In the following electrochemical cell
`Zn|Zn^(2+) ||H^(+)| pt(H_(2))`
`E^(@)_(cell) = E_(cell)`. This will be :

Options

A. `[Zn^(2+)] = [H^(+)] = 1M and pH_(2) = 1atm`
B. `[Zn^(2+)]= 0.01M, [H^(+)]= 0.1M and pH_(2) = 1atm`
C. `[Zn^(2+)]= 1M , [H^(+)= 0.1M and pH_(2) = 1atm`
D. None of the above

Explanation

To solve the problem regarding the electrochemical cell represented as `Zn|Zn^(2+) ||H^(+)| pt(H_(2))`, we need to determine the conditions under which the standard cell potential \( E^\circ_{cell} \) is equal to the cell potential \( E_{cell} \).

### Step-by-Step Solution:

1. **Identify the Half-Reactions**:
The half-reactions for the cell can be identified as follows:
- Oxidation at the anode:
\[ Zn \rightarrow Zn^{2+} + 2e^- \]
- Reduction at the cathode:
\[ 2H^+ + 2e^- \rightarrow H_2 \]

2. **Write the Cell Reaction**:
The overall cell reaction can be written by combining the two half-reactions:
\[ Zn + 2H^+ \rightarrow Zn^{2+} + H_2 \]

3. **Determine the Reaction Quotient (Q)**:
The reaction quotient \( Q \) for the cell reaction can be expressed as:
\[ Q = \frac{[Zn^{2+}]}{[H^+]^2 \cdot P_{H_2}} \]
where \( [Zn^{2+}] \) is the concentration of zinc ions, \( [H^+] \) is the concentration of hydrogen ions, and \( P_{H_2} \) is the partial pressure of hydrogen gas.

4. **Relate E_cell and E^\circ_cell**:
The Nernst equation relates the cell potential \( E_{cell} \) to the standard cell potential \( E^\circ_{cell} \):
\[ E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln Q \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the temperature in Kelvin,
- \( n \) is the number of moles of electrons transferred (2 in this case),
- \( F \) is Faraday's constant.

5. **Condition for E_cell = E^\circ_cell**:
For \( E_{cell} \) to equal \( E^\circ_{cell} \), the term \( \frac{RT}{nF} \ln Q \) must be zero. This occurs when:
\[ Q = 1 \]
Therefore, we need:
\[ [Zn^{2+}] = [H^+]^2 \cdot P_{H_2} \]

6. **Conclusion**:
The condition for the cell potential \( E_{cell} \) to equal the standard cell potential \( E^\circ_{cell} \) is when the concentrations and pressures are such that the reaction quotient \( Q \) equals 1.