Question

To dissociate a molecule into its component atoms, the energy required is 13.2 eV. The frequency of the electromagnetic radiation corresponding to this energy lies in the:

• A. infrared region
• B. visible region
• C. microwave region
• D. ultraviolet region

Hint:

To determine the region of electromagnetic radiation, convert the energy to frequency using $E = h f$, then find the wavelength using $c = f \lambda$. Alternatively, compare the energy directly to typical energy ranges for each spectrum region.

Answer 1

The Correct Option is D

Step 1: Relate the Energy to Frequency Using Planck’s Equation
The energy of a photon is related to its frequency by Planck’s equation: \[ E = h f \] Where $E = 13.2 \, \text{eV}$ is the energy, $h$ is Planck’s constant, and $f$ is the frequency. To find $f$, we need $h$ in appropriate units. Planck’s constant in eV·s is: \[ h = 4.1357 \times 10^{-15} \, \text{eV·s} \] \[ f = \frac{E}{h} = \frac{13.2} \] \[ f = \frac{13.2}{4.1357 \times 10^{-15}} \approx 3.19 \times 10^{15} \, \text{Hz} \] Step 2: Determine the Wavelength
The wavelength $\lambda$ is related to frequency by: \[ c = f \lambda \implies \lambda = \frac{c}{f} \] Where $c$ is the speed of light, $c = 3 \times 10^8 \, \text{m/s}$: \[ \lambda = \frac{3 \times 10^8}{3.19 \times 10^{15}} \approx 9.4 \times 10^{-8} \, \text{m} = 94 \, \text{nm} \] Step 3: Identify the Electromagnetic Spectrum Region
Now, compare the wavelength to the electromagnetic spectrum ranges:
Ultraviolet (UV): 10 nm to 400 nm
Visible: 400 nm to 700 nm
Infrared (IR): 700 nm to 1 mm
Microwave: 1 mm to 1 m
The wavelength of 94 nm falls within the ultraviolet region (10 nm to 400 nm). Step 4: Alternative Approach Using Energy Ranges
Photon energies for different regions:
Microwave: $10^{-5}$ to $10^{-2}$ eV
Infrared: 0.01 to 1.7 eV
Visible: 1.7 to 3.1 eV
Ultraviolet: 3.1 to 124 eV
The energy of 13.2 eV falls within the ultraviolet range (3.1 to 124 eV), confirming our finding. Step 5: Analyze Options
Option (1): infrared region. Incorrect, as 13.2 eV is much higher than the infrared range (up to 1.7 eV).
Option (2): visible region. Incorrect, as 13.2 eV exceeds the visible range (up to 3.1 eV).
Option (3): microwave region. Incorrect, as 13.2 eV is far greater than the microwave range (up to $10^{-2}$ eV).
Option (4): ultraviolet region. Correct, as 13.2 eV and 94 nm fall within the ultraviolet range.