What is the correct Nernst equation for` M^(2+) (aq) + 2e^+ → M `(s) at 45°C?

Options

A. E°`(M^(2+)/M) + 0.315log_10 (1 / [M]^(+2))`
B. E°`(M^(2+)/M) + 0.0425log_10 (1 / [M]^(+2))`
C. E°`(M^(2+)/M) + 0.0315log_10 (1 / [M]^(+2))`
D. E°`(M^(2+)/M) + 0.0326log_10 (1 / [M]^(+2))`

Answer

To derive the Nernst equation for the reaction \( M^{2+} (aq) + 2e^- \rightarrow M (s) \) at 45°C, we will follow these steps:

### Step 1: Identify the Reaction and Standard Conditions
The given reaction is:
\[ M^{2+} (aq) + 2e^- \rightarrow M (s) \]

### Step 2: Convert Temperature to Kelvin
The temperature given is 45°C. To convert this to Kelvin:
\[ T(K) = 45°C + 273.15 = 318.15 K \]

### Step 3: Determine the Number of Electrons Transferred (n)
In this reaction, 2 electrons are transferred. Therefore, we have:
\[ n = 2 \]

### Step 4: Write the Nernst Equation
The Nernst equation is given by:
\[ E = E^\circ + \frac{RT}{nF} \ln Q \]
Where:
- \( E \) = cell potential under non-standard conditions
- \( E^\circ \) = standard cell potential
- \( R \) = universal gas constant (8.314 J/(mol·K))
- \( T \) = temperature in Kelvin
- \( n \) = number of moles of electrons transferred
- \( F \) = Faraday's constant (96500 C/mol)
- \( Q \) = reaction quotient

### Step 5: Substitute Values into the Nernst Equation
For our reaction, the reaction quotient \( Q \) can be expressed as:
\[ Q = \frac{1}{[M^{2+}]} \]

Thus, the Nernst equation becomes:
\[ E = E^\circ + \frac{RT}{nF} \ln \left( \frac{1}{[M^{2+}]} \right) \]
This can be rewritten using properties of logarithms:
\[ E = E^\circ - \frac{RT}{nF} \ln [M^{2+}] \]

### Step 6: Substitute Known Values
Now substituting the known values into the equation:
- \( R = 8.314 \, \text{J/(mol·K)} \)
- \( T = 318.15 \, \text{K} \)
- \( n = 2 \)
- \( F = 96500 \, \text{C/mol} \)

Calculating the term \( \frac{RT}{nF} \):
\[ \frac{(8.314 \, \text{J/(mol·K)})(318.15 \, \text{K})}{(2)(96500 \, \text{C/mol})} = 0.0315 \, \text{V} \]

### Step 7: Final Form of the Nernst Equation
Thus, the Nernst equation simplifies to:
\[ E = E^\circ - 0.0315 \log [M^{2+}] \]

### Conclusion
The correct Nernst equation for the reaction \( M^{2+} (aq) + 2e^- \rightarrow M (s) \) at 45°C is:
\[ E = E^\circ - 0.0315 \log [M^{2+}] \]