The Lyman Series of the Hydrogen Spectrum

The Lyman series is a series of spectral lines in the hydrogen spectrum that corresponds to the transitions of electrons from higher energy levels to the ground state. These transitions result in the emission of ultraviolet (UV) light. The wavelengths of the spectral lines in the Lyman series can be calculated using the Rydberg formula.

The Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of spectral lines in the hydrogen spectrum. It is given by:

1/λ = R_H * (1/n₁² - 1/n₂²)

Where:
- λ is the wavelength of the emitted light
- R_H is the Rydberg constant for hydrogen (approximately 1.097 × 10^7 m⁻¹)
- n₁ and n₂ are positive integers representing the energy levels of the electron

Calculating the Wavelengths of the Spectral Lines

To calculate the wavelengths of the first four spectral lines in the Lyman series, we need to determine the energy levels (n₁ and n₂) involved in each transition.

1. Transition from n₁ = 2 to n₂ = 1
- Plugging the values into the Rydberg formula:
1/λ = 1.097 × 10^7 * (1/1² - 1/2²)
- Simplifying the equation:
1/λ = 1.097 × 10^7 * (1 - 1/4)
1/λ = 1.097 × 10^7 * (3/4)
- Solving for λ:
λ = 4/(3 * 1.097 × 10^7) = 91.18 nm

2. Transition from n₁ = 3 to n₂ = 1
- Plugging the values into the Rydberg formula:
1/λ = 1.097 × 10^7 * (1/1² - 1/3²)
- Simplifying the equation:
1/λ = 1.097 × 10^7 * (1 - 1/9)
1/λ = 1.097 × 10^7 * (8/9)
- Solving for λ:
λ = 9/(8 * 1.097 × 10^7) = 121.56 nm

3. Transition from n₁ = 4 to n₂ = 1
- Plugging the values into the Rydberg formula:
1/λ = 1.097 × 10^7 * (1/1² - 1/4²)
- Simplifying the equation:
1/λ = 1.097 × 10^7 * (1 - 1/16)
1/λ = 1.097 × 10^7 * (15/16)
- Solving for λ:
λ = 16/(15 * 1.097 × 10^7) = 103.2 nm

4. Transition from n₁ = 5 to n₂ = 1
- Plugging