# 17.7 Relative Strengths of Acids and Bases

## Learning Objectives

- Assess the relative strengths of acids and bases according to their ionization constants
- Rationalize trends in acid–base strength in relation to molecular structure
- Carry out equilibrium calculations for weak acid–base systems

We can rank the strengths of acids by the extent to which they ionize in aqueous solution. The reaction of an acid with water is given by the general expression:

Water is the base that reacts with the acid HA, A^{−} is the conjugate base of the acid HA, and the hydronium ion is the conjugate acid of water. A strong acid yields 100% (or very nearly so) of H_{3}O^{+} and A^{−} when the acid ionizes in water; Figure 17.7a lists several strong acids. A weak acid gives small amounts of H_{3}O^{+} and A^{−}.

The relative strengths of acids may be determined by measuring their equilibrium constants in aqueous solutions. In solutions of the same concentration, stronger acids ionize to a greater extent, and so yield higher concentrations of hydronium ions than do weaker acids. The equilibrium constant for an acid is called the . For the reaction of an acid HA:

we write the equation for the ionization constant as:

where the concentrations are those at equilibrium. Although water is a reactant in the reaction, it is the solvent as well, so we do not include [H_{2}O] in the equation. The larger the *K*_{a} of an acid, the larger the concentration of H_{3}O^{+} and A^{−} relative to the concentration of the nonionized acid, HA. Thus a stronger acid has a larger ionization constant than does a weaker acid. The ionization constants increase as the strengths of the acids increase.

Another measure of the strength of an acid is its percent ionization. The of a weak acid is the ratio of the concentration of the ionized acid to the initial acid concentration, times 100:

Because the ratio includes the initial concentration, the percent ionization for a solution of a given weak acid varies depending on the original concentration of the acid, and actually decreases with increasing acid concentration.

### Example 17.7a

#### Calculation of Percent Ionization from pH

Calculate the percent ionization of a 0.125-*M* solution of nitrous acid (a weak acid), with a pH of 2.09.

#### Solution

The percent ionization for an acid is:

The chemical equation for the dissociation of the nitrous acid is: [latex]\text{HNO}_2(aq)\;+\;\text{H}_2\text{O}(l)\;{\rightleftharpoons}\;\text{NO}_2^{\;\;-}(aq)\;+\;\text{H}_3\text{O}^{+}(aq)[/latex].

Since [latex]10^{-\text{pH}} = [\text{H}_3\text{O}^{+}][/latex], we find that 10^{−2.09} = 8.1 × 10^{−3}*M*, so that percent ionization is:

Remember, the logarithm 2.09 indicates a hydronium ion concentration with only two significant figures.

### Exercise 17.7a

Calculate the percent ionization of a 0.10-*M* solution of acetic acid with a pH of 2.89..

#### Check Your Answer^{[1]}

We can rank the strengths of bases by their tendency to form hydroxide ions in aqueous solution. The reaction of a Brønsted-Lowry base with water is given by:

Water is the acid that reacts with the base, HB^{+} is the conjugate acid of the base B, and the hydroxide ion is the conjugate base of water. A strong base yields 100% (or very nearly so) of OH^{−} and HB^{+} when it reacts with water; Figure 17.7a lists several strong bases. A weak base yields a small proportion of hydroxide ions. Soluble ionic hydroxides such as NaOH are considered strong bases because they dissociate completely when dissolved in water.

### Exercise 17.7b

**Practice using the following PhET simulation: Acid and Base Solutions**

**Activity source:**Simulation by PhET Interactive Simulations, University of Colorado Boulder, licensed under CC-BY-4.0

As we did with acids, we can measure the relative strengths of bases by measuring their in aqueous solutions. In solutions of the same concentration, stronger bases ionize to a greater extent, and so yield higher hydroxide ion concentrations than do weaker bases. A stronger base has a larger ionization constant than does a weaker base. For the reaction of a base, B:

we write the equation for the ionization constant as:

where the concentrations are those at equilibrium. Again, we do not include [H_{2}O] in the equation because water is the solvent. The chemical reactions and ionization constants of the three bases shown are:

As with acids, percent ionization can be measured for basic solutions, but will vary depending on the base ionization constant and the initial concentration of the solution.

Consider the ionization reactions for a conjugate acid-base pair, HA − A^{−}:

Adding these two chemical equations yields the equation for the autoionization for water:

As shown in the previous chapter on equilibrium, the *K* expression for a chemical equation derived from adding two or more other equations is the mathematical product of the other equations’ *K* expressions. Multiplying the mass-action expressions together and cancelling common terms, we see that:

For example, the acid ionization constant of acetic acid (CH_{3}COOH) is 1.8 × 10^{−5}, and the base ionization constant of its conjugate base, acetate ion (CH_{3}COO^{–}), is 5.6 × 10^{−10}. The product of these two constants is indeed equal to *K*_{w}:

The extent to which an acid, HA, donates protons to water molecules depends on the strength of the conjugate base, A^{−}, of the acid. If A^{−} is a strong base, any protons that are donated to water molecules are recaptured by A^{−}. Thus there is relatively little A^{−} and H_{3}O^{+} in solution, and the acid, HA, is weak. If A^{−} is a weak base, water binds the protons more strongly, and the solution contains primarily A^{−} and H_{3}O^{+}—the acid is strong. Strong acids form very weak conjugate bases, and weak acids form stronger conjugate bases (Figure 17.7b).

**Figure 17.7b**This diagram shows the relative strengths of conjugate acid-base pairs, as indicated by their ionization constants in aqueous solution (credit:

*Chemistry (OpenStax)*

*,*CC BY 4.0).

Figure 17.7c lists a series of acids and bases in order of the decreasing strengths of the acids and the corresponding increasing strengths of the bases. The acid and base in a given row are conjugate to each other.

**Figure 17.7c**The chart shows the relative strengths of conjugate acid-base pairs (credit:

*Chemistry (OpenStax)*

*,*CC BY 4.0).

The first six acids in Figure 17.7c are the most common strong acids. These acids are completely dissociated in aqueous solution. The conjugate bases of these acids are weaker bases than water. When one of these acids dissolves in water, their protons are completely transferred to water, the stronger base.

Those acids that lie between the hydronium ion and water in Figure 17.7c form conjugate bases that can compete with water for possession of a proton. Both hydronium ions and nonionized acid molecules are present in equilibrium in a solution of one of these acids. Compounds that are weaker acids than water (those found below water in the column of acids) in Figure 17.7c exhibit no observable acidic behaviour when dissolved in water. Their conjugate bases are stronger than the hydroxide ion, and if any conjugate base were formed, it would react with water to re-form the acid.

The extent to which a base forms hydroxide ion in aqueous solution depends on the strength of the base relative to that of the hydroxide ion, as shown in the last column in Figure 17.7c. A strong base, such as one of those lying below the hydroxide ion, accepts protons from water to yield 100% of the conjugate acid and hydroxide ion. Those bases lying between water and hydroxide ion accept protons from water, but a mixture of the hydroxide ion and the base results. Bases that are weaker than water (those that lie above water in the column of bases) show no observable basic behaviour in aqueous solution.

### Example 17.7b

#### The Product *K*_{a }× *K*_{b} = *K*_{w}

Use the *K*_{b} for the nitrite ion, NO_{2}^{–}, to calculate the *K*_{a} for its conjugate acid.

#### Solution

*K*_{b} for NO_{2}^{–} is given in this section as 2.17 × 10^{−11}. The conjugate acid of NO_{2}^{–} is HNO_{2}; *K*_{a} for HNO_{2} can be calculated using the relationship:

Solving for *K*_{a}, we get:

This answer can be verified by finding the *K*_{a} for HNO_{2} in Appendix I.

### Exercise 17.7c

We can determine the relative acid strengths of NH_{4}^{+} and HCN by comparing their ionization constants. The ionization constant of HCN is given in Appendix I as 4.9 × 10^{−10}. The ionization constant of NH_{4}^{+} is not listed, but the ionization constant of its conjugate base, NH_{3}, is listed as 1.8 × 10^{−5}. Determine the ionization constant of NH_{4}^{+}, and decide which is the stronger acid, HCN or NH_{4}^{+}.

#### Check Your Answer^{[2]}

## The Ionization of Weak Acids and Weak Bases

Many acids and bases are weak; that is, they do not ionize fully in aqueous solution. A solution of a weak acid in water is a mixture of the nonionized acid, hydronium ion, and the conjugate base of the acid, with the nonionized acid present in the greatest concentration. Thus, a weak acid increases the hydronium ion concentration in an aqueous solution (but not as much as the same amount of a strong acid).

Acetic acid, CH_{3}CO_{2}H, is a weak acid. When we add acetic acid to water, it ionizes to a small extent according to the equation:

giving an equilibrium mixture with most of the acid present in the nonionized (molecular) form. This equilibrium, like other equilibria, is dynamic; acetic acid molecules donate hydrogen ions to water molecules and form hydronium ions and acetate ions at the same rate that hydronium ions donate hydrogen ions to acetate ions to reform acetic acid molecules and water molecules. We can tell by measuring the pH of an aqueous solution of known concentration that only a fraction of the weak acid is ionized at any moment (Figure 17.7d). The remaining weak acid is present in the nonionized form.

For acetic acid, at equilibrium:

**Figure 17.7d**pH paper indicates that a 0.1-

*M*solution of HCl (beaker on left) has a pH of 1. The acid is fully ionized and [H

_{3}O

^{+}] = 0.1

*M*. A 0.1-

*M*solution of CH

_{3}CO

_{2}H (beaker on right) is a pH of 3 ([H

_{3}O

^{+}] = 0.001

*M*) because the weak acid CH

_{3}CO

_{2}H is only partially ionized. In this solution, [H

_{3}O

^{+}] < [CH

_{3}CO

_{2}H]. (credit: modification of work by Sahar Atwa in

*Chemistry (OpenStax)*

*,*CC BY 4.0).

Ionization Reaction | K_{a} at 25 °C |
---|---|

[latex]\text{HSO}_4^{\;\;-}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{SO}_4^{\;\;2-}[/latex] | 1.2 × 10^{−2} |

[latex]\text{HF}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{F}^{-}[/latex] | 3.5 × 10^{−4} |

[latex]\text{HNO}_2\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{NO}_2^{\;\;-}[/latex] | 4.6 × 10^{−4} |

[latex]\text{HCNO}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{NCO}^{-}[/latex] | 2 × 10^{−4} |

[latex]\text{HCO}_2\text{H}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{HCO}_2^{\;\;-}[/latex] | 1.8 × 10^{−4} |

[latex]\text{CH}_3\text{CO}_2\text{H}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{CH}_3\text{CO}_2^{\;\;-}[/latex] | 1.8 × 10^{−5} |

[latex]\text{HCIO}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{CIO}^{-}[/latex] | 2.9 × 10^{−8} |

[latex]\text{HBrO}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{BrO}^{-}[/latex] | 2.8 × 10^{−9} |

[latex]\text{HCN}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}\;+\;\text{CN}^{-}[/latex] | 4.9 × 10^{−10} |

Table 17.7a gives the ionization constants for several weak acids; additional ionization constants can be found in Appendix I.

At equilibrium, a solution of a weak base in water is a mixture of the nonionized base, the conjugate acid of the weak base, and hydroxide ion with the nonionized base present in the greatest concentration. Thus, a weak base increases the hydroxide ion concentration in an aqueous solution (but not as much as the same amount of a strong base).

For example, a solution of the weak base trimethylamine, (CH_{3})_{3}N, in water reacts according to the equation:

giving an equilibrium mixture with most of the base present as the nonionized amine. This equilibrium is analogous to that described for weak acids.

We can confirm by measuring the pH of an aqueous solution of a weak base of known concentration that only a fraction of the base reacts with water (Figure 17.7e). The remaining weak base is present as the unreacted form. The equilibrium constant for the ionization of a weak base, *K*_{b}, is called the ionization constant of the weak base, and is equal to the reaction quotient when the reaction is at equilibrium. For trimethylamine, at equilibrium:

**Figure 17.7e**pH paper indicates that a 0.1-

*M*solution of NH

_{3}(left) is weakly basic. The solution has a pOH of 3 ([OH

^{−}] = 0.001

*M*) because the weak base NH

_{3}only partially reacts with water. A 0.1-

*M*solution of NaOH (right) has a pOH of 1 because NaOH is a strong base. (credit: modification of work by Sahar Atwa,

*Chemistry (OpenStax)*

*,*CC BY 4.0).

The ionization constants of several weak bases are given in Table 17.7b and in Appendix J.

Ionization Reaction | K_{b} at 25 °C |
---|---|

[latex](\text{CH}_3)_2\text{NH}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;(\text{CH}_3)_2\text{NH}_2^{\;\;+}\;+\;\text{OH}^{-}[/latex] | 5.9 × 10^{−4} |

[latex]\text{CH}_3\text{NH}_2\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{CH}_3\text{NH}_3^{\;\;+}\;+\;\text{OH}^{-}[/latex] | 4.4 × 10^{−4} |

[latex](\text{CH}_3)_3\text{N}\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;(\text{CH}_3)_3\text{NH}^{+}\;+\;\text{OH}^{-}[/latex] | 6.3 × 10^{−5} |

[latex]\text{NH}_3\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{NH}_4^{\;\;+}\;+\;\text{OH}^{-}[/latex] | 1.8 × 10^{−5} |

[latex]\text{C}_6\text{H}_5\text{NH}_2\;+\;\text{H}_2\text{O}\;{\rightleftharpoons}\;\text{C}_6\text{N}_5\text{NH}_3^{\;\;+}\;+\;\text{OH}^{-}[/latex] | 4.3 × 10^{−10} |

### Example 17.7c

#### Determination of *K*_{a} from Equilibrium Concentrations

Acetic acid is the principal ingredient in vinegar (Figure 17.7f); that’s why it tastes sour. At equilibrium, a solution contains [CH_{3}CO_{2}H] = 0.0787 *M* and [H_{3}O^{+}] = [CH_{3}CO_{2}^{–}] = 0.00118M. What is the value of *K*_{a} for acetic acid?

**Figure 17.7f**Vinegar is a solution of acetic acid, a weak acid. (credit: modification of work by HomeSpot HQ, CC BY 2.0)

#### Solution

We are asked to calculate an equilibrium constant from equilibrium concentrations. At equilibrium, the value of the equilibrium constant is equal to the reaction quotient for the reaction:

### Exercise 17.7d

What is the equilibrium constant for the ionization of the HSO_{4}^{–} ion, the weak acid used in some household cleansers:

In one mixture of NaHSO_{4} and Na_{2}SO_{4} at equilibrium, [H_{3}O^{+}] = 0.027*M*, [HSO_{4}^{–}] = 0.29*M* and [SO_{4}^{2-}] = 0.13*M*.

#### Check Your Answer^{[3]}

### Example 17.7d

#### Determination of *K*_{b} from Equilibrium Concentrations

Caffeine, C_{8}H_{10}N_{4}O_{2} is a weak base. What is the value of *K*_{b} for caffeine if a solution at equilibrium has [C_{8}H_{10}N_{4}O_{2}] = 0.050 *M*, [C_{8}H_{10}N_{4}O_{2}H^{+} ] = 5.0 × 10^{−3}*M*, and [OH^{−}] = 2.5 × 10^{−3}*M*?

#### Solution

At equilibrium, the value of the equilibrium constant is equal to the reaction quotient for the reaction:

### Exercise 17.7e

What is the equilibrium constant for the ionization of the HPO_{4}^{2-} ion, a weak base:

In a solution containing a mixture of NaH_{2}PO_{4} and Na_{2}HPO_{4} at equilibrium, [OH^{−}] = 1.3 × 10^{−6}*M,* [H_{2}PO_{4}^{–}] = 0.042*M* and [HPO_{4}^{2-}] = 0.341*M*.

#### Check Your Answer^{[4]}

### Example 17.7e

#### Determination of *K*_{a} or *K*_{b} from pH

The pH of a 0.0516-*M* solution of nitrous acid, HNO_{2}, is 2.34. What is its *K*_{a}?

#### Solution

We determine an equilibrium constant starting with the initial concentrations of HNO_{2}, H_{3}O^{+}, and NO_{2}^{–} as well as one of the final concentrations, the concentration of hydronium ion at equilibrium. (Remember that pH is simply another way to express the concentration of hydronium ion.)

We can solve this problem with the following steps in which *x* is a change in concentration of a species in the reaction:

We can summarize the various concentrations and changes as shown here (the concentration of water does not appear in the expression for the equilibrium constant, so we do not need to consider its concentration):

To get the various values in the ICE (Initial, Change, Equilibrium) table, we first calculate [H_{3}O^{+}, the equilibrium concentration of H_{3}O^{+}, from the pH:

The change in concentration of H_{3}O^{+}, [latex]x_{[\text{H}_3\text{O}^{+}]}[/latex], is the difference between the equilibrium concentration of H_{3}O^{+}, which we determined from the pH, and the initial concentration, [H_{3}O^{+}]. The initial concentration of H_{3}O^{+}is its concentration in pure water, which is so much less than the final concentration that we approximate it as zero (~0).

The change in concentration of NO_{2}^{–} is equal to the change in concentration of H_{3}O^{+}. For each 1 mol of H_{3}O^{+} that forms, 1 mol of NO_{2}^{–} forms. The equilibrium concentration of HNO_{2} is equal to its initial concentration plus the change in its concentration.

Now we can fill in the ICE table with the concentrations at equilibrium, as shown here:

Finally, we calculate the value of the equilibrium constant using the data in the table:

### Exercise 17.7f

## Check Your Learning Exercise (Text Version)

The pH of a solution of household ammonia, a 0.950-*M* solution of NH_{3,} is 11.612. What is *K*_{b} for NH_{3}?

- 1.8 x 10
^{-5} - 1.5 x 10
^{-8} - 2.4 x 10
^{-5} - 1.8 x 10
^{-10}

#### Check Your Answer^{[5]}

**Source: **“Exercise 17.7f” is adapted from “Example 14.3-5” in General Chemistry 1 & 2, a derivative of Chemistry (Open Stax) by Paul Flowers, Klaus Theopold, Richard Langley & William R. Robinson, licensed under CC BY 4.0.

### Example 17.7f

#### Equilibrium Concentrations in a Solution of a Weak Acid

Formic acid, HCO_{2}H, is the irritant that causes the body’s reaction to ant stings (Figure 17.7g).

**Figure 17.7g**The pain of an ant’s sting is caused by formic acid. (credit: work by John Tann, CC BY 2.0)

What is the concentration of hydronium ion and the pH in a 0.534 *M* solution of formic acid?

#### Solution

*Determine x and equilibrium concentrations*. The equilibrium expression is:[latex]\text{HCO}_2\text{H}(aq)\;+\;\text{H}_2\text{O}(l)\;{\rightleftharpoons}\;\text{H}_3\text{O}^{+}(aq)\;+\;\text{HCO}_2^{\;\;-}(aq)[/latex]The concentration of water does not appear in the expression for the equilibrium constant, so we do not need to consider its change in concentration when setting up the ICE table.

The table shows initial concentrations (concentrations before the acid ionizes), changes in concentration, and equilibrium concentrations follows (the data given in the problem appear in color):

*Solve for x and the equilibrium concentrations.*At equilibrium:[latex]\begin{array}{r @{{}={}} l} K_{\text{a}} & = 1.8\;\times\;10^{-4}\\[0.5em] & = \frac{[\text{H}_3\text{O}^{+}][\text{HCO}_2^{\;\;-}]}{[\text{HCO}_2\text{H}]}\\[0.5em] & = \frac{(x)(x)}{0.534\;-\;x}\end{array}[/latex]

Now solve for *x*. Because the initial concentration of acid is reasonably large and *K*_{a} is very small, we assume that *x* << 0.534, which *permits* us to simplify the denominator term as (0.534 − *x*) = 0.534. This gives:

Solve for *x* as follows:

To check the assumption that *x* is small compared to 0.534, we calculate:

since* x* is less than 5% of the initial concentration (0.534 M), the assumption is valid.

We find the equilibrium concentration of hydronium ion in this formic acid solution from its initial concentration and the change in that concentration as indicated in the last line of the table:

The pH of the solution can be found by taking the negative log of the [latex][\text{H}_3\text{O}^{+}][/latex], so:

### Exercise 17.7g

Only a small fraction of a weak acid ionizes in aqueous solution. What is the percent ionization of acetic acid in a 0.100-*M* solution of acetic acid, CH_{3}CO_{2}H?

(Hint: Determine [CH_{3}CO_{2}^{–}] at equilibrium.) Recall that the percent ionization is the fraction of acetic acid that is ionized × 100, or

[latex]\frac{[\text{CH}_3\text{CO}_2^{\;\;-}]}{[\text{CH}_3\text{CO}_2\text{H}]_{\text{initial}}}\;\times\;100[/latex]

#### Check Your Answer^{[6]}

The following example shows that the concentration of products produced by the ionization of a weak base can be determined by the same series of steps used with a weak acid.

### Example 17.7g

#### Equilibrium Concentrations in a Solution of a Weak Base

Find the concentration of hydroxide ion in a 0.25 *M* solution of trimethylamine, a weak base:

#### Solution

This problem requires that we calculate an equilibrium concentration by determining concentration changes as the ionization of a base goes to equilibrium. The solution is approached in the same way as that for the ionization of formic acid in Example 17.7f. The reactants and products will be different and the numbers will be different, but the logic will be the same:

*Determine x and equilibrium concentrations*. The table shows the changes and concentrations

*Solve for x and the equilibrium concentrations*. At equilibrium:[latex]K_{\text{b}} = \frac{[(\text{CH}_3)_3\text{NH}^{+}][\text{OH}^{-}]}{[(\text{CH}_3)_3\text{N}]} = \frac{(x)(x)}{0.25\;-\;x} = 6.3\;\times\;10^{-5}[/latex]If we assume that

*x*is small relative to 0.25, then we can replace (0.25 −*x*) in the preceding equation with 0.25. Solving the simplified equation gives:[latex]x = 4.0\;\times\;10^{-3}[/latex]This change is less than 5% of the initial concentration (0.25), so the assumption is justified.

Recall that, for this computation,

*x*is equal to the equilibrium concentration of*hydroxide ion*in the solution (see earlier tabulation):[latex]\begin{array}{r @{{}={}} l} [\text{OH}^{-}] & = {\sim}0\;+\;x\\[0.5em] & = x \\[0.5em] & = 4.0\;\times\;10^{-3}\;M\end{array}[/latex]Then calculate pOH as follows:

[latex]\text{pOH} = -\text{log}(4.0\;\times\;10^{-3}) = 2.40[/latex]Using the relation introduced in the previous section of this chapter:

[latex]\text{pH}\;+\;\text{pOH} = \text{p}K_{\text{w}} = 14.00[/latex]permits the computation of pH:

[latex]\text{pH} = 14.00\;-\;\text{pOH} = 14.00\;-\;2.40 = 11.60[/latex]*Check the work*. A check of our arithmetic shows that*K*_{b}= 6.3 × 10^{−5}.

Some weak acids and weak bases ionize to such an extent that the simplifying assumption that *x* is small relative to the initial concentration of the acid or base is inappropriate. As we solve for the equilibrium concentrations in such cases, we will see that we cannot neglect the change in the initial concentration of the acid or base, and we must solve the equilibrium equations by using the quadratic equation.

### Example 17.7h

#### Equilibrium Concentrations in a Solution of a Weak Acid

Sodium bisulfate, NaHSO_{4}, is used in some household cleansers because it contains the HSO_{4}^{–} ion, a weak acid. What is the pH of a 0.50-*M* solution of HSO_{4}^{−}?

#### Solution

We need to determine the equilibrium concentration of the hydronium ion that results from the ionization of HSO_{4}^{−} so that we can use [H_{3}O^{+}] to determine the pH. As in the previous examples, we can approach the solution by the following steps:

*Determine x and equilibrium concentrations*. This table shows the changes and concentrations:*Solve for x and the concentrations*. As we begin solving for*x*, we will find this is more complicated than in previous examples. As we discuss these complications we should not lose track of the fact that it is still the purpose of this step to determine the value of*x*.At equilibrium:

[latex]K_{\text{a}} = 1.2\;\times\;10^{-2} = \frac{[\text{H}_3\text{O}^{+}][\text{SO}_4^{\;\;2-}]}{[\text{HSO}_4^{\;\;-}]} = \frac{(x)(x)}{0.50\;-\;x}[/latex]If we assume that

*x*is small and approximate (0.50 −*x*) as 0.50, we find:[latex]x = 7.7\;\times\;10^{-2}[/latex]When we check the assumption, we calculate:

[latex]\frac{x}{[\text{HSO}_4^{\;\;-}]}_{\text{i}} \times(100) = \frac{7.7\;\times\;10^{-2}}{0.50}\times(100) = 15\%[/latex]The value of

*x*is not less than 5% of 0.50, so the assumption is not valid. We need the quadratic formula to find*x*.The equation:

[latex]K_{\text{a}} = 1.2\;\times\;10^{-2} = \frac{(x)(x)}{0.50\;-\;x}[/latex]gives

[latex]6.0\;\times\;10^{-3}\;-\;1.2\;\times\;10^{-2}x = x^{2}[/latex]or

[latex]x^{2}\;+\;1.2\;\times\;10^{-2}x\;-\;6.0\;\times\;10^{-3} = 0[/latex]This equation can be solved using the quadratic formula. For an equation of the form

[latex]ax^{2}\;+\;bx\;+\;c = 0[/latex],*x*is given by the equation:[latex]x = \frac{-b\;{\pm}\;\sqrt{b^{2}\;-\;4\text{ac}}}{2a}[/latex]In this problem,

*a*= 1,*b*= 1.2 × 10^{−3}, and*c*= −6.0 × 10^{−3}.Solving for

*x*gives a negative root (which cannot be correct since concentration cannot be negative) and a positive root:[latex]x = 7.2\;\times\;10^{-2}[/latex]Now determine the hydronium ion concentration and the pH:

[latex][\text{H}_3\text{O}^{+}] = {\sim}0\;+\;x = 0\;+\;7.2\;\times\;10^{-2}\;M[/latex][latex]= 7.2\;\times\;10^{-2}\;M[/latex]The pH of this solution is:

[latex]\text{pH} = -\text{log}[\text{H}_3\text{O}^{+}] = -\text{log}\;7.2\;\times\;10^{-2} = 1.14[/latex]

### Exercise 17.7h

Calculate the pH in a 0.010-*M* solution of caffeine, a weak base:

(Hint: It will be necessary to convert [OH^{−}] to [H_{3}O^{+}] or pOH to pH toward the end of the calculation.)

#### Check Your Answer^{[7]}

## Key Equations

- [latex]K_{\text{a}} = \frac{[\text{H}_3\text{O}^{+}][\text{A}^{-}]}{[\text{HA}]}[/latex]
- [latex]K_{\text{b}} = \frac{[\text{HB}^{+}][\text{OH}^{-}]}{[\text{B}]}[/latex]
- [latex]K_{\text{a}}\;\times\;K_{\text{b}} = 1.0\;\times\;10^{-14} = K_{\text{w}}[/latex]
- [latex]\text{Percent ionization} = \frac{[\text{H}_3\text{O}^{+}]_{\text{eq}}}{[\text{HA}]_{0}}\;\times\;100[/latex]

### Attribution & References

*General Chemistry 1 & 2*by Rice University, a derivative of

*Chemistry (Open Stax)*by Paul Flowers, Klaus Theopold, Richard Langley & William R. Robinson and is licensed under CC BY 4.0. Access for free at

*Chemistry (OpenStax)*

equilibrium constant for the ionization of a weak acid

ratio of the concentration of the ionized acid to the initial acid concentration, times 100

equilibrium constant for the ionization of a weak base