Problem 100 - Chemistry An Atoms First Approach

Question

Consider the dissociation of a weak acid HA \(\left(K_{\mathrm{a}}=4.5 \times 10^{-3}\right)\) in water: $$\mathrm{HA}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{A}^{-}(a q)$$ Calculate \(\Delta G^{\circ}\) for this reaction at \(25^{\circ} \mathrm{C}.\)

Step-by-Step Solution

Answer

The standard Gibbs free energy change, ΔG°, for the dissociation of the weak acid HA in water is approximately -147,989 J/mol.

Step 1: Convert the temperature to Kelvin

To convert the temperature from Celsius to Kelvin, we need to add 273.15 to the Celsius temperature. In this case, the given temperature is 25°C. So the temperature in Kelvin is: \(T = 25 + 273.15 = 298.15 K\)

Step 2: Calculate the equilibrium constant, K

For weak acids, the acid dissociation constant, Ka, can be used as the equilibrium constant, K. So in this case, we have: \(K = K_a = 4.5 \times 10^{-3}\)

Step 3: Calculate ΔG° using the equation ΔG° = -RT ln(K)

Now we can calculate the standard Gibbs free energy change, ΔG°, using the equation: \(\Delta G^{\circ} = -RT \ln{K}\) Here, R is the universal gas constant (8.314 J/mol·K) and T is the temperature in Kelvin (298.15 K), which we calculated in Step 1. Substitute the values into the equation: \(\Delta G^{\circ} = -(8.314 \, J/mol \cdot K) \times (298.15 \, K) \times \ln{(4.5 \times 10^{-3})}\) \(\Delta G^{\circ} \approx 27406.4 \, J/mol \times (-5.40)\) \(\Delta G^{\circ} \approx -147989 \, J/mol\) Therefore, the standard Gibbs free energy change, ΔG°, for the dissociation of the weak acid HA in water is approximately -147,989 J/mol.

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. This means that the concentrations of the reactants and products remain constant over time, not that they are necessarily equal. In other words, once a system reaches equilibrium, no further change in the quantity of reactants and products occurs without external influence.

For the equation \(\mathrm{HA}(aq) \rightleftharpoons \mathrm{H}^{+}(aq) + \mathrm{A}^{-}(aq)\), the point of chemical equilibrium is achieved when the rate at which the weak acid (HA) dissociates into its ions (H+ and A-) is equal to the rate at which these ions recombine to form the weak acid. Understanding chemical equilibrium is fundamental when studying the behavior of reactions, such as the dissociation of weak acids, as it helps predict the concentrations of different species involved in the reaction under specific conditions.

Gibbs Free Energy

Gibbs free energy (G) is a thermodynamic quantity that measures the maximum amount of work that can be performed by a process at constant temperature and pressure. It is also an indicator of the spontaneity of a reaction; if \(\Delta G^\circ\) is negative, the process occurs spontaneously, while a positive \(\Delta G^\circ\) suggests that the reaction is non-spontaneous.

The relationship between Gibbs free energy and chemical equilibrium is expressed by the equation \(\Delta G^\circ = -RT \ln K\), where \(\Delta G^\circ\) is the standard change in Gibbs free energy, R is the universal gas constant, T is the temperature in Kelvin, and K is the equilibrium constant. This equation is derived from combining the concepts of enthalpy, entropy, and temperature to predict the thermodynamic favorability of a reaction.

Weak Acid Dissociation

Weak acid dissociation refers to the partial ionization of a weak acid in aqueous solution to form hydrogen ions (H+) and the corresponding anions (A-). Unlike strong acids, which dissociate completely, weak acids only dissociate partially. This behavior is characterized by the acid's dissociation constant \(K_a\), which is a measure of the strength of the acid.

A larger \(K_a\) value indicates a stronger acid (more dissociation), whereas a smaller \(K_a\) value indicates a weaker acid (less dissociation). In the case of the weak acid HA, the dissociation can be shown as \(HA(aq) \rightleftharpoons H^{+}(aq) + A^{-}(aq)\), with \(K_a = 4.5 \times 10^{-3}\). This implies that at equilibrium, the concentration of ionized species is a small fraction of the undissociated acid.

Equilibrium Constant Calculation

The equilibrium constant, denoted as K, quantifies the extent of a reaction at equilibrium. For the dissociation reaction \(HA \rightleftharpoons H^+ + A^-\), the equilibrium constant \(K_a\) for the weak acid can be used to compute the extent of dissociation. It's calculated based on the concentrations of the products and reactants at equilibrium.

Calculating \(K_a\)

Using the expression for the dissociation of a weak acid: \(K_a = \frac{[H^+][A^-]}{[HA]}\), where the square brackets denote the molar concentrations of the ions and the undissociated acid at equilibrium. For a given weak acid at a specific temperature, \(K_a\) is constant and can be used to determine the equilibrium concentrations of all species involved or to calculate related thermodynamic quantities, such as \(\Delta G^\circ\), which provides insights into the Gibbs free energy change associated with the acid dissociation process.