Buffer capacity of water

What is the formula describing the buffer capacity of water?

Consider a solution containing water and a strong base with the following equilibria:

H_2O(l)\rightleftharpoons H^+(aq)+OH^-(aq)

NaOH(aq)\rightleftharpoons Na^+(aq)+OH^-(aq)

With reference to the above equilibria, the sum of the number of moles of H+ and Na+ must equal to that of OH(see this article for details). As the volume is of the solution is common to all ions,

[H^+]+[Na^+]=[OH^-]\; \; \; \; \; \; \; \; (3)

Substituting cb = [Na+] and Kw = [H+][OH] in eq3


\frac{dc_b}{dpH}=\frac{dK_w[H^+]^{-1}}{dpH}-\frac{d[H^+]}{dpH}\; \; \; \; \; \; \; \; (4)

Substitute eq1 and pH = –log[H+] in eq4

\beta_w=-\frac{dK_w[H^+]^{-1}}{dlog[H^+]}+\frac{d[H^+]}{dlog[H^+]}\; \; \; \; \; \; \; \; (4a)

\beta_w=ln10\left (\frac{K_w}{[H^+]}+[H^+] \right )\; \; \; \; \; \; \; \; (5)

To determine the maximum or minimum buffer capacity of water,

\frac{d\beta_w}{d[H^+]}=ln10\left (\frac{dK_w[H^+]^{-1}}{d[H^+]}+\frac{d[H^+]}{d[H^+]} \right )

\frac{d\beta_w}{d[H^+]}=ln10\left ( -\frac{K_w}{[H^+]^2}+1 \right )\; \; \; \; \; \; \; \; (6)

Let \frac{d\beta_w}{d[H^+]}=0


Substitute Kw = [H+][OH] in the above equation


This means that a stationary point for the function in eq5 occurs when [OH] = [H+] , i.e. at pH = 7. Let’s investigate the nature of this stationary point by differentiating eq6.

\frac{d^{\, 2}\beta_w}{d[H^+]^2}=2ln10\frac{K_w}{[H^+]^3}

Substitute Kw = [H+][OH] in the above equation, noting that [OH] = [H+]

\frac{d^{\, 2}\beta_w}{d[H^+]^2}=\frac{2ln10}{[H^+]}

\frac{d^{\, 2}\beta_w}{d[H^+]^2}> 0

If we repeat the above steps and consider the case of the addition of a strong acid to water, we end up with the same conclusion. Hence, water has minimum buffer capacity at pH = 7.

The buffer capacity of water over the entire range of pH can be found by substituting [H+] = 10-pH in eq5 to give

\beta_w=ln10\left ( \frac{K_w}{10^{-pH}}+10^{-pH} \right )\; \; \; \; \; \; \; \; (6a)

and plotting the above equation with βw as the vertical axis and pH as the horizontal axis to give:

From the graph, the buffering capacity of water increases infinitesimally from a minimum of 0 at pH 7, to 0.023 at pH 2 on the left and pH 12 on the right, and then drastically to 2.303 at pH 0 on the left and pH 14 on the right. This means that water is only relatively effective as a buffer at extreme pH levels.


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