Electrolysis of dilute aqueous `NaCl` solution was carried out by passing `10mA` current. The time required to liberate `0.01mol` of `H_(2)` gas at the cathode is `(1F=96500C mol ^(-1))`
(a)`9.65xx10^(4)s`
(b)`19.3xx10^(4)s`
(c)`28.95xx10^(4)s`
(d)`38.6xx10^(4)s`

Options

A. `9.65xx10^(4)s`
B. `19.3xx10^(4)s`
C. `28.95xx10^(4)s`
D. `38.6xx10^(4)s`

Explanation

To solve the problem of determining the time required to liberate 0.01 moles of hydrogen gas at the cathode during the electrolysis of a dilute aqueous NaCl solution with a current of 10 mA, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Reaction**:
- The electrolysis of water produces hydrogen gas (H₂) at the cathode. The half-reaction at the cathode can be written as:
\[ 2H^+ + 2e^- \rightarrow H_2 \]
- This shows that 2 moles of electrons are required to produce 1 mole of hydrogen gas.

2. **Calculate Moles of Electrons**:
- Since we are liberating 0.01 moles of hydrogen gas, the moles of electrons required can be calculated as:
\[ \text{Moles of electrons} = 0.01 \, \text{mol H}_2 \times 2 = 0.02 \, \text{mol e}^- \]

3. **Calculate Total Charge (Q)**:
- Using Faraday's constant (F = 96500 C/mol), the total charge (Q) required can be calculated:
\[ Q = \text{Moles of electrons} \times F = 0.02 \, \text{mol} \times 96500 \, \text{C/mol} = 1930 \, \text{C} \]

4. **Use the Formula for Charge**:
- The relationship between charge (Q), current (I), and time (t) is given by:
\[ Q = I \times t \]
- Rearranging this equation to solve for time (t) gives:
\[ t = \frac{Q}{I} \]

5. **Convert Current to Amperes**:
- The current is given as 10 mA, which can be converted to amperes:
\[ I = 10 \, \text{mA} = 10 \times 10^{-3} \, \text{A} = 0.01 \, \text{A} \]

6. **Calculate Time**:
- Now substituting the values of Q and I into the equation for time:
\[ t = \frac{1930 \, \text{C}}{0.01 \, \text{A}} = 193000 \, \text{s} \]
- To express this in scientific notation:
\[ t = 19.3 \times 10^4 \, \text{s} \]

7. **Final Answer**:
- The time required to liberate 0.01 moles of hydrogen gas at the cathode is:
\[ \text{(b) } 19.3 \times 10^4 \, \text{s} \]