`Pt(Cl_(2))(p_(1))|HCl(0.1M)|(Cl_(2))(p_(2)),Pt` cell reaction will be endergonic if

Options

A. `p_(1)=p_(2)`
B. `p_(1)gtp_(2)`
C. `p_(2)gtp_(1)`
D. `p_(1)=p_(2)=1atm`

Explanation

To determine when the cell reaction for the given electrochemical cell will be endergonic, we need to analyze the conditions under which the Gibbs free energy change (ΔG) is greater than zero. This can be related to the cell potential (E_cell) being less than zero.

### Step-by-Step Solution:

1. **Identify the Cell Reaction**:
The cell is represented as:
\[ \text{Pt(Cl}_2)(p_1) | \text{HCl}(0.1M) | (\text{Cl}_2)(p_2), \text{Pt} \]
Here, Cl2 at pressure \(p_1\) is oxidized at the anode, and Cl- ions from HCl are reduced at the cathode to form Cl2 at pressure \(p_2\).

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

3. **Net Reaction**:
Combining the half-reactions gives:
\[ \text{Cl}_2(p_1) \rightarrow \text{Cl}_2(p_2) \]

4. **Nernst Equation**:
The Nernst equation relates the cell potential to the concentrations (or pressures) of the reactants and products:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log \left( \frac{p_2}{p_1} \right) \]
where \(n\) is the number of moles of electrons transferred (in this case, \(n = 2\)).

5. **Condition for Endergonic Reaction**:
For the reaction to be endergonic, we need:
\[ \Delta G > 0 \implies E_{\text{cell}} < 0 \]
Substituting into the Nernst equation:
\[ E^\circ_{\text{cell}} - \frac{0.0591}{2} \log \left( \frac{p_2}{p_1} \right) < 0 \]

6. **Analyzing the Logarithmic Term**:
Rearranging gives:
\[ - \frac{0.0591}{2} \log \left( \frac{p_2}{p_1} \right) < -E^\circ_{\text{cell}} \]
This implies:
\[ \log \left( \frac{p_2}{p_1} \right) > \frac{2E^\circ_{\text{cell}}}{0.0591} \]
For \(E_{\text{cell}} < 0\), we need \(p_2 < p_1\).

7. **Conclusion**:
Therefore, the reaction will be endergonic if:
\[ p_2 > p_1 \]
This corresponds to option 3 from the given choices.

### Final Answer:
The cell reaction will be endergonic if \(p_2 > p_1\).

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