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 the conditions under which the cell reaction is spontaneous, we can follow these steps:

### Step 1: Identify the Cell Components
We have a cell with gas electrodes at pressures \( P_1 \) and \( P_2 \). The components are:
- At \( P_1 \): \( \text{Pt} | \text{Cl}_2 | \text{HCl (1M)} \)
- At \( P_2 \): \( \text{Pt} | \text{Cl}_2 \)

### Step 2: Determine the Half-Reactions
- **At the Cathode (Reduction)**: Chlorine gas (\( \text{Cl}_2 \)) is reduced to chloride ions (\( \text{Cl}^- \)):
\[ \text{Cl}_2 + 2e^- \rightarrow 2\text{Cl}^- \]

- **At the Anode (Oxidation)**: Chloride ions (\( \text{Cl}^- \)) are oxidized to chlorine gas (\( \text{Cl}_2 \)):
\[ 2\text{Cl}^- \rightarrow \text{Cl}_2 + 2e^- \]

### Step 3: Write the Nernst Equation
The Nernst equation for the cell can be expressed as:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log Q \]
Where:
- \( n = 2 \) (number of electrons transferred)
- \( Q \) is the reaction quotient.

For our cell, since it is a concentration cell, \( E^\circ_{\text{cell}} = 0 \). Thus, the equation simplifies to:
\[ E_{\text{cell}} = - \frac{0.0591}{2} \log \left( \frac{P_1}{P_2} \right) \]

### Step 4: Rearranging the Equation
We can rearrange the equation to express it in terms of \( P_2 \) and \( P_1 \):
\[ E_{\text{cell}} = \frac{0.0591}{2} \log \left( \frac{P_2}{P_1} \right) \]

### Step 5: Condition for Spontaneity
For the cell reaction to be spontaneous, \( E_{\text{cell}} \) must be positive:
\[ \frac{0.0591}{2} \log \left( \frac{P_2}{P_1} \right) > 0 \]
This implies:
\[ \log \left( \frac{P_2}{P_1} \right) > 0 \]
Which means:
\[ \frac{P_2}{P_1} > 1 \quad \Rightarrow \quad P_2 > P_1 \]

### Conclusion
The cell reaction is spontaneous if \( P_2 > P_1 \).

### Final Answer
The correct condition for spontaneity is when \( P_2 \) is greater than \( P_1 \).

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