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Chapter 20: Problem 52

If the equilibrium constant for a one-electron redox reaction at \(298 \mathrm{~K}\) is \(8.7 \times 10^{4}\), calculate the corresponding \(\Delta G^{\circ}\) and \(E_{\text {cell }}^{0}\)

Short Answer

Expert verified
The standard Gibbs free energy change (螖G掳) for the redox reaction is -44.39 kJ/mol, and the standard cell potential (E掳cell) is 0.44 V.

Step by step solution

01

Find 螖G掳

: First, we'll use the formula 饾毇G掳 = -RT lnK to find the Gibbs free energy change (螖G掳): 螖G掳 = -(8.314 J/mol路K) 脳 (298 K) 脳 ln(8.7 脳 10^4) 螖G掳 鈮� -44387.6 J/mol Since the result is typically expressed in kJ/mol, we'll convert it: 螖G掳 鈮� -44.39 kJ/mol
02

Calculate E掳cell

: Now, we'll use the formula E掳cell = -螖G掳 / nF to find the standard cell potential: E掳cell = -(-44.39 kJ/mol) / (1 脳 96,485 C/mol) E掳cell 鈮� 0.44 V So, the standard Gibbs free energy change (螖G掳) for the redox reaction is -44.39 kJ/mol, and the standard cell potential (E掳cell) is 0.44 V.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gibbs Free Energy
Gibbs free energy, often denoted as \( \Delta G \), is a thermodynamic quantity that helps predict the spontaneity of a chemical reaction.

When calculating \( \Delta G^{\circ} \), the formula used is:
  • \( \Delta G^{\circ} = -RT \ln K \)
  • \( R \) is the universal gas constant (8.314 J/mol路K)
  • \( T \) is the temperature in Kelvin
  • \( K \) is the equilibrium constant of the reaction
This formula shows the relationship between Gibbs free energy and the equilibrium constant. A negative \( \Delta G^{\circ} \) indicates a spontaneous reaction under standard conditions, meaning the reaction will proceed without external energy input.

In our case, substituting the values gives \( \Delta G^{\circ} \approx -44.39 \, \text{kJ/mol} \), indicating the reaction is spontaneous.
Redox Reaction
Redox, short for reduction-oxidation, reactions are chemical reactions where electrons are transferred between substances. These reactions are crucial in many natural processes and technological applications.

Each redox reaction consists of two half-reactions:
  • Oxidation: Loss of electrons
  • Reduction: Gain of electrons
These processes must occur simultaneously. The electrons lost in the oxidation process are gained by another species in the reduction process.

In the provided exercise, the reaction involves a one-electron redox process. The equilibrium constant helps us understand how far the reaction proceeds, while the associated Gibbs free energy tells us about spontaneity.
Standard Cell Potential
Standard cell potential \( (E_{\text{cell}}^{0}) \) is a measure of the voltage or electric potential difference of a cell under standard conditions. It reflects the cell's ability to do electrical work.

To find \( E_{\text{cell}}^{0} \), we use the formula:
  • \( E_{\text{cell}}^{0} = -\frac{\Delta G^{\circ}}{nF} \)
  • \( n \) is the number of moles of electrons transferred in the reaction
  • \( F \) is Faraday鈥檚 constant (96,485 C/mol)
In this scenario, since \( \Delta G^{\circ} \approx -44.39 \, \text{kJ/mol} \) and \( n = 1 \), we calculate \( E_{\text{cell}}^{0} \approx 0.44 \, \text{V} \).

A positive standard cell potential indicates that the cell reaction is spontaneous and can produce electrical energy. This aligns with our Gibbs free energy calculation, confirming the reaction's spontaneity.

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Most popular questions from this chapter

A student designs an ammeter (a device that measures electrical current) that is based on the electrolysis of water into hydrogen and oxygen gases. When electrical current of unknown magnitude is run through the device for \(2.00 \mathrm{~min}, 12.3 \mathrm{~mL}\) of water-saturated \(\mathrm{H}_{2}(\mathrm{~g})\) is collected. The temperature of the system is \(25.5^{\circ} \mathrm{C}\), and the atmospheric pressure is 768 torr. What is the magnitude of the current in amperes?

If the equilibrium constant for a two-electron redox reaction at \(298 \mathrm{~K}\) is \(1.5 \times 10^{-4}\), calculate the corresponding \(\Delta G^{\circ}\) and \(E_{\text {cell }}^{\circ}\).

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 \(25^{\circ} \mathrm{C}\).

(a) What is a standard reduction potential? (b) What is the standard reduction potential of a standard hydrogen electrode?

(a) Based on standard reduction potentials, would you expect copper metal to oxidize under standard conditions in the presence of oxygen and hydrogen ions? (b) When the Statue of Liberty was refurbished, Teflon spacers were placed between the iron skeleton and the copper metal on the surface of the statue. What role do these spacers play?
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