Question

Consider a weak acid, HX. If a \(0.10 M\) solution of HX has a 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

The standard Gibbs Free Energy change (ΔG°) for the dissociation of the weak acid HX at 25°C is 56.196 kJ/mol.

Step-by-step solution

Convert the pH to the concentration of H+ ions

We first need to find the concentration of H+ ions from the given pH value: pH = - log[H+] Or [H+] = 10^(-pH) Given pH = 5.83, we can determine the concentration of H+ ions: [H+] = 10^(-5.83) = 1.47 × 10^(-6) M

Write the chemical equation for the dissociation of HX

The chemical equation for the dissociation of the weak acid HX is: HX(aq) ⇌ H+(aq) + X^-(aq)

Calculate the equilibrium constant (Ka) for the reaction

Using the reaction above, we can now determine the equilibrium constant (Ka). Since the dissociation is at equilibrium, we can write the expression for Ka: \[ K_a = \frac{[H+][X^-]}{[HX]} \] We know that [H+] = 1.47 × 10^(-6) M, and [HX] at the start is 0.10 M. Since the weak acid dissociates partially, at equilibrium, the concentration of X⁻ will also be equal to [H+]. Therefore, [X⁻] = 1.47 × 10^(-6) M. Now, we can calculate Ka: \[ K_a = \frac{(1.47 \times 10^{-6})(1.47 \times 10^{-6})}{0.10} \] Ka = 2.15 × 10^(-12)

Determine the standard reaction Gibbs energy change (ΔG°) using the relationship between ΔG° and Ka

Finally, we need to find the standard Gibbs Free Energy change (ΔG°) using the following formula: \[ \Delta G^{\circ} = -RT \ln(K_a) \] Where R is the gas constant (8.314 J/(mol·K)) and T is the temperature in Kelvin (25°C = 298.15 K) \[ \Delta G^{\circ} = -(8.314)(298.15) \ln(2.15 \times 10^{-12}) \] ΔG° = 56,196 J/mol = 56.196 kJ/mol Therefore, the standard Gibbs Free Energy change (ΔG°) for the dissociation of the weak acid HX at 25°C is 56.196 kJ/mol.

Key concepts

pH calculation

The concept of pH is central to understanding acid-base reactions. It measures the acidity or basicity of a solution. The pH scale ranges from 0 to 14, with 7 being neutral. Acidic solutions have pH values less than 7, while basic solutions are above 7.
To find the concentration of hydrogen ions ([H+]) from a given pH, we use the formula:\[\text{pH} = -\log[\text{H}^+]\]
Reversing this gives the concentration as:\[[\text{H}^+] = 10^{-\text{pH}}\]
In our example, a pH of 5.83 means the concentration of H+ ions is \(1.47 \times 10^{-6}\ M\). This calculation helps in determining how strongly an acid dissociates in solution.

Equilibrium constant (Ka)

The equilibrium constant \(K_a\) is a vital parameter for weak acids. It quantifies the extent of dissociation of an acid in water. For a weak acid like HX, which dissociates into H+ and X-, the expression is given by:
\[ K_a = \frac{[\text{H}^+][\text{X}^-]}{[\text{HX}]} \]
In our scenario, we know that at equilibrium, [H+] = [X-], both equal to \(1.47 \times 10^{-6}\ M\). The initial concentration of HX is \(0.10\ M\).
By substituting these values into the expression, we find:\[ K_a = \frac{(1.47 \times 10^{-6})(1.47 \times 10^{-6})}{0.10} \approx 2.15 \times 10^{-12} \]
This small \(K_a\) value indicates that HX is a weak acid, dissociating only slightly in solution.

Gibbs Free Energy (ΔG°)

Gibbs Free Energy change (\(\Delta G^\circ\)) provides insight into the favorability of a reaction. It's a thermodynamic quantity that predicts whether a reaction will occur spontaneously.
For chemical reactions, \(\Delta G^\circ\) is related to the equilibrium constant (\(K_a\)) using:
\[ \Delta G^{\circ} = -RT \ln(K_a) \]
Here, \(R\) represents the gas constant (8.314 J/(mol·K)), and \(T\) is the temperature in Kelvin.
At 25°C or 298.15 K, substituting \(K_a = 2.15 \times 10^{-12}\), we find:
\[ \Delta G^{\circ} = -(8.314)(298.15) \ln(2.15 \times 10^{-12}) \approx 56.196\ \text{kJ/mol} \]
This positive \(\Delta G^\circ\) suggests the dissociation of HX into H+ and X- is not spontaneous.

Chemical equilibrium

Chemical equilibrium is a state where the forward and reverse reactions occur at the same rate, resulting in constant concentrations of products and reactants.
For the dissociation of a weak acid like HX:
\[ \text{HX} (aq) \rightleftharpoons \text{H}^+ (aq) + \text{X}^- (aq) \]
Equilibrium is reached when the concentration of HX, H+, and X- stabilize. At this point, the rates of dissociation and recombination of HX equal each other.
This balance is described quantitatively by the equilibrium constant \(K_a\). It helps determine how much an acid will dissociate under specific conditions.
In weak acids, the position of equilibrium is toward the reactants side, indicating limited dissociation, which is why \(K_a\) values are usually small.
This concept is crucial for understanding not only acid-base chemistry, but overall chemical reactions and their behaviors in various conditions.