The equilibrium constant of the reaction; Cu(s)+2Ag + (aq)⟶Cu 2+ (aq)+2Ag(s) E o =0.46V at 298K

Explanation

To solve the problem, we will use the Nernst equation to find the equilibrium constant (K) for the given reaction. The Nernst equation relates the standard cell potential (E°) to the equilibrium constant.

### Step-by-Step Solution:

1. **Write the Nernst Equation**:
The Nernst equation is given by:
\[ E = E^\circ - \frac{RT}{nF} \ln Q \]
At equilibrium, \(E = 0\) and \(Q = K\) (the equilibrium constant), so we can rewrite the equation as:
\[ 0 = E^\circ - \frac{RT}{nF} \ln K \]

2. **Rearrange the Equation**:
Rearranging the equation to solve for \(K\):
\[ E^\circ = \frac{RT}{nF} \ln K \]
\[ \ln K = \frac{nFE^\circ}{RT} \]
\[ K = e^{\frac{nFE^\circ}{RT}} \]

3. **Identify the Variables**:
- Given \(E^\circ = 0.46 \, \text{V}\)
- The number of electrons transferred \(n = 2\) (from the balanced reaction)
- Faraday's constant \(F = 96500 \, \text{C/mol}\)
- Gas constant \(R = 8.314 \, \text{J/(mol K)}\)
- Temperature \(T = 298 \, \text{K}\)

4. **Substitute the Values**:
Substitute the known values into the equation:
\[ K = e^{\frac{(2)(96500)(0.46)}{(8.314)(298)}} \]

5. **Calculate the Exponent**:
First, calculate the numerator:
\[ (2)(96500)(0.46) = 88940 \]
Then calculate the denominator:
\[ (8.314)(298) = 2477.572 \]
Now, compute the exponent:
\[ \frac{88940}{2477.572} \approx 35.9 \]

6. **Find \(K\)**:
Now calculate \(K\):
\[ K = e^{35.9} \]
Using a calculator, we find:
\[ K \approx 4.4 \times 10^{15} \]

### Final Answer:
The equilibrium constant \(K\) for the reaction is approximately \(4.4 \times 10^{15}\).

---