Cost of electricity for the production of X litre `H_(2)` at STP at cathode is Rs. X, then cost of electricity for the production of X litre `O_(2)` gas at STP at anode will be :(assume 1 "mole" of electrons as one unit of electricity)

Options

A. 2X
B. 4X
C. 16 X
D. 32 X

Answer

To solve the problem of calculating the cost of electricity for the production of X liters of O₂ gas at STP at the anode, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Electrolysis Reaction**:
- The electrolysis of water can be represented by the following reactions:
- At the cathode: \( 2H_2O + 2e^- \rightarrow H_2 + 2OH^- \)
- At the anode: \( 2H_2O \rightarrow O_2 + 4H^+ + 4e^- \)

2. **Calculating Moles of Hydrogen (H₂)**:
- Given that the cost of producing X liters of H₂ is Rs. X, we need to determine how many moles of H₂ this corresponds to.
- At STP, 1 mole of gas occupies 22.4 liters.
- Therefore, the number of moles of H₂ produced is:
\[ \text{Moles of } H_2 = \frac{X}{22.4} \]

3. **Calculating Moles of Electrons for H₂ Production**:
- From the cathode reaction, producing 1 mole of H₂ requires 2 moles of electrons.
- Thus, for \( \frac{X}{22.4} \) moles of H₂, the moles of electrons required are:
\[ \text{Moles of electrons} = 2 \times \frac{X}{22.4} = \frac{2X}{22.4} \]

4. **Cost of Electricity for H₂ Production**:
- The cost of electricity for producing the above amount of H₂ is Rs. X.
- Since 1 mole of electrons is considered as 1 unit of electricity, the cost per mole of electrons is:
\[ \text{Cost per mole of electrons} = \frac{X}{\frac{2X}{22.4}} = \frac{22.4}{2} = 11.2 \]

5. **Calculating Moles of Oxygen (O₂)**:
- Now, we need to find the cost of producing X liters of O₂ at the anode.
- The number of moles of O₂ produced is also:
\[ \text{Moles of } O_2 = \frac{X}{22.4} \]

6. **Calculating Moles of Electrons for O₂ Production**:
- From the anode reaction, producing 1 mole of O₂ requires 4 moles of electrons.
- Thus, for \( \frac{X}{22.4} \) moles of O₂, the moles of electrons required are:
\[ \text{Moles of electrons} = 4 \times \frac{X}{22.4} = \frac{4X}{22.4} \]

7. **Cost of Electricity for O₂ Production**:
- The cost of electricity for producing the above amount of O₂ can be calculated as:
\[ \text{Cost for O}_2 = \left(\text{Cost per mole of electrons}\right) \times \left(\text{Moles of electrons for O}_2\right) \]
- Substituting the values:
\[ \text{Cost for O}_2 = 11.2 \times \frac{4X}{22.4} \]
- Simplifying gives:
\[ \text{Cost for O}_2 = 2X \]

### Final Answer:
The cost of electricity for the production of X liters of O₂ gas at STP at the anode will be Rs. 2X.

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