Chapter 8: Stoichiometry of Chemical Reactions

# 8.5 Solutions and Molarity

### Learning Outcomes

• Describe the fundamental properties of solutions
• Calculate solution concentrations using molarity
• Perform dilution calculations using the dilution equation

In preceding sections, we focused on the composition of substances: samples of matter that contain only one type of element or compound. However, mixtures—samples of matter containing two or more substances physically combined—are more commonly encountered in nature than are pure substances. Similar 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’s 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 “alloy”) 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 Figure 8.5.1). In this section, we will describe one of the most common ways in which the relative compositions of mixtures may be quantified.

## Solutions

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.

The relative amount of a given solution component is known as its concentration. 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 solvent and may be viewed as the medium in which the other components are dispersed, or dissolved. 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 aqueous solution.

A solute 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 dilute (of relatively low concentration) and concentrated (of relatively high concentration).

Concentrations may be quantitatively assessed using a wide variety of measurement units, each convenient for particular applications. Molarity (M) 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:

$\displaystyle{M}=\dfrac{\text{mol solute}}{\text{L solution}}$

### Example 8.5.1: Calculating Molar Concentrations

A 355-mL soft drink sample contains 0.133 mol of sucrose (table sugar). What is the molar concentration of sucrose in the beverage?

Show Solution

Since 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:

$M=\dfrac{\text{mol solute}}{\text{L solution}}=\dfrac{0.133\text{mol}}{355\text{mL}\times \frac{1\text{L}}{1000\text{mL}}}=0.375M$

### Example 8.5.2: Deriving Moles and Volumes from Molar Concentrations

How much sugar (mol) is contained in a modest sip $(\text{~}10\text{ mL})$ of the soft drink from Example 1?

Show Solution

In 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 Example 8.5.1, 0.375 M:

$\begin{array}{rcl}\\ M&=&\dfrac{\text{mol solute}}{\text{L solution}}\\ \text{mol solute}&=&M\times \text{L solution}\\ \text{mol solute}&=&0.375\dfrac{\text{mol sugar}}{\text{L}}\times \left(10\text{mL}\times \dfrac{1\text{L}}{1000\text{mL}}\right)=0.004\text{mol sugar}\end{array}$

### Example 8.5.3: Calculating Molar Concentrations from the Mass of Solute

Distilled white vinegar (Figure 8.5.2) is a solution of acetic acid, $\ce{CH3CO2H}$, 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?

Show Solution

As in previous examples, 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’s molar mass to obtain the amount of solute in moles:

$M=\dfrac{\text{mol solute}}{\text{L solution}}=\dfrac{25.2 {\text{ g CH}}_{2}{\text{CO}}_{2}\text{H}\times \frac{1{\text{mol CH}}_{2}{\text{CO}}_{2}\text{H}}{{\text{60.052 g CH}}_{2}{\text{CO}}_{2}\text{H}}}{\text{0.500 L solution}}=0.839M$

$\begin{array}{l}\\ M=\dfrac{\text{mol solute}}{\text{L solution}}=0.839M\\ M=\dfrac{0.839\text{mol solute}}{1.00\text{L solution}}\end{array}$

### Example 8.5.4: Determining the Mass of Solute in a Given Volume of Solution

How many grams of $\ce{NaCl}$ are contained in 0.250 L of a 5.30-M solution?

Show Solution

The volume and molarity of the solution are specified, so the amount (mol) of solute is easily computed as demonstrated in Example 8.5.2:

$\begin{array}{rcl}\\ M&=&\dfrac{\text{mol solute}}{\text{L solution}}\\ \text{mol solute}&=&M\times \text{L solution}\\ \text{mol solute}&=&5.30\dfrac{\text{mol NaCl}}{\text{L}}\times 0.250\text{L}=1.325\text{mol NaCl}\end{array}$

Finally, this molar amount is used to derive the mass of $\ce{NaCl}$:

$\text{1.325 mol NaCl}\times \dfrac{58.44\text{g NaCl}}{\text{mol NaCl}}=77.4\text{g NaCl}$

When performing calculations stepwise, as in Example 8.5.4, it is important to refrain from rounding any intermediate calculation results, which can lead to rounding errors in the final result. In Example 8.5.4, the molar amount of $\ce{NaCl}$ 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 $\ce{NaCl}$ would have been 77.1 g, a difference of 0.3 g.

In addition to retaining a guard digit for intermediate calculations, we can also avoid rounding errors by performing computations in a single step (see Example 8.5.5). This eliminates intermediate steps so that only the final result is rounded.

### Example 8.5.5: Determining the Volume of Solution Containing a Given Mass of Solute

In Example 8.5.3, we found the typical concentration of vinegar to be 0.839 M. What volume of vinegar contains 75.6 g of acetic acid?

Show Solution

First, use the molar mass to calculate moles of acetic acid from the given mass: $\text{g solute}\times \dfrac{\text{mol solute}}{\text{g solute}}=\text{mol solute}$

Then, use the molarity of the solution to calculate the volume of solution containing this molar amount of solute:

$\text{mol solute}\times \dfrac{\text{L solution}}{\text{mol solute}}=\text{L solution}$

Combining these two steps into one yields:

$\text{g solute}\times \dfrac{\text{mol solute}}{\text{g solute}}\times \dfrac{\text{L solution}}{\text{mol solute}}=\text{L solution}$

$75.6\text{g}{\text{CH}}_{3}{\text{CO}}_{2}\text{H}\left(\dfrac{\text{mol}{\text{CH}}_{3}{\text{CO}}_{2}\text{H}}{60.05\text{g}}\right)\left(\dfrac{\text{L solution}}{0.839\text{mol}{\text{CH}}_{3}{\text{CO}}_{2}\text{H}}\right)=1.50\text{L solution}$

## Dilution of Solutions

Dilution 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 (Figure 8.5.3).

Dilution is also a common means of preparing solutions of a desired concentration. By adding solvent to a measured portion of a more concentrated stock solution, 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 (Figure 8.5.4).

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’s molarity and its volume in liters:

$n=ML$

Expressions like these may be written for a solution before and after it is diluted:

${n}_{1}={M}_{1}{L}_{1}$
${n}_{2}={M}_{2}{L}_{2}$

where the subscripts “1” and “2” refer to the solution before and after the dilution, respectively. Since the dilution process does not change the amount of solute in the solution, $n_{1}= n_{2}$. Thus, these two equations may be set equal to one another:

${M}_{1}{L}_{1}={M}_{2}{L}_{2}$

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:

${C}_{1}{V}_{1}={C}_{2}{V}_{2}$

where $C$ and $V$ are concentration and volume, respectively.

### Example 8.5.6: Determining the Concentration of a Diluted Solution

If 0.850 L of a 5.00-M solution of copper nitrate, $\ce{Cu(NO3)2}$, is diluted to a volume of 1.80 L by the addition of water, what is the molarity of the diluted solution?

Show Solution

We are given the volume and concentration of a stock solution, V1 and C1, and the volume of the resultant diluted solution, V2. We need to find the concentration of the diluted solution, C2. We thus rearrange the dilution equation in order to isolate C2:

$\begin{array}{c}{C}_{1}{V}_{1}={C}_{2}{V}_{2}\\{C}_{2}=\dfrac{{C}_{1}{V}_{1}}{{V}_{2}}\end{array}$

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’s concentration to be less than one-half 5 M. 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:

${C}_{2}=\dfrac{0.850\text{L}\times 5.00\frac{\text{mol}}{\text{L}}}{1.80 L}=2.36M$

This result compares well to our ballpark estimate (it’s a bit less than one-half the stock concentration, 5 M).

### Example 8.5.7: Volume of a Diluted Solution

What volume of 0.12 M HBr can be prepared from 11 mL (0.011 L) of 0.45 M HBr?

Show Solution

We are given the volume and concentration of a stock solution, V1 and C1, and the concentration of the resultant diluted solution, C2. We need to find the volume of the diluted solution, V2. We thus rearrange the dilution equation in order to isolate V2:

$\begin{array}{c}{C}_{1}{V}_{1}={C}_{2}{V}_{2}\\{V}_{2}=\dfrac{{C}_{1}{V}_{1}}{{C}_{2}}\end{array}$

Since the diluted concentration (0.12 M) is slightly more than one-fourth the original concentration (0.45 M), we would expect the volume of the diluted solution to be roughly four times the original concentration, or around 44 mL. Substituting the given values and solving for the unknown volume yields:

$\begin{array}{c}\\ {V}_{2}=\dfrac{\left(0.45M\right)\left(0.011\text{L}\right)}{\left(0.12M\right)}\\ {V}_{2}=0.041\text{L}\end{array}$

The volume of the 0.12-M solution is 0.041 L (41 mL). The result is reasonable and compares well with our rough estimate.

### Example 8.5.8: Volume of a Concentrated Solution Needed for Dilution

What volume of 1.59 M $\ce{KOH}$ is required to prepare 5.00 L of 0.100 M $\ce{KOH}$?

Show Solution

We are given the concentration of a stock solution, C1, and the volume and concentration of the resultant diluted solution, V2 and C2. We need to find the volume of the stock solution, V1. We thus rearrange the dilution equation in order to isolate V1:

$\begin{array}{c}{C}_{1}{V}_{1}={C}_{2}{V}_{2}\\ {V}_{1}=\dfrac{{C}_{2}{V}_{2}}{{C}_{1}}\end{array}$

Since the concentration of the diluted solution 0.100 M is roughly one-sixteenth that of the stock solution (1.59 M), 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:

$\begin{array}{c}{V}_{1}=\dfrac{\left(0.100M\right)\left(5.00\text{L}\right)}{1.59M}\\ {V}_{1}=0.314\text{L}\end{array}$

Thus, we would need 0.314 L of the 1.59-M solution to prepare the desired solution. This result is consistent with our rough estimate.

### Key Concepts and Summary

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.

#### Key Equations

• $M=\dfrac{\text{mol solute}}{\text{L solution}}$
• ${C}_{1}{V}_{1}={C}_{2}{V}_{2}$

### Try It

1. What information do we need to calculate the molarity of a sulfuric acid solution?
2. Determine the molarity for each of the following solutions:
1. 0.444 mol of $\ce{CoCl2}$ in 0.654 L of solution
2. 98.0 g of phosphoric acid, $\ce{H3PO4}$, in 1.00 L of solution
3. 0.2074 g of calcium hydroxide, $\ce{Ca(OH)2}$, in 40.00 mL of solution
4. 10.5 kg of $\ce{Na2SO4} \cdot \ce{10H2O}$ in 18.60 L of solution
5. $7.0\times {10}^{-3}\text{mol}$ of $\ce{I2}$ in 100.0 mL of solution
6. $1.8\times {10}^{4}\text{mg}$ of $\ce{HCl}$ in 0.075 L of solution
3. Calculate the number of moles and the mass of the solute in each of the following solutions:
1. 2.00 L of $18.5M\text{ H}_{2}\text{SO}_{4}$, concentrated sulfuric acid
2. 100.0 mL of $3.8\times {10}^{-5}M\text{ NaCN,}$ the minimum lethal concentration of sodium cyanide in blood serum
3. 5.50 L of $13.3M\text{ H}_{2}\text{CO}$, the formaldehyde used to “fix” tissue samples
4. 325 mL of $1.8\times {10}^{-6}M\text{ FeSO}_{4}$, the minimum concentration of iron sulfate detectable by taste in drinking water
4. What is the molarity of the diluted solution when each of the following solutions is diluted to the given final volume?
1. 1.00 L of a 0.250-M solution of $\ce{Fe(NO3)3}$ is diluted to a final volume of 2.00 L
2. 0.5000 L of a 0.1222-M solution of $\ce{C3H7OH}$ is diluted to a final volume of 1.250 L
3. 2.35 L of a 0.350-M solution of $\ce{H3PO4}$ is diluted to a final volume of 4.00 L
4. 22.50 mL of a 0.025-M solution of $\ce{C_{12}H_{22}O_{11}}$ is diluted to 100.0 mL
5. What volume of a 0.20-M $\ce{K2SO4}$ solution contains 57 g of $\ce{K2SO4}$?
Show Selected Solutions
1. We need to know the number of moles of sulfuric acid dissolved in the solution and the volume of the solution.
2. The molarity of each solution is as follows:
1. $\dfrac{0.444\text{mol}}{0.654\text{L}}=0.679{\text{mol L}}^{-1}=0.679M$
2. First convert mass in grams to moles, and then substitute the proper terms into the definition.
Molar mass of H3PO4 = 97.995 g/mol
${\text{mol (H}}_{3}{\text{PO}}_{4}\text{)}=98.0\text{g}\times \dfrac{1\text{mol}}{97.995\text{g}}=1.00\text{mol}$
$M=\dfrac{1.00\text{mol}}{1.00\text{L}}=1.00M$
3. Molar mass [Ca(OH)2] = 79.09 g/mol
$0.2074\cancel{\text{g}}\times \dfrac{\text{1 mol}}{74.09\cancel{\text{g}}}=0.002799\text{mol}{\text{Ca(OH)}}_{2}$
$\dfrac{0.002799\text{mol}}{0.0400\text{L}}=0.06998{\text{mol L}}^{-1}=0.06998M$
4. Molar mass (Na2SO4•10H2O) = 322.20 g/mol
$10,500\times \dfrac{1\text{mol}}{322.20\text{g}}=32.6\text{mol}$
$\dfrac{32.6\text{mol}}{18.60\text{L}}=1.75M$
5. $M=\dfrac{\text{millimoles solute}}{\text{volume of solution in milliliters}}$
$\dfrac{{\text{7.00 mmol I}}_{2}}{\text{100 mL}}=0.070M$
6. Molar mass (HCl) = 36.46 g/mol
$\text{mass (HCl)}=1.8\times {10}^{1}\text{g HCl}\times \dfrac{\text{1 mol}}{\text{36.46 g}}=\text{0.49 mol HCl}$
$\dfrac{\text{0.49 mol HCl}}{\text{0.075 L}}=\text{6.6}M$
3. The molarity must be converted to moles of solute, which is then converted to grams of solute:

$M=\dfrac{\text{mol}}{\text{liter}}\,\,\,\,\text{or}\,\,\,\,\text{mol}=M\times \text{liter}$

1. $\begin{array}{l}\\ {\text{mol H}}_{2}{\text{SO}}_{4}=2.00\cancel{\text{L}}\times \dfrac{18.5\text{mol}}{\cancel{\text{L}}}=37.0\text{mol}{\text{H}}_{2}{\text{SO}}_{4}\\ 37.0\cancel{{\text{mol H}}_{2}{\text{SO}}_{4}}\times \dfrac{98.08{\text{g H}}_{2}{\text{SO}}_{4}}{1\cancel{{\text{mol H}}_{2}{\text{SO}}_{4}}}=3.63\times {10}^{3}{\text{g H}}_{2}{\text{SO}}_{4}\end{array}$
2. $\begin{array}{l}\\ \text{mol NaCN}=0.1000\cancel{\text{L}}\times \dfrac{3.8\times {10}^{-5}\text{mol}}{\cancel{\text{L}}}=3.8\times {10}^{-6}\text{mol NaCN}\\ 3.8\times {10}^{-5}\cancel{\text{mol NaCN}}\times \dfrac{49.01\text{g}}{1\cancel{\text{mol NaCN}}}=1.9\times {\text{10}}^{-4}\text{g NaCN}\end{array}$
3. $\begin{array}{l}\\ {\text{mol H}}_{2}\text{CO}=5.50\cancel{\text{L}}\times \dfrac{13.3\text{mol}}{\cancel{\text{L}}}=73.2\text{mol}{\text{H}}_{2}\text{CO}\\ 73.2\cancel{\text{mol}{\text{H}}_{2}\text{CO}}\times \dfrac{30.026\text{g}}{1\cancel{{\text{mol H}}_{2}\text{CO}}}=2198\text{g}{\text{H}}_{2}\text{CO}=2.20\text{kg}{\text{H}}_{2}\text{CO}\end{array}$
4. $\begin{array}{l}\\ {\text{mol FeSO}}_{4}=0.325\cancel{\text{L}}\times \dfrac{1.8\times {10}^{-6}\text{mol}}{\cancel{L}}=5.9\times {10}^{-7}{\text{mol FeSO}}_{4}\\ 5.85\times {10}^{-7}\cancel{{\text{mol FeSO}}_{4}}\times \dfrac{151.9\text{g}}{1\cancel{{\text{mol FeSO}}_{4}}}=8.9\times {10}^{-5}{\text{g FeSO}}_{4}\end{array}$
4. The molarity for each diluted solution is as follows:
1. ${C}_{2}=\dfrac{{V}_{1}\times {C}_{1}}{{V}_{2}}=1.00\cancel{\text{L}}\times \dfrac{0.250M}{2.00\cancel{\text{L}}}=0.125M$
2. ${C}_{2}=\dfrac{{V}_{1}\times {C}_{1}}{{V}_{2}}=0.5000\cancel{\text{L}}\times \dfrac{0.1222M}{1.250\cancel{\text{L}}}=0.04888M$
3. ${C}_{2}=\dfrac{{V}_{1}\times {C}_{1}}{{V}_{2}}=2.35\cancel{\text{L}}\times \dfrac{0.350M}{4.00\cancel{\text{L}}}=0.206M$
4. ${C}_{2}=\dfrac{{V}_{1}\times {C}_{1}}{{V}_{2}}=0.02250\cancel{\text{mL}}\times \dfrac{0.025M}{0.100\cancel{\text{mL}}}=0.0056M$
5. $57\text{g}{\text{K}}_{2}{\text{SO}}_{4}\times \dfrac{1\text{mol}}{174.26\text{g}}\times \dfrac{1\text{L}}{0.20\text{mol}}=1.6\text{L}$

## Glossary

aqueous solution: solution for which water is the solvent

concentrated: qualitative term for a solution containing solute at a relatively high concentration

concentration: quantitative measure of the relative amounts of solute and solvent present in a solution

dilute: qualitative term for a solution containing solute at a relatively low concentration

dilution: process of adding solvent to a solution in order to lower the concentration of solutes

dissolved: describes the process by which solute components are dispersed in a solvent

molarity (M): unit of concentration, defined as the number of moles of solute dissolved in 1 liter of solution

solute: solution component present in a concentration less than that of the solvent

solvent: solution component present in a concentration that is higher relative to other components

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