Find the thickness of the electro silver if the surface area over which deposition occurred was `100 cm^(2)` and a current of `0.2A` flowed
for `1 hr` with the cathode efficiency of `80%` . Density of `Ag=10 g//c c (Ag=108)`.

Options

A. `6.4xx10^(-5)cm`
B. `6.4xx10^(-4)cm`
C. `6.4xx10^(-7)cm`
D. `6.4xx10^(-8)cm`

Explanation

To find the thickness of the electro-deposited silver, we can follow these steps:

### Step 1: Understand the Given Data
- Surface area (A) = 100 cm²
- Current (I) = 0.2 A
- Time (t) = 1 hour = 3600 seconds
- Cathode efficiency = 80% = 0.8
- Density of silver (D) = 10 g/cm³
- Atomic weight of silver (Ag) = 108 g/mol
- Equivalent weight of silver (n = 1, as it gains one electron)

### Step 2: Calculate the Total Charge (Q)
Using the formula:
\[ Q = I \times t \]
\[ Q = 0.2 \, \text{A} \times 3600 \, \text{s} = 720 \, \text{C} \]

### Step 3: Calculate the Effective Charge Considering Cathode Efficiency
The effective charge (Q_eff) is given by:
\[ Q_{\text{eff}} = Q \times \text{Efficiency} \]
\[ Q_{\text{eff}} = 720 \, \text{C} \times 0.8 = 576 \, \text{C} \]

### Step 4: Calculate the Mass of Silver Deposited (W)
Using Faraday's law:
\[ W = \frac{Q_{\text{eff}} \times \text{Equivalent Weight}}{96500} \]
Where the equivalent weight of silver is:
\[ \text{Equivalent Weight} = \frac{\text{Atomic Weight}}{n} = \frac{108 \, \text{g/mol}}{1} = 108 \, \text{g/mol} \]

Thus:
\[ W = \frac{576 \, \text{C} \times 108 \, \text{g/mol}}{96500} \]

Calculating W:
\[ W = \frac{62208}{96500} \approx 0.645 \, \text{g} \]

### Step 5: Calculate the Volume of Silver Deposited (V)
Using the formula:
\[ V = \frac{W}{D} \]
Where D is the density of silver:
\[ V = \frac{0.645 \, \text{g}}{10 \, \text{g/cm}^3} = 0.0645 \, \text{cm}^3 \]

### Step 6: Calculate the Thickness (X)
The volume of the deposited silver can also be expressed as:
\[ V = A \times X \]
Where A is the surface area. Rearranging gives:
\[ X = \frac{V}{A} \]
Substituting the values:
\[ X = \frac{0.0645 \, \text{cm}^3}{100 \, \text{cm}^2} = 0.000645 \, \text{cm} \]

### Step 7: Convert Thickness to Scientific Notation
\[ X = 6.45 \times 10^{-4} \, \text{cm} \]

### Final Answer
The thickness of the electro-deposited silver is approximately:
\[ \text{Thickness} = 6.45 \times 10^{-4} \, \text{cm} \]

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