$\mathrm{pH} = -\log[\ce{H+}]$ while $\mathrm{pOH} = -\log[\ce{OH-}]$

Why does $\mathrm p$ represent $-\log$ of something? Is it due to some historical reason or due to some scientific reason?

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    $\begingroup$ @TanMath Today, the symbol $\mathrm p$ is interpreted as an operator ($\mathrm px=-\lg x$). If you are interested in the etymology of this use, you might get better answers on hsm.stackexchange.com. $\endgroup$ – Loong Sep 25 '15 at 19:48
  • $\begingroup$ @Loong can you give me evidence for your claim? I cant find anything about p being an operator (except for physics stuff) $\endgroup$ – TanMath Sep 25 '15 at 19:51
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    $\begingroup$ @TanMath You can find this definition in the international standard ISO 80000-9 Quantities and units – Part 9: Physical chemistry and molecular physics as well as in the IUPAC Green Book. $\endgroup$ – Loong Sep 25 '15 at 19:57
  • $\begingroup$ @Loong is that online? $\endgroup$ – TanMath Sep 25 '15 at 19:59
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    $\begingroup$ @TanMath The Green Book is available online: iupac.org/fileadmin/user_upload/publications/e-resources/… $\endgroup$ – Loong Sep 25 '15 at 20:06

According to the international standard ISO 80000 Quantities and units – Part 9: Physical chemistry and molecular physics, the symbol $\mathrm p$ is interpreted as an operator $(\mathrm px=-\lg x)$. It is used in particular (but not only) in the definition of the quantity $\mathrm{pH}$:

$$\mathrm{pH}=\mathrm pa_{\ce{H+}}=-\lg\left( a_{\ce{H+}}\right)=-\lg\left(m_{\ce{H+}}\gamma_{\mathrm m,\ce{H+}}/m^\circ\right).$$

where $a_{\ce{H+}}$ is the activity of $\ce{H+}$ in solution and $\gamma_{\mathrm m,\ce{H+}}$ is the activity coefficient of $\ce{H+}$ on the molality basis at molality $m_{\ce{H+}}$. The standard molality $m^\circ$ is chosen to be equal to $1\ \mathrm{mol\cdot kg^{-1}}$.

The definition of $\mathrm p$ (and $\mathrm{pH}$) given by ISO is actually quoted from IUPAC Quantities, Units and Symbols in Physical Chemistry (Green Book).

In its current form, this definition can be traced back to a recommendation given in
R. G. Bates and E. A. Guggenheim
Report on the standardization of pH and related terminology
Pure Appl. Chem., 1960, Vol. 1, No. 1, pp. 163–168
which reads

There already exists international agreement that $\mathrm{pH}$ should be written and printed on line in roman type. We recommend that, with the unique exception of $\mathrm{pH}$, the operator $\mathrm p$ (printed in roman) should denote $-\log_{10}$.
For example
$\mathrm p m_\mathrm H$ means $-\log_{10}m_\mathrm H$
where $m$ denotes molality (…)


$[{\ce{H+}}] = 10^{-\text{pH}}$. In other words $[{\ce{H+}}]$ is $10$ to the power of $-\text{pH}$. Actually, this definition was published in a German journal, but luckily power in the mathematical sense is written as Potenz.

Note that the correct definition of $\text{pH}$ is now $\text{pH} = -\lg a_{\ce{H+}}$ where $a_{\ce{H+}}$ is the activity of $\ce{H+}$


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