Problem 78 - Chemistry: Principles and Reactions

Question

At \(25^{\circ} \mathrm{C}\), a \(0.13 \mathrm{M}\) solution of a weak acid, \(\mathrm{HB}\), has a \(\mathrm{pH}\) of \(3.71\). What is \(\Delta G^{\circ}\) for $$ \mathrm{H}^{+}(a q)+\mathrm{B}^{-}(a q) \rightleftharpoons \mathrm{HB}(a q) $$

Step-by-Step Solution

Answer

Question: Calculate the ΔG° for the reaction involving a weak acid, given its pH is 3.71, the concentration of HB is 0.13 M, and the temperature is 25°C. Answer: The ΔG° for the reaction is 8.30 kJ/mol.

Step 1: Calculate the concentration of H+ ions

Given the pH of the solution is 3.71, we can calculate the concentration of H+ ions using the formula: $$ [\mathrm{H}^{+}] = 10^{-\mathrm{pH}} $$ Plugging in the given pH value: $$ [\mathrm{H}^{+}] = 10^{-3.71} $$ $$ [\mathrm{H}^{+}] = 1.93 \times 10^{-4} \mathrm{M} $$

Step 2: Calculate the concentration of B- ions

Since the concentration of HB is given (0.13 M), and we know the concentration of H+ ions, we can calculate the concentration of B- ions using conservation of mass: $$ [\mathrm{B}^{-}] = [\mathrm{HB}]_{\text{initial}} - [\mathrm{H}^{+}] $$ Plugging in the values, we get: $$ [\mathrm{B}^{-}] = 0.13 - 1.93 \times 10^{-4} $$ $$ [\mathrm{B}^{-}] = 0.1298 \mathrm{M} $$

Step 3: Find the equilibrium constant (K)

Now that we have the concentrations of H+, B-, and HB, we can find the equilibrium constant (K) using the formula: $$ K = \frac{[\mathrm{H}^{+}][\mathrm{B}^{-}]}{[\mathrm{HB}]} $$ Plugging in the concentrations, we get: $$ K = \frac{(1.93 \times 10^{-4})(0.1298)}{0.130 - 1.93 \times 10^{-4}} $$ $$ K = 1.92 \times 10^{-4} $$

Step 4: Calculate ΔG° using the relationship between K and ΔG°

We can now find ΔG° using the relationship between K and ΔG°: $$ \Delta G^{\circ} = -RT \ln{K} $$ Given the temperature is 25°C or 298 K, and the gas constant R = 8.314 J/(mol K), we can plug in the values to find ΔG°: $$ \Delta G^{\circ} = -(8.314)(298)\ln{(1.92 \times 10^{-4})} $$ $$ \Delta G^{\circ} = 8297.68 \mathrm{J/mol} $$ So, the ΔG° for the reaction is 8.30 kJ/mol (rounded to 2 decimal places).

Key Concepts

pH Calculation

Understanding pH calculation is essential for many chemistry and biology topics, as pH is a measure of the acidity or basicity of a solution. The pH scale typically ranges from 0 to 14, with 7 being neutral, values less than 7 indicating an acidic solution, and values greater than 7 indicating a basic solution.
In the given exercise, the pH of a weak acid solution is provided, and from this value, we must calculate the hydrogen ion concentration \( [H^+] \), since pH is defined as the negative logarithm of \( [H^+] \):

\( pH = -\log [H^+] \)

From this equation, we infer that as pH decreases, the acidity of the solution increases because \( [H^+] \) increases. It is this relationship that allows us to determine the \( [H^+] \) from a given pH, which is crucial in finding the concentration of other species in a solution, and in this exercise, helps us to understand the dissociation of the weak acid \( HB \).
To solidify this concept, remember that a pH of 3.71 corresponds to a hydrogen ion concentration of \( 1.93 \times 10^{-4} \) M, calculated as \( 10^{-pH} \). Lower pH values correspond to higher hydrogen ion concentrations, indicating greater acidity.

Equilibrium Constant (K)

The equilibrium constant (K) is a fundamental concept in chemistry that reveals the extent of a chemical reaction at equilibrium. For the reaction of a weak acid \( HB \), the equilibrium constant can be expressed as:

\( K = \frac{[H^+][B^-]}{[HB]} \)

In this reaction, \( K \) helps us quantify the ratio of the product concentrations to the reactant concentrations at equilibrium. The value of \( K \) provides insight into whether the reaction favors the formation of products (\( K > 1 \)), is balanced between reactants and products (\( K \approx 1 \)), or favors the reactants (\( K < 1 \)).
In our exercise, the equilibrium constant is calculated from the known concentrations of \( H^+ \) and \( B^- \) ions, as well as the initial concentration of \( HB \). A \( K \) value of \( 1.92 \times 10^{-4} \) suggests that the dissociation of \( HB \) is not extensive, as is typical with weak acids, which do not completely ionize in solution. Understanding how to calculate and interpret \( K \) is vital for predicting the behavior of reactions and designing experiments accordingly.

Gibbs Free Energy (ΔG°)

The concept of Gibbs free energy (ΔG°) is crucial in physical chemistry and thermodynamics as it pertains to the spontaneity of a reaction. It is defined as the maximum amount of non-expansion work that can be extracted from a closed system; this potential is realized when a system evolves from an initial state to equilibrium.
Gibbs free energy is determined by the formula:

\( \Delta G^\circ = -RT \ln{K} \)

where \( R \) is the universal gas constant, \( T \) is the temperature in Kelvins, and \( K \) is the equilibrium constant. If \( \Delta G^\circ < 0 \) the process is spontaneous, and if \( \Delta G^\circ > 0 \) the process is non-spontaneous as it requires energy input.
In our example, by calculating the equilibrium constant \( K \) of the weak acid dissociation reaction and using the given temperature, we determine \( \Delta G^\circ \) to be 8.30 kJ/mol. It is positive, indicating that the formation of \( HB \) from \( H^+ \) and \( B^- \) ions is not spontaneous under standard conditions. Understanding the relationship between \( K \) and \( \Delta G^\circ \) empowers students to predict reaction tendencies and equilibria, deepening their knowledge of chemical dynamics.