What pressure of `H_2` would be required to make emf of the hydrogen electrode zero in pure water at `25^(@)C`?

Options

A. `10^(-7)` atm
B. `10^(-14)` atm
C. 1 atm
D. 0.5 atm

Answer

To determine the pressure of hydrogen gas (`H2`) required to make the electromotive force (emf) of the hydrogen electrode zero in pure water at `25°C`, we can use the Nernst equation. Here’s a step-by-step solution:

### Step 1: Understand the Reaction
The half-reaction for the hydrogen electrode can be written as:
\[ 2H^+ + 2e^- \leftrightarrow H_2(g) \]

### Step 2: Write the Nernst Equation
The Nernst equation for this reaction is given by:
\[ E = E^0 - \frac{0.0591}{n} \log \left( \frac{[H_2]}{[H^+]^2} \right) \]
Where:
- \( E \) is the cell potential (emf).
- \( E^0 \) is the standard reduction potential (0 V for the hydrogen electrode).
- \( n \) is the number of electrons transferred (2 for this reaction).

### Step 3: Set the EMF to Zero
To find the pressure of hydrogen gas that makes the emf zero:
\[ 0 = 0 - \frac{0.0591}{2} \log \left( \frac{[H_2]}{[H^+]^2} \right) \]
This simplifies to:
\[ 0 = - \frac{0.0591}{2} \log \left( \frac{[H_2]}{[H^+]^2} \right) \]

### Step 4: Solve for the Logarithmic Term
Rearranging the equation gives:
\[ \log \left( \frac{[H_2]}{[H^+]^2} \right) = 0 \]
This implies:
\[ \frac{[H_2]}{[H^+]^2} = 1 \]

### Step 5: Substitute the Concentration of \( H^+ \)
In pure water at `25°C`, the concentration of hydrogen ions \( [H^+] \) is:
\[ [H^+] = 10^{-7} \, \text{M} \]
Thus:
\[ [H^+]^2 = (10^{-7})^2 = 10^{-14} \]

### Step 6: Find the Pressure of \( H_2 \)
Substituting this into the equation gives:
\[ [H_2] = 10^{-14} \, \text{M} \]
Since we are looking for pressure, we can relate concentration to pressure using the ideal gas law. At standard conditions (1 atm and 25°C), the concentration of a gas can be approximated as:
\[ P = [H_2] \cdot RT \]
However, for our purposes, we can directly state:
\[ P_{H_2} = 10^{-14} \, \text{atm} \]

### Final Answer
The pressure of hydrogen gas required to make the emf of the hydrogen electrode zero in pure water at `25°C` is:
\[ P_{H_2} = 10^{-14} \, \text{atm} \]

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