Calculating the pH of a weak acid is a bit more complicated than determining the pH of a strong acid because weak acids don't completely dissociate in water. Fortunately, the formula for calculating pH is simple. Here's what you do.

### Key Takeaways: pH of a Weak Acid

- Finding the pH of a weak acid is a bit more complicated than finding pH of a strong acid because the acid does not fully dissociate into its ions.
- The pH equation is still the same (pH = -log[H
^{+}]), but you need to use the acid dissociation constant (K_{a}) to find [H^{+}]. - There are two main methods of solving for hydrogen ion concentration. One involves the quadratic equation. The other assumes the weak acid barely dissociates in water and approximates the pH. Which one you choose depends on how accurate you need the answer to be. For homework, use the quadratic equation. For a quick estimate in the lab, use the approximation.

### pH of a Weak Acid Problem

What is the pH of a 0.01 M benzoic acid solution?

Given: benzoic acid K_{a}= 6.5 x 10^{-5}

#### Solution

Benzoic acid dissociates in water as:

C_{6}H_{5}COOH → H^{+} + C_{6}H_{5}COO^{-}

The formula for K_{a} is:

**K _{a} = [H^{+}][B^{-}]/[HB]**

where:

[H^{+}] = concentration of H^{+} ions

[B^{-}] = concentration of conjugate base ions

[HB] = concentration of undissociated acid molecules

for a reaction HB → H^{+} + B^{-}

Benzoic acid dissociates one H^{+} ion for every C_{6}H_{5}COO^{-} ion, so [H^{+}] = [C_{6}H_{5}COO^{-}].

Let x represent the concentration of H^{+} that dissociates from HB, then [HB] = C - x where C is the initial concentration.

Enter these values into the K_{a} equation:

K_{a} = x · x / (C -x)**K _{a} = x²/(C - x)**

(C - x)K

_{a}= x²

x² = CK

_{a}- xK

_{a}

x² + K

_{a}x - CK

_{a}= 0

Solve for x using the quadratic equation:

**x = [-b ± (b² - 4ac) ^{½}]/2a**

x = [-K_{a} + (K_{a}² + 4CK_{a})^{½}]/2

**Note** Technically, there are two solutions for x. Since x represents a concentration of ions in solution, the value for x cannot be negative.

Enter values for K_{a} and C:

K_{a} = 6.5 x 10^{-5}

C = 0.01 M

x = {-6.5 x 10^{-5} + [(6.5 x 10^{-5})² + 4(0.01)(6.5 x 10^{-5})]^{½}}/2

x = (-6.5 x 10^{-5} + 1.6 x 10^{-3})/2

x = (1.5 x 10^{-3})/2

x = 7.7 x 10^{-4}

Find pH:

pH = -log[H^{+}]

pH = -log(x)

pH = -log(7.7 x 10^{-4})

pH = -(-3.11)

pH = 3.11

#### Answer

The pH of a 0.01 M benzoic acid solution is 3.11.

### Solution: Quick and Dirty Method to Find Weak Acid pH

Most weak acids barely dissociate in solution. In this solution we found the acid only dissociated by 7.7 x 10^{-4} M. The original concentration was 1 x 10^{-2} or 770 times stronger than the dissociated ion concentration.

Values for C - x then, would be very close to C to seem unchanged. If we substitute C for (C - x) in the K_{a} equation,

K_{a} = x²/(C - x)

K_{a} = x²/C

With this, there is no need to use the quadratic equation to solve for x:

x² = K_{a}·C

x² = (6.5 x 10^{-5})(0.01)

x² = 6.5 x 10^{-7}

x = 8.06 x 10^{-4}

Find pH

pH = -log[H^{+}]

pH = -log(x)

pH = -log(8.06 x 10^{-4})

pH = -(-3.09)

pH = 3.09

Note the two answers are nearly identical with only 0.02 difference. Also notice the difference between the first method's x and the second method's x is only 0.000036 M. For most laboratory situations, the second method is "good enough" and much simpler.

Check your work before reporting a value. The pH of a weak acid should be less than 7 (not neutral) and it's usually less than the value for a strong acid. Note there are exceptions. For example, the pH of hydrochloric acid is 3.01 for a 1 mM solution, while the pH of hydrofluoric acid is also low, with a value of 3.27 for a 1 mM solution.

### Sources

- Bates, Roger G. (1973).
*Determination of pH: theory and practice*. Wiley. - Covington, A. K.; Bates, R. G.; Durst, R. A. (1985). "Definitions of pH scales, standard reference values, measurement of pH, and related terminology".
*Pure Appl. Chem*. 57 (3): 531–542. doi:10.1351/pac198557030531 - Housecroft, C. E.; Sharpe, A. G. (2004).
*Inorganic Chemistry*(2nd ed.). Prentice Hall. ISBN 978-0130399137. - Myers, Rollie J. (2010). "One-Hundred Years of pH".
*Journal of Chemical Education*. 87 (1): 30–32. doi:10.1021/ed800002c - Miessler G. L.; Tarr D .A. (1998).
*Inorganic Chemistry (*2nd ed.). Prentice-Hall. ISBN 0-13-841891-8.