Answer :
To solve this problem, we need to analyze the wavelengths of photons produced by the given transitions of electrons between energy levels. We can use the Rydberg formula to calculate the wavelength of light emitted during a transition between two energy levels in a hydrogen atom.
The Rydberg formula is given by:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength,
- [tex]\(R\)[/tex] is the Rydberg constant ([tex]\(1.097 \times 10^7 \, \text{m}^{-1}\)[/tex]),
- [tex]\(n_i\)[/tex] is the initial energy level,
- [tex]\(n_f\)[/tex] is the final energy level.
Given three transitions:
1. From [tex]\(n = 2\)[/tex] to [tex]\(n = 1\)[/tex],
2. From [tex]\(n = 3\)[/tex] to [tex]\(n = 2\)[/tex],
3. From [tex]\(n = 4\)[/tex] to [tex]\(n = 3\)[/tex],
we can use the Rydberg formula to determine the wavelengths for each transition.
### Calculated Wavelengths
#### Transition from [tex]\(n=2\)[/tex] to [tex]\(n=1\)[/tex]:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \][/tex]
[tex]\[ \lambda = \frac{1}{R \left( \frac{3}{4} \right)} = \frac{4}{3R} \approx 1.215 \times 10^{-7} \, \text{m} \][/tex]
#### Transition from [tex]\(n=3\)[/tex] to [tex]\(n=2\)[/tex]:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{5}{36} \right) \][/tex]
[tex]\[ \lambda = \frac{1}{R \left( \frac{5}{36} \right)} = \frac{36}{5R} \approx 6.563 \times 10^{-7} \, \text{m} \][/tex]
#### Transition from [tex]\(n=4\)[/tex] to [tex]\(n=3\)[/tex]:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = R \left( \frac{1}{9} - \frac{1}{16} \right) = R \left( \frac{7}{144} \right) \][/tex]
[tex]\[ \lambda = \frac{1}{R \left( \frac{7}{144} \right)} = \frac{144}{7R} \approx 1.875 \times 10^{-6} \, \text{m} \][/tex]
### Comparative Results:
- Wavelengths: [tex]\([1.215 \times 10^{-7} \, \text{m}, 6.563 \times 10^{-7} \, \text{m}, 1.875 \times 10^{-6} \, \text{m}]\)[/tex]
- Shortest Wavelength: [tex]\(1.215 \times 10^{-7} \, \text{m}\)[/tex]
- Longest Wavelength: [tex]\(1.875 \times 10^{-6} \, \text{m}\)[/tex]
### Spectrum Classification:
We classify each wavelength according to its corresponding part of the electromagnetic spectrum:
1. Ultraviolet (UV): Wavelengths shorter than [tex]\(400 \times 10^{-9} \, \text{m}\)[/tex].
2. Visible: Wavelengths between [tex]\(400 \times 10^{-9} \, \text{m}\)[/tex] and [tex]\(700 \times 10^{-9} \, \text{m}\)[/tex].
3. Infrared (IR): Wavelengths longer than [tex]\(700 \times 10^{-9} \, \text{m}\)[/tex].
- [tex]\(1.215 \times 10^{-7} \, \text{m}\)[/tex] is in the Ultraviolet (UV) part of the spectrum.
- [tex]\(6.563 \times 10^{-7} \, \text{m}\)[/tex] is in the Visible part of the spectrum (specifically, red light).
- [tex]\(1.875 \times 10^{-6} \, \text{m}\)[/tex] is in the Infrared (IR) part of the spectrum.
### Summary of Results:
- The transition from [tex]\(n = 2\)[/tex] to [tex]\(n = 1\)[/tex] produces photons with the shortest wavelength, [tex]\(1.215 \times 10^{-7} \, \text{m}\)[/tex], which is in the ultraviolet (UV) part of the spectrum.
- The transition from [tex]\(n = 4\)[/tex] to [tex]\(n = 3\)[/tex] produces photons with the longest wavelength, [tex]\(1.875 \times 10^{-6} \, \text{m}\)[/tex], which is in the infrared (IR) part of the spectrum.
- The transition from [tex]\(n = 3\)[/tex] to [tex]\(n = 2\)[/tex] produces photons with a wavelength of [tex]\(6.563 \times 10^{-7} \, \text{m}\)[/tex], which is in the visible part of the spectrum.
The Rydberg formula is given by:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \][/tex]
where:
- [tex]\(\lambda\)[/tex] is the wavelength,
- [tex]\(R\)[/tex] is the Rydberg constant ([tex]\(1.097 \times 10^7 \, \text{m}^{-1}\)[/tex]),
- [tex]\(n_i\)[/tex] is the initial energy level,
- [tex]\(n_f\)[/tex] is the final energy level.
Given three transitions:
1. From [tex]\(n = 2\)[/tex] to [tex]\(n = 1\)[/tex],
2. From [tex]\(n = 3\)[/tex] to [tex]\(n = 2\)[/tex],
3. From [tex]\(n = 4\)[/tex] to [tex]\(n = 3\)[/tex],
we can use the Rydberg formula to determine the wavelengths for each transition.
### Calculated Wavelengths
#### Transition from [tex]\(n=2\)[/tex] to [tex]\(n=1\)[/tex]:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = R \left( 1 - \frac{1}{4} \right) = R \left( \frac{3}{4} \right) \][/tex]
[tex]\[ \lambda = \frac{1}{R \left( \frac{3}{4} \right)} = \frac{4}{3R} \approx 1.215 \times 10^{-7} \, \text{m} \][/tex]
#### Transition from [tex]\(n=3\)[/tex] to [tex]\(n=2\)[/tex]:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = R \left( \frac{1}{4} - \frac{1}{9} \right) = R \left( \frac{5}{36} \right) \][/tex]
[tex]\[ \lambda = \frac{1}{R \left( \frac{5}{36} \right)} = \frac{36}{5R} \approx 6.563 \times 10^{-7} \, \text{m} \][/tex]
#### Transition from [tex]\(n=4\)[/tex] to [tex]\(n=3\)[/tex]:
[tex]\[ \frac{1}{\lambda} = R \left( \frac{1}{3^2} - \frac{1}{4^2} \right) = R \left( \frac{1}{9} - \frac{1}{16} \right) = R \left( \frac{7}{144} \right) \][/tex]
[tex]\[ \lambda = \frac{1}{R \left( \frac{7}{144} \right)} = \frac{144}{7R} \approx 1.875 \times 10^{-6} \, \text{m} \][/tex]
### Comparative Results:
- Wavelengths: [tex]\([1.215 \times 10^{-7} \, \text{m}, 6.563 \times 10^{-7} \, \text{m}, 1.875 \times 10^{-6} \, \text{m}]\)[/tex]
- Shortest Wavelength: [tex]\(1.215 \times 10^{-7} \, \text{m}\)[/tex]
- Longest Wavelength: [tex]\(1.875 \times 10^{-6} \, \text{m}\)[/tex]
### Spectrum Classification:
We classify each wavelength according to its corresponding part of the electromagnetic spectrum:
1. Ultraviolet (UV): Wavelengths shorter than [tex]\(400 \times 10^{-9} \, \text{m}\)[/tex].
2. Visible: Wavelengths between [tex]\(400 \times 10^{-9} \, \text{m}\)[/tex] and [tex]\(700 \times 10^{-9} \, \text{m}\)[/tex].
3. Infrared (IR): Wavelengths longer than [tex]\(700 \times 10^{-9} \, \text{m}\)[/tex].
- [tex]\(1.215 \times 10^{-7} \, \text{m}\)[/tex] is in the Ultraviolet (UV) part of the spectrum.
- [tex]\(6.563 \times 10^{-7} \, \text{m}\)[/tex] is in the Visible part of the spectrum (specifically, red light).
- [tex]\(1.875 \times 10^{-6} \, \text{m}\)[/tex] is in the Infrared (IR) part of the spectrum.
### Summary of Results:
- The transition from [tex]\(n = 2\)[/tex] to [tex]\(n = 1\)[/tex] produces photons with the shortest wavelength, [tex]\(1.215 \times 10^{-7} \, \text{m}\)[/tex], which is in the ultraviolet (UV) part of the spectrum.
- The transition from [tex]\(n = 4\)[/tex] to [tex]\(n = 3\)[/tex] produces photons with the longest wavelength, [tex]\(1.875 \times 10^{-6} \, \text{m}\)[/tex], which is in the infrared (IR) part of the spectrum.
- The transition from [tex]\(n = 3\)[/tex] to [tex]\(n = 2\)[/tex] produces photons with a wavelength of [tex]\(6.563 \times 10^{-7} \, \text{m}\)[/tex], which is in the visible part of the spectrum.
