Calculate the wavelength of light required to break the bond between two chlorine atoms in a chlorine molecule. The Cl- Cl bond energy is 243 kJ `mol^(-1) (h=6.6 xx10^(-34)Js,c=3xx10^8m//s, ` Avogadro's number `=6.02xx10^(23)mol^(-1)` )

Options

A. `8.18xx10^(-31)m `
B. `6.26xx10^(-21)m `
C. `4.91xx10^(-7)m `
D. `4.1xx10^(-6)m `

Answer

To calculate the wavelength of light required to break the bond between two chlorine atoms in a chlorine molecule, we can follow these steps:

### Step 1: Convert Bond Energy to Joules
The bond energy given is 243 kJ/mol. We need to convert this energy into joules.

\[ \text{Energy (E)} = 243 \, \text{kJ/mol} \times 1000 \, \text{J/kJ} = 243000 \, \text{J/mol} \]

### Step 2: Calculate Energy per Molecule
Since we need the energy required to break the bond for a single molecule, we will divide the energy per mole by Avogadro's number.

\[ E_{\text{single}} = \frac{243000 \, \text{J/mol}}{6.02 \times 10^{23} \, \text{mol}^{-1}} \]

Calculating this gives:

\[ E_{\text{single}} = \frac{243000}{6.02 \times 10^{23}} \approx 4.03 \times 10^{-19} \, \text{J} \]

### Step 3: Use the Energy to Find Wavelength
We can use the formula that relates energy to wavelength:

\[ E = \frac{hc}{\lambda} \]

Rearranging this formula to solve for wavelength (\(\lambda\)) gives:

\[ \lambda = \frac{hc}{E} \]

### Step 4: Substitute Known Values
Now, we substitute the values for Planck's constant (\(h = 6.63 \times 10^{-34} \, \text{J s}\)), the speed of light (\(c = 3 \times 10^8 \, \text{m/s}\)), and the energy we calculated for a single molecule.

\[ \lambda = \frac{(6.63 \times 10^{-34} \, \text{J s}) \times (3 \times 10^8 \, \text{m/s})}{4.03 \times 10^{-19} \, \text{J}} \]

Calculating this gives:

\[ \lambda \approx \frac{1.989 \times 10^{-25}}{4.03 \times 10^{-19}} \approx 4.93 \times 10^{-7} \, \text{m} \]

### Step 5: Convert to Nanometers
To express the wavelength in nanometers, we convert meters to nanometers (1 m = \(10^9\) nm):

\[ \lambda \approx 4.93 \times 10^{-7} \, \text{m} \times 10^9 \, \text{nm/m} \approx 493 \, \text{nm} \]

### Final Answer
The wavelength of light required to break the bond between two chlorine atoms in a chlorine molecule is approximately **493 nm**.

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