Question

For the overall reaction $$ \mathrm{Cu}^{2+}\left(\mathrm{C}_{1} \mathrm{aq}\right)+\mathrm{Zn}(\mathrm{s}) \rightarrow \mathrm{Zn}^{2+}\left(C_{2} \mathrm{aq}\right)+\mathrm{Cu}(\mathrm{s}) $$ (a) \(\ln C_{1}\) (b) \(\ln \left(C_{2} / C_{1}\right)\) (c) \(\ln C_{2}\) (d) \(\ln \left(C_{1}+C_{2}\right)\) of an electrochemical cell, the change in the free energy \(\Delta G\) at a given temperature is a function of (a) \(\ln C_{1}\) (b) \(\ln \left(C_{2} / C_{1}\right)\) (c) \(\ln C_{2}\) (d) \(\ln \left(C_{1}+C_{2}\right)\)

Short answer

The change in the free energy ΔG at a given temperature of an electrochemical cell is a function of ln(C2/C1)

Step-by-step solution

Identify the Reaction Quotient (Q)

Start by identifying the reaction quotient for the given reaction. The reaction quotient Q in this context is \(Q = [Zn^{2+}] / [Cu^{2+}]\), where each species inside the square brackets represents its molar concentration (C).

Apply the Nernst Equation

Next, apply the Nernst Equation, which relates the free energy change, ΔG, to the reaction quotient, Q. Here, ΔG = -RTln(Q). In this equation, R is the ideal gas constant, T is the absolute temperature, and ln indicates the natural logarithm.

Substitute the Reaction Quotient

Substitute the expression for Q into the Nernst Equation. This substitution gives ΔG = -RTln([Zn^{2+}]/[Cu^{2+}]) = -RT(ln C2 - ln C1). From this, it's clear that the change in free energy ΔG is a function of ln C1 and ln C2, which means option (b), ln(C2/C1), correctly describes this relationship.

Key concepts

Electrochemical Cell

An electrochemical cell is a device capable of either generating electrical energy from chemical reactions or facilitating chemical reactions through the introduction of electrical energy. An electrochemical cell is composed of two half-cells that include electrodes and an electrolyte.

In the context of our copper-zinc cell:

• The anode is where oxidation occurs—in this case, zinc oxidizes to zinc ions ( ext{Zn} ightarrow ext{Zn}^{2+} +2 ext{e}^- ).
• The cathode is the site of reduction, where copper ions are reduced to copper ( ext{Cu}^{2+} +2 ext{e}^- ightarrow ext{Cu} ).

This movement of ions and electrons creates an electrical circuit, providing continuous energy until the reactants are depleted. Electrochemical cells are fundamental for devices like batteries and are pivotal in understanding energy conversion processes.

Free Energy Change

The free energy change ( ext{ΔG} ) in a reaction indicates the spontaneity of the process. A negative ext{ΔG} suggests a process is spontaneous under constant temperature and pressure conditions.

The Nernst Equation establishes the link between ext{ΔG} and the reaction quotient ( Q ):

• ext{ΔG}= -RT ext{ln}(Q) .

Here, R is the universal gas constant, T is the temperature in Kelvin, and ext{ln} indicates a natural logarithm. This equation helps predict how changes in concentration affect the direction and extent of the reaction.

• A small value of Q leading to a negative ext{ΔG} implies the reaction proceeds towards equilibrium, favorably converting reactants to products.

Reaction Quotient

The reaction quotient ( Q ) is a measure used to determine the progress of a reaction at any given point compared to its equilibrium state. Q is expressed using the concentrations of the reactants and products:

• For the reaction ext{Cu}^{2+} + ext{Zn} ightarrow ext{Zn}^{2+} + ext{Cu} , Q = [ ext{Zn}^{2+}]/[ ext{Cu}^{2+}] .

This quotient provides information on the direction a reaction must move to reach equilibrium:

• If Q
• If Q=K_{eq} , the system is at equilibrium.
• If Q>K_{eq} , the reaction will proceed left, forming more reactants.

Understanding Q helps in deducing whether a reaction will go towards the right, creating more products, or the left, reverting to the original reactants, impacting the electrochemical cell's ability to supply energy.