Which of the following changes will cause the free energy of a cell reaction to decrease ?
`Zn|ZnSO_(4)(aq)(x_(1)M)||HCl(aq)(x_(2)M)|H_(2)(g),Pt`.

Options

A. Increase in the volume of `HCl` solution from `100mL` to` 200mL`
B. Increase in the pressure of hydrogen from `1 atm ` to `2 atm`
C. Increase in molarity `x_(2)` from `0.1 ` to `1M`
D. Increase in molarity `x_(1)` from `1M` to `0.1M`.

Explanation

To determine which changes will cause the free energy of the cell reaction to decrease, we need to analyze the electrochemical cell provided in the question:

**Cell Representation:**
`Zn | ZnSO4(aq)(x1 M) || HCl(aq)(x2 M) | H2(g), Pt`

### Step-by-Step Solution:

1. **Identify the Half-Reactions:**
- At the anode (oxidation):
\[ \text{Zn} \rightarrow \text{Zn}^{2+} + 2e^- \]
- At the cathode (reduction):
\[ 2\text{H}^+ + 2e^- \rightarrow \text{H}_2(g) \]

2. **Determine the Standard Cell Potential (E°):**
- The standard reduction potential for Zn is -0.76 V and for H⁺/H₂ is 0.00 V.
- Therefore, the standard cell potential (E°) can be calculated as:
\[ E°_{\text{cell}} = E°_{\text{cathode}} - E°_{\text{anode}} = 0.00 - (-0.76) = 0.76 \, \text{V} \]

3. **Apply the Nernst Equation:**
- The Nernst equation relates the cell potential to the concentrations of the reactants and products:
\[ E_{\text{cell}} = E°_{\text{cell}} - \frac{0.059}{n} \log \left( \frac{[\text{Zn}^{2+}]}{[\text{H}^+]^2 \cdot P_{\text{H}_2}} \right) \]
- Here, \( n = 2 \) (number of electrons transferred).

4. **Calculate Gibbs Free Energy (ΔG):**
- Gibbs free energy change is given by:
\[ \Delta G = -nF E_{\text{cell}} \]
- To decrease ΔG, we need to increase E_cell (make it more positive).

5. **Analyze the Changes:**
- **Increasing [H⁺]:** If we increase the concentration of H⁺ (x2), it will decrease the logarithmic term, thus increasing E_cell and decreasing ΔG.
- **Decreasing [Zn²⁺]:** If we decrease the concentration of Zn²⁺ (x1), it will also decrease the logarithmic term, increasing E_cell and decreasing ΔG.

6. **Conclusion:**
- Therefore, the changes that will cause the free energy of the cell reaction to decrease are:
- Increasing the concentration of H⁺ (option C).
- Decreasing the concentration of Zn²⁺ (option D).

### Final Answer:
The correct options that will cause the free energy of the cell reaction to decrease are **C and D**.

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