(a) How do the relative magnitudes of \(Q\) and \(K\) relate to the signs of \(\Delta G\) and \(E_{\text {cell }} ?\) Explain. (b) Can a cell do work when \(Q / K>1\) or \(Q / K<1 ?\) Explain. 21.53 A voltaic cell consists of \(\mathrm{A} / \mathrm{A}^{+}\) and \(\mathrm{B} / \mathrm{B}^{+}\) half-cells, where A and \(B\) are metals and the A electrode is negative. The initial \(\left[\mathrm{A}^{+}\right] /\left[\mathrm{B}^{+}\right]\) is such that \(E_{\text {cell }}>E_{\text {cell }}^{\circ}\) (a) How do \(\left[\mathrm{A}^{+}\right]\) and \(\left[\mathrm{B}^{+}\right]\) change as the cell operates? (b) How does \(E_{\text {cell }}\) change as the cell operates? (c) What is \(\left[\mathrm{A}^{+}\right] /\left[\mathrm{B}^{+}\right]\) when \(E_{\text {cell }}=E_{\text {cell }}^{\circ} ?\) Explain. (d) Is it possible for \(E_{\text {cell }}\) to be less than \(E_{\text {cell }}^{\circ} ?\) Explain.

Step by step solution

01

Understanding the Relationship between Q and K, and their Effects on \Delta G and \E_{cell}

To relate the relative magnitudes of the reaction quotient \(Q\) and the equilibrium constant \(K\) to the signs of \Delta G\ and \E_{cell}\, one must consider the Gibbs free energy equation \Delta G = \Delta G^0 + RT \ln Q\ and the Nernst equation \E_{cell} = E_{cell}^0 - \frac{RT}{nF} \ln Q\. When \(Q < K\) (reaction shifts to the right), \Delta G < 0\ indicating a spontaneous reaction, and \E_{cell} > 0\. When \(Q > K\) (reaction shifts to the left), \Delta G > 0\ indicating a non-spontaneous reaction, and \E_{cell} < 0\.

02

Determining if the Cell Can Do Work Based on Q/K Ratio

A voltaic cell does work when the cell potential \E_{cell} > 0\. If \(Q/K < 1\), the cell potential is positive, hence the cell can do work. Conversely, if \(Q/K > 1\), \E_{cell} < 0\ and the cell cannot do work.

03

Effect of Cell Operation on \A^{+} and \B^{+} Concentrations Over Time

Given that electrode A is negative, it means that \(A\) undergoes oxidation and \A^{+}\ concentration increases. Conversely, \B\ undergoes reduction and \B^{+}\ concentration decreases.

04

Change in \E_{cell} as the Cell Operates

As the cell operates, \E_{cell}\ will decrease because the reaction tends towards equilibrium where \(Q\) approaches \(K\).

05

Determining \A^{+} / \B^{+} Ratio When \E_{cell} = E_{cell}^0

When \E_{cell} = E_{cell}^0\, the reaction quotient \(Q\) becomes equal to the equilibrium constant \(K\). Hence, \frac{[\text{A}^+]}{[\text{B}^+]} = K\.

06

Possibility of \E_{cell} Being Less than \E_{cell}^0

Yes, it is possible for \E_{cell} < E_{cell}^0\ if \(Q > K\), which indicates that the reaction mixture has an excess of products, and the cell is closer to equilibrium.

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Key Concepts

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

Gibbs Free Energy

In electrochemical cells, Gibbs Free Energy (\( \text{Δ}G \)) is a measure of the maximum amount of work that can be obtained from a chemical reaction. The relationship between Gibbs Free Energy and cell potential is given by the equation:
\[ \text{Δ}G = \text{Δ}G^{\text{0}} + RT \text{ln} Q \]
Here,

• \text{Δ}G is the Gibbs Free Energy change
• \text{Δ}G^{\text{0}} is the standard Gibbs Free Energy change
• R is the gas constant
• T is the temperature in Kelvin
• Q is the reaction quotient

When a reaction is spontaneous, \text{Δ}G < 0, and this aligns with a positive cell potential (\text{E}_{\text{cell}} > 0). Conversely, when a reaction is non-spontaneous, \text{Δ}G > 0, meaning the cell potential is negative (\text{E}_{\text{cell}} < 0). Essentially, a negative Gibbs Free Energy indicates that the reaction can perform work on the surroundings, which is ideal for a voltaic cell.

Nernst Equation

The Nernst Equation is crucial in calculating the cell potential under non-standard conditions. The equation is represented as:
\[ \text{E}_{\text{cell}} = \text{E}_{\text{cell}}^{\text{0}} - \frac{RT}{nF} \text{ln} Q \]
where:

• \text{E}_{\text{cell}} is the cell potential under non-standard conditions
• \text{E}_{\text{cell}}^{\text{0}} is the standard cell potential
• R is the gas constant
• T is the temperature in Kelvin
• n is the number of moles of electrons exchanged
• F is the Faraday constant
• Q is the reaction quotient

The Nernst Equation shows how the cell potential (\text{E}_{\text{cell}}) depends on the concentrations of the reactants and products. When the reaction quotient Q is less than the equilibrium constant K (\text{Q < K}), the cell potential is positive, meaning the cell can perform electrical work. Conversely, when \text{Q > K}, the cell potential becomes negative, indicating the reaction is moving towards non-spontaneity.

Reaction Quotient

The reaction quotient (Q) is a measure of the relative amounts of products and reactants present during a reaction at any given point in time. It is defined similarly to the equilibrium constant (K) but is applicable to non-equilibrium conditions. The formula for Q is:
\[ Q = \frac{[\text{Products}]}{[\text{Reactants}]} \]
In the context of electrochemical cells, Q helps in determining how far the reaction has proceeded and in what direction it must shift to reach equilibrium. Consider the changes in concentration of ionic species in half-cells:

• When Q < K: The reaction shifts to the right, producing more products from reactants, and indicates spontaneity (\text{Δ}G < 0 and \text{E}_{\text{cell}} > 0).
• When Q > K: The reaction shifts to the left, converting products back into reactants, and the reaction is non-spontaneous (\text{Δ}G > 0 and \text{E}_{\text{cell}} < 0).

Understanding Q helps predict how the concentrations of reactants and products evolve over time and how the cell potential changes. It is a fundamental part of determining the efficiency and directionality of electrochemical reactions.