Suppose the energy liberated in the in the recombination of a hole-electron pair is converted into electromagnetic radiation.If the maximum wavelength emmited is 820nm,What is the band gap?

Options

A. 0.5 e V
B. 2.5 e V
C. 1.5 e V
D. 15 e V

Explanation

To find the band gap energy (E_g) from the maximum wavelength emitted (820 nm), we can follow these steps:

### Step-by-Step Solution:

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,
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength in meters.

2. **Convert the wavelength from nanometers to meters**:
Given the wavelength \( \lambda = 820 \, \text{nm} \):
\[ \lambda = 820 \times 10^{-9} \, \text{m} \]

3. **Substitute the values into the energy formula**:
\[ E = \frac{(6.626 \times 10^{-34} \, \text{J s})(3 \times 10^8 \, \text{m/s})}{820 \times 10^{-9} \, \text{m}} \]

4. **Calculate the energy**:
\[ E = \frac{(6.626 \times 3) \times 10^{-34 + 8}}{820 \times 10^{-9}} \]
\[ E = \frac{19.878 \times 10^{-26}}{820 \times 10^{-9}} \]
\[ E = \frac{19.878}{820} \times 10^{-26 + 9} \]
\[ E = 0.0242 \times 10^{-17} \, \text{J} \]
\[ E \approx 2.42 \times 10^{-19} \, \text{J} \]

5. **Convert the energy from Joules to electron volts**:
To convert Joules to electron volts, use the conversion factor \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \):
\[ E_g = \frac{2.42 \times 10^{-19} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \approx 1.51 \, \text{eV} \]

6. **Final Result**:
The band gap energy \( E_g \) is approximately \( 1.51 \, \text{eV} \).