How long a current of `3A` has to be passed through a solution of `AgNO_(3)`, to coat a metal surface of `80 cm^(2)` with `5 mum` thick layer? Density of sikver `= 10.8 g//cm^(3)`

Explanation

To solve the problem of how long a current of 3 A must be passed through a solution of AgNO3 to coat a metal surface of 80 cm² with a 5 µm thick layer, we can follow these steps:

### Step 1: Calculate the volume of silver to be deposited
First, we need to find the volume of silver that will be deposited on the surface. The volume (V) can be calculated using the formula:

\[ V = \text{Area} \times \text{Thickness} \]

Given:
- Area = 80 cm²
- Thickness = 5 µm = \( 5 \times 10^{-4} \) cm

So,

\[ V = 80 \, \text{cm}^2 \times 5 \times 10^{-4} \, \text{cm} = 0.04 \, \text{cm}^3 \]

### Step 2: Calculate the mass of silver to be deposited
Next, we can find the mass (m) of silver using the density (d) of silver. The mass can be calculated using the formula:

\[ m = \text{Density} \times \text{Volume} \]

Given:
- Density of silver = 10.8 g/cm³

So,

\[ m = 10.8 \, \text{g/cm}^3 \times 0.04 \, \text{cm}^3 = 0.432 \, \text{g} \]

### Step 3: Calculate the equivalent weight of silver
The equivalent weight (E) of silver (Ag) can be calculated using the formula:

\[ E = \frac{\text{Molar mass}}{n} \]

Where:
- Molar mass of silver (Ag) = 107.87 g/mol
- Valency (n) of silver = 1 (since it forms Ag⁺ ions)

So,

\[ E = \frac{107.87 \, \text{g/mol}}{1} = 107.87 \, \text{g/equiv} \]

### Step 4: Use Faraday's law to find the time
According to Faraday's law, the weight of the substance deposited is given by:

\[ \frac{w}{E} = \frac{I \cdot t}{F} \]

Where:
- \( w \) = mass of silver deposited = 0.432 g
- \( E \) = equivalent weight of silver = 107.87 g/equiv
- \( I \) = current = 3 A
- \( t \) = time in seconds
- \( F \) = Faraday's constant = 96500 C/equiv

Rearranging the formula to solve for time \( t \):

\[ t = \frac{w \cdot F}{E \cdot I} \]

Substituting the values:

\[ t = \frac{0.432 \, \text{g} \times 96500 \, \text{C/equiv}}{107.87 \, \text{g/equiv} \times 3 \, \text{A}} \]

Calculating:

\[ t = \frac{41664}{323.61} \approx 128.92 \, \text{seconds} \]

### Final Answer
The time required to pass a current of 3 A through the solution to coat the metal surface is approximately **128.92 seconds**.

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