The wavelength of the spectral line that corresponds to a transition in hydrogen atom from `n = 10` to ground state would be [In which part of electromagnetic spectrum this line liues?]`

Options

A. `92.25 nm`, ultraviolet
B. `92.25 nm`, infrared
C. `86.95 nm`, ultraviolet
D. `97.65 nm`, ultraviolet

Explanation

To find the wavelength of the spectral line corresponding to a transition in a hydrogen atom from \( n = 10 \) to the ground state (\( n = 1 \)), we can use the Rydberg formula for hydrogen, which relates the wavelengths of spectral lines to the principal quantum numbers involved in the transition.

### Step-by-Step Solution:

1. **Identify the Energy Levels**:
The transition is from \( n_2 = 10 \) to \( n_1 = 1 \).

2. **Use the Energy Formula**:
The energy difference (\( \Delta E \)) between two energy levels in a hydrogen atom can be calculated using the formula:
\[ \Delta E = 13.6 \, \text{eV} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Substituting the values:
\[ \Delta E = 13.6 \, \text{eV} \left( \frac{1}{1^2} - \frac{1}{10^2} \right) = 13.6 \, \text{eV} \left( 1 - \frac{1}{100} \right) = 13.6 \, \text{eV} \left( \frac{99}{100} \right) \]
\[ \Delta E = 13.6 \times 0.99 = 13.464 \, \text{eV} \]

3. **Relate Energy to Wavelength**:
The energy of a photon is also related to its wavelength (\( \lambda \)) by the equation:
\[ E = \frac{hc}{\lambda} \]
Rearranging this gives:
\[ \lambda = \frac{hc}{E} \]
Where \( h \) (Planck's constant) is approximately \( 4.1357 \times 10^{-15} \, \text{eV s} \) and \( c \) (speed of light) is \( 3 \times 10^8 \, \text{m/s} \).

4. **Calculate the Wavelength**:
Substituting the values of \( h \), \( c \), and \( E \):
\[ \lambda = \frac{(4.1357 \times 10^{-15} \, \text{eV s})(3 \times 10^8 \, \text{m/s})}{13.464 \, \text{eV}} \]
\[ \lambda = \frac{1.241 \times 10^{-6} \, \text{eV m}}{13.464 \, \text{eV}} \approx 9.22 \times 10^{-8} \, \text{m} = 92.2 \, \text{nm} \]

5. **Determine the Part of the Electromagnetic Spectrum**:
The calculated wavelength of approximately \( 92.2 \, \text{nm} \) lies in the ultraviolet (UV) region of the electromagnetic spectrum.

### Final Answer:
The wavelength of the spectral line corresponding to the transition from \( n = 10 \) to the ground state is approximately \( 92.2 \, \text{nm} \), which lies in the ultraviolet region of the electromagnetic spectrum.

---