Calculate the wavelength of light required to break the bond between two Cl atoms in `Cl_(2)` molecules `((BE)_(Cl-Cl)= 243 kJ mol^(-1))`

Options

A. `4.19 xx10^(-7) m`
B. `4.11 xx 10^(-6) `m
C. `8.81 xx10^(-31)`m
D. `6.26 xx10^(-21)` m

Answer

To calculate the wavelength of light required to break the bond between two Cl atoms in Cl₂ molecules, we can follow these steps:

### Step 1: Understand the relationship between energy and wavelength
The energy (E) required to break a bond can be related to the wavelength (λ) of light using the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \))
- \( c \) is the speed of light (\( 3.00 \times 10^{8} \, \text{m/s} \))

### Step 2: Convert bond energy from kJ/mol to J/atom
The bond energy for Cl₂ is given as \( 243 \, \text{kJ/mol} \). We need to convert this to joules per atom:
1. Convert kJ to J:
\[ 243 \, \text{kJ/mol} = 243 \times 10^{3} \, \text{J/mol} \]
2. Divide by Avogadro's number (\( N_A = 6.022 \times 10^{23} \, \text{atoms/mol} \)) to find the energy per bond (per Cl atom):
\[ E_{\text{per atom}} = \frac{243 \times 10^{3} \, \text{J/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}} \]

### Step 3: Calculate the energy per Cl atom
Now, performing the calculation:
\[ E_{\text{per atom}} = \frac{243 \times 10^{3}}{6.022 \times 10^{23}} \approx 4.03 \times 10^{-19} \, \text{J} \]

### Step 4: Rearrange the energy-wavelength equation to find λ
Now we can rearrange the equation to solve for wavelength \( \lambda \):
\[ \lambda = \frac{hc}{E} \]

### Step 5: Substitute the values into the equation
Substituting the known values:
- \( h = 6.626 \times 10^{-34} \, \text{J s} \)
- \( c = 3.00 \times 10^{8} \, \text{m/s} \)
- \( E = 4.03 \times 10^{-19} \, \text{J} \)

\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.00 \times 10^{8} \, \text{m/s})}{4.03 \times 10^{-19} \, \text{J}} \]

### Step 6: Calculate the wavelength
Now, performing the calculation:
\[ \lambda \approx \frac{1.9878 \times 10^{-25}}{4.03 \times 10^{-19}} \approx 4.92 \times 10^{-7} \, \text{m} \]

### Step 7: Convert to nanometers
To convert meters to nanometers (1 m = \( 10^{9} \) nm):
\[ \lambda \approx 4.92 \times 10^{-7} \, \text{m} = 492 \, \text{nm} \]

### Final Answer
The wavelength of light required to break the bond between two Cl atoms in Cl₂ molecules is approximately **492 nm**.

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