For the following cell with gas electrodes at `p_(1) and p_(2)` as shown:
`underset(at P1)(PtCl_(2))underset(1M)(abs(HCl)) underset(at P2)(Pt (Cl_(2))` cell reaction is spontaneous if :

Options

A. `P_(1)=P_(2)`
B. `P_(1) gt P_(2)`
C. `P_(1) lt P_(2)`
D. `P_(2) = 0`

Answer

To determine when the cell reaction is spontaneous for the given electrochemical cell involving PtCl₂ and HCl, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Electrochemical Cell Components**:
- The cell consists of two gas electrodes at pressures P1 and P2.
- The left side (P1) contains PtCl₂ in 1M HCl and the right side (P2) contains PtCl₂.

2. **Determine the Cell Reaction**:
- The oxidation reaction occurs at the anode (left side):
\[ 2Cl^- \rightarrow Cl_2 + 2e^- \]
- The reduction reaction occurs at the cathode (right side):
\[ Cl_2 + 2e^- \rightarrow 2Cl^- \]

3. **Use the Nernst Equation**:
- The Nernst equation relates the cell potential (E) to the standard cell potential (E°) and the concentrations (or pressures) of the reactants and products:
\[ E = E° - \frac{0.0591}{n} \log \left( \frac{[Products]}{[Reactants]} \right) \]
- For this concentration cell, E° = 0 because the same species are involved on both sides.

4. **Substituting Values into the Nernst Equation**:
- Since E° = 0, the equation simplifies to:
\[ E = - \frac{0.0591}{2} \log \left( \frac{P_{Cl_2} \text{ (right)}}{P_{Cl_2} \text{ (left)}} \right) \]
- Substitute \( P_{Cl_2} \) with P2 (right side) and P1 (left side):
\[ E = - \frac{0.0591}{2} \log \left( \frac{P2}{P1} \right) \]

5. **Determine Conditions for Spontaneity**:
- For the cell reaction to be spontaneous, the cell potential (E) must be positive:
\[ E > 0 \implies - \frac{0.0591}{2} \log \left( \frac{P2}{P1} \right) > 0 \]
- This implies:
\[ \log \left( \frac{P2}{P1} \right) < 0 \implies \frac{P2}{P1} < 1 \implies P2 < P1 \]

6. **Conclusion**:
- Therefore, the cell reaction will be spontaneous when \( P2 > P1 \).

### Final Answer:
The cell reaction is spontaneous if \( P2 > P1 \).