Using the Rydberg formula, calculate the wavelength of the first four spectral lines in the Lyman series of the hydrogen spectrum.

The Rydberg formula is a powerful tool for calculating the wavelengths of spectral lines in hydrogen. The formula is given by:

Rydberg Formula

The formula is expressed as:

1/λ = R_H (1/n1² - 1/n2²)

Where:

• λ = wavelength of the emitted light
• R_H = Rydberg constant (approximately 1.097 x 10^7 m^-1)
• n1 = lower energy level (for Lyman series, n1 = 1)
• n2 = higher energy level (n2 = 2, 3, 4, ...)

Calculating Wavelengths for the Lyman Series

For the Lyman series, the transitions occur from higher energy levels down to n1 = 1. The first four lines correspond to:

• Transition from n2 = 2 to n1 = 1
• Transition from n2 = 3 to n1 = 1
• Transition from n2 = 4 to n1 = 1
• Transition from n2 = 5 to n1 = 1

Calculations

Now, let’s calculate the wavelengths:

1. For n2 = 2

1/λ = R_H (1/1² - 1/2²)

1/λ = 1.097 x 10^7 (1 - 1/4) = 1.097 x 10^7 (3/4)

λ = 4.86 x 10^-7 m or 486 nm

2. For n2 = 3

1/λ = R_H (1/1² - 1/3²)

1/λ = 1.097 x 10^7 (1 - 1/9) = 1.097 x 10^7 (8/9)

λ = 3.66 x 10^-7 m or 366 nm

3. For n2 = 4

1/λ = R_H (1/1² - 1/4²)

1/λ = 1.097 x 10^7 (1 - 1/16) = 1.097 x 10^7 (15/16)

λ = 2.73 x 10^-7 m or 273 nm

4. For n2 = 5

1/λ = R_H (1/1² - 1/5²)

1/λ = 1.097 x 10^7 (1 - 1/25) = 1.097 x 10^7 (24/25)

λ = 2.18 x 10^-7 m or 218 nm

Summary of Wavelengths

The wavelengths of the first four spectral lines in the Lyman series are:

• n2 = 2: 486 nm
• n2 = 3: 366 nm
• n2 = 4: 273 nm
• n2 = 5: 218 nm

These wavelengths correspond to the ultraviolet region of the electromagnetic spectrum.