To find the energy of the photon with the critical wavelength that can dissociate molecular chlorine, we can follow these steps:
### Step 1: Understand the relationship between energy and wavelength
The energy of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( E \) is the energy of the photon,
- \( h \) 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 2: Calculate the energy required for dissociation
Given that the dissociation of molecular chlorine has an enthalpy change (\( \Delta H \)) of \( 243.6 \, \text{kJ/mol} \), we need to convert this value into joules:
\[ \Delta H = 243.6 \, \text{kJ/mol} \times 1000 \, \text{J/kJ} = 243600 \, \text{J/mol} \]
### Step 3: Calculate the energy of a single photon
To find the energy of a single photon that can dissociate one mole of chlorine molecules, we use Avogadro's number (\( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)):
\[ E_{\text{photon}} = \frac{\Delta H}{N_A} = \frac{243600 \, \text{J/mol}}{6.022 \times 10^{23} \, \text{mol}^{-1}} \]
Calculating this gives:
\[ E_{\text{photon}} = \frac{243600}{6.022 \times 10^{23}} \approx 4.05 \times 10^{-19} \, \text{J} \]
### Step 4: Calculate the critical wavelength
Now that we have the energy of the photon, we can find the critical wavelength using the energy-wavelength relationship:
\[ \lambda = \frac{hc}{E} \]
Substituting the values:
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.00 \times 10^8 \, \text{m/s})}{4.05 \times 10^{-19} \, \text{J}} \]
Calculating this gives:
\[ \lambda \approx 4.91 \times 10^{-7} \, \text{m} \]
or
\[ \lambda \approx 491 \, \text{nm} \]
### Final Answer
The energy of the photon with the critical wavelength required to dissociate molecular chlorine is approximately \( 4.05 \times 10^{-19} \, \text{J} \) and the corresponding critical wavelength is approximately \( 491 \, \text{nm} \).
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