#### Learning Objective

- Solve acid-base equilibrium problems for weak acids.

#### Key Points

- The dissociation of weak acids, which are the most popular type of acid, can be calculated mathematically and applied in experimental work.
- If the concentration and K
_{a}of a weak acid are known, the pH of the entire solution can be calculated. The exact method of calculation varies according to what assumptions and simplifications can be made. - Weak acids and weak bases are essential for preparing buffer solutions, which have important experimental uses.

#### Terms

- weak acidone that dissociates incompletely, donating only some of its hydrogen ions into solution
- conjugate basethe species created after donating a proton.
- conjugate acidthe species created when a base accepts a proton

A weak acid is one that does not dissociate completely in solution; this means that a weak acid does not donate all of its hydrogen ions (H^{+}) in a solution. Weak acids have very small values for K_{a} (and therefore higher values for pK_{a}) compared to strong acids, which have very large K_{a} values (and slightly negative pK_{a} values).

The majority of acids are weak. On average, only about 1 percent of a weak acid solution dissociates in water in a 0.1 mol/L solution. Therefore, the concentration of H^{+} ions in a weak acid solution is always less than the concentration of the undissociated species, HA. Examples of weak acids include acetic acid (CH_{3}COOH), which is found in vinegar, and oxalic acid (H_{2}C_{2}O_{4}), which is found in some vegetables.

## Dissociation

Weak acids ionize in a water solution only to a very moderate extent. The generalized dissociation reaction is given by:

[latex]HA(aq) \rightleftharpoons H^+ (aq) + A^- (aq)[/latex]

where HA is the undissociated species and A^{–} is the conjugate base of the acid. The strength of a weak acid is represented as either an equilibrium constant or a percent dissociation. The equilibrium concentrations of reactants and products are related by the acid dissociation constant expression, K_{a}:

[latex]K_a = \frac{[H^+][A^-]}{[HA]}[/latex]

The greater the value of K_{a}, the more favored the H^{+} formation, which makes the solution more acidic; therefore, a high K_{a} value indicates a lower pH for a solution. The K_{a} of weak acids varies between 1.8×10^{−16} and 55.5. Acids with a K_{a} less than 1.8×10^{−16} are weaker acids than water.

If acids are polyprotic, each proton will have a unique K_{a}. For example, H_{2}CO_{3} has two K_{a} values because it has two acidic protons. The first K_{a} refers to the first dissociation step:

[latex]H_2CO_3 + H_2O \rightarrow HCO_3^{-} + H_3O^+[/latex]

This K_{a }value is 4.46×10^{−7} (pK_{a1} = 6.351). The second K_{a} is 4.69×10^{−11} (pK_{a2} = 10.329) and refers to the second dissociation step:

[latex]HCO_3^- + H_2O \rightarrow CO_3^{2- } + H_3O^+[/latex]

## Calculating the pH of a Weak Acid Solution

The K_{a} of acetic acid is [latex]1.8\times 10^{-5}[/latex]. What is the pH of a solution of 1 M acetic acid?

In this case, you can find the pH by solving for concentration of H^{+} (*x*) using the acid’s concentration (*F*) and K_{a}. Assume that the concentration of H^{+} in this simple case is equal to the concentration of A^{–}, since the two dissociate in a 1:1 mole ratio:

[latex]K_a = \frac{[H^+][C_2H_3O_2^-]}{[HA]} = \frac{x^2}{(F-x)}[/latex]

This quadratic equation can be manipulated and solved. A common assumption is that *x* is small; we can justify assuming this for calculations involving weak acids and bases, because we know that these compounds only dissociate to a very small extent. Therefore, our above equation simplifies to:

[latex]K_a=1.8\times 10^{-5}=\frac{x^2}{F-x}\approx \frac{x^2}{F}=\frac{x^2}{\text{1 M}}[/latex]

[latex]1.8\times 10^{-5}=x^2[/latex]

[latex]x=3.9 \times 10^{-3}\text{ M}[/latex]

[latex]pH=-log[H^+]=-log(3.9\times 10^{-3})=2.4[/latex]

Although it is only a weak acid, a concentrated enough solution of acetic acid can still be quite acidic.