For a cell reaction : `M^(n+)(aq)+"ne"^(-) to M(s)`, the Nernst equation for electrode potential at any concertation measured with respect to standard hydrogen electrode is represented as

Options

A. `E_((M^(n+)//M))=E_((M^(n+)//M))^@-"RT"/"nF"ln (1/([M^(n+)]))`
B. `E_((M//M^(n+)))=E_((M//M^(n+)))^@-"RT"/"nF"ln (([M^(n+)])/([M]))`
C. `E_((M^(n+)//M))=E_((M^(n+)//M))^@-"RT"/"nF" "log" 1/([M])`
D. `E_((M^(n+)//M))=E_((M^(n+)//M^(n+)))^@-"RT"/"nF"ln [M^(n+)]`

Answer

To derive the Nernst equation for the given cell reaction \( M^{n+}(aq) + ne^- \rightarrow M(s) \), we will follow these steps:

### Step 1: Write the Nernst Equation
The general form of the Nernst equation is:
\[ E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln \left( \frac{a_{products}}{a_{reactants}} \right) \]
where:
- \( E_{cell} \) is the electromotive force (EMF) of the cell.
- \( E^0_{cell} \) is the standard EMF of the cell.
- \( R \) is the universal gas constant (8.314 J/(mol·K)).
- \( T \) is the temperature in Kelvin.
- \( n \) is the number of moles of electrons transferred in the reaction.
- \( F \) is Faraday's constant (approximately 96485 C/mol).
- \( a_{products} \) and \( a_{reactants} \) are the activities of the products and reactants, respectively.

### Step 2: Identify Products and Reactants
In the given reaction:
- The product is \( M(s) \) (solid metal).
- The reactant is \( M^{n+}(aq) \) (metal ion in aqueous solution).

### Step 3: Determine Activities
For solids, the activity is defined as 1. Therefore, the activity of the product \( M(s) \) is:
\[ a_{products} = 1 \]
The activity of the reactant \( M^{n+}(aq) \) is equal to its concentration, which we denote as \( [M^{n+}] \).

### Step 4: Substitute into the Nernst Equation
Substituting the activities into the Nernst equation gives:
\[ E_{cell} = E^0_{cell} - \frac{RT}{nF} \ln \left( \frac{1}{[M^{n+}]} \right) \]

### Step 5: Simplify the Equation
Using the property of logarithms, we can rewrite the equation:
\[ E_{cell} = E^0_{cell} + \frac{RT}{nF} \ln \left( [M^{n+}] \right) \]
This is because \( \ln \left( \frac{1}{x} \right) = -\ln(x) \).

### Final Equation
Thus, the Nernst equation for the electrode potential at any concentration measured with respect to the standard hydrogen electrode is:
\[ E_{M^{n+} \rightarrow M} = E^0_{M^{n+} \rightarrow M} - \frac{RT}{nF} \ln \left( [M^{n+}] \right) \]