Problem 93 - Chemistry: An Atoms-Focused Approach

Question

Starting with Equation \(14.19\left(\Delta G=\Delta G^{*}+R T \ln Q\right)\) and \(\Delta G_{\text {cll }}^{n}=-n F E_{\text {call }}^{\circ}\) derive an cquation relating \(E_{\text {cell and }}\) the equilibrium constant ( \(K\) ) of an electrochemical cell reaction. Hint: Recall that \(Q=K\) and \(\Delta G=0\) in a reaction mixture at chemical equilibrium.

Step-by-Step Solution

Answer

Answer: The equation relating the cell potential (\(E_{\text{cell}}\)) and the equilibrium constant (\(K\)) of an electrochemical cell reaction is \(E_{\text{cell}}^{\circ} = \frac{RT}{nF}\ln K\).

Step 1: Write down the given equations

We have the following equations given to us: 1. \(\Delta G = \Delta G^{*} + RT\ln Q\) 2. \(\Delta G_{\text{cll}}^{n} = -nFE_{\text{cell}}^{\circ}\) And at chemical equilibrium, \(Q = K\) and \(\Delta G = 0\).

Step 2: Analyze the equations at chemical equilibrium

At chemical equilibrium, \(\Delta G = 0\). We can now substitute this into equation (1): 0 = \(\Delta G^{*} + RT\ln K\)

Step 3: Solve for \(\Delta G^{*}\)

Re-arrange equation derived in step 2 to solve for \(\Delta G^{*}\): \(\Delta G^{*} = -RT\ln K\)

Step 4: Substitute \(\Delta G^{*}\) in equation (2)

Now we can substitute the expression for \(\Delta G^{*}\) we found in step 3 into equation (2): 0 = \(-nFE_{\text{cell}}^{\circ} - RT\ln K\)

Step 5: Re-arrange the equation to solve for \(E_{\text{cell}}^{\circ}\)

Re-arrange the equation derived in step 4 to solve for \(E_{\text{cell}}^{\circ}\): \(E_{\text{cell}}^{\circ} = \frac{RT}{nF}\ln K\)

Step 6: Final Equation

We have derived the equation relating \(E_{\text{cell}}\) and the equilibrium constant \(K\) of an electrochemical cell reaction: \(E_{\text{cell}}^{\circ} = \frac{RT}{nF}\ln K\)

Key Concepts

Gibbs Free Energy

Gibbs Free Energy, denoted by the symbol \( \Delta G \), is a measure of the maximum reversible work that a thermodynamic system can perform at constant temperature and pressure. It indicates the spontaneity of a chemical reaction. If \( \Delta G < 0 \), the reaction is spontaneous, meaning it can occur without any input of energy. If \( \Delta G > 0 \), the reaction is non-spontaneous and external energy is needed for it to proceed.

Key Points to Note:

• \( \Delta G = \Delta H - T\Delta S \), where \( \Delta H \) is the change in enthalpy, \( T \) is the temperature, and \( \Delta S \) is the change in entropy.
• At chemical equilibrium, \( \Delta G = 0 \). This means that the system is in a state where both forward and reverse reactions occur at equal rates, leading to no net change.

In the context of electrochemical cells, Gibbs Free Energy is related to the cell's voltage through the equation \( \Delta G = -nFE_{\text{cell}} \), where \( n \) is the number of moles of electrons transferred, \( F \) is Faraday's constant, and \( E_{\text{cell}} \) is the cell potential.

Understanding Gibbs Free Energy helps in predicting the feasibility and spontaneity of electrochemical reactions and establishing a relation with other thermodynamic parameters.

Equilibrium Constant

The Equilibrium Constant, denoted as \( K \), is a key concept in understanding chemical reactions at equilibrium. For a given chemical reaction, it represents the ratio of the concentrations of the products to the reactants, each raised to the power of their stoichiometric coefficients.

Important Aspects of \( K \):

• \( K \) is constant at a given temperature, meaning it depends only on temperature and not on initial concentrations.
• The magnitude of \( K \) gives insight into the extent of the reaction: \( K >> 1 \) implies products are favored, while \( K << 1 \) implies reactants are favored.

In electrochemical cells, \( K \) is closely related to the cell potential at equilibrium. When the reaction in a cell is at equilibrium, no net electron flow occurs, and thus the cell's Gibbs Free Energy change is zero (\( \Delta G = 0 \)).

This is crucial for deriving relationships between the cell potential and the equilibrium constant, as used in deriving equations like \( E_{\text{cell}}^{\circ} = \frac{RT}{nF} \ln K \). Here, \( R \) is the universal gas constant, \( T \) is the temperature, \( n \) is the number of moles of electrons transferred, and \( F \) is Faraday's constant.

Nernst Equation

The Nernst Equation provides a mathematical way to determine the cell potential of an electrochemical cell under non-standard conditions. It is a powerful tool in electrochemistry, allowing us to calculate the potential of a cell at any given concentration of reactants and products.

The standard form of the Nernst Equation is:

• \( E = E^{\circ} - \frac{RT}{nF} \ln Q \)

Where:

• \( E \) is the cell potential under non-standard conditions,
• \( E^{\circ} \) is the standard cell potential,
• \( R \) is the universal gas constant,
• \( T \) is the temperature in Kelvin,
• \( n \) is the number of moles of electrons exchanged,
• \( F \) is Faraday's constant, and
• \( Q \) is the reaction quotient, which is the ratio of the concentrations of the products to the reactants at any point in time.

At equilibrium, \( Q = K \), and the cell potential, \( E \), will be zero because no net reaction occurs. Therefore, understanding the Nernst Equation helps us connect the measurable electric potential with the concentrations of reacting species.

The Nernst Equation not only bridges the gap between chemical thermodynamics and electrochemistry but also emphasizes the relationship between the equilibrium constant and cell potential by demonstrating how changes in reactant concentrations affect the overall electrochemical reaction.