The time required to coat aluminum metal on the surface of a square plate of length 20 cm about 5 mm thickness on both sides by using molten `AlCl_3` solution and 10 A current is nearly [Specific gravity of `Al = 1.8 g mL^(-1)`]

Options

A. 107.2 hr
B. 214.4 hr
C. 53.6 hr
D. 71.5 hr

Answer

To solve the problem of determining the time required to coat aluminum metal on the surface of a square plate using molten AlCl₃ solution and a 10 A current, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Volume of the Square Plate:**
- The length of the square plate is given as 20 cm.
- The thickness on both sides is 5 mm, which is equivalent to 0.5 cm (since 5 mm = 0.5 cm).
- The volume (V) of the plate can be calculated using the formula:
\[ V = \text{Length} \times \text{Width} \times \text{Thickness} \]
Since it is a square plate, Length = Width = 20 cm.
\[ V = 20 \, \text{cm} \times 20 \, \text{cm} \times 0.5 \, \text{cm} = 200 \, \text{cm}^3 \]

2. **Convert Volume to Mass:**
- The specific gravity of aluminum is given as 1.8 g/mL. Since 1 cm³ = 1 mL, the mass (m) can be calculated as:
\[ m = \text{Density} \times \text{Volume} = 1.8 \, \text{g/mL} \times 200 \, \text{mL} = 360 \, \text{g} \]

3. **Use Faraday's Law to Calculate Time:**
- According to Faraday's law, the weight (W) of the substance deposited is given by:
\[ W = Z \times I \times T \]
- Where:
- \( Z \) is the electrochemical equivalent,
- \( I \) is the current (10 A),
- \( T \) is the time in seconds.
- The electrochemical equivalent \( Z \) can be calculated using:
\[ Z = \frac{\text{Molar Mass}}{n \times 96500} \]
- For AlCl₃, the molar mass is:
\[ \text{Molar Mass of AlCl}_3 = 27 + (3 \times 35.5) = 133.5 \, \text{g/mol} \]
- The n-factor for Al in AlCl₃ is 3 (since it can lose 3 electrons).
- Therefore:
\[ Z = \frac{133.5}{3 \times 96500} = \frac{133.5}{289500} \approx 0.00046 \, \text{g/C} \]

4. **Substituting Values into Faraday's Law:**
- Rearranging Faraday's law to solve for time \( T \):
\[ T = \frac{W}{Z \times I} \]
- Substituting the values:
\[ T = \frac{360 \, \text{g}}{0.00046 \, \text{g/C} \times 10 \, \text{A}} = \frac{360}{0.0046} \approx 78260.87 \, \text{s} \]

5. **Convert Time to Hours:**
- To convert seconds to hours:
\[ T_{\text{hours}} = \frac{78260.87 \, \text{s}}{3600 \, \text{s/hour}} \approx 21.73 \, \text{hours} \]

### Final Answer:
The time required to coat aluminum metal on the surface of the square plate is approximately **21.73 hours**.