Ionic Product of Water and pH

The pH scale measures the acidity or alkalinity of a solution:

$\text{pH} = -\log[\text{H}_3\text{O}^+]$

$\text{pOH} = -\log[\text{OH}^-]$

For pure water at 25°C:

pH = 7, pOH = 7

Note:

• In acidic solutions, $[\text{H}_3\text{O}^+] > [\text{OH}^-]$.
• In basic solutions, $[\text{OH}^-] > [\text{H}_3\text{O}^+]$.

The auto-ionization of water is endothermic, meaning increasing the temperature shifts the equilibrium to favor more ion production:

1. $K_w$ increases with temperature.
2. pH decreases with increasing temperature, but water remains neutral as $[\text{H}_3\text{O}^+] = [\text{OH}^-]$.

Example: At physiological temperature (37°C), $K_w = 2.5 \times 10^{-14}$. The pH of neutral water is:

$\text{pH} = -\log(\sqrt{K_w}) = 6.81$

Strong acids and bases ionize completely in water. For example, HCl in water produces $\text{H}_3\text{O}^+$, and NaOH produces $\text{OH}^-$. The self-ionization of water is suppressed due to the common ion effect.

Example 1

If 0.001 M NaOH is added to water, the concentration of $\text{OH}^- = 0.001\ \text{M}$. The concentration of $\text{H}_3\text{O}^+$ is:

$[\text{H}_3\text{O}^+] = \frac{K_w}{[\text{OH}^-]} = \frac{1 \times 10^{-14}}{0.001} = 1 \times 10^{-11}\ \text{M}$

Neutralization reactions

Neutralization occurs when $\text{H}_3\text{O}^+$ from an acid reacts with $\text{OH}^-$ from a base to form water. The final solution may be acidic, basic, or neutral depending on the amounts of $\text{H}_3\text{O}^+$ and $\text{OH}^-$ remaining after the reaction.

Example 2

What is the pH of a solution obtained by mixing 100 cm³ of 0.015 M HCl and 50 cm³ of 0.01 M $\text{Ba(OH)}_2$?

Solution:

Moles of $\text{H}_3\text{O}^+$ from HCl = $0.015 \times 0.1 = 0.0015$ moles

Moles of $\text{OH}^-$ from $\text{Ba(OH)}_2$ = $0.01 \times 0.05 \times 2 = 0.001$ moles

Excess $\text{H}_3\text{O}^+ = 0.0015 - 0.001 = 0.0005$ moles

Concentration of $\text{H}_3\text{O}^+ = \frac{0.0005}{0.1 + 0.05} = 0.0033\ \text{M}$

$\text{pH} = -\log(0.0033) \approx 2.48$

The solution is acidic.