Improved Henderson-Hasselbalch equation

In deriving the Henderson-Hasselbalch equation, we made the following three assumptions:

    • [H+] from water is negligible, i.e. H+ in the flask containing the weak acid is solely due to that of the acid, [H+]a as the dissociation of water is again suppressed at this stage.
    • For a weak acid, [HA] is approximately equal to the concentration of the undissociated acid[HA]ud, i.e. the dissociation of the weak acid HA is negligible and the remaining concentration of HA after the addition of base is [HA][HA]ud – [A]s.
    • [A] is approximately equal to the concentration of the salt, [A]s, in the solution, i.e. Ais completely attributed to the salt formed, with no contribution from the further dissociation of HA.

However, the Henderson-Hasselbalch equation is unreliable when Ka ≥ 10-3. If we disregard the 2nd and 3rd assumptions, the equilibrium expression is:

K_a=\frac{[H^+]_a\left ( [A^-]+[H^+]_a \right )}{[HA]-[H^+]_a}

Rearranging the above equation and solving the quadratic equation in [H+]a,

[H^+]_a=\frac{-\left ( [A^-]+K_a \right )+\sqrt{\left ( [A^-]+K_a \right )^2+4K_a[HA]}}{2}

pH=-log\frac{-\left ( [A^-]+K_a \right )+\sqrt{\left ( [A^-]+K_a \right )^2+4K_a[HA]}}{2}

pH=-log\frac{-\left ( \frac{C_bV_b}{V_a+V_b}+K_a \right )+\sqrt{\left ( \frac{C_bV_b}{V_a+V_b}+K_a \right )^2+4K_a\frac{C_aV_a-C_bV_b}{V_a+V_b}}}{2}\; \; \; \; \; \; \; \; (6)

where Ca and Cb are the analytical concentrations of the acid and base respectively (i.e. the concentrations of the acid and base before any dissociation occurs). Eq6 is the improved version of the Henderson-Hasselbalch equation.

The diagram below shows the superimposition of eq6 (green curve) over the complete pH titration curve (red curve) that disregard all three assumptions, for the titration of 10 cm3 of 0.200 M of CH3COOH (Ka = 1.75 x 10-5) with 0.100 M of NaOH.

The fit is much better than the Henderson-Hasselbalch curve even though the two curves (i.e. eq6 and the complete pH titration curve) do not actually coalesce and are still two separate curves when the axes of the plot are scaled to a very high resolution. Furthermore, eq6 doesn’t become invalid when Ka ≥ 10-3, for example, the diagram below is the superimposition of eq6 on the complete pH titration curve for the titration of 10 cm3 of 0.200 M of an acid (Ka = 1.75 x 10-2) with 0.100 M of NaOH:

If we further superimpose the Henderson-Hasselbalch curve (purple curve) onto the above diagram for the same titration, we have,

Clearly, the Henderson-Hasselbalch equation breaks down when Ka ≥ 10-3.


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