# What is the activity of pure water?

May 29, 2017

I assume you mean the activity $a$ of liquid water... which is just the "real life" version of concentration.

In general, we define activity as:

$\textcolor{b l u e}{{a}_{A} = {\chi}_{A} {\gamma}_{A}} = \frac{{\chi}_{A} {\gamma}_{A} {P}_{A}^{\circ}}{{P}_{A}^{\circ}} = \textcolor{b l u e}{{P}_{A} / {P}_{A}^{\circ}}$,

where:

• ${a}_{A}$ is the activity of substance $A$.
• ${\gamma}_{A}$ is the activity coefficient of substance $A$.
• ${\chi}_{A} = {n}_{A} / \left({\sum}_{i = 1}^{N} {n}_{i}\right)$ is the mol fraction of substance $A$ and ${n}_{i}$ is the mols of substance $i$.
• ${P}_{A}$ is the partial vapor pressure of substance $A$.
• ${P}_{A}^{\circ}$ is the vapor pressure of pure $A$ under the same conditions.
• ${P}_{A} = {\chi}_{A} {\gamma}_{A} {P}_{A}^{\circ}$ is the real-life version of Raoult's law (i.e. for nonideal solutions).

You can find a more tailored definition here, but we can cover this in general using water as an example.

Let's say we had a pure water solution of $\text{pH}$ $7$ at ${25}^{\circ} \text{C}$ and $\text{1 atm}$, with concentrations ["H"^(+)] = 10^(-7) "M" and ["OH"^(-)] = 10^(-7) "M". In a $\text{1 L}$ solution, we thus have:

${n}_{{H}^{+}} = {10}^{- 7} \text{mols}$
${n}_{O {H}^{-}} = {10}^{- 7} \text{mols}$
${n}_{{H}_{2} O} = \cancel{\text{1 L" xx (997.0749 cancel"g")/cancel"L" xx "1 mol water"/(18.015 cancel"g") = "55.34 mols}}$

As a result, the mol fraction of water in water is:

${\chi}_{{H}_{2} O} = {n}_{{H}_{2} O} / \left({n}_{{H}^{+}} + {n}_{O {H}^{-}} + {n}_{{H}_{2} O}\right)$

= "55.34 mols"/(10^(-7) "mols" + 10^(-7) "mols" + "55.34 mols")

$= 0.9999999964 \cdots \approx 1$

It is known that as ${\chi}_{A} \to 1$, ${\gamma}_{A} \to 1$. Since ${\chi}_{A} \approx 1$, it follows that ${a}_{A} \approx 1$, and the activity of water in water is $\boldsymbol{1}$.

Another way to recognize this is to realize that, and this is a redundant description, but water by itself can be treated as "water in water", so:

${P}_{A} / {P}_{A}^{\circ} = 1$

since the vapor pressure of water in this pure water "solution", ${P}_{A}$, is (effectively) not reduced by any solutes relative to the vapor pressure of pure water, ${P}_{A}^{\circ}$, both at the same temperature and pressure.