The oxidation potential of hydrogen half-cell will be negative if:

Options

A. `p(H_2)=1atm "and" [H^+]=1M`
B. `p(H_2)=1atm "and" [H^+]=2M`
C. `p(H_2)=0.2atm "and" [H^+]=1M`
D. `p(H_2)=0.2atm "and" [H^+]=0.2M`

Answer

To determine when the oxidation potential of the hydrogen half-cell will be negative, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Reaction**: The oxidation of hydrogen gas (H2) can be represented as:
\[ \text{H}_2 \rightarrow 2\text{H}^+ + 2e^- \]
This reaction shows that hydrogen gas is oxidized to hydrogen ions (H⁺) and electrons.

2. **Nernst Equation**: The potential of the half-cell can be calculated using the Nernst equation:
\[ E_{\text{cell}} = E^\circ_{\text{cell}} - \frac{0.0591}{n} \log \left( \frac{[\text{Products}]}{[\text{Reactants}]} \right) \]
For our reaction, the products are H⁺ ions and the reactant is H₂ gas.

3. **Identify Standard Potentials**: The standard reduction potential (E°) for the hydrogen half-cell is defined as 0 V. Therefore, the standard oxidation potential (E°) is also 0 V.

4. **Substituting Values**: In our case, the number of electrons (n) involved is 2. Thus, the equation becomes:
\[ E_{\text{cell}} = 0 - \frac{0.0591}{2} \log \left( \frac{[\text{H}^+]^2}{P_{\text{H}_2}} \right) \]

5. **Condition for Negative Potential**: For the oxidation potential to be negative, the term inside the logarithm must be greater than 1:
\[ \frac{[\text{H}^+]^2}{P_{\text{H}_2}} > 1 \]
This implies that:
\[ [\text{H}^+]^2 > P_{\text{H}_2} \]
or equivalently:
\[ [\text{H}^+] > \sqrt{P_{\text{H}_2}} \]

6. **Analyzing Given Options**: We need to evaluate the given options to see which one satisfies this condition:
- **Option A**: \( P_{\text{H}_2} = 1 \, \text{atm}, [\text{H}^+] = 1 \, \text{M} \) → \( 1 < 1 \) (not valid)
- **Option B**: \( P_{\text{H}_2} = 1 \, \text{atm}, [\text{H}^+] = 2 \, \text{M} \) → \( 2 > 1 \) (valid)
- **Option C**: \( P_{\text{H}_2} = 0.2 \, \text{atm}, [\text{H}^+] = 1 \, \text{M} \) → \( 1 > 0.447 \) (valid)
- **Option D**: Both are equal → \( 1 = 1 \) (not valid)

7. **Conclusion**: The oxidation potential of the hydrogen half-cell will be negative when the concentration of H⁺ ions is greater than the partial pressure of hydrogen gas. The correct answer is **Option B**.