Question

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 \(25^{\circ} \mathrm{C} ?\)

Short answer

Calculating the acid dissociation constant (\(K_a\)) using the given pH and initial concentration of HX, we get \(K_a = \frac{(1.48 \times 10^{-6})^2}{(0.10 - 1.48 \times 10^{-6})}\). Next, we use the formula \(\Delta G^\circ = -RT \ln K_a\) to find the change in Gibbs free energy at 25°C. The result will indicate whether the reaction is spontaneous or not under standard conditions.

Step-by-step solution

Determine the acid dissociation constant (Ka)

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, \(K_a = \frac{(1.48 \times 10^{-6})^2}{(0.10 - 1.48 \times 10^{-6})}\).

Calculate the standard Gibbs free energy change

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: \(\Delta G^\circ = -8.314 \times 298 \times \ln K_a\).

Interpret the Result

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.

Key concepts

Acid Dissociation Constant

The acid dissociation constant (\(K_a\)) is a crucial concept when working with weak acids. It measures how completely an acid can dissociate into its ions in a solution. A higher \(K_a\) indicates a stronger acid, which dissociates more completely in water. In our exercise, the weak acid \(HX\) only partially dissociates in water.
To find \(K_a\), begin by calculating the concentration of hydrogen ions \([H^+]\) using the provided pH value. Remember the formula, \(\text{pH} = -\log [H^+]\), which lets us solve for \([H^+]\). Plugging in the pH of 5.83 gives us \([H^+] = 10^{-5.83}\).
Using the formula for the dissociation constant:
\[K_a = \frac{[H^+][X^-]}{[HX]}\] we can calculate it once the ion concentrations are known. Understanding \(K_a\) not only helps in calculating Gibbs free energy but also provides insight into the strength of the acid being analyzed.

Weak Acid

Weak acids like \(HX\) do not fully dissociate in solutions, unlike strong acids. This means a weak acid in water will establish an equilibrium between the non-dissociated and dissociated forms. When a weak acid dissociates:

• It only partially releases hydrogen ions.
• Equilibrium is reached quickly, and concentrations stabilize.

In our scenario, knowing that \(HX\) is a weak acid allows us to assume that not all of the initial \(0.10\) M concentration will convert to \([H^+]\) and \([X^-]\) ions. The initial concentration minus the small portion that dissociates gives us the equilibrium concentration remaining of the acid.

pH Calculation

Calculating pH is fundamental when analyzing acids and bases. The pH scale, ranging from 0 to 14, indicates how acidic or basic a solution is. A lower pH means higher acidity.
For the weak acid \(HX\), which has a pH of 5.83, we calculate \([H^+]\) using the formula:\[ \text{pH} = -\log [H^+] \]
By rearranging, we find \([H^+] = 10^{-5.83}\), leading to \([H^+] \approx 1.48 \times 10^{-6}\) M.
This value is integral for finding the acid dissociation constant and understanding the acid's behavior. Because pH directly relates to hydrogen ion concentration, it serves as a bridge to calculating other important properties, such as \(K_a\) and \(\Delta G^\circ \).