`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)=1 atm`

Explanation

To determine the conditions under which the cell reaction \( Pt(Cl_2)(p_1)|HCl(0.1M)|(Cl_2)(p_2),Pt \) will be endergonic, we can follow these steps:

### Step 1: Understand the Concept of Endergonic Reactions
An endergonic reaction is one that requires energy input to proceed, which is indicated by a positive change in Gibbs free energy (\( \Delta G > 0 \)). The relationship between Gibbs free energy and cell potential (\( E \)) is given by the equation:
\[ \Delta G = -nFE \]
where:
- \( n \) = number of moles of electrons transferred
- \( F \) = Faraday's constant (approximately 96485 C/mol)
- \( E \) = cell potential

### Step 2: Determine Conditions for Endergonic Reaction
For the reaction to be endergonic, we need \( \Delta G \) to be positive:
\[ \Delta G > 0 \implies -nFE > 0 \implies E < 0 \]
This means that the cell potential \( E \) must be negative.

### Step 3: Analyze the Cell Reaction
In the given cell, we have:
- At the anode: \( 2Cl^- \rightarrow Cl_2 + 2e^- \) (oxidation)
- At the cathode: \( Cl_2 + 2e^- \rightarrow 2Cl^- \) (reduction)

### Step 4: Write the Equilibrium Constant Expression
The equilibrium constant \( K \) for the reaction can be expressed as:
\[ K = \frac{P_{Cl_2}}{[Cl^-]^2} \]
Given that the concentration of \( Cl^- \) is \( 0.1 \, M \), we can express \( K \) in terms of the pressures:
\[ K = \frac{p_2}{(0.1)^2} \]

### Step 5: Relate \( K \) to the Cell Potential
Using the Nernst equation:
\[ E = E^\circ - \frac{0.0591}{n} \log K \]
Since \( E^\circ = 0 \) (as the same species is oxidized and reduced), we have:
\[ E = -\frac{0.0591}{2} \log K \]
For \( E \) to be negative, \( \log K \) must be positive, which implies:
\[ K > 1 \]

### Step 6: Conclude the Condition for Endergonic Reaction
Since \( K = \frac{p_2}{(0.1)^2} \) and for \( K > 1 \):
\[ p_2 > (0.1)^2 \implies p_2 > 0.01 \, atm \]
Thus, for the cell reaction to be endergonic, the pressure \( p_1 \) must be greater than \( p_2 \).

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

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