I'm making some graphs and I have to label the axes. I want to be extra careful and put the units in even though the meaning of $\text{pH}$ is well known. But I have a problem (though a simple one): $\text{pH}$ is a minus logarithm (base 10) of concentration of hydrogen ions (or rather their activity). What is the unit then, is it $[-\log(\text{mol}/\text{L})]$? What should I write, could you help me?
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7$\begingroup$ That formulation of the definition is nonsensical -- you can't take the logarithm of a concentration since it has a non trivial unit. You first need to divide the concentration by a "standard concentration" such as $1 \mathrm{mol}/l$. Had some trouble with my Chemistry teacher who taught that sloppy definition. $\endgroup$– CodesInChaosApr 17, 2014 at 15:58
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$\begingroup$ Some useful information about the units of transcendental functions can be found here. The paragraphs around equation 11 are relevant to your question, although they sadly ignore @phillipp's comments about activity. $\endgroup$– bobthechemistApr 17, 2014 at 19:23
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2$\begingroup$ Why is "pH" not a sufficient label? $\endgroup$– user5219Apr 17, 2014 at 20:53
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1$\begingroup$ Relevant question on physics.SE $\endgroup$– leftaroundaboutApr 17, 2014 at 23:07
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2$\begingroup$ @CodesInChaos: At least that was “only” a teacher. I once failed to explain to a full professor why I had problems with taking the logarithm of something other than a scalar. $\endgroup$– WrzlprmftApr 18, 2014 at 9:41
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2 Answers
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The real definition of the $\text{pH}$ is not in terms of concentration but in terms of the activity of a proton,
\begin{equation} \text{pH} = - \log a_{\ce{H+}} \ , \end{equation}
and the activity is a dimensionless quantity. You can think of the activity as a generalization of the mole fraction that takes into account deviations from the ideal behaviour in real solutions. By introducing the (dimensionless) activity coefficient $\gamma_{\ce{H+}}$, which represents the effect of the deviations from the ideal behaviour on the concentration, you can link the activity to the concentration via
\begin{equation} a_{\ce{H+}} = \frac{\gamma_{\ce{H+}} c_{\ce{H+}}}{c^0} \ , \end{equation}
where $c^0$ is the standard concentration of $1 \, \text{mol}/\text{L}$. If you ignore the non-ideal contributions you can approximately express the $\text{pH}$ in terms of the normalized proton concentration
\begin{equation} \text{pH} \approx - \log \frac{c_{\ce{H+}}}{c^0} \ . \end{equation}
In general, there can be no logarithm of a quantity bearing a unit. If however you encounter such a case it is usually due to sloppy notation: either the argument of the logarithm is implicitly understood to be normalized and thus becomes unitless or the units in the logarithm's argument originate from using the mathematical properties of logarithms to divide the logarithm of a product which is by itself unitless into a sum of logarithms: $\log(a \cdot b) = \log(a) + \log(b)$.
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Unless you have very good reason to do otherwise, treat pH as dimensionless.
