# Understanding how the definitions of pH and activity are compatible

At $$T=\pu{25 ^\circ C}$$ and $$P=\pu{1bar}$$, the density of water is $$\pu{0.997 kg L-1}$$.

Neglecting nonideality of the mixture $$\ce{H2O(l), H+}$$ and $$\ce{OH-}$$ (all activity coefficients are close to $$1$$), we have the following for $$\ce{H+}$$:

• amount concentration (molarity): $$C =\pu{1 \times 10^{-7} mol L-1}$$.
• mole fraction: $$x = 1.807 \times10^{-9}$$

An old definition for pH is $$\mathrm{pH} = −\log_{10}(C)$$, which gives the familiar $$pH=7$$ for pure water.

A more modern definition is $$\mathrm{pH} = -\log_{10}(a)$$, where $$a$$ is activity.

This definition would make sense if the activity of a component in an ideal solution were equal to its molarity. However, isn't the activity defined as

$$a = \exp\left(\frac{\mu - \mu_0}{RT}\right)$$,

where $$R$$ is the gas constant, $$T$$ the thermodynamic temperature, $$\mu$$ the chemical potential and $$\mu_0$$ the standard chemical potential of reference, in which case it is equal the mole fraction in an ideal solution?

My impression is that the definition of $$\mathrm{pH}$$ was changed in order to incorporate the effect of nonideality, in which case I would expect it to be redefined as $$\mathrm{pH} = -\log_{10}(\gamma C)$$, where $$\gamma$$ is the activity coefficient. Maybe there is an alternative definition for activity such that $$a = \gamma C$$.