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. `65.6`
C. `69.2`
D. `63.5`

Explanation

To calculate the cell entropy change (\( \Delta S \)) for the given electrochemical cell, we can use the relationship between the entropy change, the cell potential (EMF), and the temperature coefficient. The steps are as follows:

### Step 1: Identify the given values
- EMF of the cell (\( E \)) = 0.045 V
- Temperature coefficient (\( \frac{dE}{dT} \)) = \( 3.4 \times 10^{-4} \, V/K \)
- Faraday's constant (\( F \)) = 96500 C/mol
- Number of electrons transferred (\( n \)) = 2 (since \( Hg_2Cl_2 \) accepts 2 electrons)

### Step 2: Use the formula for entropy change
The formula relating entropy change to EMF and temperature is given by:
\[ \Delta S = -nF \frac{dE}{dT} \]

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ \Delta S = -2 \times 96500 \, C/mol \times (3.4 \times 10^{-4} \, V/K) \]

### Step 4: Calculate the entropy change
Calculating the above expression:
\[ \Delta S = -2 \times 96500 \times 3.4 \times 10^{-4} \]
\[ \Delta S = -2 \times 96500 \times 0.00034 \]
\[ \Delta S = -65.62 \, J/K \]

### Step 5: Interpret the result
Since entropy change is typically expressed as a positive value in thermodynamics, we take the absolute value:
\[ \Delta S \approx 65.62 \, J/K \]

### Final Answer
Thus, the cell entropy change is approximately \( 65.62 \, J/K \).

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