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

Answer

To solve the problem of how long a current of 2A has to be passed through a solution of AgNO₃ to coat a metal surface of 80 cm² with a 5 µm thick layer, we will follow these steps:

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

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

Calculating the volume:
\[ \text{Volume} = 80 \, \text{cm}^2 \times 5 \times 10^{-4} \, \text{cm} = 0.04 \, \text{cm}^3 \]

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

Given:
- Density of silver = 10.8 g/cm³

Calculating the mass:
\[ \text{Mass} = 10.8 \, \text{g/cm}^3 \times 0.04 \, \text{cm}^3 = 0.432 \, \text{g} \]

### Step 3: Calculate the equivalent weight of silver
The equivalent weight of silver can be calculated using its atomic weight and n-factor:
\[ \text{Equivalent weight} = \frac{\text{Atomic weight}}{n} \]

Given:
- Atomic weight of silver (Ag) = 108 g/mol
- n-factor for Ag = 1 (since Ag⁺ is produced)

Calculating the equivalent weight:
\[ \text{Equivalent weight} = \frac{108}{1} = 108 \, \text{g/equiv} \]

### Step 4: Calculate the number of equivalents of silver
Using the mass and equivalent weight, we can find the number of equivalents:
\[ \text{Number of equivalents} = \frac{\text{Mass}}{\text{Equivalent weight}} \]
Calculating the number of equivalents:
\[ \text{Number of equivalents} = \frac{0.432 \, \text{g}}{108 \, \text{g/equiv}} = 0.004 \, \text{equiv} \]

### Step 5: Use Faraday's law to calculate time
According to Faraday's law:
\[ \text{Number of equivalents} = \frac{I \times t}{96500} \]
Where:
- \(I\) = current in amperes
- \(t\) = time in seconds
- 96500 = Faraday's constant (C/equiv)

Rearranging the formula to solve for time:
\[ t = \frac{\text{Number of equivalents} \times 96500}{I} \]

Substituting the values:
\[ t = \frac{0.004 \times 96500}{2} \]
Calculating time:
\[ t = \frac{386}{2} = 193 \, \text{seconds} \]

### Final Answer
The time required to coat the metal surface is **193 seconds**.

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