Cell Potential, Gibbs Free Energy, and Equilibrium Constant (Electrochemistry)

Cell Potential, Gibbs Free Energy, and Equilibrium Constant

Relationship Between Ecell, ΔG, and K

In electrochemistry, the spontaneity of a reaction is determined by the cell potential (Ecell), Gibbs free energy change (ΔG), and the equilibrium constant (K). These three quantities are interrelated and can be used to predict the direction and extent of chemical reactions.

• Spontaneous reactions have Ecell > 0, ΔG < 0, and K > 1.

• The following equations relate these quantities:

Key Equations:

• Gibbs Free Energy and Cell Potential: $\Delta G = -n F E_{cell}$ where n = number of moles of electrons transferred, F = Faraday's constant (96,485 C/mol).

• Gibbs Free Energy and Equilibrium Constant: $\Delta G^\circ = -RT \ln K$ where R = gas constant (8.314 J/mol·K), T = temperature in Kelvin, K = equilibrium constant.

• Cell Potential and Equilibrium Constant: $E_{cell}^\circ = \frac{RT}{nF} \ln K$ At 25°C (298 K): $E_{cell}^\circ = \frac{0.0257}{n} \ln K$ or $E_{cell}^\circ = \frac{0.0592}{n} \log K$

Triangular Relationship: The diagram in the notes shows how these three quantities are connected, allowing conversion from one to another using the above equations.

Example: Calculating Standard Cell Potential

To calculate the standard cell potential for a reaction where 10 moles of electrons are transferred:

• Given half-reactions and their standard reduction potentials:

• Use the formula: $E_{cell}^\circ = E_{cathode}^\circ - E_{anode}^\circ$

• Example calculation yields $+3.59$ V.

Application: This calculation is essential for determining the voltage a galvanic cell can produce under standard conditions.

Practice: Calculating Cell Potential from Standard Reduction Potentials

Given standard reduction potentials, determine the cell potential for a reaction:

• Write the half-reactions and identify the cathode and anode.

• Apply the formula for standard cell potential.

• Example: For Hg2Cl2 and Hg, $E_{cell}^\circ = +0.271$ V.

Practice: Calculating Cell Potential Under Non-Standard Conditions

For reactions not at standard conditions, use the Nernst equation:

• $E_{cell} = E_{cell}^\circ - \frac{0.0592}{n} \log Q$

• Where Q is the reaction quotient, calculated from the concentrations of reactants and products.

• Example: For a 4-electron transfer reaction with given concentrations, calculate $E_{cell}$.

Practice: Calculating Gibbs Free Energy from Cell Potential

Given a reaction and its cell potential, calculate the Gibbs Free Energy change:

• Use $\Delta G = -n F E_{cell}$

• Insert the number of electrons transferred and the cell potential.

• Example: For a reaction at 30°C, calculate $\Delta G$ using the provided cell potential.

Summary Table: Relationships Among Ecell, ΔG, and K

Quantity	Equation	Physical Meaning
Gibbs Free Energy (ΔG)	$\Delta G = -n F E_{cell}$	Energy available to do work; negative for spontaneous reactions
Equilibrium Constant (K)	$\Delta G^\circ = -RT \ln K$	Extent to which a reaction proceeds to completion
Cell Potential (Ecell)	$E_{cell}^\circ = \frac{RT}{nF} \ln K$	Voltage produced by a cell; positive for spontaneous reactions

Additional info: The notes also include practice problems that reinforce the calculation of cell potential, Gibbs free energy, and equilibrium constant, which are central concepts in electrochemistry (Ch.20 - Electrochemistry).