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Chapter 20: Problem 101

Derive an equation that directly relates the standard emf of a redox reaction to its equilibrium constant.

Short Answer

Expert verified
The equation that directly relates the standard emf of a redox reaction (E掳) to its equilibrium constant (K_eq) can be derived by combining the Nernst Equation with the relationships between standard Gibbs free energy change (螖G掳) and equilibrium constants. The final equation is: \(E掳 = -\frac{RT}{nF} * ln(K_{eq})\) where R is the gas constant, T is temperature in Kelvin, n is the number of electrons transferred in the redox reaction, and F is Faraday's constant.

Step by step solution

01

Recall the Nernst Equation

The Nernst Equation is used to calculate the emf of an electrochemical cell under non-standard conditions. For a redox reaction, it is given as: \(E = E掳 - \frac{RT}{nF} * ln(Q)\) where: - E is the emf of the reaction under non-standard conditions - E掳 is the standard emf - R is the gas constant - T is temperature (in Kelvin) - n is the number of electrons transferred in the redox reaction - F is Faraday's constant - Q is the reaction quotient
02

Recall the relationship between 螖G掳 and K_eq

The standard Gibbs free energy change (螖G掳) of a reaction is related to its equilibrium constant (K_eq) by the equation: \(螖G掳 = -RT * ln(K_{eq})\)
03

Recall the relationship between 螖G and E

The relationship between Gibbs free energy change (螖G) and emf (E) is given by: \(螖G = -nFE\) The relationship between standard Gibbs free energy change (螖G掳) and standard emf (E掳) is given by: \(螖G掳 = -nFE掳\)
04

Derive the equation relating E掳 and K_eq

Using the relationships from Step 2 and Step 3, we have: \(螖G掳 = -nFE掳 = -RT * ln(K_{eq})\) Now we can solve for E掳: \(E掳 = -\frac{RT}{nF} * ln(K_{eq})\) This equation directly relates the standard emf of a redox reaction to its equilibrium constant.

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Most popular questions from this chapter

The following quotation is taken from an article dealing with corrosion of electronic materials: "Sulfur dioxide, its acidic oxidation products, and moisture are well established as the principal causes of outdoor corrosion of many metals." Using \(\mathrm{Ni}\) as an example, explain why the factors cited affect the rate of corrosion. Write chemical equations to illustrate your points. (Note: \(\mathrm{NiO}(s)\) is soluble in acidic solution.)

(a) Which electrode of a voltaic cell, the cathode or the anode, corresponds to the higher potential energy for the electrons? (b) What are the units for electrical potential? How does this unit relate to energy expressed in joules? (c) What is special about a standard cell potential?

A voltaic cell utilizes the following reaction: $$ 4 \mathrm{Fe}^{2+}(a q)+\mathrm{O}_{2}(g)+4 \mathrm{H}^{+}(a q)-\cdots \mathrm{Fe}^{3+}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l) $$ (a) What is the emf of this cell under standard conditions? (b) What is the emf of this cell when \(\left[\mathrm{Fe}^{2+}\right]=1.3 \mathrm{M}\), \(\left[\mathrm{Fe}^{3+}\right]=0.010 \mathrm{M}, P_{\mathrm{O}_{2}}=0.50 \mathrm{~atm}\), and the \(\mathrm{pH}\) of the so- lution in the cathode compartment is \(3.50 ?\)

A \(1 M\) solution of \(\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}\) is placed in a beaker with a strip of Cu metal. A \(1 M\) solution of \(\mathrm{SnSO}_{4}\) is placed in a second beaker with a strip of Sn metal. A salt bridge connects the two beakers, and wires to a voltmeter link the two metal electrodes. (a) Which electrode serves as the anode, and which as the cathode? (b) Which electrode gains mass and which loses mass as the cell reaction proceeds? (c) Write the equation for the overall cell reaction. (d) What is the emf generated by the cell under standard conditions?

A voltaic cell is based on \(\mathrm{Ag}^{+}(a q) / \mathrm{Ag}(\mathrm{s})\) and \(\mathrm{Fe}^{3+}(a q) / \mathrm{Fe}^{2+}(a q)\) half-cells. (a) What is the standard emf of the cell? (b) Which reaction occurs at the cathode, and which at the anode of the cell? (c) Use \(S^{\circ}\) values in Appendix \(C\) and the relationship between cell potential and free-energy change to predict whether the standard cell potential increases or decreases when the temperature is raised above \(25^{\circ} \mathrm{C}\).
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