For the given cell reaction of an electrochemical cell, the change in free energy at a given temperature is a function of `Cu^(2+)(C_(1)aq)+Zn(s) rarrZn^(2+)(C_2aq)+Cu(s)`

Options

A. In `(C_1)`
B. In `(C_2)`
C. In `(C_1+C_2)`
D. In `(C_2//C_1)`

Answer

To determine the change in free energy (ΔG) for the given electrochemical cell reaction, we can follow these steps:

### Step 1: Understand the Reaction
The cell reaction provided is:
\[ \text{Cu}^{2+}(C_1) + \text{Zn}(s) \rightarrow \text{Zn}^{2+}(C_2) + \text{Cu}(s) \]

### Step 2: Identify the Relevant Equations
For an electrochemical cell, the relationship between the change in free energy (ΔG) and the cell potential (E_cell) is given by:
\[ \Delta G = -nFE_{\text{cell}} \]
where:
- \( n \) = number of moles of electrons transferred in the reaction
- \( F \) = Faraday's constant (approximately 96500 C/mol)
- \( E_{\text{cell}} \) = cell potential

### Step 3: Relate E_cell to Reaction Quotient (Q)
The cell potential can also be expressed in terms of the standard cell potential (E°) and the reaction quotient (Q) using the Nernst equation:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q \]
where:
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature in Kelvin
- \( Q \) = reaction quotient, which for this reaction is given by:
\[ Q = \frac{[\text{Cu}^{2+}]}{[\text{Zn}^{2+}]} = \frac{C_1}{C_2} \]

### Step 4: Substitute E_cell into ΔG Equation
Substituting the expression for \( E_{\text{cell}} \) into the ΔG equation, we get:
\[ \Delta G = -nF\left(E^\circ_{\text{cell}} - \frac{RT}{nF} \ln Q\right) \]

### Step 5: Simplify the Expression
This simplifies to:
\[ \Delta G = -nFE^\circ_{\text{cell}} + RT \ln Q \]

### Step 6: Identify the Function of ΔG
From the equation \( \Delta G = -nFE^\circ_{\text{cell}} + RT \ln Q \), we can see that ΔG is a function of:
- The standard cell potential \( E^\circ_{\text{cell}} \) (a constant for a given reaction)
- The temperature \( T \)
- The reaction quotient \( Q \) (which depends on the concentrations of the reactants and products)

### Conclusion
Thus, the change in free energy (ΔG) at a given temperature for the electrochemical cell reaction is a function of the reaction quotient \( Q \) and temperature \( T \).