Consider a weak acid, HX. If a 0.10-M solution of HX has a \(\mathrm{pH}\) of 5.83 at \(25^{\circ} \mathrm{C},\) what is \(\Delta G^{\circ}\) for the acid's dissociation reaction at \(25^{\circ} \mathrm{C} ?\)

First, use the given pH to determine the concentration of \(H^+\) ions. Remember, \(\text{pH} = -\log [H^+]\), so rearranging for \([H^+]\) gives \([H^+]=10^{-\text{pH}}\). Plugging in the given pH of 5.83, we find \([H^+] = 10^{-5.83}\) or \(1.48 \times 10^{-6}\) M. For a weak acid HX, it partially dissociates in water: \[HX <=> H^+ + X^-\]. At equilibrium, the concentration of \(H^+\) and \(X^-\) ions are equal, so \([X^-] = 1.48 \times 10^{-6}\) M. The dissociation constant of the acid, \(K_a\), is given by \(K_a = \frac{[H^+][X^-]}{[HX]}\). So using the given initial concentration of the acid \([HX]_{\text{initial}} = 0.10\) M and knowing that the \(HX\) that dissociates is equal to \([H^+]\), we get \([HX]_{\text{equilibrium}} = [HX]_{\text{initial}} - [H^+]\). Therefore, \(K_a = \frac{(1.48 \times 10^{-6})^2}{(0.10 - 1.48 \times 10^{-6})}\).

After calculating \(K_a\), the next step is to use it to calculate the change in Gibbs free energy (\(\Delta G^\circ\)). The relationship between these two variables at standard conditions is given by the equation \(\Delta G^\circ = -RT \ln K_a\), where \(\Delta G^\circ\) is the standard Gibbs free energy, \(R\) is the ideal gas constant (equal to 8.314 J/(mol K) when \(G\) is expressed in Joules), and \(T\) is the temperature in Kelvin. First, convert the given temperature from Celsius to Kelvin by adding 273 to it (thus 25°C = 298 K). Then substitute these values into the equation: \(\Delta G^\circ = -8.314 \times 298 \times \ln K_a\).

The value obtained for \(\Delta G^\circ\) gives the change in Gibbs free energy under standard conditions for the dissociation reaction of the weak acid HX. If \(\Delta G^\circ\) is negative, it means the reaction is spontaneous under standard conditions. If \(\Delta G^\circ\) is positive, the reaction is non-spontaneous under standard conditions.