Reference value for the ka of acetic acid
The ka of acetic acid (acid dissociation constant for CH3COOH in water) at \(25^\circ\text{C}\) and dilute conditions is commonly reported as approximately \(K_a \approx 1.8 \times 10^{-5}\), corresponding to \(pK_a \approx 4.74\).
| Quantity | Meaning in acid–base equilibrium | Typical value at \(25^\circ\text{C}\) (dilute aqueous) |
|---|---|---|
| \(K_a\) | Equilibrium constant for proton transfer from acetic acid to water | \(\approx 1.8 \times 10^{-5}\) |
| \(pK_a\) | Logarithmic measure of weak-acid strength, \(pK_a=-\log_{10}(K_a)\) | \(\approx 4.74\) |
| Conjugate base | Acetate ion paired with acetic acid in a conjugate acid–base pair | \(\mathrm{CH_3COO^-}\) |
Meaning of \(K_a\) for acetic acid in water
Acetic acid is a weak acid because only a small fraction of dissolved molecules donate a proton to water. The equilibrium is
\[ \mathrm{CH_3COOH(aq) + H_2O(l) \rightleftharpoons H_3O^+(aq) + CH_3COO^-(aq)} \]
\[ K_a=\frac{[\mathrm{H_3O^+}]\,[\mathrm{CH_3COO^-}]}{[\mathrm{CH_3COOH}]} \]
Brackets \([\;]\) denote equilibrium molar concentrations. The magnitude of \(K_a\) quantifies the position of equilibrium: smaller \(K_a\) indicates less ionization and a weaker acid.
Equilibrium (ICE-table) algebra used with \(K_a\)
A dilute acetic acid solution with initial concentration \(C\) has equilibrium concentrations that follow the same weak-acid pattern. With \(x=[\mathrm{H_3O^+}]\) produced at equilibrium (and \([\mathrm{CH_3COO^-}]=x\)), the remaining acetic acid is \([\mathrm{CH_3COOH}]=C-x\). Substitution into the definition of \(K_a\) gives
\[ K_a=\frac{x^2}{C-x} \]
The quadratic form follows directly: \[ x^2 + K_a x - K_a C = 0 \] and the physically meaningful root is \[ x=\frac{-K_a+\sqrt{K_a^2+4K_aC}}{2} \]
Worked example using the ka of acetic acid
An aqueous solution prepared with \(C=0.10\ \mathrm{mol\,L^{-1}}\) acetic acid at \(25^\circ\text{C}\) illustrates the standard weak-acid calculation.
The weak-acid approximation applies when \(x \ll C\), producing \(C-x \approx C\). Under that condition, \[ K_a \approx \frac{x^2}{C} \quad\Rightarrow\quad x \approx \sqrt{K_a C} \]
With \(K_a = 1.8 \times 10^{-5}\) and \(C=0.10\), \[ x \approx \sqrt{(1.8 \times 10^{-5})(0.10)}=\sqrt{1.8 \times 10^{-6}} \approx 1.34 \times 10^{-3}\ \mathrm{mol\,L^{-1}} \] \[ \mathrm{pH}=-\log_{10}([\mathrm{H_3O^+}]) \approx -\log_{10}(1.34 \times 10^{-3}) \approx 2.87 \] \[ \%\ \text{ionization}=\frac{x}{C}\times 100\% \approx \frac{1.34 \times 10^{-3}}{0.10}\times 100\% \approx 1.34\% \]
The approximation check is satisfied because \(1.34\%\) is well below the common \(5\%\) guideline. A quadratic evaluation typically shifts the pH only slightly for these values.
Common checks and pitfalls
Temperature and solution conditions matter: reported \(K_a\) values assume a stated temperature (often \(25^\circ\text{C}\)) and low ionic strength, while higher ionic strength can alter activities relative to concentrations.
Logarithms require base 10 in the \(pK_a\) definition: \[ pK_a=-\log_{10}(K_a) \] and \(K_a\) must be dimensionless in the equilibrium expression, with concentrations referenced to standard conditions.
For moderately concentrated weak acids, the approximation \(C-x \approx C\) loses accuracy, and the quadratic form for \(x\) provides a consistent equilibrium concentration for \([\mathrm{H_3O^+}]\), \([\mathrm{CH_3COO^-}]\), and \([\mathrm{CH_3COOH}]\).