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 = pKa + log(1) = pKa.

The Henderson-Hasselbalch equation can be used to calculate the pH of a buffer solution, given the Ka 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 The approximation that x is at least 100 times smaller than [HA]0 and [A]0 is no longer valid, and hence you cannot use the Henderson-Hasselbalch equation anymore.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 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.