Question

The equilibrium constant for the reaction $$ 2 \mathrm{Fe}^{3+}(a q)+\mathrm{Hg}_{2}^{2+}(a q) \rightleftharpoons 2 \mathrm{Fe}^{2+}(a q)+2 \mathrm{Hg}^{2+}(a q) $$ is \(K_{c}=9.1 \times 10^{-6}\) at \(298 \mathrm{~K}\) (a) What is \(\Delta G^{\circ}\) at this temperature? (b) If standard-state concentrations of the reactants and products are mixed, in which direction does the reaction proceed? (c) Calculate \(\Delta G\) when \(\left[\mathrm{Fe}^{3+}\right]=0.20 M,\left[\mathrm{Hg}_{2}^{2+}\right]=0.010 \mathrm{M}\) \(\left[\mathrm{Fe}^{2+}\right]=0.010 \mathrm{M},\) and \(\left[\mathrm{Hg}^{2+}\right]=0.025 \mathrm{M} .\) In which direction will the reaction proceed to achieve equilibrium?

Short answer

a) \(\Delta G^{\circ} = 28.75 \backslash text { kJ/mol }\), b) Reverse direction, c) \(\Delta G = 2.42 \backslash text { kJ/mol }\), reverse direction

Step-by-step solution

Calculate \(\backslash Delta G^{\circ}\) using the equilibrium constant

The relationship between the Gibbs free energy change at standard conditions \((\Delta G^{\circ})\) and the equilibrium constant \((K_{c})\) is given by the formula: \[\Delta G^{\circ} = -RT \ln K_{c}\] Here, \(R\) is the universal gas constant (8.314 J/(mol·K)), \(T\) is the temperature in Kelvin, and \(K_{c}\) is the equilibrium constant. Plug the values into the equation: \[\Delta G^{\circ} = -(8.314 \text{ J/(mol·K)}) (298 \text{ K}) \ln (9.1 \times 10^{-6})\]

Solve for \(\backslash Delta G^{\backslash circ}\)

First, calculate the natural logarithm: \[\backslash ln(9.1 \times 10^{-6}) = -11.608\] Then, multiply the constants: \[\Delta G^{\circ} = - (8.314 \text{ J/(mol·K)}) (298 \text{ K}) (-11.608) = 28747.9 \text{ J/mol} \approx 28.75 \text{ kJ/mol}\]

Determine the direction of the reaction at standard-state

If \(\Delta G^{\circ} \) is positive, it means the reaction is not spontaneous in the forward direction at standard-state conditions, and the reaction proceeds in the reverse direction.

Calculate \(\backslash Delta G\) using given concentrations

The Gibbs free energy change \((\Delta G)\) at non-standard conditions can be found using the equation: \[\Delta G = \Delta G^{\circ} + RT \ln Q\] where \(Q\) is the reaction quotient: \[Q = \frac{[\text{Fe}^{2+}]^{2} [\text{Hg}^{2+}]^{2}}{[\text{Fe}^{3+}]^{2} [\text{Hg}_{2}^{2+}]} = \frac{(0.010)^{2} (0.025)^{2}}{(0.20)^{2} (0.010)}\]

Solve for \(\backslashDelta G\)

First, calculate the reaction quotient \(Q\): \[Q = \frac{(0.010)^{2} (0.025)^{2}}{(0.20)^{2} (0.010)} = 1.5625 \times 10^{-5}\] Then, use the Gibbs free energy equation: \[\Delta G = 28747.9 \backslash text { J/mol } + (8.314 \backslash text { J/(mol·K)}) (298 \backslash text { K }) \backslash ln (1.5625 \times 10^{-5})\] Evaluate the natural logarithm: \[\backslash ln (1.5625 \times 10^{-5}) = -11.062\] Finally, calculate: \[\Delta G = 28747.9 + (8.314)(298)(-11.062) = 28747.9 - 27323.3 = 2424.6\]

Determine the direction of the reaction when specific concentrations are given

If \(\backslashDelta G\) is positive, the reaction proceeds in the reverse direction to achieve equilibrium.

Key concepts

Equilibrium Constant

The equilibrium constant (denoted as \(K_{c}\)) is crucial in understanding the extent to which a reaction proceeds to reach equilibrium. It is a value that expresses the ratio of concentrations of products to reactants at equilibrium for a given reaction. In the exercise example, the equilibrium constant for the provided reaction at 298 K is given as \(K_c = 9.1 \times 10^{-6}\). This small value of \(K_{c}\) suggests that the equilibrium mixture contains mostly reactants and very few products.

Reaction Quotient

The reaction quotient (denoted as \(Q\)) is similar to the equilibrium constant but can be calculated at any point in time, not just at equilibrium. For a given reaction, \(Q\) is defined using the same expression as \(K_{c}\), but with the current concentrations:

\[ Q = \frac{[\text{Products}]^{coefficients}}{[\text{Reactants}]^{coefficients}} \]

In the exercise, given the concentrations \(\text{[Fe}^{3+}] = 0.20 \text{ M}\), \(\text{[Hg}_{2}^{2+}] = 0.010 \text{ M}\), \(\text{[Fe}^{2+}] = 0.010 \text{ M}\), and \(\text{[Hg}^{2+}] = 0.025 \text{ M}\), we compute

• \(Q\) = \(\frac{(0.010)^2 (0.025)^2}{(0.20)^2 (0.010)} = 1.5625 \times 10^{-5}\)

After calculating \(Q\), the relationship between \(Q\) and \(K\) helps predict the direction in which the reaction will proceed to achieve equilibrium. If \(Q < K\), the reaction will move forward (towards products), whereas if \(Q > K\), it will move in the reverse direction (towards reactants).

Standard-State Conditions

Standard-state conditions refer to a set of specific conditions used as a reference to measure and compare the properties of substances. Under standard-state conditions for reactions involving solutions, concentrations of all reactants and products are typically 1 M (molar) at a pressure of 1 bar and a temperature of 298 K (25°C).

When calculating Gibbs free energy \(\backslashDelta G^{\backslashcirc}\) in the exercise, these standard-state conditions are assumed. If actual concentrations deviate from these standard conditions, we need to account for this difference using the reaction quotient \(Q\) when computing \(\Delta G\).

Natural Logarithm

The natural logarithm (denoted \(\ln\)) is a mathematical function that is important in various calculations in chemistry. It uses the base \(e\) (approximately 2.718) and is commonly used to describe rates of exponential growth or decay processes.

In the exercise, we use the natural logarithm to link the equilibrium constant \(K_{c}\) and Gibbs free energy change at standard conditions \(\backslashDelta G^{\circ}\) with the equation:

\[\Delta G^{\circ} = -RT \backslashln (K_{c}) \]

This equation explicitly shows how the spontaneity of a reaction is affected by the equilibrium constant and temperature, where R is the universal gas constant (8.314 J/(mol⋅K)). For our specific case with \(K_{c} = 9.1 \times 10^{-6}\), we calculate:

\[\ln(9.1 \times 10^{-6}) \approx -11.608 \]

Therefore, the natural logarithm is a fundamental part of the calculations needed to determine Gibbs free energy changes and understand reaction dynamics.