The logarithm of the equilibriium constant of the cell reaction corresponding to the cell
`X(s)|x^(2+)(aq)||Y^(+)(aq)|Y(s)` with standard cell potential `E_(cell)^(@)=1.2V` given by

Options

A. 12.5
B. 21.5
C. 40.6
D. 47.2

Explanation

To find the logarithm of the equilibrium constant (K) for the cell reaction corresponding to the given cell, we can use the Nernst equation. Here’s a step-by-step solution:

### Step 1: Understand the Nernst Equation
The Nernst equation relates the standard cell potential (E°) to the cell potential (E) and the reaction quotient (Q):
\[ E = E° - \frac{RT}{nF} \ln Q \]
At equilibrium, E = 0, and Q becomes the equilibrium constant K. Thus, we can rewrite the equation as:
\[ 0 = E° - \frac{RT}{nF} \ln K \]

### Step 2: Rearranging the Equation
Rearranging the equation gives:
\[ \ln K = \frac{nFE°}{RT} \]

### Step 3: Substitute Values
We know:
- Standard cell potential, \( E° = 1.2 \, V \)
- The number of electrons transferred, \( n = 2 \) (as inferred from the reaction)
- The constants:
- \( F = 96485 \, C/mol \) (Faraday's constant)
- \( R = 8.314 \, J/(mol \cdot K) \)
- Temperature \( T \) is typically taken as 298 K (standard conditions).

### Step 4: Calculate \( \ln K \)
Substituting the values into the equation:
\[ \ln K = \frac{(2)(96485)(1.2)}{(8.314)(298)} \]

### Step 5: Perform the Calculation
Calculating the numerator:
\[ (2)(96485)(1.2) = 231564 \]

Calculating the denominator:
\[ (8.314)(298) = 2477.572 \]

Now, calculate \( \ln K \):
\[ \ln K = \frac{231564}{2477.572} \approx 93.5 \]

### Step 6: Convert \( \ln K \) to \( K \)
To find K, we use:
\[ K = e^{93.5} \]

### Step 7: Find \( \log K \)
To find \( \log K \), we can use the conversion from natural logarithm to base 10 logarithm:
\[ \log K = \frac{\ln K}{\ln 10} \]
Where \( \ln 10 \approx 2.303 \):
\[ \log K \approx \frac{93.5}{2.303} \approx 40.6 \]

### Final Answer
The logarithm of the equilibrium constant \( \log K \) for the cell reaction is approximately **40.6**.

---