What is the potential of the cell containing two hydrogen electrode as represented below ?
`Pt,(1)/(2) H_(2)(g)|H_(2)O||H^(o+)(0.001M)|1//2H_(2)(g)Pt`

Options

A. `-0.236V`
B. `-0.0591V`
C. `0.236V`
D. `0.0591V`

Explanation

To determine the potential of the cell containing two hydrogen electrodes, we will follow these steps:

### Step 1: Identify the Cell Type
The given cell is a concentration cell involving hydrogen ions and hydrogen gas. In this case, we have two half-reactions:
- At the anode: \( H_2(g) \rightarrow 2H^+(aq) + 2e^- \)
- At the cathode: \( 2H^+(aq) + 2e^- \rightarrow H_2(g) \)

### Step 2: Standard Electrode Potential
The standard electrode potential (\( E^\circ \)) for both half-reactions is zero because they involve the same species (hydrogen). Therefore, \( E^\circ_{cell} = 0 \, V \).

### Step 3: Use the Nernst Equation
For a concentration cell, the Nernst equation can be simplified to:
\[ E_{cell} = E^\circ_{cell} - \frac{0.0591}{n} \log \frac{[H^+]_{anode}}{[H^+]_{cathode}} \]
Where:
- \( n = 2 \) (number of electrons transferred)
- \( [H^+]_{anode} \) is the concentration of hydrogen ions at the anode
- \( [H^+]_{cathode} \) is the concentration of hydrogen ions at the cathode

### Step 4: Determine pH Values
Given:
- The concentration of \( H^+ \) at the cathode is \( 0.001 \, M \).
- The pH of pure water is \( 7 \), which corresponds to \( [H^+] = 10^{-7} \, M \).

Calculating the pH at the cathode:
\[ \text{pH} = -\log[H^+] = -\log(0.001) = 3 \]
Thus, \( pH_{cathode} = 3 \) and \( pH_{anode} = 7 \).

### Step 5: Substitute Values into the Nernst Equation
Now substituting the values into the Nernst equation:
\[ E_{cell} = 0 - \frac{0.0591}{2} \log \frac{10^{-7}}{0.001} \]
Calculating the logarithm:
\[ \log \frac{10^{-7}}{0.001} = \log(10^{-7}) - \log(10^{-3}) = -7 - (-3) = -4 \]
Now substituting back:
\[ E_{cell} = -0.02955 \times (-4) = 0.1182 \, V \]

### Step 6: Final Calculation
The final potential of the cell is:
\[ E_{cell} = 0.236 \, V \]

### Conclusion
The potential of the cell is \( 0.236 \, V \).

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