The ratio of the wavelengths of the lines in the Lyman series of a hydrogen atom can be determined using the Rydberg formula. According to the Rydberg formula, the inverse of the wavelength (1/λ) of a spectral line in the hydrogen atom can be calculated as:
1/λ = R * (1/n₁² - 1/n₂²)
Where R is the Rydberg constant and n₁ and n₂ are the principal quantum numbers for the energy levels involved in the transition. For the Lyman series, the initial energy level (n₁) is 1, and the final energy level (n₂) is greater than 1.
To find the ratio between the wavelengths of the first and second lines of the Lyman series, we can calculate the ratio of their respective inverse wavelengths:
(1/λ₁) / (1/λ₂) = (R * (1/n₁² - 1/n₂²)) / (R * (1/n₁² - 1/n₃²))
Since n₂ > n₃, we can simplify the equation to:
(1/λ₁) / (1/λ₂) = (1/n₃²) / (1/n₂²)
The principal quantum numbers for the first and second lines of the Lyman series are n₃ = 3 and n₂ = 2, respectively. Plugging in these values, we have:
(1/λ₁) / (1/λ₂) = (1/3²) / (1/2²) = 4/9
Therefore, the ratio of the wavelength of the first line (λ₁) to the second line (λ₂) in the Lyman series of a hydrogen atom is 4/9.