In which case `(E_("cell")-E_("cell")^(@))` is zero

Options

A. `Cu|Cu^(2+) (0.01 M)||Ag^(+) (0.1M)|Ag`
B. `Pt(H_(2))PH=1||Zn^(2+) (0.01M)|Zn`
C. `Pt(H_(2))PH=1||Zn^(2+)(1M)|Zn`
D. `Pt(H_(2))|H^(+) (0.01M)||Zn^(2+) (0.01M)|Zn`

Answer

To determine when \( E_{\text{cell}} - E_{\text{cell}}^{\circ} = 0 \), we can analyze the conditions under which the cell potential equals the standard cell potential. Here’s a step-by-step solution:

### Step 1: Understand the Nernst Equation
The Nernst equation is given by:
\[ E_{\text{cell}} = E_{\text{cell}}^{\circ} - \frac{0.0591}{n} \log Q \]
where:
- \( E_{\text{cell}} \) is the cell potential.
- \( E_{\text{cell}}^{\circ} \) is the standard cell potential.
- \( n \) is the number of moles of electrons transferred in the reaction.
- \( Q \) is the reaction quotient.

### Step 2: Set Up the Condition
We want to find when:
\[ E_{\text{cell}} - E_{\text{cell}}^{\circ} = 0 \]
This implies:
\[ E_{\text{cell}} = E_{\text{cell}}^{\circ} \]

### Step 3: Substitute into the Nernst Equation
From the Nernst equation, we can substitute:
\[ E_{\text{cell}}^{\circ} = E_{\text{cell}}^{\circ} - \frac{0.0591}{n} \log Q \]
Setting the left side to zero gives:
\[ 0 = - \frac{0.0591}{n} \log Q \]

### Step 4: Analyze the Logarithmic Term
For the above equation to hold true, the logarithmic term must equal zero:
\[ \log Q = 0 \]
This occurs when:
\[ Q = 1 \]

### Step 5: Determine the Reaction Quotient \( Q \)
The reaction quotient \( Q \) is defined as:
\[ Q = \frac{[\text{products}]}{[\text{reactants}]} \]
For \( Q \) to equal 1, the concentrations of products and reactants must be such that their ratio equals 1.

### Conclusion
Thus, \( E_{\text{cell}} - E_{\text{cell}}^{\circ} = 0 \) when the reaction quotient \( Q = 1 \), which occurs at equilibrium or when the concentrations of products and reactants are equal in a balanced reaction.