`Na-` amalgam is prepared by electrolysis of `NaCl` solution using liquid `Hg` as cathode . How long should the current of `10 amp`. Is passed to produce `10% Na-Hg` on a cathode of `10gm Hg`. ( atomic mass of `Na=23)`.

Options

A. `7.77 `min
B. `9.44` min.
C. `5.24 mi n.`
D. `11.39 mi n`.

Explanation

To solve the problem of how long a current of 10 amperes should be passed to produce a 10% sodium amalgam on a cathode of 10 g of mercury, we can follow these steps:

### Step 1: Determine the mass of sodium in the amalgam
Since the amalgam is 10% sodium, we can calculate the mass of sodium in 10 g of mercury.

\[ \text{Mass of Na} = \frac{10\% \text{ of } 10 \text{ g}}{90\%} \times 10 \text{ g} = \frac{10}{90} \times 10 = \frac{100}{90} = \frac{10}{9} \text{ g} \]

### Step 2: Calculate the equivalent mass of sodium
The equivalent mass of sodium (Na) can be calculated using its atomic mass. The atomic mass of sodium is given as 23 g/mol, and since sodium has a valency of 1 (it gains one electron), the equivalent mass is:

\[ \text{Equivalent mass of Na} = \frac{23 \text{ g}}{1} = 23 \text{ g} \]

### Step 3: Use Faraday's law of electrolysis
According to Faraday's law, the relationship between the mass of the substance deposited (W), the current (I), time (T), and Faraday's constant (F) is given by:

\[ \frac{W}{\text{Equivalent mass}} = \frac{I \cdot T}{F} \]

Where:
- \( W = \frac{10}{9} \text{ g} \)
- Equivalent mass = 23 g
- Current \( I = 10 \text{ A} \)
- Faraday's constant \( F = 96500 \text{ C} \)

### Step 4: Substitute the values into the equation
Substituting the known values into the equation:

\[ \frac{\frac{10}{9}}{23} = \frac{10 \cdot T}{96500} \]

### Step 5: Solve for time (T)
Cross-multiplying to solve for T:

\[ \frac{10}{9} \cdot 96500 = 10 \cdot T \cdot 23 \]

\[ T = \frac{\frac{10}{9} \cdot 96500}{10 \cdot 23} \]

Calculating the right-hand side:

\[ T = \frac{10 \cdot 96500}{9 \cdot 10 \cdot 23} = \frac{96500}{9 \cdot 23} \]

Calculating the denominator:

\[ 9 \cdot 23 = 207 \]

Now substituting back:

\[ T = \frac{96500}{207} \approx 466.62 \text{ seconds} \]

### Step 6: Convert seconds to minutes
To convert seconds to minutes:

\[ T \text{ (in minutes)} = \frac{466.62}{60} \approx 7.77 \text{ minutes} \]

### Final Answer
The time required to produce 10% sodium amalgam on a cathode of 10 g of mercury with a current of 10 A is approximately **7.77 minutes**.

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