D28.2 Henderson-Hasselbalch Equation

John Moore; Jia Zhou; Etienne Garand

D28.2 Henderson-Hasselbalch Equation

The ionization constant expression for a weak acid HA is:

$K_a = \dfrac{[\text{H}_3\text{O}^{+}]_e[\text{A}^{-}]_e}{[\text{HA}]_e}$

Rearranging gives:

$[\text{H}_3\text{O}^{+}]_e = K_a\;\times\;\dfrac{[\text{HA}]_e}{[\text{A}^{-}]_e}$

Taking the negative logarithm of both sides, we have:

$\begin{array}{rcl} -\text{log}[\text{H}_3\text{O}^{+}]_e &=& -\text{log}\;K_a\;-\;\text{log}\dfrac{[\text{HA}]_e}{[\text{A}^{-}]_e} \\[1 em] \text{pH} &=& \text{p}K_a\;-\;\text{log}\dfrac{[\text{HA}]_e}{[\text{A}^{-}]_e} \\[1 em] \text{pH} &=& \text{p}K_a\;+\;\text{log}\dfrac{[\text{A}^{-}]_e}{[\text{HA}]_e} \end{array}$

It is much more convenient to do calculations with the initial concentrations of the weak acid and weak base used to prepare a buffer solution. (The initial concentration is the amount of weak acid and weak base added to the solution mixture divide by the volume.) Using the initial concentrations give us:

$\text{pH} = \text{p}K_a\;+\;\text{log}\dfrac{[\text{A}^{-}]_0\;+\;x}{[\text{HA}]_0\;-\;x}$

Here, “x” is the change in [H₃O⁺] as the solution reaches equilibrium and [HA]₀ and [A⁻]₀ are the initial concentrations of weak acid and weak base used to prepare the buffer solution.

When the approximation that x is at least 100 times smaller than [HA]₀ and [A⁻]₀ is valid, we have the Henderson-Hasselbalch equation:

$\text{pH} = \text{p}K_a\;+\;\text{log}\dfrac{[\text{A}^{-}]_0}{[\text{HA}]_0}$

Note that when [A⁻]₀ = [HA]₀, pH = pK_a + log(1) = pK_a.

The Henderson-Hasselbalch equation can be used to calculate the pH of a buffer solution, given the K_a and the initial concentrations, or it can be used to determine the ratio of initial concentrations of weak acid and weak base needed to achieve a desired pH.

The Henderson-Hasselbalch equation applies only to buffer solutions in which the ratio $\dfrac{[\text{A}^{-}]_0}{[\text{HA}]_0}$ is between 0.1 and 10. If enough strong acid or strong base is added to the buffer solution so that the weak-base-to-weak-acid ratio falls outside of this range, the the solution is no longer a buffer solution and its pH will begin to change significantly. The approximation that x is at least 100 times smaller than [HA]₀ and [A⁻]₀ is no longer valid, and hence you cannot use the Henderson-Hasselbalch equation anymore.

Exercise: Using the Henderson-Hasselbalch Equation to Calculate pH

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Chemistry 109 Fall 2021 Copyright © by John Moore; Jia Zhou; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.