What is the pH of HCl solution when the hydrogen gas electrode shows a potential of -0.22 V at standard temperature and pressure?

Options

A. 2.17
B. 2.98
C. 3.73
D. 3.14

Answer

To find the pH of the HCl solution when the hydrogen gas electrode shows a potential of -0.22 V at standard temperature and pressure, we can use the Nernst equation. Here are the steps to solve the problem:

### Step 1: Write the electrode reaction
The electrode reaction for the hydrogen gas electrode is:
\[ \text{2H}^+ + 2e^- \rightarrow \text{H}_2 \]

### Step 2: Apply the Nernst equation
The Nernst equation for the hydrogen electrode can be expressed as:
\[ E = E^\circ - \frac{0.059}{n} \log \left( \frac{1}{[\text{H}^+]^2} \right) \]
Where:
- \( E \) is the electrode potential (-0.22 V)
- \( E^\circ \) is the standard electrode potential (0 V for the hydrogen electrode)
- \( n \) is the number of electrons transferred (2 for the hydrogen reaction)

### Step 3: Substitute values into the Nernst equation
Substituting the known values into the equation:
\[ -0.22 = 0 - \frac{0.059}{2} \log \left( \frac{1}{[\text{H}^+]^2} \right) \]
This simplifies to:
\[ -0.22 = -0.0295 \log \left( \frac{1}{[\text{H}^+]^2} \right) \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ 0.22 = 0.0295 \log \left( [\text{H}^+]^2 \right) \]

### Step 5: Solve for \([\text{H}^+]\)
Taking the logarithm:
\[ \log \left( [\text{H}^+]^2 \right) = \frac{0.22}{0.0295} \]
Calculating the right side:
\[ \log \left( [\text{H}^+]^2 \right) \approx 7.46 \]
Now, using the property of logarithms:
\[ 2 \log \left( [\text{H}^+] \right) = 7.46 \]
Thus:
\[ \log \left( [\text{H}^+] \right) = \frac{7.46}{2} \approx 3.73 \]

### Step 6: Calculate \([\text{H}^+]\)
Now, we can find \([\text{H}^+]\):
\[ [\text{H}^+] \approx 10^{-3.73} \]

### Step 7: Calculate pH
The pH is defined as:
\[ \text{pH} = -\log [\text{H}^+] \]
Thus:
\[ \text{pH} \approx 3.73 \]

### Final Answer
The pH of the HCl solution is approximately **3.73**.

---