Electromagnetic radiation of wavelength 242 nm is just sufficient to ionise the sodium atom . Calculate the ionisation energy of sodium in kJ `mol^(-1)`.

Options

A. 494 KJ
B. 8169.5 KJ
C. 5.85 KJ
D. 8.214 KJ

Explanation

To calculate the ionization energy of sodium in kilojoules per mole from the given wavelength of electromagnetic radiation, we can follow these steps:

### Step 1: Convert the wavelength from nanometers to meters
The wavelength (λ) is given as 242 nm. We need to convert this to meters for our calculations.

\[ \text{Wavelength in meters} = 242 \, \text{nm} \times 10^{-9} \, \text{m/nm} = 242 \times 10^{-9} \, \text{m} \]

### Step 2: Use the energy formula for electromagnetic radiation
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) = \( 3 \times 10^{8} \, \text{m/s} \)
- \( \lambda \) = \( 242 \times 10^{-9} \, \text{m} \)

### Step 3: Substitute the values into the formula
Now we can substitute the values into the energy formula:

\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s}) \times (3 \times 10^{8} \, \text{m/s})}{242 \times 10^{-9} \, \text{m}} \]

### Step 4: Calculate the energy
Perform the calculation:

\[ E = \frac{1.9878 \times 10^{-25} \, \text{J m}}{242 \times 10^{-9} \, \text{m}} \approx 8.21 \times 10^{-19} \, \text{J} \]

### Step 5: Convert the energy from joules to kilojoules per mole
To find the ionization energy in kilojoules per mole, we need to multiply the energy of one photon by Avogadro's number (\( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)):

\[ \text{Energy per mole} = E \times N_A = (8.21 \times 10^{-19} \, \text{J}) \times (6.022 \times 10^{23} \, \text{mol}^{-1}) \]

### Step 6: Perform the calculation
Calculating this gives:

\[ \text{Energy per mole} \approx 494000 \, \text{J/mol} = 494 \, \text{kJ/mol} \]

### Final Answer
Thus, the ionization energy of sodium is approximately:

\[ \text{Ionization Energy} \approx 494 \, \text{kJ/mol} \]