Show that when [H3 O^+] is reduced to half its original...

Show that when $\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]$ is reduced to half its original value, the $\mathrm{pH}$ of a solution increases by 0.30 unit, regardless of the initial $p H .$ Is it also true that when any solution is diluted to half its original concentration, the $\mathrm{pH}$ increases by 0.30 unit? Explain.

Key Concepts

Chemical Equilibrium and Buffer Systems

For solutions that involve weak acids, weak bases, or buffer systems, the simple dilution effect does not necessarily translate into a pH change of exactly 0.30 units. This is because dilution in an equilibrated system can disturb the equilibrium, prompting a shift in the ionization balance according to Le Châtelier’s principle. As a result, the expected pH change upon halving the concentration might differ from the direct application of the logarithmic rule due to additional equilibrium adjustments.

Logarithmic pH Scale

pH is defined as the negative logarithm (base 10) of the hydronium ion concentration. This means that the pH scale is logarithmic, so a proportional change in [H?O?] leads to an additive change in pH. For example, halving the hydronium ion concentration increases the pH by log(2), which is approximately 0.30 units. This relation holds true as long as the concentration of H?O? is directly linked to the amount of an acid or base that fully dissociates.

Effect of Dilution on Concentration

Dilution reduces the concentration of all solutes in a solution by a constant factor. In the case of a strong acid where complete dissociation occurs, halving the total amount of acid results in halving the hydronium ion concentration. Due to the logarithmic relationship in the pH definition, this specific dilution causes the pH to increase by about 0.30 units. This straightforward relationship applies when the system’s components are directly diluted without any other interactions.

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