Ionization of Water and the pH Scale

Even the purest water conducts electricity very slightly, because water itself undergoes a tiny

degree of self-ionization: $\text{H}_2\text{O}(l) \rightleftharpoons \text{H}^+(aq) + \text{OH}^-(aq)$

(more precisely written with a second water molecule accepting the proton to form

$\text{H}_3\text{O}^+$, but the simpler notation is used throughout this chapter). Applying the

equilibrium law and treating the concentration of the vast excess of un-ionized water as constant

gives the ionic product of water,

At $298\ \text{K}$, careful measurement gives $K_w = 1.0 \times 10^{-14}\ \text{mol}^2\,\text{L}^{-2}$.

In pure water, $[\text{H}^+]$ and $[\text{OH}^-]$ must be equal (each water molecule that ionizes

produces exactly one of each), so $[\text{H}^+] = [\text{OH}^-] = \sqrt{K_w} = 1.0 \times 10^{-7}\ \text{M}$ — an extremely small but non-zero concentration. Crucially, $K_w$ is a genuine equilibrium

constant: its numerical value applies not only to pure water but to any aqueous solution at that

temperature, acidic, neutral or basic alike, so that once $[\text{H}^+]$ is known, $[\text{OH}^-]$

can always be found from $[\text{OH}^-] = K_w / [\text{H}^+]$, and vice versa.

Because $[\text{H}^+]$ in ordinary aqueous solutions spans a huge range — from about $1\ \text{M}$ in

a strong acid down to $10^{-14}\ \text{M}$ in a strong base — Sørensen introduced a compressed

logarithmic scale, the pH scale, defined as

so that pH values run conveniently from about 0 to 14 at room temperature. Taking $-\log_{10}$ of the

$K_w$ expression gives the companion relation $\text{pH} + \text{pOH} = 14$ (at $298\ \text{K}$),

where $\text{pOH} = -\log_{10}[\text{OH}^-]$. A solution is neutral when $\text{pH} = 7$ (equal

$[\text{H}^+]$ and $[\text{OH}^-]$, as in pure water), acidic when $\text{pH} < 7$ (excess

$\text{H}^+$), and basic when $\text{pH} > 7$ (excess $\text{OH}^-$).

For a strong acid or strong base, which ionizes essentially completely, $[\text{H}^+]$ or

$[\text{OH}^-]$ can be read off directly from the stated concentration (accounting for the number of …