Chromium plating is applied by electrolysis to objects suspended in a dichlromate solution , according to following `(` unbalanced `)` hald reaction `:`
`Cr_(2)O_(7)^(2-)(aq) +e^(-) +H^(o+)(aq) rarr Cr(s)+H_(2)O(l)`
How many hours would it take to apply a chromium plating of thickness `2.0xx10^(-2)mm` to a car bumper of suface area `0.25m^(2)` in an electrolysis cell carrying a current of `75.0A?`
`[` Density of chromium is `7.19g cm^(-3)]`

Options

A. `2.2h`
B. `1.5h`
C. `3.0h`
D. `0.25h`

Explanation

To solve the problem of how many hours it would take to apply a chromium plating of thickness \(2.0 \times 10^{-2} \, \text{mm}\) to a car bumper with a surface area of \(0.25 \, \text{m}^2\) using an electrolysis cell carrying a current of \(75.0 \, \text{A}\), we can follow these steps:

### Step 1: Convert the thickness from mm to cm
Given thickness:
\[ \text{Thickness} = 2.0 \times 10^{-2} \, \text{mm} = 2.0 \times 10^{-2} \times 10^{-1} \, \text{cm} = 2.0 \times 10^{-3} \, \text{cm} \]

### Step 2: Calculate the volume of chromium to be deposited
Using the formula:
\[ \text{Volume} = \text{Area} \times \text{Thickness} \]
Convert the area from \(m^2\) to \(cm^2\):
\[ \text{Area} = 0.25 \, \text{m}^2 = 0.25 \times 10^4 \, \text{cm}^2 = 2500 \, \text{cm}^2 \]
Now, calculate the volume:
\[ \text{Volume} = 2500 \, \text{cm}^2 \times 2.0 \times 10^{-3} \, \text{cm} = 5.0 \, \text{cm}^3 \]

### Step 3: Calculate the mass of chromium using its density
Using the formula:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]
Given density of chromium:
\[ \text{Density} = 7.19 \, \text{g/cm}^3 \]
Now, calculate the mass:
\[ \text{Mass} = 7.19 \, \text{g/cm}^3 \times 5.0 \, \text{cm}^3 = 35.95 \, \text{g} \]

### Step 4: Calculate the number of moles of chromium
Using the molar mass of chromium (\( \text{Cr} \approx 52 \, \text{g/mol} \)):
\[ \text{Moles of Cr} = \frac{\text{Mass}}{\text{Molar Mass}} = \frac{35.95 \, \text{g}}{52 \, \text{g/mol}} \approx 0.691 \, \text{mol} \]

### Step 5: Write the balanced half-reaction
The half-reaction for chromium plating can be written as:
\[ \text{Cr}_2\text{O}_7^{2-} + 14 \text{H}^+ + 6 e^- \rightarrow 2 \text{Cr} + 7 \text{H}_2\text{O} \]
From the balanced equation, we see that 2 moles of chromium require 6 moles of electrons.

### Step 6: Calculate the total moles of electrons required
For \(0.691 \, \text{mol}\) of chromium:
\[ \text{Moles of electrons} = 0.691 \, \text{mol Cr} \times \frac{6 \, \text{mol e}^-}{2 \, \text{mol Cr}} = 2.073 \, \text{mol e}^- \]

### Step 7: Calculate the total charge required
Using Faraday's constant (\(F = 96500 \, \text{C/mol}\)):
\[ Q = \text{Moles of e}^- \times F = 2.073 \, \text{mol} \times 96500 \, \text{C/mol} \approx 199,000 \, \text{C} \]

### Step 8: Calculate the time required using the current
Using the formula:
\[ Q = I \times t \implies t = \frac{Q}{I} \]
Substituting the values:
\[ t = \frac{199,000 \, \text{C}}{75.0 \, \text{A}} \approx 2653.33 \, \text{s} \]

### Step 9: Convert time from seconds to hours
\[ t \approx \frac{2653.33 \, \text{s}}{3600 \, \text{s/h}} \approx 0.737 \, \text{h} \approx 0.74 \, \text{h} \]

### Conclusion
The time required to apply a chromium plating of thickness \(2.0 \times 10^{-2} \, \text{mm}\) to a car bumper of surface area \(0.25 \, \text{m}^2\) in an electrolysis cell carrying a current of \(75.0 \, \text{A}\) is approximately \(0.74 \, \text{hours}\).

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