em> solution of H<sub>3<\/sub>PO<sub>4<\/sub> is diluted to a final volume of 4.00 L<\/p>\r\n<p id=\"fs-idm26629440\">d) 22.50 mL of a 0.025-<em>M<\/em> solution of C<sub>12<\/sub>H<sub>22<\/sub>O<sub>11<\/sub> is diluted to 100.0 mL<\/p>\r\n11. A 2.00-L bottle of a solution of concentrated HCl was purchased for the general chemistry laboratory. The solution contained 868.8 g of HCl. What is the molarity of the solution?\r\n\r\n12. What volume of a 0.20-<em>M<\/em> K<sub>2<\/sub>SO<sub>4<\/sub> solution contains 57 g of K<sub>2<\/sub>SO<sub>4<\/sub>?\r\n\r\n&nbsp;\r\n\r\n<strong>Answers\u00a0<\/strong>\r\n<p id=\"fs-idm76389120\">1. We need to know the number of moles of sulfuric acid dissolved in the solution and the volume of the solution.<\/p>\r\n<p id=\"fs-idm73057968\">2. a) 0.679 <em>M<\/em>; \u00a0 \u00a0 \u00a0b) 1.00 <em>M<\/em>; \u00a0 \u00a0 \u00a0 \u00a0c) 0.06998 <em>M<\/em>;\r\nd) 1.75 <em>M<\/em>; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0e) 0.070 <em>M<\/em>; \u00a0 \u00a0 \u00a0f) 6.6 <em>M<\/em><\/p>\r\n<p id=\"fs-idm73585728\">3. a) determine the number of moles of glucose in 0.500 L of solution; determine the molar mass of glucose; determine the mass of glucose from the number of moles and its molar mass;<\/p>\r\nb) 27 g\r\n<p id=\"fs-idm96721040\">4. a) 37.0 mol H<sub>2<\/sub>SO<sub>4<\/sub>; \u00a0\u00a03.63 \u00d7 10<sup>3<\/sup> g H<sub>2<\/sub>SO<sub>4<\/sub>;\r\nb) 3.8 \u00d7 10<sup>\u22126<\/sup> mol NaCN; \u00a0 \u00a01.9 \u00d7 10<sup>\u22124<\/sup> g NaCN;\r\nc) 73.2 mol H<sub>2<\/sub>CO; \u00a0\u00a02.20 kg H<sub>2<\/sub>CO;\r\nd) 5.9 \u00d7 10<sup>\u22127<\/sup> mol FeSO<sub>4<\/sub>; \u00a0 \u00a08.9 \u00d7 10<sup>\u22125<\/sup> g FeSO<sub>4<\/sub><\/p>\r\n<p id=\"fs-idm26504256\">5. a) Determine the molar mass of KMnO<sub>4<\/sub>; determine the number of moles of KMnO<sub>4<\/sub> in the solution; from the number of moles and the volume of solution, determine the molarity;<\/p>\r\nb) 1.15 \u00d7 10<sup>\u22123<\/sup><em>M<\/em>\r\n<p id=\"fs-idm26572160\">6. a) 5.04 \u00d7 10<sup>\u22123<\/sup><em>M<\/em>; \u00a0 \u00a0 \u00a0b) 0.499 <em>M<\/em>;\r\nc) 9.92 <em>M<\/em>; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0d) 1.1 \u00d7 10<sup>\u22123<\/sup><em>M<\/em><\/p>\r\n<p id=\"fs-idp35203600\">7. 0.025 <em>M<\/em><\/p>\r\n<p id=\"fs-idp171131440\">8. 0.5000 L<\/p>\r\n<p id=\"fs-idp223045936\">9. 1.9 mL<\/p>\r\n<p id=\"fs-idm26627008\">10. a) 0.125 <em>M<\/em>; \u00a0 \u00a0 \u00a0b) 0.04888 <em>M<\/em>;\r\nc) 0.206 <em>M<\/em>; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0d) 0.0056 <em>M<\/em><\/p>\r\n<p id=\"fs-idm27372304\">11. 11.9 <em>M<\/em><\/p>\r\n<p id=\"fs-idm49632864\">12. 1.6 L<\/p>\r\n\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<strong>aqueous solution:\u00a0<\/strong>solution for which water is the solvent\r\n\r\n<strong>concentrated:\u00a0<\/strong>qualitative term for a solution containing solute at a relatively high concentration\r\n\r\n<strong>concentration:\u00a0<\/strong>quantitative measure of the relative amounts of solute and solvent present in a solution\r\n\r\n<strong>dilute:\u00a0<\/strong>qualitative term for a solution containing solute at a relatively low concentration\r\n\r\n<strong>dilution:\u00a0<\/strong>process of adding solvent to a solution in order to lower the concentration of solutes\r\n\r\n<strong>dissolved:\u00a0<\/strong>describes the process by which solute components are dispersed in a solvent\r\n\r\n<strong>molarity (<em>M<\/em>):\u00a0<\/strong>unit of concentration, defined as the number of moles of solute dissolved in 1 liter of solution\r\n\r\n<strong>solute:\u00a0<\/strong>solution component present in a concentration less than that of the solvent\r\n\r\n<strong>solvent:\u00a0<\/strong>solution component present in a concentration that is higher relative to other components\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Describe the fundamental properties of solutions<\/li>\n<li>Calculate solution concentrations using molarity<\/li>\n<li>Perform dilution calculations using the dilution equation<\/li>\n<\/ul>\n<\/div>\n<p>In preceding sections, we focused on the composition of substances: samples of matter that contain only one type of element or compound. However, mixtures\u2014samples of matter containing two or more substances physically\u00a0combined\u2014are more commonly encountered in nature than are pure substances.<\/p>\n<section id=\"fs-idm10227280\">S<span style=\"font-family: Tinos, Georgia, serif;font-size: 16px\">imilar to a pure substance, the relative composition of a mixture plays an important role in determining its properties. The relative amount of oxygen in a planet\u2019s atmosphere determines its ability to sustain aerobic life. The relative amounts of iron, carbon, nickel, and other elements in steel (a mixture known as an \u201calloy\u201d) determine its physical strength and resistance to corrosion. The relative amount of the active ingredient in a medicine determines its effectiveness in achieving the desired pharmacological effect. The relative amount of sugar in a beverage determines its sweetness (see <\/span><a href=\"#CNX_Chem_03_03_espresso\" class=\"autogenerated-content\" style=\"font-family: Tinos, Georgia, serif;font-size: 16px\">Figure 1<\/a><span style=\"font-family: Tinos, Georgia, serif;font-size: 16px\">). In this section, we will describe one of the most common ways in which the relative compositions of mixtures may be quantified.<\/span><\/section>\n<figure id=\"CNX_Chem_03_03_espresso\">\n<figure style=\"width: 281px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/opentextbc.ca\/chemistry\/wp-content\/uploads\/sites\/150\/2016\/05\/CNX_Chem_03_03_espresso.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_03_03_espresso-2.jpg\" alt=\"A picture is shown of sugar being poured from a spoon into a cup.\" width=\"281\" height=\"246\" class=\"\" \/><\/a><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong> Sugar is one of many components in the complex mixture known as coffee. The amount of sugar in a given amount of coffee is an important determinant of the beverage\u2019s sweetness. (credit: Jane Whitney)<\/figcaption><\/figure>\n<\/figure>\n<section id=\"fs-idm10227280\">\n<h2>\u00a0Solutions<\/h2>\n<p id=\"fs-idm60561504\">We have previously defined solutions as homogeneous mixtures, meaning that the composition of the mixture (and therefore its properties) is uniform throughout its entire volume. Solutions occur frequently in nature and have also been implemented in many forms of manmade technology. We will explore a more thorough treatment of solution properties in the chapter on solutions and colloids, but here we will introduce some of the basic properties of solutions.<\/p>\n<p id=\"fs-idm635664\">The relative amount of a given solution component is known as its <strong>concentration<\/strong>. Often, though not always, a solution contains one component with a concentration that is significantly greater than that of all other components. This component is called the <strong>solvent<\/strong> and may be viewed as the medium in which the other components are dispersed, or <strong>dissolved<\/strong>. Solutions in which water is the solvent are, of course, very common on our planet. A solution in which water is the solvent is called an <strong>aqueous solution<\/strong>.<\/p>\n<p id=\"fs-idm64279024\">A <strong>solute<\/strong> is a component of a solution that is typically present at a much lower concentration than the solvent. Solute concentrations are often described with qualitative terms such as <strong>dilute<\/strong> (of relatively low concentration) and <strong>concentrated<\/strong> (of relatively high concentration).<\/p>\n<p id=\"fs-idm62212352\">Concentrations may be quantitatively assessed using a wide variety of measurement units, each convenient for particular applications. <strong>Molarity (<em>M<\/em>)<\/strong> is a useful concentration unit for many applications in chemistry. Molarity is defined as the number of moles of solute in exactly 1 liter (1 L) of the solution:<\/p>\n<div class=\"equation\" id=\"fs-idm57520096\" style=\"text-align: center\">[latex]M = \\frac{\\text{mol solute}}{\\text{L solution}}[\/latex]<\/div>\n<div class=\"textbox shaded\" id=\"fs-idm98982768\">\n<h3>Example 1<\/h3>\n<p id=\"fs-idm10181424\">A 355-mL soft drink sample contains 0.133 mol of sucrose (table sugar). What is the molar concentration of sucrose in the beverage?<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm60197152\"><strong>Solution<\/strong><br \/>\nSince the molar amount of solute and the volume of solution are both given, the molarity can be calculated using the definition of molarity. Per this definition, the solution volume must be converted from mL to L:<\/p>\n<div class=\"equation\" id=\"fs-idm77939616\" style=\"text-align: center\">[latex]M = \\frac{\\text{mol solute}}{\\text{L solution}} = \\frac{0.133 \\;\\text{mol}}{355 \\;\\text{mL} \\times \\frac{1 \\;\\text{L}}{1000 \\;\\text{mL}}} = 0.375 \\; M[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm883648\"><em><strong>Test Yourself<\/strong><\/em><br \/>\nA teaspoon of table sugar contains about 0.01 mol sucrose. What is the molarity of sucrose if a teaspoon of sugar has been dissolved in a cup of tea with a volume of 200 mL?<\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>Answer<\/strong><\/em><\/p>\n<p>0.05 M<\/p>\n<\/div>\n<div class=\"textbox shaded\" id=\"fs-idm64107376\">\n<h3>Example 2<\/h3>\n<p id=\"fs-idm58170960\">How much sugar (mol) is contained in a modest sip (~10 mL) of the soft drink from <a href=\"#fs-idm98982768\" class=\"autogenerated-content\">Example 1<\/a>?<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm768\"><strong>Solution<\/strong><br \/>\nIn this case, we can rearrange the definition of molarity to isolate the quantity sought, moles of sugar. We then substitute the value for molarity that we derived in <a href=\"#fs-idm98982768\" class=\"autogenerated-content\">Example 1<\/a>, 0.375 <em>M<\/em>:<\/p>\n<div class=\"equation\" id=\"fs-idm72962928\">\n<p style=\"text-align: center\">[latex]M = \\frac{\\text{mol solute}}{\\text{L solution}}[\/latex]<br \/>\n[latex]\\text{mol solute} = M \\times \\text{L solution}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\text{mol solute} = 0.375 \\;\\frac{\\text{mol sugar}}{\\text{L}} \\times (10 \\;\\text{mL} \\times \\frac{1 \\text{L}}{1000 \\;\\text{mL}}) = 0.004 \\;\\text{mol sugar}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm97768960\"><em><strong>Test Yourself<\/strong><\/em><br \/>\nWhat volume (mL) of the sweetened tea described in <a href=\"#fs-idm98982768\" class=\"autogenerated-content\">Example 1<\/a> contains the same amount of sugar (mol) as 10 mL of the soft drink in this example?<\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>Answer<\/strong><\/em><\/p>\n<p>80 mL<\/p>\n<\/div>\n<div class=\"textbox shaded\" id=\"fs-idm81897840\">\n<h3>Example 3<\/h3>\n<p id=\"fs-idm98918048\">Distilled white vinegar (<a href=\"#CNX_Chem_03_04_vinegar\" class=\"autogenerated-content\">Figure 2<\/a>) is a solution of acetic acid, CH<sub>3<\/sub>CO<sub>2<\/sub>H, in water. A 0.500-L vinegar solution contains 25.2 g of acetic acid. What is the concentration of the acetic acid solution in units of molarity?<\/p>\n<figure id=\"attachment_1726\" aria-describedby=\"caption-attachment-1726\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2-300x286.jpg\" alt=\"\" width=\"300\" height=\"286\" class=\"size-medium wp-image-1726\" srcset=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2-300x286.jpg 300w, https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2-65x62.jpg 65w, https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2-225x214.jpg 225w, https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2-350x333.jpg 350w, https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_14_03_Vinegar-2.jpg 650w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><figcaption id=\"caption-attachment-1726\" class=\"wp-caption-text\"><strong>Figure 2.<\/strong> Distilled white vinegar is a solution of acetic acid in water.<\/figcaption><\/figure>\n<figure id=\"CNX_Chem_03_04_vinegar\"><\/figure>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm82387696\"><strong>Solution<\/strong><br \/>\nAs in previous textbox shaded, the definition of molarity is the primary equation used to calculate the quantity sought. In this case, the mass of solute is provided instead of its molar amount, so we must use the solute\u2019s molar mass to obtain the amount of solute in moles:<\/p>\n<div class=\"equation\" id=\"fs-idm81742128\" style=\"text-align: center\">[latex]M = \\frac{\\text{mol solute}}{\\text{L solution}} = \\frac{25.2 \\;\\text{g CH}_3\\text{CO}_2\\text{H} \\times \\frac{1 \\;\\text{mol CH}_3\\text{CO}_2\\text{H}}{60.052 \\;\\text{g CH}_3\\text{CO}_2\\text{H}}}{0.500 \\;\\text{L solution}} = 0.839 \\; M[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm61725600\"><em><strong>Test Yourself<\/strong><\/em><br \/>\nCalculate the molarity of 6.52 g of CoCl<sub>2<\/sub> (128.9 g\/mol) dissolved in an aqueous solution with a total volume of 75.0 mL.<\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>Answer<\/strong><\/em><\/p>\n<p>0.674 M<\/p>\n<\/div>\n<div class=\"textbox shaded\" id=\"fs-idm104693104\">\n<h3>Example 4<\/h3>\n<p id=\"fs-idm2448752\">How many grams of NaCl are contained in 0.250 L of a 5.30-<em>M<\/em> solution?<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm122143856\"><strong>Solution<\/strong><br \/>\nThe volume and molarity of the solution are specified, so the amount (mol) of solute is easily computed as demonstrated in <a href=\"#fs-idm64107376\" class=\"autogenerated-content\">Example 2<\/a>:<\/p>\n<div class=\"equation\" id=\"fs-idm85599968\" style=\"text-align: center\">\n<p>[latex]M = \\;\\frac{\\text{mol solute}}{\\text{L solution}}[\/latex]<br \/>\n[latex]\\text{mol solute} = M \\times \\text{L solution}[\/latex]<br \/>\n[latex]\\text{mol solute} = 5.30 \\;\\frac{\\text{mol NaCl}}{\\text{L}} \\times 0.250 \\;\\text{L} = \\underline{1.32}50 \\;\\text{mol NaCl with 3 sig figs}[\/latex]<\/p>\n<p id=\"fs-idm108465360\" style=\"text-align: left\">Finally, this molar amount is used to derive the mass of NaCl:<\/p>\n<div class=\"equation\" id=\"fs-idm1818176\">[latex]\\underline{1.32}50 \\;\\text{mol NaCl} \\times \\frac{58.4425 \\;\\text{g NaCl}}{\\text{mol NaCl}} = 77.4 \\;\\text{g NaCl}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm122705296\" style=\"text-align: left\"><em><strong>Test Yourself<\/strong><\/em><br \/>\nHow many grams of CaCl<sub>2<\/sub> (110.98 g\/mol) are contained in 250.0 mL of a 0.200-<em>M<\/em> solution of calcium chloride?<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: left\"><em><strong>Answer<\/strong><\/em><\/p>\n<p style=\"text-align: left\">5.55 g CaCl<sub>2<\/sub><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-idm75726960\">When performing calculations stepwise, as in <a href=\"#fs-idm104693104\" class=\"autogenerated-content\">Example 4<\/a>, it is important to refrain from rounding any intermediate calculation results, which can lead to rounding errors in the final result. In <a href=\"#fs-idm104693104\" class=\"autogenerated-content\">Example 4<\/a>, the molar amount of NaCl computed in the first step, 1.325 mol, would be properly rounded to 1.32 mol if it were to be reported; however, although the last digit (5) is not significant, it must be retained as a guard digit in the intermediate calculation. If we had not retained this guard digit, the final calculation for the mass of NaCl would have been 77.1 g, a difference of 0.3 g.<\/p>\n<p id=\"fs-idm105202592\">In addition to retaining a guard digit for intermediate calculations, we can also avoid rounding errors by performing computations in a single step (see <a href=\"#fs-idm88061984\" class=\"autogenerated-content\">Example 5<\/a>). This eliminates intermediate steps so that only the final result is rounded.<\/p>\n<div class=\"textbox shaded\" id=\"fs-idm88061984\">\n<h3>Example 5<\/h3>\n<p id=\"fs-idm111967328\">In <a href=\"#fs-idm81897840\" class=\"autogenerated-content\">Example 3<\/a>, we found the typical concentration of vinegar to be 0.839 <em>M<\/em>. What volume of vinegar contains 75.6 g of acetic acid?<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm108289568\"><strong>Solution<\/strong><br \/>\nFirst, use the molar mass to calculate moles of acetic acid from the given mass:<\/p>\n<div class=\"equation\" id=\"fs-idm112801328\" style=\"text-align: center\">[latex]\\text{g solute} \\times \\frac{\\text{mol solute}}{\\text{g solute}} = \\text{mol solute}[\/latex]<\/div>\n<p id=\"fs-idm67563520\">Then, use the molarity of the solution to calculate the volume of solution containing this molar amount of solute:<\/p>\n<div class=\"equation\" id=\"fs-idm79868224\" style=\"text-align: center\">[latex]\\text{mol solute} \\times \\frac{\\text{L solution}}{\\text{mol solute}} = \\text{L solution}[\/latex]<\/div>\n<p id=\"fs-idm141610240\">Combining these two steps into one yields:<\/p>\n<div class=\"equation\" id=\"fs-idm104886656\">\n<p style=\"text-align: center\">[latex]\\text{g solute} \\times \\frac{\\text{mol solute}}{\\text{g solute}} \\times \\frac{\\text{L solution}}{\\text{mol solute}} = \\text{L solution}[\/latex][latex]75.6 \\;\\text{g CH}_3\\text{CO}_2\\text{H} (\\frac{\\text{mol CH}_3\\text{CO}_2\\text{H}}{60.053 \\;\\text{g}}) (\\frac{\\text{L solution}}{0.839 \\;\\text{mol CH}_3\\text{CO}_2\\text{H}}) = 1.50 \\;\\text{L solution}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm59437344\"><em><strong>Test Yourself<\/strong><\/em><br \/>\nWhat volume of a 1.50-<em>M<\/em> KBr solution contains 66.0 g KBr?<\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>Answer<\/strong><\/em><\/p>\n<p>0.370 L<\/p>\n<\/div>\n<\/section>\n<section id=\"fs-idm97949648\">\n<h2>Dilution of Solutions<\/h2>\n<p id=\"fs-idm134700400\"><strong>Dilution<\/strong> is the process whereby the concentration of a solution is lessened by the addition of solvent. For example, we might say that a glass of iced tea becomes increasingly diluted as the ice melts. The water from the melting ice increases the volume of the solvent (water) and the overall volume of the solution (iced tea), thereby reducing the relative concentrations of the solutes that give the beverage its taste (<a href=\"#CNX_Chem_03_04_dilution\" class=\"autogenerated-content\">Figure 3<\/a>).<\/p>\n<figure id=\"CNX_Chem_03_04_dilution\">\n<figure style=\"width: 422px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/opentextbc.ca\/chemistry\/wp-content\/uploads\/sites\/150\/2016\/05\/CNX_Chem_03_04_dilution.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_03_04_dilution-2.jpg\" alt=\"This figure shows two graduated cylinders side-by-side. The first has about half as much blue liquid as the second. The blue liquid is darker in the first cylinder than in the second.\" width=\"422\" height=\"250\" class=\"\" \/><\/a><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong> Both solutions contain the same mass of copper nitrate. The solution on the right is more dilute because the copper nitrate is dissolved in more solvent. (credit: Mark Ott)<\/figcaption><\/figure>\n<\/figure>\n<p id=\"fs-idm77995840\">Dilution is also a common means of preparing solutions of a desired concentration. By adding solvent to a measured portion of a more concentrated <em>stock solution<\/em>, we can achieve a particular concentration. For example, commercial pesticides are typically sold as solutions in which the active ingredients are far more concentrated than is appropriate for their application. Before they can be used on crops, the pesticides must be diluted. This is also a very common practice for the preparation of a number of common laboratory reagents (<a href=\"#CNX_Chem_03_04_solution\" class=\"autogenerated-content\">Figure 4<\/a>).<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"CNX_Chem_03_04_solution\"><figcaption>\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/opentextbc.ca\/chemistry\/wp-content\/uploads\/sites\/150\/2016\/05\/CNX_Chem_03_04_solution.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/CNX_Chem_03_04_solution-2.jpg\" alt=\"This figure shows two photos. In the first, there is an empty glass container, 4.75 g of K M n O subscript 4 powder on a white circle, and a bottle of distilled water. In the second photo the powder and about half the water have been added to the glass container. The liquid in the glass container is almost black in color.\" width=\"975\" height=\"405\" \/><\/a><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong> A solution of KMnO<sub>4<\/sub> is prepared by mixing water with 4.74 g of KMnO<sub>4<\/sub> in a flask. (credit: modification of work by Mark Ott)<\/figcaption><\/figure>\n<\/figcaption><\/figure>\n<p id=\"fs-idm77996832\">A simple mathematical relationship can be used to relate the volumes and concentrations of a solution before and after the dilution process. According to the definition of molarity, the molar amount of solute in a solution is equal to the product of the solution\u2019s molarity and its volume in liters:<\/p>\n<div class=\"equation\" id=\"fs-idm32193760\" style=\"text-align: center\">[latex]n = ML[\/latex]<\/div>\n<p id=\"fs-idm104760288\">Expressions like these may be written for a solution before and after it is diluted:<\/p>\n<div class=\"equation\" id=\"fs-idm18261728\" style=\"text-align: center\">[latex]n_1 = M_1L_1[\/latex]<\/div>\n<div class=\"equation\" id=\"fs-idm61402048\" style=\"text-align: center\">[latex]n_2 = M_2L_2[\/latex]<\/div>\n<p id=\"fs-idm123215808\">where the subscripts \u201c1\u201d and \u201c2\u201d refer to the solution before and after the dilution, respectively. Since the dilution process <em>does not change the amount of solute in the solution,<\/em><em>n<\/em><sub>1<\/sub> = <em>n<\/em><sub>2<\/sub>. Thus, these two equations may be set equal to one another:<\/p>\n<div class=\"equation\" id=\"fs-idp188032944\" style=\"text-align: center\">[latex]M_1L_1 = M_2L_2[\/latex]<\/div>\n<p id=\"fs-idm103311152\">This relation is commonly referred to as the dilution equation. Although we derived this equation using molarity as the unit of concentration and liters as the unit of volume, other units of concentration and volume may be used, so long as the units properly cancel per the factor-label method. Reflecting this versatility, the dilution equation is often written in the more general form:<\/p>\n<div class=\"equation\" id=\"fs-idm69146864\" style=\"text-align: center\">[latex]C_1V_1 = C_2V_2[\/latex]<\/div>\n<p id=\"fs-idm81143040\">where <em>C<\/em> and <em>V<\/em> are concentration and volume, respectively.<\/p>\n<div id=\"fs-idm98324208\" class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-content\/uploads\/sites\/387\/2018\/04\/OSC_Interactive_200-7-2.png\" alt=\"\u00a0\" width=\"138\" height=\"86\" class=\"alignleft\" \/><\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm105278864\">Use the <a href=\"http:\/\/openstaxcollege.org\/l\/16Phetsolvents\">simulation<\/a> to explore the relations between solute amount, solution volume, and concentration and to confirm the dilution equation.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox shaded\" id=\"fs-idm81737600\">\n<h3>Example 6<\/h3>\n<p id=\"fs-idm65542512\">If 0.850 L of a 5.00-<em>M<\/em> solution of copper nitrate, Cu(NO<sub>3<\/sub>)<sub>2<\/sub>, is diluted to a volume of 1.80 L by the addition of water, what is the molarity of the diluted solution?<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm63250768\"><strong>Solution<\/strong><br \/>\nWe are given the volume and concentration of a stock solution, <em>V<\/em><sub>1<\/sub> and <em>C<\/em><sub>1<\/sub>, and the volume of the resultant diluted solution, <em>V<\/em><sub>2<\/sub>. We need to find the concentration of the diluted solution, <em>C<\/em><sub>2<\/sub>. We thus rearrange the dilution equation in order to isolate <em>C<\/em><sub>2<\/sub>:<\/p>\n<div class=\"equation\" id=\"fs-idm82973104\" style=\"text-align: center\">[latex]C_1V_1 = C_2V_2[\/latex]<br \/>\n[latex]C_2 = \\frac{C_1V_1}{V_2}[\/latex]<\/div>\n<p id=\"fs-idm81759792\">Since the stock solution is being diluted by more than two-fold (volume is increased from 0.85 L to 1.80 L), we would expect the diluted solution\u2019s concentration to be less than one-half 5 <em>M<\/em>. We will compare this ballpark estimate to the calculated result to check for any gross errors in computation (for example, such as an improper substitution of the given quantities). Substituting the given values for the terms on the right side of this equation yields:<\/p>\n<div class=\"equation\" id=\"fs-idm61572480\" style=\"text-align: center\">[latex]C_2 = \\frac{0.850 \\;\\text{L} \\times 5.00 \\frac{\\text{mol}}{\\text{L}}}{1.80 \\;\\text{L}} = 2.36 \\;M[\/latex]<\/div>\n<p id=\"fs-idm125598048\">This result compares well to our ballpark estimate (it\u2019s a bit less than one-half the stock concentration, 5 <em>M<\/em>).<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm105016944\"><em><strong>Test Yourself<\/strong><\/em><br \/>\nWhat is the concentration of the solution that results from diluting 25.0 mL of a 2.04-<em>M<\/em> solution of CH<sub>3<\/sub>OH to 500.0 mL?<\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>Answer<\/strong><\/em><\/p>\n<p>0.102 <em>M<\/em> CH<sub>3<\/sub>OH<\/p>\n<\/div>\n<div class=\"textbox shaded\" id=\"fs-idm81033056\">\n<h3>Example 7<\/h3>\n<p id=\"fs-idm90576336\">What volume of 0.12 <em>M<\/em> HBr can be prepared from 11 mL (0.011 L) of 0.45 <em>M<\/em> HBr?<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm75791728\"><strong>Solution<\/strong><br \/>\nWe are given the volume and concentration of a stock solution, <em>V<\/em><sub>1<\/sub> and <em>C<\/em><sub>1<\/sub>, and the concentration of the resultant diluted solution, <em>C<\/em><sub>2<\/sub>. We need to find the volume of the diluted solution, <em>V<\/em><sub>2<\/sub>. We thus rearrange the dilution equation in order to isolate <em>V<\/em><sub>2<\/sub>:<\/p>\n<div class=\"equation\" id=\"fs-idm80850240\" style=\"text-align: center\">[latex]C_1V_1 = C_2V_2[\/latex]<br \/>\n[latex]V_2 = \\frac{C_1V_1}{C_2}[\/latex]<\/div>\n<p id=\"fs-idm61209504\">Since the diluted concentration (0.12 <em>M<\/em>) is slightly more than one-fourth the original concentration (0.45 <em>M<\/em>), we would expect the volume of the diluted solution to be roughly four times the original volume, or around 44 mL. Substituting the given values and solving for the unknown volume yields:<\/p>\n<div class=\"equation\" id=\"fs-idm82787888\" style=\"text-align: center\">[latex]V_2 = \\frac{(0.45\\;M)(0.011 \\;\\text{L})}{0.12 \\; M}[\/latex]<br \/>\n[latex]V_2 = 0.041 \\;\\text{L}[\/latex]<\/div>\n<p id=\"fs-idm80652096\">The volume of the 0.12-<em>M<\/em> solution is 0.041 L (41 mL). The result is reasonable and compares well with our rough estimate.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm78684320\"><em><strong>Test Yourself<\/strong><\/em><br \/>\nA laboratory experiment calls for 0.125 <em>M<\/em> HNO<sub>3<\/sub>. What volume of 0.125 <em>M<\/em> HNO<sub>3<\/sub> can be prepared from 0.250 L of 1.88 <em>M<\/em> HNO<sub>3<\/sub>?<\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>Answer<\/strong><\/em><\/p>\n<p>3.76 L<\/p>\n<\/div>\n<div class=\"textbox shaded\" id=\"fs-idm60422464\">\n<h3>Example 8<\/h3>\n<p id=\"fs-idm58713072\">What volume of 1.59 <em>M<\/em> KOH is required to prepare 5.00 L of 0.100 <em>M<\/em> KOH?<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm72581456\"><strong>Solution<\/strong><br \/>\nWe are given the concentration of a stock solution, <em>C<\/em><sub>1<\/sub>, and the volume and concentration of the resultant diluted solution, <em>V<\/em><sub>2<\/sub> and <em>C<\/em><sub>2<\/sub>. We need to find the volume of the stock solution, <em>V<\/em><sub>1<\/sub>. We thus rearrange the dilution equation in order to isolate <em>V<\/em><sub>1<\/sub>:<\/p>\n<div class=\"equation\" id=\"fs-idm108330752\" style=\"text-align: center\">[latex]C_1V_1 = C_2V_2[\/latex]<br \/>\n[latex]V_2 = \\frac{C_2V_2}{C_2}[\/latex]<\/div>\n<p id=\"fs-idm108418736\">Since the concentration of the diluted solution 0.100 <em>M<\/em> is roughly one-sixteenth that of the stock solution (1.59 <em>M<\/em>), we would expect the volume of the stock solution to be about one-sixteenth that of the diluted solution, or around 0.3 liters. Substituting the given values and solving for the unknown volume yields:<\/p>\n<div class=\"equation\" id=\"fs-idm129889216\" style=\"text-align: center\">[latex]V_1 = \\frac{(0.100\\;M)(5.00 \\;\\text{L})}{1.59 \\; M}[\/latex]<br \/>\n[latex]V_1 = 0.314 \\;\\text{L}[\/latex]<\/div>\n<p id=\"fs-idm85673088\">Thus, we would need 0.314 L of the 1.59-<em>M<\/em> solution to prepare the desired solution. This result is consistent with our rough estimate.<\/p>\n<p>&nbsp;<\/p>\n<p id=\"fs-idm61586320\"><em><strong>Test Yourself<\/strong><\/em><br \/>\nWhat volume of a 0.575-<em>M<\/em> solution of glucose, C<sub>6<\/sub>H<sub>12<\/sub>O<sub>6<\/sub>, can be prepared from 50.00 mL of a 3.00-<em>M<\/em> glucose solution?<\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>Answer<\/strong><\/em><\/p>\n<p>0.261 L<\/p>\n<\/div>\n<\/section>\n<section id=\"fs-idm102255792\" class=\"summary\">\n<h2>Key Concepts and Summary<\/h2>\n<p id=\"fs-idm67554336\">Solutions are homogeneous mixtures. Many solutions contain one component, called the solvent, in which other components, called solutes, are dissolved. An aqueous solution is one for which the solvent is water. The concentration of a solution is a measure of the relative amount of solute in a given amount of solution. Concentrations may be measured using various units, with one very useful unit being molarity, defined as the number of moles of solute per liter of solution. The solute concentration of a solution may be decreased by adding solvent, a process referred to as dilution. The dilution equation is a simple relation between concentrations and volumes of a solution before and after dilution.<\/p>\n<\/section>\n<section id=\"fs-idm26459312\" class=\"key-equations\">\n<h2>Key Equations<\/h2>\n<ul id=\"fs-idm26458432\">\n<li>[latex]M = \\frac{\\text{mol solute}}{\\text{L solution}}[\/latex]<\/li>\n<li><em>C<\/em><sub>1<\/sub><em>V<\/em><sub>1<\/sub> = <em>C<\/em><sub>2<\/sub><em>V<\/em><sub>2<\/sub><\/li>\n<\/ul>\n<\/section>\n<section id=\"fs-idm92499152\" class=\"exercises\">\n<div class=\"bcc-box bcc-info\">\n<h3>Exercises<\/h3>\n<p>1. What information do we need to calculate the molarity of a sulfuric acid solution?<\/p>\n<p>2. Determine the molarity for each of the following solutions:<\/p>\n<p id=\"fs-idm72574192\">a) 0.444 mol of CoCl<sub>2<\/sub> in 0.654 L of solution<\/p>\n<p id=\"fs-idm1278048\">b) 98.0 g of phosphoric acid, H<sub>3<\/sub>PO<sub>4<\/sub>, in 1.00 L of solution<\/p>\n<p id=\"fs-idm76759104\">c) 0.2074 g of calcium hydroxide, Ca(OH)<sub>2<\/sub>, in 40.00 mL of solution<\/p>\n<p id=\"fs-idm62727632\">d) 10.5 kg of Na<sub>2<\/sub>SO<sub>4<\/sub>\u00b710H<sub>2<\/sub>O in 18.60 L of solution<\/p>\n<p id=\"fs-idm62190816\">e) 7.0 \u00d7 10<sup>\u22123<\/sup> mol of I<sub>2<\/sub> in 100.0 mL of solution<\/p>\n<p id=\"fs-idm76892960\">f) 1.8 \u00d7 10<sup>4<\/sup> mg of HCl in 0.075 L of solution<\/p>\n<p>3. Consider this question: What is the mass of the solute in 0.500 L of 0.30 <em>M<\/em> glucose, C<sub>6<\/sub>H<sub>12<\/sub>O<sub>6<\/sub>, used for intravenous injection?<\/p>\n<p id=\"fs-idm75876656\">a) Outline the steps necessary to answer the question.<\/p>\n<p id=\"fs-idp41676320\">b) Answer the question.<\/p>\n<p>4. Calculate the number of moles and the mass of the solute in each of the following solutions:<\/p>\n<p id=\"fs-idm102145824\">a) 2.00 L of 18.5 <em>M<\/em> H<sub>2<\/sub>SO<sub>4<\/sub>, concentrated sulfuric acid<\/p>\n<p id=\"fs-idm94203104\">b) 100.0 mL of 3.8 \u00d7 10<sup>\u22125<\/sup><em>M<\/em> NaCN, the minimum lethal concentration of sodium cyanide in blood serum<\/p>\n<p id=\"fs-idm60008496\">c) 5.50 L of 13.3 <em>M<\/em> H<sub>2<\/sub>CO, the formaldehyde used to \u201cfix\u201d tissue samples<\/p>\n<p id=\"fs-idm91658464\">d) 325 mL of 1.8 \u00d7 10<sup>\u22126<\/sup><em>M<\/em> FeSO<sub>4<\/sub>, the minimum concentration of iron sulfate detectable by taste in drinking water<\/p>\n<p>5. Consider this question: What is the molarity of KMnO<sub>4<\/sub> in a solution of 0.0908 g of KMnO<sub>4<\/sub> in 0.500 L of solution?<\/p>\n<p id=\"fs-idm39482304\">a) Outline the steps necessary to answer the question.<\/p>\n<p id=\"fs-idm44707952\">b) Answer the question.<\/p>\n<p>6. Calculate the molarity of each of the following solutions:<\/p>\n<p id=\"fs-idm26577024\">a) 0.195 g of cholesterol, C<sub>27<\/sub>H<sub>46<\/sub>O, in 0.100 L of serum, the average concentration of cholesterol in human serum<\/p>\n<p id=\"fs-idm26575872\">b) 4.25 g of NH<sub>3<\/sub> in 0.500 L of solution, the concentration of NH<sub>3<\/sub> in household ammonia<\/p>\n<p id=\"fs-idm26574720\">c) 1.49 kg of isopropyl alcohol, C<sub>3<\/sub>H<sub>7<\/sub>OH, in 2.50 L of solution, the concentration of isopropyl alcohol in rubbing alcohol<\/p>\n<p id=\"fs-idm26573568\">d) 0.029 g of I<sub>2<\/sub> in 0.100 L of solution, the solubility of I<sub>2<\/sub> in water at 20 \u00b0C<\/p>\n<p>7. There is about 1.0 g of calcium, as Ca<sup>2+<\/sup>, in 1.0 L of milk. What is the molarity of Ca<sup>2+<\/sup> in milk?<\/p>\n<p>8. If 0.1718 L of a 0.3556-<em>M<\/em> C<sub>3<\/sub>H<sub>7<\/sub>OH solution is diluted to a concentration of 0.1222 <em>M<\/em>, what is the volume of the resulting solution?<\/p>\n<p>9. What volume of a 0.33-<em>M<\/em> C<sub>12<\/sub>H<sub>22<\/sub>O<sub>11<\/sub> solution can be diluted to prepare 25 mL of a solution with a concentration of 0.025 <em>M<\/em>?<\/p>\n<p>10. What is the molarity of the diluted solution when each of the following solutions is diluted to the given final volume?<\/p>\n<p id=\"fs-idm27671600\">a) 1.00 L of a 0.250-<em>M<\/em> solution of Fe(NO<sub>3<\/sub>)<sub>3<\/sub> is diluted to a final volume of 2.00 L<\/p>\n<p id=\"fs-idm26632768\">b) 0.5000 L of a 0.1222-<em>M<\/em> solution of C<sub>3<\/sub>H<sub>7<\/sub>OH is diluted to a final volume of 1.250 L<\/p>\n<p id=\"fs-idm26631104\">c) 2.35 L of a 0.350-<em>M<\/em> solution of H<sub>3<\/sub>PO<sub>4<\/sub> is diluted to a final volume of 4.00 L<\/p>\n<p id=\"fs-idm26629440\">d) 22.50 mL of a 0.025-<em>M<\/em> solution of C<sub>12<\/sub>H<sub>22<\/sub>O<sub>11<\/sub> is diluted to 100.0 mL<\/p>\n<p>11. A 2.00-L bottle of a solution of concentrated HCl was purchased for the general chemistry laboratory. The solution contained 868.8 g of HCl. What is the molarity of the solution?<\/p>\n<p>12. What volume of a 0.20-<em>M<\/em> K<sub>2<\/sub>SO<sub>4<\/sub> solution contains 57 g of K<sub>2<\/sub>SO<sub>4<\/sub>?<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Answers\u00a0<\/strong><\/p>\n<p id=\"fs-idm76389120\">1. We need to know the number of moles of sulfuric acid dissolved in the solution and the volume of the solution.<\/p>\n<p id=\"fs-idm73057968\">2. a) 0.679 <em>M<\/em>; \u00a0 \u00a0 \u00a0b) 1.00 <em>M<\/em>; \u00a0 \u00a0 \u00a0 \u00a0c) 0.06998 <em>M<\/em>;<br \/>\nd) 1.75 <em>M<\/em>; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0e) 0.070 <em>M<\/em>; \u00a0 \u00a0 \u00a0f) 6.6 <em>M<\/em><\/p>\n<p id=\"fs-idm73585728\">3. a) determine the number of moles of glucose in 0.500 L of solution; determine the molar mass of glucose; determine the mass of glucose from the number of moles and its molar mass;<\/p>\n<p>b) 27 g<\/p>\n<p id=\"fs-idm96721040\">4. a) 37.0 mol H<sub>2<\/sub>SO<sub>4<\/sub>; \u00a0\u00a03.63 \u00d7 10<sup>3<\/sup> g H<sub>2<\/sub>SO<sub>4<\/sub>;<br \/>\nb) 3.8 \u00d7 10<sup>\u22126<\/sup> mol NaCN; \u00a0 \u00a01.9 \u00d7 10<sup>\u22124<\/sup> g NaCN;<br \/>\nc) 73.2 mol H<sub>2<\/sub>CO; \u00a0\u00a02.20 kg H<sub>2<\/sub>CO;<br \/>\nd) 5.9 \u00d7 10<sup>\u22127<\/sup> mol FeSO<sub>4<\/sub>; \u00a0 \u00a08.9 \u00d7 10<sup>\u22125<\/sup> g FeSO<sub>4<\/sub><\/p>\n<p id=\"fs-idm26504256\">5. a) Determine the molar mass of KMnO<sub>4<\/sub>; determine the number of moles of KMnO<sub>4<\/sub> in the solution; from the number of moles and the volume of solution, determine the molarity;<\/p>\n<p>b) 1.15 \u00d7 10<sup>\u22123<\/sup><em>M<\/em><\/p>\n<p id=\"fs-idm26572160\">6. a) 5.04 \u00d7 10<sup>\u22123<\/sup><em>M<\/em>; \u00a0 \u00a0 \u00a0b) 0.499 <em>M<\/em>;<br \/>\nc) 9.92 <em>M<\/em>; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0d) 1.1 \u00d7 10<sup>\u22123<\/sup><em>M<\/em><\/p>\n<p id=\"fs-idp35203600\">7. 0.025 <em>M<\/em><\/p>\n<p id=\"fs-idp171131440\">8. 0.5000 L<\/p>\n<p id=\"fs-idp223045936\">9. 1.9 mL<\/p>\n<p id=\"fs-idm26627008\">10. a) 0.125 <em>M<\/em>; \u00a0 \u00a0 \u00a0b) 0.04888 <em>M<\/em>;<br \/>\nc) 0.206 <em>M<\/em>; \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0d) 0.0056 <em>M<\/em><\/p>\n<p id=\"fs-idm27372304\">11. 11.9 <em>M<\/em><\/p>\n<p id=\"fs-idm49632864\">12. 1.6 L<\/p>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<p><strong>aqueous solution:\u00a0<\/strong>solution for which water is the solvent<\/p>\n<p><strong>concentrated:\u00a0<\/strong>qualitative term for a solution containing solute at a relatively high concentration<\/p>\n<p><strong>concentration:\u00a0<\/strong>quantitative measure of the relative amounts of solute and solvent present in a solution<\/p>\n<p><strong>dilute:\u00a0<\/strong>qualitative term for a solution containing solute at a relatively low concentration<\/p>\n<p><strong>dilution:\u00a0<\/strong>process of adding solvent to a solution in order to lower the concentration of solutes<\/p>\n<p><strong>dissolved:\u00a0<\/strong>describes the process by which solute components are dispersed in a solvent<\/p>\n<p><strong>molarity (<em>M<\/em>):\u00a0<\/strong>unit of concentration, defined as the number of moles of solute dissolved in 1 liter of solution<\/p>\n<p><strong>solute:\u00a0<\/strong>solution component present in a concentration less than that of the solvent<\/p>\n<p><strong>solvent:\u00a0<\/strong>solution component present in a concentration that is higher relative to other components<\/p>\n<\/div>\n","protected":false},"author":330,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"7.3 Molarity","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by"},"chapter-type":[],"contributor":[],"license":[50],"class_list":["post-1399","chapter","type-chapter","status-publish","hentry","license-cc-by"],"part":1405,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/pressbooks\/v2\/chapters\/1399","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/wp\/v2\/users\/330"}],"version-history":[{"count":15,"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/pressbooks\/v2\/chapters\/1399\/revisions"}],"predecessor-version":[{"id":4766,"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/pressbooks\/v2\/chapters\/1399\/revisions\/4766"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/pressbooks\/v2\/parts\/1405"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/pressbooks\/v2\/chapters\/1399\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/wp\/v2\/media?parent=1399"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/pressbooks\/v2\/chapter-type?post=1399"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/wp\/v2\/contributor?post=1399"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/chem1114langaracollege\/wp-json\/wp\/v2\/license?post=1399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}