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Chapter 20: Problem 53

If the equilibrium constant for a one-electron redox reaction at $298 \mathrm{~K}\( is \)2.2 \times 10^{-5},\( calculate the corresponding \)\Delta G^{\circ}\( and \)E^{\circ}$.

Short Answer

Expert verified
The standard electrode potential (E°) for the one-electron redox reaction at 298 K is -0.028 V, and the corresponding change in Gibbs free energy (ΔG°) is +2700 J/mol.

Step by step solution

01

Determine the relationship between K and Q

Since we are given K, which is the equilibrium constant, we need to express this in terms of the reaction quotient (Q). Whenever a reaction reaches equilibrium, the following relationship holds: \[K = Q\]
02

Determine the number of electrons

Since the reaction is a one-electron redox reaction, the number of electrons involved in the reaction is n = 1.
03

Use the Nernst Equation to find E°

At equilibrium, the Nernst Equation becomes: \[\Delta E = E - E^\circ = 0\] Now, plug in the values for n, R, T, and F, and rearrange the equation to solve for E°: \[E^\circ = \frac{RT}{nF}\ln{K}\] Where: - R = the gas constant = 8.314 J/mol·K - T = the temperature = 298 K - n = the number of electrons transferred = 1 - F = the Faraday constant = 96,485 C/mol - K = the equilibrium constant = 2.2 × 10⁻⁵ Now, substitute the given values into the equation: \[E^\circ = \frac{(8.314 \;\text{J/mol·K})(298 \;\text{K})}{(1)(96,485\; \text{C/mol})}\ln{(2.2 × 10^{-5})}\] Calculate E°: \[E^\circ = -0.028 \;\text{V}\]
04

Calculate ΔG° using the Gibbs free energy equation

Now that we have the value of E°, we can use the Gibbs free energy equation to find ΔG°: \[\Delta G^\circ = -nFE^\circ\] Plug in the values for n, F, and E°: \[\Delta G^\circ = -(1)(96,485 \;\text{C/mol})(-0.028 \;\text{V})\] Calculate ΔG°: \[\Delta G^\circ = +2700 \;\text{J/mol}\]
05

State the final results

The standard electrode potential for the one-electron redox reaction at 298 K is E° = -0.028 V, and the corresponding change in Gibbs free energy is ΔG° = +2700 J/mol.

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Most popular questions from this chapter

If the equilibrium constant for a two-electron redox reaction at $298 \mathrm{~K}\( is \)2.2 \times 10^{5},\( calculate the corresponding \)\Delta G^{\circ}\( and \)E^{\circ}$.

A voltaic cell is constructed that is based on the following reaction: $$ \mathrm{Sn}^{2+}(a q)+\mathrm{Pb}(s) \longrightarrow \mathrm{Sn}(s)+\mathrm{Pb}^{2+}(a q) $$ (a) If the concentration of \(\mathrm{Sn}^{2+}\) in the cathode half-cell is \(1.00 M\) and the cell generates an emf of \(+0.22 \mathrm{~V},\) what is the concentration of \(\mathrm{Pb}^{2+}\) in the anode half-cell? \((\mathbf{b})\) If the anode half-cell contains \(\left[\mathrm{SO}_{4}^{2-}\right]=1.00 M\) in equilibrium with \(\mathrm{PbSO}_{4}(s),\) what is the \(K_{s p}\) of \(\mathrm{PbSO}_{4} ?\)

(a) In the Nernst equation, what is the numerical value of the reaction quotient, \(Q,\) under standard conditions? (b) Can the Nernst equation be used at temperatures other than room temperature?

(a) What conditions must be met for a reduction potential to be a standard reduction potential? (b) What is the standard reduction potential of a standard hydrogen electrode? (c) Why is it impossible to measure the standard reduction potential of a single half-reaction?

Elemental calcium is produced by the electrolysis of molten \(\mathrm{CaCl}_{2}\). (a) What mass of calcium can be produced by this process if a current of \(7.5 \times 10^{3} \mathrm{~A}\) is applied for $48 \mathrm{~h}\( ? Assume that the electrolytic cell is \)68 \%$ efficient. (b) What is the minimum voltage needed to cause the electrolysis?
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