# What scale indicates the relative concentrations of hydrogen and hydroxide ions in a solution?

Sep 29, 2016

In aqueous solution, i.e. in water solution, I think you refer to the $p H$ scale.

#### Explanation:

$p H$ is short for $\text{pouvoir hydrogen}$ i.e. $\text{the power of hydrogen}$ or $\text{hydrogen potential}$. This is applied to aqueous solutions in the following way.

In water, the following reaction takes place, and this is referred to as $\text{autoprotolysis}$:

$2 {H}_{2} O r i g h t \le f t h a r p \infty n s {H}_{3} {O}^{+} + H {O}^{-}$

As with any equilibrium, we can write the following equilibrium expression:

$\frac{\left[{H}_{3} {O}^{+}\right] \left[H {O}^{-}\right]}{{\left[{H}_{2} O\right]}^{2}}$ $=$ $K$.

Because $\left[{H}_{2} O\right]$ is large and effectively constant, this simplifies to:

$\left[{H}_{3} {O}^{+}\right] \left[H {O}^{-}\right]$ $=$ ${K}_{w}$.

This equilibrium has been measured meticulously, and at $298$ $K$, we can give a value for ${K}_{w}$:

${K}_{w}$ $=$ ${10}^{- 14}$ $=$ $\left[{H}_{3} {O}^{+}\right] \left[H {O}^{-}\right]$

We can take ${\log}_{10}$ of both sides to give:

${\log}_{10} \left({10}^{- 14}\right)$ $=$ ${\log}_{10} \left[{H}_{3} {O}^{+}\right] + {\log}_{10} \left[H {O}^{-}\right]$

But ${\log}_{10} \left({10}^{- 14}\right)$ $=$ $- 14$ by definition of the log function.

And so, $- 14 = {\log}_{10} \left[{H}_{3} {O}^{+}\right] + {\log}_{10} \left[H {O}^{-}\right]$ or

$14 = - {\log}_{10} \left[{H}_{3} {O}^{+}\right] - {\log}_{10} \left[H {O}^{-}\right]$

Now if we define $- {\log}_{10} \left[{H}_{3} {O}^{+}\right] = p H$ and $- {\log}_{10} \left[H {O}^{-}\right] = p O H$, then:

$14 = p H + p O H$.

And this given a value of $p H$, I can find $\left[{H}_{3} {O}^{+}\right]$ or $\left[H {O}^{-}\right]$ by taking antilogarithms.

What is the $p H$ of a solution that is $0.5$ $m o l \cdot {L}^{-} 1$ with respect to ${H}_{3} {O}^{+}$? What is the corresponding $p O H$?