The energy required to break one mole of `Cl-Cl` bonds in `Cl_2` is `242 kJ mol^-1`. The longest wavelength of light capable of breaking a since `Cl-Cl` bond is

Options

A. 594 nm
B. 640 nm
C. 700 nm
D. 494 nm

Answer

To find the longest wavelength of light capable of breaking a single Cl-Cl bond in Cl2, we can follow these steps:

### Step 1: Convert the bond energy from kJ/mol to J/mol
The energy required to break one mole of Cl-Cl bonds is given as 242 kJ/mol. We need to convert this to joules:

\[ 242 \, \text{kJ/mol} = 242 \times 10^3 \, \text{J/mol} = 242000 \, \text{J/mol} \]

### Step 2: Calculate the energy required to break a single Cl-Cl bond
Since one mole contains Avogadro's number of molecules (\(N_A = 6.022 \times 10^{23}\)), we can find the energy required to break a single Cl-Cl bond by dividing the total energy by Avogadro's number:

\[ E_{\text{single}} = \frac{242000 \, \text{J/mol}}{6.022 \times 10^{23} \, \text{molecules/mol}} \]

Calculating this gives:

\[ E_{\text{single}} \approx 40.18 \times 10^{-20} \, \text{J} \]

### Step 3: Use the energy-wavelength relationship
The relationship between energy (E) and wavelength (\(\lambda\)) is given by 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 (\(3.00 \times 10^8 \, \text{m/s}\))

Rearranging this equation to solve for wavelength gives:

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

### Step 4: Substitute the known values
Now, we can substitute the values of \(h\), \(c\), and \(E_{\text{single}}\) into the equation:

\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3.00 \times 10^8 \, \text{m/s})}{40.18 \times 10^{-20} \, \text{J}} \]

Calculating this gives:

\[ \lambda \approx 0.494 \times 10^{-6} \, \text{m} \]

### Step 5: Convert meters to nanometers
To express the wavelength in nanometers (1 nm = \(10^{-9}\) m):

\[ \lambda \approx 494 \, \text{nm} \]

### Final Answer
Thus, the longest wavelength of light capable of breaking a single Cl-Cl bond is approximately **494 nm**.

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