the heat of atomization of methane and ethane are 360 KJ/mol and 620 KJ/mol , respectively . The longest Wavelength of light capable of breaking . The c-c bond is :
(Avogadro number`=6.02xx10^(23),h=6.62xx10^(-34)Js)`

Options

A. `2.48 xx 10^(-3)` nm
B. `1.49 xx 10^(3)` nm
C. `2.49 xx 10^(4)` nm
D. `2.48 xx 10^(4)` nm

Explanation

To find the longest wavelength of light capable of breaking the carbon-carbon bond in ethane, we can follow these steps:

### Step 1: Determine the energy required to break the C-C bond.
Given that the heat of atomization of ethane is 620 kJ/mol and that it consists of 6 C-H bonds and 1 C-C bond, we can set up the equation to find the energy of the C-C bond.

1. The energy of one C-H bond is given as 90 kJ/mol.
2. Therefore, the total energy for the 6 C-H bonds is:
\[ \text{Energy of 6 C-H bonds} = 6 \times 90 \text{ kJ/mol} = 540 \text{ kJ/mol} \]
3. The total energy for the atomization of ethane is 620 kJ/mol, which includes the energy of the C-C bond (let's denote it as \( x \)):
\[ 540 \text{ kJ/mol} + x = 620 \text{ kJ/mol} \]
4. Solving for \( x \):
\[ x = 620 \text{ kJ/mol} - 540 \text{ kJ/mol} = 80 \text{ kJ/mol} \]
Thus, the energy required to break the C-C bond is 80 kJ/mol.

### Step 2: Convert the energy from kJ/mol to Joules.
Since we need the energy in Joules for the calculation of wavelength, we convert kJ/mol to J:
\[ 80 \text{ kJ/mol} = 80 \times 10^3 \text{ J/mol} \]

### Step 3: Calculate the energy per bond.
To find the energy per bond, we divide by Avogadro's number:
\[ \text{Energy per bond} = \frac{80 \times 10^3 \text{ J/mol}}{6.02 \times 10^{23} \text{ mol}^{-1}} \approx 1.327 \times 10^{-19} \text{ J} \]

### Step 4: Use the energy-wavelength relationship.
The relationship between energy (E) and wavelength (λ) is given by:
\[ E = \frac{hc}{\lambda} \]
Where:
- \( h = 6.62 \times 10^{-34} \text{ J s} \) (Planck's constant)
- \( c = 3.00 \times 10^8 \text{ m/s} \) (speed of light)

Rearranging for wavelength gives:
\[ \lambda = \frac{hc}{E} \]

### Step 5: Substitute the values into the equation.
Substituting the values we have:
\[ \lambda = \frac{(6.62 \times 10^{-34} \text{ J s})(3.00 \times 10^8 \text{ m/s})}{1.327 \times 10^{-19} \text{ J}} \]

### Step 6: Calculate the wavelength.
Calculating this gives:
\[ \lambda \approx \frac{1.986 \times 10^{-25} \text{ J m}}{1.327 \times 10^{-19} \text{ J}} \approx 1.496 \times 10^{-6} \text{ m} \]

### Step 7: Convert meters to nanometers.
To convert meters to nanometers:
\[ 1.496 \times 10^{-6} \text{ m} = 1.496 \times 10^{3} \text{ nm} \approx 1496 \text{ nm} \]

### Conclusion:
The longest wavelength of light capable of breaking the C-C bond in ethane is approximately **1496 nm**.