What is the cell entropy change `(` in `J K ^(-1))` of the following cell `:`
`Pt(s)|underset(p=1atm)(H_(2)(g))|CH_(2)underset(0.1M)(COOH,HCl)|underset(0.1M)(KCl(aq))|Hg_(2)Cl_(2)(s)|Hg`
The `EMF` of the cell is found to be `0.045 V` at `298K` and temperature coefficient if `3.4xxx10^(-4)V K^(-1)`
`(` Given `:K_(a(CH_(3)COOH))=10^(-5)M)`

Options

A. 60
B. 65.2
C. 69.2
D. 63.5

Explanation

To find the cell entropy change (ΔS) for the given electrochemical cell, we can use the formula:

\[ \Delta S = \frac{nF \cdot dE}{dT} \]

Where:
- \( n \) = number of moles of electrons transferred in the cell reaction
- \( F \) = Faraday's constant (\( 96500 \, \text{C/mol} \))
- \( dE \) = change in EMF (in volts)
- \( dT \) = change in temperature (in Kelvin)

### Step 1: Identify the number of electrons transferred (n)
From the cell reaction, we can determine the number of electrons involved. The reaction can be represented as:

\[ \text{H}_2 + \text{Hg}_2^{2+} \rightarrow 2 \text{H}^+ + 2 \text{Hg} \]

Here, 2 electrons are transferred in the reaction. Thus, \( n = 2 \).

### Step 2: Use the given values
From the question, we have:
- \( dE = 0.045 \, \text{V} \)
- \( dT = 1 \, \text{K} \) (since we are considering the change per Kelvin)
- \( F = 96500 \, \text{C/mol} \)
- The temperature coefficient \( \frac{dE}{dT} = 3.4 \times 10^{-4} \, \text{V/K} \)

### Step 3: Substitute the values into the formula
Now we can substitute the values into the formula for ΔS:

\[ \Delta S = \frac{nF \cdot dE}{dT} \]

Substituting the known values:

\[ \Delta S = \frac{2 \cdot 96500 \cdot 3.4 \times 10^{-4}}{1} \]

### Step 4: Calculate ΔS
Calculating the above expression:

\[ \Delta S = 2 \cdot 96500 \cdot 3.4 \times 10^{-4} \]
\[ \Delta S = 2 \cdot 96500 \cdot 0.00034 \]
\[ \Delta S = 2 \cdot 32.81 \]
\[ \Delta S = 65.62 \, \text{J/K} \]

### Step 5: Round to the nearest option
The calculated value of ΔS is approximately \( 65.62 \, \text{J/K} \). The closest answer from the options provided is:

**Option 2: 65.2 J/K**

### Final Answer
Thus, the cell entropy change (ΔS) is approximately \( 65.2 \, \text{J/K} \).

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