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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 $a = \gamma C$.

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