A brass plate having surface area `200 cm^2` on one side is electroplated with `0.10 mm` thick silver layers on both sides using a `15 A` current. Find the time taken to do the job. The specific gravity of silver is `10.5 ` and its atomic weight is `107.9 g mol^(-1)`

Explanation

To solve the problem of determining the time taken to electroplate a brass plate with silver, we will follow these steps:

### Step 1: Calculate the Volume of Silver Deposited
The thickness of the silver layer is given as 0.10 mm, which we need to convert to centimeters:
\[ \text{Thickness} = 0.10 \, \text{mm} = 0.01 \, \text{cm} \]
The surface area of the brass plate is given as 200 cm². Since silver is deposited on both sides, the total surface area for deposition is:
\[ \text{Total Surface Area} = 200 \, \text{cm}^2 \times 2 = 400 \, \text{cm}^2 \]
Now we can calculate the volume of silver deposited:
\[ \text{Volume of Silver} = \text{Total Surface Area} \times \text{Thickness} = 400 \, \text{cm}^2 \times 0.01 \, \text{cm} = 4 \, \text{cm}^3 \]

### Step 2: Calculate the Mass of Silver Deposited
To find the mass of silver deposited, we use the specific gravity of silver. The specific gravity of silver is given as 10.5, which means:
\[ \text{Density of Silver} = 10.5 \, \text{g/cm}^3 \]
Now, we can calculate the mass of silver:
\[ \text{Mass of Silver} = \text{Volume} \times \text{Density} = 4 \, \text{cm}^3 \times 10.5 \, \text{g/cm}^3 = 42 \, \text{g} \]

### Step 3: Calculate the Electrochemical Equivalent (Z)
The electrochemical equivalent (Z) can be calculated using the atomic weight of silver and Faraday's constant. The atomic weight of silver is given as 107.9 g/mol and the Faraday's constant is approximately 96500 C/mol. The electrochemical equivalent is given by:
\[ Z = \frac{\text{Atomic Weight}}{n \times F} \]
Where \( n \) (the number of electrons transferred) for silver is 1 (since Ag⁺ + e⁻ → Ag). Thus:
\[ Z = \frac{107.9 \, \text{g/mol}}{1 \times 96500 \, \text{C/mol}} \approx 0.001115 \, \text{g/C} \]

### Step 4: Apply Faraday's Law
According to Faraday's law, the mass deposited (m) is related to the current (I), time (t), and electrochemical equivalent (Z) by the equation:
\[ m = Z \cdot I \cdot t \]
Rearranging for time (t):
\[ t = \frac{m}{Z \cdot I} \]
Substituting the values:
\[ t = \frac{42 \, \text{g}}{0.001115 \, \text{g/C} \times 15 \, \text{A}} = \frac{42}{0.016725} \approx 2504.1 \, \text{s} \]

### Step 5: Convert Time to Minutes
To convert seconds into minutes:
\[ t \approx \frac{2504.1 \, \text{s}}{60} \approx 41.74 \, \text{minutes} \approx 42 \, \text{minutes} \]

### Final Answer
The time taken to electroplate the brass plate with silver is approximately **42 minutes**.

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