D36.1 pH and Ratio of Conjugate Base to Conjugate Acid

Jia Zhou; John Moore

D36.1 pH and Ratio of Conjugate Base to Conjugate Acid

The ionization constant expression for a weak acid HA is:

$K_a = \dfrac{[\text{H}_3\text{O}^{+}]_e[\text{A}^{-}]_e}{[\text{HA}]_e}$

Rearranging gives:

$[\text{H}_3\text{O}^{+}]_e = K_a\;\times\;\dfrac{[\text{HA}]_e}{[\text{A}^{-}]_e}$

This equation shows that the ratio of concentration of conjugate acid over concentration of conjugate base determines the hydrogen-ion concentration of a solution. The bigger the concentration of conjugate acid relative to conjugate base, the bigger the hydrogen-ion concentration is and the more acidic the solution is.

Conversely, if we know the hydrogen-ion concentration of a solution, the ratio of concentration of conjugate acid over concentration of conjugate base can be calculated, as this rearranged version of the equation shows.

$\dfrac{[\text{HA}]_e}{[\text{A}^{-}]_e} \;=\;\dfrac{[\text{H}_3\text{O}^{+}]_e}{K_a}$

These equations can be rearranged to show how pH is related to composition of a conjugate acid-base pair. Taking the negative logarithm of both sides of the first equation shows that the higher the pH is the greater the ratio of concentration of conjugate base over concentration of conjugate acid:

$\begin{array}{rcl} -\text{log}[\text{H}_3\text{O}^{+}]_e &=& -\text{log}\;K_a\;-\;\text{log}\dfrac{[\text{HA}]_e}{[\text{A}^{-}]_e} \\[1 em] \text{pH} &=& \text{p}K_a\;-\;\text{log}\dfrac{[\text{HA}]_e}{[\text{A}^{-}]_e} \\[1 em] \text{pH} &=& \text{p}K_a\;+\;\text{log}\dfrac{[\text{A}^{-}]_e}{[\text{HA}]_e} \end{array}$

Similarly, this equation can be rearranged to show that the ratio of concentration of conjugate base over concentration of conjugate acid is determined by the pH:

$\dfrac{[\text{A}^{-}]_e}{[\text{HA}]_e}\;=\;10^{(\text{p}\text{H}-\text{p}K_{\text{a}})}$

This equation shows that for an aqueous solution that contains a weak acid and its conjugate base, raising the pH two units above pK_a converts 99% of the weak acid to conjugate base. Conversely, lowering pH two units below pK_a results in a solution that is at least 99% weak acid. In other words, raising the solution pH shifts the weak-acid ionization equilibrium toward the conjugate base and lowering the solution pH shifts the equilibrium toward the weak acid. (Such pH changes could be made by adding strong base or strong acid to the solution.)

raise pH →

$\text{HA(aq)}\ \ + \ \ \text{H}_2\text{O}(\ell)\ \ \rightleftharpoons \ \ \text{A}^-\text{(aq)}\ \ +\ \ \text{H}_3\text{O}^+\text{(aq})$

← lower pH

The equation verifies quantitatively the prediction of Le Chatelier’s principle that adding strong base would react away hydronium ions, shifting the equilibrium toward products.

Exercise: Acid/base Ratio and pH

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