Wavelength for high energy EMR transition in H-atom is 91 nm. What energy is needed for this transition?

Options

A. `1.36` eV
B. `1240` eV
C. `13` eV
D. `13.6` eV

Answer

To calculate the energy needed for the high energy electromagnetic radiation (EMR) transition in a hydrogen atom with a given wavelength of 91 nm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Wavelength (λ) = 91 nm
- Planck's constant (h) = \(6.626 \times 10^{-34}\) Joule seconds
- Speed of light (c) = \(3 \times 10^{8}\) meters/second

2. **Convert Wavelength from Nanometers to Meters**:
- Since \(1 \text{ nm} = 10^{-9} \text{ m}\), we convert:
\[ \lambda = 91 \text{ nm} = 91 \times 10^{-9} \text{ m} = 9.1 \times 10^{-8} \text{ m} \]

3. **Use the Energy Formula**:
- The energy (E) of the photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]

4. **Substitute the Values into the Formula**:
\[ E = \frac{(6.626 \times 10^{-34} \text{ J s}) \times (3 \times 10^{8} \text{ m/s})}{91 \times 10^{-9} \text{ m}} \]

5. **Calculate the Energy**:
- First, calculate the numerator:
\[ hc = 6.626 \times 10^{-34} \times 3 \times 10^{8} = 1.9878 \times 10^{-25} \text{ J m} \]
- Now, divide by the wavelength:
\[ E = \frac{1.9878 \times 10^{-25}}{91 \times 10^{-9}} = 2.18 \times 10^{-18} \text{ J} \]

6. **Convert Energy from Joules to Electron Volts**:
- To convert Joules to electron volts (eV), use the conversion factor \(1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}\):
\[ E = \frac{2.18 \times 10^{-18} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 13.6 \text{ eV} \]

7. **Final Answer**:
- The energy needed for this transition is approximately **13.6 eV**.