# 53 Molarity

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### LEARNING OBJECTIVES

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

Preceding sections of this chapter 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. This section will describe one of the most common ways in which the relative compositions of mixtures may be quantified.

### Solutions

Solutions have previously been defined 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.

The relative amount of a given solution component is known as its . Solute concentrations are often described with qualitative terms such as (of relatively low concentration) and  (of relatively high concentration).

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

$M=\frac{ \text{mol solute} }{ \text{L solution} }$

### EXAMPLE 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?

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:

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

A 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?

0.05 M

### EXAMPLE 2

Deriving Moles and Volumes from Molar Concentrations
How much sugar (mol) is contained in a modest sip (~10 mL) of the soft drink from Example 1?

Solution
Rearrange the definition of molarity to isolate the quantity sought, moles of sugar, then substitute the value for molarity derived in Example 1, 0.375 M:

$M = \frac{ \text{mol solute} }{ \text{L solution} }$

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

$\text{mol solute = 0.375 } \frac{ \text{mol sugar} }{ \text{L} } \times \text{(10 mL} \times \frac{ \text{1 L} }{ \text{1000 mL} }) = \text{0.004 mol sugar}$

What volume (mL) of the sweetened tea described in Example 1 contains the same amount of sugar (mol) as 10 mL of the soft drink in this example?

80 mL

### EXAMPLE 3

Calculating Molar Concentrations from the Mass of Solute
Distilled white vinegar is a solution of acetic acid, $\text{CH}_3\text{CO}_2\text{H}$, 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?

Solution
As in previous examples, the definition of molarity is the primary equation used to calculate the quantity sought. Since the mass of solute is provided instead of its molar amount, use the solute’s molar mass to obtain the amount of solute in moles:

$M = \frac{ \text{mol solute} }{ \text{L solution} } = \frac{ \text{25.2 g CH}_3\text{CO}_2\text{H} \times \frac{ \text{1 mol CH}_3\text{CO}_2\text{H} }{ \text{60.052 g CH}_3\text{CO}_2\text{H} }}{ \text{0.500 L solution} } = \text{0.839 M}$

$M = \frac{ \text{mol solute} }{ \text{L solution} } = \text{0.839 M}$

$M = \frac{ \text{0.839 mol solute} }{ \text{1.00L solution} }$

Calculate the molarity of 6.52 g of $\text{CoCl}_2$ (128.9 g/mol) dissolved in an aqueous solution with a total volume of 75.0 mL.

0.674 M

### EXAMPLE 4

Determining the Mass of Solute in a Given Volume of Solution
How many grams of $\text{NaCl}$ are contained in 0.250 L of a 5.30-M solution?

Solution
The volume and molarity of the solution are specified, so the amount (mol) of solute is easily computed as demonstrated in Example 2:

$\text{M = \frac{ \text{mol solute} }{ \text{L solution} } }$

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

$\text{mol solute = 5.30 } \frac{ \text{mol NaCl} }{ \text{L} } \times \text{0.250 L} = \text{1.325 mol NaCl}$

Finally, this molar amount is used to derive the mass of NaCl:

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

How many grams of $\text{CaCl}_2$ (110.98 g/mol) are contained in 250.0 mL of a 0.200-M solution of calcium chloride?

5.55 g $\text{CaCl}_2$

### EXAMPLE 5

Determining the Volume of Solution Containing a Given Mass of Solute
In Example 3, the concentration of acetic acid in white vinegar was determined to be 0.839 M. What volume of vinegar contains 75.6 g of acetic acid?

Solution
First, use the molar mass to calculate moles of acetic acid from the given mass:

$\text{g solute} \times \frac{ \text{mol solute} }{ \text{g solute} } = \text{mol solute}$

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

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

Combining these two steps into one yields:

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

$\text{75.6 g CH}_3\text{CO}_2\text{H } \left( \frac{ \text{mol CH}_3\text{CO}_2\text{H} }{ \text{60.05 g} } \right) \left( \frac{ \text{L solution} }{ \text{0.839 mol CH}_3\text{CO}_2\text{H} } \right) = \text{1.50 L solution}$

What volume of a 1.50-M $\text{KBr}$ solution contains 66.0 g $\text{KBr}$?

0.370 L

### Dilution of Solutions

is the process whereby the concentration of a solution is lessened by the addition of solvent. For example, 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.

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, a solution of lesser concentration may be prepared. 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 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 (n) is equal to the product of the solution’s molarity (M) and its volume in liters (L):

$n = ML$

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, n1 = n2. 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 this equation uses molarity as the unit of concentration and liters as the unit of volume, other units of concentration and volume may be used as 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.

Use the simulation to explore the relations between solute amount, solution volume, and concentration and to confirm the dilution equation.

### EXAMPLE 6

Determining the Concentration of a Diluted Solution
If 0.850 L of a 5.00-M solution of copper nitrate, $\text{Cu(NO}_3)_2$, is diluted to a volume of 1.80 L by the addition of water, what is the molarity of the diluted solution?

Solution
The stock concentration, C1, and volume, V1, are provided as well as the volume of the diluted solution, V2. Rearrange the dilution equation to isolate the unknown property, the concentration of the diluted solution, C2:

$C_1V_1 = C_2V_2$

$C_2 = \frac{C_1V_1}{V_2}$

Since the stock solution is being diluted by more than two-fold (volume is increased from 0.85 L to 1.80 L), the diluted solution’s concentration is expected to be less than one-half 5 M. This ballpark estimate will be compared 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:

$C_2 = \frac{ \text{0.850 L } \times \text{ 5.00 } \frac{ \text{mol} }{ \text{L} }}{ \text{1.80 L} } = \text{2.36 } M$

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

What is the concentration of the solution that results from diluting 25.0 mL of a 2.04-M solution of $\text{CH}_3\text{OH}$ to 500.0 mL?

0.102 M $\text{CH}_3\text{OH}$

### EXAMPLE 7

Volume of a Diluted Solution
What volume of 0.12 M $\text{HBr}$ can be prepared from 11 mL (0.011 L) of 0.45 M $\text{HBr}$?

Solution
Provided are the volume and concentration of a stock solution, V1 and C1, and the concentration of the resultant diluted solution, C2. Find the volume of the diluted solution, V2 by rearranging the dilution equation to isolate V2:

$C_{1}V_{1} = C_{2}V_{2}$

$V_{2} = \frac{C_{1}V_{1}}{C_{2}}$

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

$V_{2} = \frac{ \text{(0.45 M)(0.011 L)} }{ \text{(0.12 M)} }$

$V_{2} = \text{0.041 L}$

The volume of the 0.12-M solution is 0.041 L (41 mL). The result is reasonable and compares well with the rough estimate.
A laboratory experiment calls for 0.125 M $\text{HNO}_3$. What volume of 0.125 M $\text{HNO}_3$ can be prepared from 0.250 L of 1.88 M $\text{HNO}_3$?

3.76 L

### EXAMPLE 8

Volume of a Concentrated Solution Needed for Dilution
What volume of 1.59 M $\text{KOH}$ is required to prepare 5.00 L of 0.100 M $\text{KOH}$?

Solution
Given are the concentration of a stock solution, C1, and the volume and concentration of the resultant diluted solution, V2 and C2. Find the volume of the stock solution, V1 by rearranging the dilution equation to isolate V1:

$C_1V_1 = C_2V_2$

$V_1 = \frac{ C_2V_2 }{ C_1 }$

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

$V_1 = \frac{ \text{(0.100 M)(5.00 L)} }{ \text{1.59 M} }$

$V_1 = \text{0.314 L}$

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

What volume of a 0.575-M solution of glucose, $\text{C}_6\text{H}_{12}\text{O}_6$, can be prepared from 50.00 mL of a 3.00-M glucose solution?

0.261 L

### KEY TAKEAWAYS

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.

### Chemistry End of Chapter Exercises

1. Explain what changes and what stays the same when 1.00 L of a solution of $\text{NaCl}$ is diluted to 1.80 L.
2. What information is needed to calculate the molarity of a sulfuric acid solution?
We need to know the number of moles of sulfuric acid dissolved in the solution and the volume of the solution.
3. A 200-mL sample and a 400-mL sample of a solution of salt have the same molarity. In what ways are the two samples identical? In what ways are these two samples different?
4. Determine the molarity for each of the following solutions:      (a) 0.444 mol of $\text{CoCl}_2$ in 0.654 L of solution      (b) 98.0 g of phosphoric acid, $\text{H}_3\text{PO}_4$, in 1.00 L of solution      (c) 0.2074 g of calcium hydroxide, $\text{Ca(OH)}_2$, in 40.00 mL of solution      (d) 10.5 kg of $\text{Na}_2\text{SO}_4 \cdot \text{10H}_2\text{O}$ in 18.60 L of solution      (e) 7.0 × 10−3 mol of $\text{I}_2$ in 100.0 mL of solution      (f) 1.8 × 104 mg of $\text{HCl}$ in 0.075 L of solution
(a) 0.679 M; (b) 1.00 M; (c) 0.06998 M; (d) 1.75 M; (e) 0.070 M; (f) 6.6 M
5. Determine the molarity of each of the following solutions:     (a) 1.457 mol $\text{KCl}$ in 1.500 L of solution      (b) 0.515 g of $\text{H}_2\text{SO}_4$ in 1.00 L of solution      (c) 20.54 g of $\text{Al(NO}_3\text{)}_3$ in 1575 mL of solution       (d) 2.76 kg of $\text{CuSO}_4 \cdot \text{5H}_2\text{O}$ in 1.45 L of solution      (e) 0.005653 mol of $\text{Br}_2$ in 10.00 mL of solution      (f) 0.000889 g of glycine, $\text{C}_2\text{H}_5\text{NO}_2$, in 1.05 mL of solution
6. Consider this question: What is the mass of the solute in 0.500 L of 0.30 M glucose, $\text{C}_6\text{H}_{12}\text{O}_6$, used for intravenous injection?      (a) Outline the steps necessary to answer the question.     (b) Answer the question.
(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; (b) 27 g
7. Consider this question: What is the mass of solute in 200.0 L of a 1.556-M solution of $\text{KBr}$?      (a) Outline the steps necessary to answer the question.      (b) Answer the question.
8. Calculate the number of moles and the mass of the solute in each of the following solutions:      (a) 2.00 L of 18.5 M $\text{H}_2\text{SO}_4$, concentrated sulfuric acid      (b) 100.0 mL of 3.8 × 10−5 M $\text{NaCN}$, the minimum lethal concentration of sodium cyanide in blood serum      (c) 5.50 L of 13.3 M $\text{H}_2\text{CO}$, the formaldehyde used to “fix” tissue samples       (d) 325 mL of 1.8 × 10−6 M $\text{FeSO}_4$, the minimum concentration of iron sulfate detectable by taste in drinking water
(a) 37.0 mol $\text{H}_2\text{SO}_4$, 3.63 × 103 g $\text{H}_2\text{SO}_4$; (b) 3.8 × 10−6 mol $\text{NaCN}$, 1.9 × 10−4 g $\text{NaCN}$; (c) 73.2 mol $\text{H}_2\text{CO}$, 2.20 kg $\text{H}_2\text{CO}$; (d) 5.9 × 10−7 mol $\text{FeSO}_4$, 8.9 × 10−5 g $\text{FeSO}_4$
9. Calculate the number of moles and the mass of the solute in each of the following solutions:      (a) 325 mL of 8.23 × 10−5 M $\text{KI}$, a source of iodine in the diet      (b) 75.0 mL of 2.2 × 10−5 M $\text{H}_2\text{SO}_4$, a sample of acid rain       (c) 0.2500 L of 0.1135 M $\text{K}_2\text{CrO}_4$, an analytical reagent used in iron assays      (d) 10.5 L of 3.716 M $\text{(NH}_4\text{)}_2\text{SO}_4$, a liquid fertilizer
10. Consider this question: What is the molarity of $\text{KMnO}_4$ in a solution of 0.0908 g of $\text{KMnO}_4$ in 0.500 L of solution?      (a) Outline the steps necessary to answer the question.      (b) Answer the question.
(a) Determine the molar mass of $\text{KMnO}_4$; determine the number of moles of $\text{KMnO}_4$ in the solution; from the number of moles and the volume of solution, determine the molarity; (b) 1.15 × 10−3 M
11. Consider this question: What is the molarity of $\text{HCl}$ if 35.23 mL of a solution of $\text{HCl}$ contain 0.3366 g of $\text{HCl}$?      (a) Outline the steps necessary to answer the question.      (b) Answer the question.
12. Calculate the molarity of each of the following solutions:      (a) 0.195 g of cholesterol, $\text{C}_{27}\text{H}_{46}\text{O}$, in 0.100 L of serum, the average concentration of cholesterol in human serum      (b) 4.25 g of $\text{NH}_3$ in 0.500 L of solution, the concentration of $\text{NH}_3$ in household ammonia      (c) 1.49 kg of isopropyl alcohol, $\text{C}_3\text{H}_7\text{OH}$, in 2.50 L of solution, the concentration of isopropyl alcohol in rubbing alcohol     (d) 0.029 g of $\text{I}_2$ in 0.100 L of solution, the solubility of $\text{I}_2$ in water at 20 °C
(a) 5.04 × 10−3 M; (b) 0.499 M; (c) 9.92 M; (d) 1.1 × 10−3 M
13. Calculate the molarity of each of the following solutions:      (a) 293 g $\text{HCl}$ in 666 mL of solution, a concentrated $\text{HCl}$ solution      (b) 2.026 g $\text{FeCl}_3$ in 0.1250 L of a solution used as an unknown in general chemistry laboratories      (c) 0.001 mg $\text{Cd}^{2+}$ in 0.100 L, the maximum permissible concentration of cadmium in drinking water      (d) 0.0079 g $\text{C}_7\text{H}_5\text{SNO}_3$ in one ounce (29.6 mL), the concentration of saccharin in a diet soft drink.
14. There is about 1.0 g of calcium, as $\text{Ca}^{2+}$, in 1.0 L of milk. What is the molarity of $\text{Ca}^{2+}$ in milk?
0.025 M
15. What volume of a 1.00-M $\text{Fe(NO}_3\text{)}_3$ solution can be diluted to prepare 1.00 L of a solution with a concentration of 0.250 M?
16. If 0.1718 L of a 0.3556-M $\text{C}_3\text{H}_7\text{OH}$ solution is diluted to a concentration of 0.1222 M, what is the volume of the resulting solution?
0.5000 L
17. If 4.12 L of a 0.850 M–$\text{H}_3\text{PO}_4$ solution is be diluted to a volume of 10.00 L, what is the concentration of the resulting solution?
18. What volume of a 0.33-M $\text{C}_{12}\text{H}_{22}\text{O}_{11}$ solution can be diluted to prepare 25 mL of a solution with a concentration of 0.025 M?
1.9 mL
19. What is the concentration of the $\text{NaCl}$ solution that results when 0.150 L of a 0.556-M solution is allowed to evaporate until the volume is reduced to 0.105 L?
20. What is the molarity of the diluted solution when each of the following solutions is diluted to the given final volume?     (a) 1.00 L of a 0.250-M solution of $\text{Fe(NO}_3\text{)}_3$ is diluted to a final volume of 2.00 L     (b) 0.5000 L of a 0.1222-M solution of $\text{C}_3\text{H}_7\text{OH}$ is diluted to a final volume of 1.250 L     (c) 2.35 L of a 0.350-M solution of $\text{H}_3\text{PO}_4$ is diluted to a final volume of 4.00 L     (d) 22.50 mL of a 0.025-M solution of $\text{C}_{12}\text{H}_{22}\text{O}_{11}$ is diluted to 100.0 mL
(a) 0.125 M; (b) 0.04888 M; (c) 0.206 M; (d) 0.0056 M
21. What is the final concentration of the solution produced when 225.5 mL of a 0.09988-M solution of $\text{Na}_2\text{CO}_3$ is allowed to evaporate until the solution volume is reduced to 45.00 mL?
22. A 2.00-L bottle of a solution of concentrated $\text{HCl}$ was purchased for the general chemistry laboratory. The solution contained 868.8 g of $\text{HCl}$. What is the molarity of the solution?
11.9 M
23. An experiment in a general chemistry laboratory calls for a 2.00-M solution of $\text{HCl}$. How many mL of 11.9 M $\text{HCl}$ would be required to make 250 mL of 2.00 M $\text{HCl}$?
24. What volume of a 0.20-M $\text{K}_2\text{SO}_4$ solution contains 57 g of $\text{K}_2\text{SO}_4$?
1.6 L
25. The US Environmental Protection Agency (EPA) places limits on the quantities of toxic substances that may be discharged into the sewer system. Limits have been established for a variety of substances, including hexavalent chromium, which is limited to 0.50 mg/L. If an industry is discharging hexavalent chromium as potassium dichromate ($\text{K}_2\text{Cr}_2\text{O}_7$), what is the maximum permissible molarity of that substance?

### Key Equations

• $M = \frac{ \text{mol solute} }{ \text{L solution} }$
• C1V1 = C2V2

### Glossary

solution for which water is the solvent
qualitative term for a solution containing solute at a relatively high concentration
quantitative measure of the relative amounts of solute and solvent present in a solution
qualitative term for a solution containing solute at a relatively low concentration
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
solution component present in a concentration less than that of the solvent
solution component present in a concentration that is higher relative to other components