What is the formula describing the buffer capacity of water?

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

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,

^{– }Substituting *c _{b}* = [

*Na*] and

^{+}*K*= [

_{w }*H*][

^{+}*OH*] in eq3

^{–}Substitute eq1 and *pH* = –*log*[*H ^{+}*] in eq4

To determine the maximum or minimum buffer capacity of water,

Let

Substitute *K _{w }*= [

*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.

^{+}Substitute *K _{w }*= [

*H*][

^{+}*OH*] in the above equation, noting that [

^{–}*OH*] = [

^{–}*H*]

^{+}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*in eq5 to give

^{-pH }and plotting the above equation with *β _{w }*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.