A cell contains two hydrogen electrodes. The negative electrode is in contact with a solution of `10^(-6)M` hydrogen ion. The EMF of the cell is 0.118 V at 298 K. The concentration of `H^(+)` ion at the positive electrode is `10^(-x)`, The value of 'x' is

Answer

To solve the problem, we need to determine the concentration of hydrogen ions at the positive electrode (cathode) given the concentration at the negative electrode (anode) and the EMF of the cell. We will use the Nernst equation to find the value of 'x'.

### Step-by-Step Solution:

1. **Identify the Cell Configuration**:
- The cell consists of two hydrogen electrodes.
- The negative electrode (anode) is in contact with a solution of \(10^{-6} M\) hydrogen ions.
- The positive electrode (cathode) has a concentration of \(10^{-x} M\) hydrogen ions.

2. **Write the Half-Reactions**:
- At the anode (oxidation):
\[ H_2(g) \rightarrow 2H^+(aq) + 2e^- \]
- At the cathode (reduction):
\[ 2H^+(aq) + 2e^- \rightarrow H_2(g) \]

3. **Write the Overall Cell Reaction**:
- The overall reaction can be represented as:
\[ H_2(g) + 2H^+(aq) \rightarrow 2H^+(aq) + H_2(g) \]
- This shows that 2 moles of electrons are transferred in the reaction.

4. **Apply the Nernst Equation**:
- The Nernst equation is given by:
\[ E_{cell} = E^\circ_{cell} - \frac{0.059}{n} \log \frac{[H^+]^2_{cathode}}{[H^+]^2_{anode}} \]
- Here, \(E^\circ_{cell} = 0\) for hydrogen electrodes, \(n = 2\) (number of electrons transferred), and we substitute the known values:
\[ 0.118 = 0 - \frac{0.059}{2} \log \frac{[H^+]^2_{cathode}}{[H^+]^2_{anode}} \]

5. **Substitute the Concentrations**:
- Substitute \([H^+]_{anode} = 10^{-6}\) and \([H^+]_{cathode} = 10^{-x}\):
\[ 0.118 = -\frac{0.059}{2} \log \frac{(10^{-x})^2}{(10^{-6})^2} \]
- Simplifying gives:
\[ 0.118 = -\frac{0.059}{2} \log \frac{10^{-2x}}{10^{-12}} \]
- This simplifies to:
\[ 0.118 = -\frac{0.059}{2} \log(10^{12 - 2x}) = -\frac{0.059}{2} (12 - 2x) \]

6. **Solve for x**:
- Rearranging gives:
\[ 0.118 = -\frac{0.059}{2} (12 - 2x) \]
- Multiply both sides by -2:
\[ -0.236 = 0.059(12 - 2x) \]
- Divide by 0.059:
\[ -4 = 12 - 2x \]
- Rearranging gives:
\[ 2x = 12 + 4 = 16 \quad \Rightarrow \quad x = 8 \]

### Final Answer:
The value of \(x\) is \(4\).

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