Calculate the pH of a buffer when the reduction potential of hydrogen electrode placed in the buffer solution is found to be `-0.413V`.

Options

A. 10
B. 4
C. 7
D. 12

Answer

To calculate the pH of a buffer solution based on the reduction potential of a hydrogen electrode, we can follow these steps:

### Step 1: Understand the Given Data
We are given the reduction potential (E_cell) of the hydrogen electrode in the buffer solution, which is -0.413 V.

### Step 2: Identify the Standard Reduction Potential
The standard reduction potential (E°) for the hydrogen electrode is 0 V. This is a reference value from the electrochemical series.

### Step 3: Use the Nernst Equation
The Nernst equation is given by:
\[ E_{cell} = E^{\circ}_{cell} - \frac{0.0591}{n} \log Q \]
where:
- \(E_{cell}\) is the cell potential (-0.413 V),
- \(E^{\circ}_{cell}\) is the standard cell potential (0 V),
- \(n\) is the number of electrons transferred (for the hydrogen electrode, \(n = 2\)),
- \(Q\) is the reaction quotient.

### Step 4: Set Up the Reaction Quotient
For the hydrogen electrode reaction:
\[ \text{H}_2 \rightleftharpoons 2\text{H}^+ + 2e^- \]
The reaction quotient \(Q\) can be expressed as:
\[ Q = \frac{[\text{H}^+]^2}{P_{\text{H}_2}} \]
Assuming the partial pressure of hydrogen gas (\(P_{\text{H}_2}\)) is 1 atm, we simplify \(Q\) to:
\[ Q = [\text{H}^+]^2 \]

### Step 5: Substitute Values into the Nernst Equation
Substituting the known values into the Nernst equation:
\[ -0.413 = 0 - \frac{0.0591}{2} \log [\text{H}^+]^2 \]
This simplifies to:
\[ -0.413 = -0.02955 \log [\text{H}^+]^2 \]

### Step 6: Solve for \([\text{H}^+]\)
Rearranging gives:
\[ \log [\text{H}^+]^2 = \frac{0.413}{0.02955} \]
Calculating the right side:
\[ \log [\text{H}^+]^2 \approx 13.97 \]
Taking the antilogarithm:
\[ [\text{H}^+]^2 \approx 10^{13.97} \]
Thus:
\[ [\text{H}^+] \approx 10^{6.985} \approx 1.0 \times 10^{-7} \text{ M} \]

### Step 7: Calculate the pH
Using the relationship between pH and \([\text{H}^+]\):
\[ \text{pH} = -\log [\text{H}^+] \]
Substituting the concentration:
\[ \text{pH} \approx -\log(1.0 \times 10^{-7}) \approx 7 \]

### Final Answer
The pH of the buffer solution is approximately **7**.

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