`E^(@)` of a cell `aA+bBtocC+dD` is

Options

A. `-(RT)/(nF)"log"([C]^(C)[D]^(d))/([A]^(a)[B]^(b))`
B. `-Rt"log"([a]^(A)[b]^(B))/([a]^(C)[d]^(D))`
C. `-(RT)/(nF)"log"([C]^(c)[d]^(D))/([A]^(a)[B]^(b))`
D. `-(RT)/(nF)"log"([C]^(c)[d]^(D))/([a]^(A)[B]^(b))`

Answer

To determine the standard electrode potential \( E^\circ \) of a cell for the reaction \( aA + bB \rightarrow cC + dD \), we use the Nernst equation. The Nernst equation allows us to calculate the cell potential under non-standard conditions. The standard form of the Nernst equation is:

\[ E = E^\circ - \frac{RT}{nF} \ln Q \]

where:
- \( E \) is the cell potential under non-standard conditions,
- \( E^\circ \) is the standard cell potential,
- \( R \) is the universal gas constant (\( 8.314 \, \text{J/mol·K} \)),
- \( T \) is the temperature in Kelvin,
- \( n \) is the number of moles of electrons transferred in the reaction,
- \( F \) is the Faraday constant (\( 96485 \, \text{C/mol} \)),
- \( Q \) is the reaction quotient.

For the reaction \( aA + bB \rightarrow cC + dD \), the reaction quotient \( Q \) is given by:

\[ Q = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]

Now, substituting \( Q \) into the Nernst equation, we get:

\[ E = E^\circ - \frac{RT}{nF} \ln \left( \frac{[C]^c [D]^d}{[A]^a [B]^b} \right) \]

However, the question specifically asks for the standard electrode potential \( E^\circ \) of the cell. The standard electrode potential \( E^\circ \) is a constant value that is determined under standard conditions (1 M concentration for solutions, 1 atm pressure for gases, and 298 K temperature).

Therefore, the standard electrode potential \( E^\circ \) for the cell reaction \( aA + bB \rightarrow cC + dD \) is simply \( E^\circ \).

### Step-by-Step Solution:

1. **Identify the cell reaction:**
The given cell reaction is \( aA + bB \rightarrow cC + dD \).

2. **Write the Nernst equation:**
The Nernst equation is:
\[ E = E^\circ - \frac{RT}{nF} \ln Q \]

3. **Determine the reaction quotient \( Q \):**
For the reaction \( aA + bB \rightarrow cC + dD \), the reaction quotient \( Q \) is:
\[ Q = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]

4. **Substitute \( Q \) into the Nernst equation:**
\[ E = E^\circ - \frac{RT}{nF} \ln \left( \frac{[C]^c [D]^d}{[A]^a [B]^b} \right) \]

5. **Identify the standard electrode potential \( E^\circ \):**
The standard electrode potential \( E^\circ \) is the value of the cell potential under standard conditions.

Therefore, the standard electrode potential \( E^\circ \) for the cell reaction \( aA + bB \rightarrow cC + dD \) is \( E^\circ \).