To determine the pressure of hydrogen gas ($ \text{H}_2 $) required to make the electromotive force (emf) of the hydrogen electrode zero in pure water at 25°C, we can apply the Nernst equation.

### 1. ### Break Down the Problem
We need to solve for the partial pressure of hydrogen gas that will cause the emf of the hydrogen electrode to equal zero. The general reaction for the hydrogen electrode can be represented as:
$$
\text{H}_2 \rightleftharpoons 2\text{H}^+ + 2e^-
$$

### 2. ### Relevant Concepts
The Nernst equation is given by:
$$
E = E^0 - \frac{RT}{nF} \ln Q
$$
Where:
- $ E $ = emf of the cell
- $ E^0 $ = standard electrode potential
- $ R $ = universal gas constant $ (8.314 \, \text{J/mol·K}) $
- $ T $ = temperature in Kelvin
- $ n $ = number of moles of electrons transferred (for hydrogen, $ n = 2 $)
- $ F $ = Faraday constant $ (96485 \, \text{C/mol}) $
- $ Q $ = reaction quotient

At 25°C, the temperature in Kelvin is:
$$
T = 25 + 273.15 = 298.15 \, \text{K}
$$

The standard reduction potential for the hydrogen half-reaction is defined as:
$$
E^0 = 0 \, \text{V}
$$

### 3. ### Analysis and Detail
Since we want the emf to be zero ($ E = 0 $):
$$
0 = 0 - \frac{RT}{nF} \ln Q
$$
This implies:
$$
\frac{RT}{nF} \ln Q = 0
$$
This means $ \ln Q $ must also equal zero, leading to:
$$
Q = 1
$$
The reaction quotient $ Q $ for the hydrogen electrode in pure water is given by:
$$
Q = \frac{[\text{H}^+]^2}{P_{\text{H}_2}}
$$

In pure water at 25°C, the concentration of hydrogen ions ($ [\text{H}^+] $) is:
$$
[\text{H}^+] = 10^{-7} \, \text{mol/L}
$$

Substituting into the expression for $ Q $:
$$
1 = \frac{(10^{-7})^2}{P_{\text{H}_2}}
$$
Solving for $ P_{\text{H}_2} $:
$$
P_{\text{H}_2} = (10^{-7})^2 = 10^{-14} \, \text{bar}
$$

### 4. ### Verify and Summarize
The calculations check out, ensuring that the Nernst equation was applied correctly with the correct values substituted.

### Final Answer
The pressure of $ \text{H}_2 $ required to make the emf of the hydrogen electrode zero in pure water at 25°C is:
$$
P_{\text{H}_2} = 10^{-14} \, \text{bar}
$$

Question

To determine the pressure of hydrogen gas ($ \text{H}_2 $) required to make the electromotive force (emf) of the hydrogen electrode zero in pure water at 25°C, we can apply the Nernst equation.

### 1. ### Break Down the Problem
We need to solve for the partial pressure of hydrogen gas that will cause the emf of the hydrogen electrode to equal zero. The general reaction for the hydrogen electrode can be represented as:
$$
\text{H}_2 \rightleftharpoons 2\text{H}^+ + 2e^-
$$

### 2. ### Relevant Concepts
The Nernst equation is given by:
$$
E = E^0 - \frac{RT}{nF} \ln Q
$$
Where:
- $ E $ = emf of the cell
- $ E^0 $ = standard electrode potential
- $ R $ = universal gas constant $ (8.314 \, \text{J/mol·K}) $
- $ T $ = temperature in Kelvin
- $ n $ = number of moles of electrons transferred (for hydrogen, $ n = 2 $)
- $ F $ = Faraday constant $ (96485 \, \text{C/mol}) $
- $ Q $ = reaction quotient

At 25°C, the temperature in Kelvin is:
$$
T = 25 + 273.15 = 298.15 \, \text{K}
$$

The standard reduction potential for the hydrogen half-reaction is defined as:
$$
E^0 = 0 \, \text{V}
$$

### 3. ### Analysis and Detail
Since we want the emf to be zero ($ E = 0 $):
$$
0 = 0 - \frac{RT}{nF} \ln Q
$$
This implies:
$$
\frac{RT}{nF} \ln Q = 0
$$
This means $ \ln Q $ must also equal zero, leading to:
$$
Q = 1
$$
The reaction quotient $ Q $ for the hydrogen electrode in pure water is given by:
$$
Q = \frac{[\text{H}^+]^2}{P_{\text{H}_2}}
$$

In pure water at 25°C, the concentration of hydrogen ions ($ [\text{H}^+] $) is:
$$
[\text{H}^+] = 10^{-7} \, \text{mol/L}
$$

Substituting into the expression for $ Q $:
$$
1 = \frac{(10^{-7})^2}{P_{\text{H}_2}}
$$
Solving for $ P_{\text{H}_2} $:
$$
P_{\text{H}_2} = (10^{-7})^2 = 10^{-14} \, \text{bar}
$$

### 4. ### Verify and Summarize
The calculations check out, ensuring that the Nernst equation was applied correctly with the correct values substituted.

### Final Answer
The pressure of $ \text{H}_2 $ required to make the emf of the hydrogen electrode zero in pure water at 25°C is:
$$
P_{\text{H}_2} = 10^{-14} \, \text{bar}
$$

Knowee AI · Accepted Answer