Calculate the shortest and longest wavelengths in the hydrogen spectrum of the Lyman series.

The Lyman series in the hydrogen spectrum consists of transitions of electrons from higher energy levels down to the first energy level (n=1). The wavelengths of these transitions can be calculated using the Rydberg formula.

Rydberg Formula

The formula is given by:

1/λ = R_H (1/n1² - 1/n2²)

Where:

• λ = wavelength
• 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, ...)

Shortest Wavelength

The shortest wavelength occurs when the transition is from n=2 to n=1:

1/λ = R_H (1/1² - 1/2²)

Calculating this gives:

1/λ = R_H (1 - 1/4) = R_H (3/4)

Thus, λ = 4/3R_H.

Substituting the value of R_H:

λ = 4/(3 × 1.097 x 10^7) ≈ 1.216 x 10^-7 m or 121.6 nm.

Longest Wavelength

The longest wavelength occurs when the transition is from n=∞ to n=1:

1/λ = R_H (1/1² - 1/∞²)

This simplifies to:

1/λ = R_H (1 - 0) = R_H

Thus, λ = 1/R_H.

Calculating this gives:

λ = 1/(1.097 x 10^7) ≈ 9.128 x 10^-8 m or 91.2 nm.

Summary

In the Lyman series of the hydrogen spectrum:

• Shortest wavelength: 121.6 nm
• Longest wavelength: 91.2 nm