To calculate the ionization energy of a sodium atom in kJ per mole given that electromagnetic radiation of wavelength 100 nm is sufficient to ionize it, we can follow these steps:
### Step 1: Convert Wavelength to Meters
The wavelength (λ) is given as 100 nm. We need to convert this into meters for our calculations.
\[
\text{Wavelength in meters} = 100 \, \text{nm} = 100 \times 10^{-9} \, \text{m} = 1 \times 10^{-7} \, \text{m}
\]
**Hint:** Remember that 1 nm = \(10^{-9}\) m.
### Step 2: Use the Photon Energy Formula
The energy (E) of a photon can be calculated using the formula:
\[
E = \frac{hc}{\lambda}
\]
Where:
- \(h\) = Planck's constant = \(6.626 \times 10^{-34} \, \text{J s}\)
- \(c\) = speed of light = \(3 \times 10^{8} \, \text{m/s}\)
- \(\lambda\) = wavelength in meters
Substituting the values:
\[
E = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3 \times 10^{8} \, \text{m/s})}{1 \times 10^{-7} \, \text{m}}
\]
### Step 3: Calculate the Energy
Now, we perform the calculation:
\[
E = \frac{(6.626 \times 10^{-34}) \times (3 \times 10^{8})}{1 \times 10^{-7}} = 1.9878 \times 10^{-18} \, \text{J}
\]
**Hint:** Ensure you multiply the constants correctly and pay attention to the powers of ten.
### Step 4: Convert Energy to kJ per Mole
To find the ionization energy in kJ per mole, we need to multiply the energy per photon by Avogadro's number (\(N_A = 6.022 \times 10^{23} \, \text{mol}^{-1}\)):
\[
\text{Ionization Energy per mole} = E \times N_A
\]
\[
\text{Ionization Energy per mole} = (1.9878 \times 10^{-18} \, \text{J}) \times (6.022 \times 10^{23} \, \text{mol}^{-1})
\]
Calculating this gives:
\[
\text{Ionization Energy per mole} = 1192.8 \, \text{J/mol}
\]
### Step 5: Convert Joules to Kilojoules
Finally, we convert the energy from joules to kilojoules:
\[
\text{Ionization Energy in kJ/mol} = \frac{1192.8 \, \text{J}}{1000} = 1.1928 \, \text{kJ/mol}
\]
**Hint:** Remember that to convert from joules to kilojoules, you divide by 1000.
### Final Answer
The ionization energy of the sodium atom is approximately **1.1928 kJ/mol**.
