To solve the question regarding the dissociation of molecular chlorine and the effect of light on this process, we will follow these steps:
### Step 1: Understand the Endothermic Process
The dissociation of molecular chlorine (Cl₂) is an endothermic reaction, meaning it requires energy input to break the bonds. The given enthalpy change (ΔH) for the dissociation is 243.6 kJ/mol.
### Step 2: Relate Energy to Wavelength
The energy (E) required for the dissociation can be related to the wavelength (λ) of the light using the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy in joules,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3.00 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength in meters.
### Step 3: Convert ΔH to Joules
Since ΔH is given in kJ/mol, we need to convert it to joules:
\[ \Delta H = 243.6 \, \text{kJ/mol} = 243.6 \times 10^3 \, \text{J/mol} \]
### Step 4: Calculate the Critical Wavelength
Using the relationship between energy and wavelength, we can rearrange the equation to find the critical wavelength:
\[ \lambda = \frac{hc}{E} \]
Substituting \( E \) with ΔH per mole and using Avogadro's number (Na = \( 6.022 \times 10^{23} \, \text{mol}^{-1} \)):
\[ E = \frac{\Delta H}{N_a} = \frac{243.6 \times 10^3 \, \text{J/mol}}{6.022 \times 10^{23} \, \text{mol}^{-1}} \]
Now substituting this value into the wavelength equation:
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.00 \times 10^8 \, \text{m/s})}{\frac{243.6 \times 10^3}{6.022 \times 10^{23}}} \]
### Step 5: Perform the Calculation
Calculating the energy:
\[ E = \frac{243.6 \times 10^3}{6.022 \times 10^{23}} \approx 4.05 \times 10^{-19} \, \text{J} \]
Now substituting back to find λ:
\[ \lambda = \frac{(6.626 \times 10^{-34})(3.00 \times 10^8)}{4.05 \times 10^{-19}} \approx 4.91 \times 10^{-7} \, \text{m} = 491 \, \text{nm} \]
### Step 6: Analyze the Wavelength
The critical wavelength calculated is 491 nm. For dissociation to occur, the wavelength of the light must be less than this critical wavelength (λ < 491 nm).
### Conclusion
Thus, light with a wavelength smaller than 491 nm can cause the dissociation of molecular chlorine. Light with a wavelength larger than 491 nm will not have enough energy to dissociate the chlorine molecules.
