How long current of 3A has to be passed through a solution of silver nitrate to coat a metal surface of `80cm^(2)` with a 0.005mm thick layer?
Density of silver is 10.5 `g//cm^(3)`.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the Volume of Silver Required
To find the volume of silver needed to coat the surface, we use the formula:
\[ \text{Volume} = \text{Area} \times \text{Thickness} \]

Given:
- Area = \( 80 \, \text{cm}^2 \)
- Thickness = \( 0.005 \, \text{mm} = 0.0005 \, \text{cm} \)

Now, substituting the values:
\[ \text{Volume} = 80 \, \text{cm}^2 \times 0.0005 \, \text{cm} = 0.04 \, \text{cm}^3 \]

### Step 2: Calculate the Mass of Silver Required
Using the density of silver, we can find the mass:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]

Given:
- Density of silver = \( 10.5 \, \text{g/cm}^3 \)

Now, substituting the values:
\[ \text{Mass} = 10.5 \, \text{g/cm}^3 \times 0.04 \, \text{cm}^3 = 0.42 \, \text{g} \]

### Step 3: Calculate the Equivalent Weight of Silver
The equivalent weight of silver (Ag) can be calculated using its atomic weight and the number of electrons involved in the reduction:
- Atomic weight of silver (Ag) = \( 108 \, \text{g/mol} \)
- Number of electrons involved (n factor) = 1 (since Ag goes from +1 to 0)

Thus, the equivalent weight of silver is:
\[ \text{Equivalent weight} = \frac{\text{Atomic weight}}{n} = \frac{108 \, \text{g/mol}}{1} = 108 \, \text{g} \]

### Step 4: Use Faraday's Law to Calculate Time
According to Faraday's law, the relationship can be expressed as:
\[ \text{Weight} = \frac{\text{Current} \times \text{Time} \times \text{Equivalent weight}}{96500} \]

Rearranging the formula to find time:
\[ \text{Time} = \frac{\text{Weight} \times 96500}{\text{Current} \times \text{Equivalent weight}} \]

Substituting the known values:
- Weight = \( 0.42 \, \text{g} \)
- Current = \( 3 \, \text{A} \)
- Equivalent weight = \( 108 \, \text{g} \)

Now, substituting:
\[ \text{Time} = \frac{0.42 \, \text{g} \times 96500}{3 \, \text{A} \times 108 \, \text{g}} \]

Calculating:
\[ \text{Time} = \frac{40530}{324} \approx 125 \, \text{seconds} \]

### Final Answer
The time required to coat the metal surface is approximately **125 seconds**.

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