In the spectrum of hydrogen, the ratio of the longest wavelength in the Lyman series to the longest wavelength in the Balmer series is:

The Rydberg formula for the wavelength of light emitted during electronic transitions in a hydrogen atom is given by:

1/λ = R * (1/n₁² - 1/n₂²)

where:

• λ is the wavelength of the emitted light
• R is the Rydberg constant (approximately 1.097 x 10⁷ m⁻¹)
• n₁ and n₂ are integers representing the principal quantum numbers of the initial and final energy levels (n₂ > n₁)

For the Lyman series, n₁ = 1. The longest wavelength in this series corresponds to the transition from n₂ = 2 to n₁ = 1.

1/λ₁ = R * (1/1² - 1/2²) = R * (1 - 1/4) = (3/4)R

λ₁ = 4/(3R)

For the Balmer series, n₁ = 2. The longest wavelength in this series corresponds to the transition from n₂ = 3 to n₁ = 2.

1/λ₂ = R * (1/2² - 1/3²) = R * (1/4 - 1/9) = (5/36)R

λ₂ = 36/(5R)

The ratio of the longest wavelength in the Lyman series to the longest wavelength in the Balmer series is:

λ₁/λ₂ = (4/(3R)) / (36/(5R)) = (4/(3R)) * (5R/36) = 20/108 = 5/27

Therefore, the ratio is 5/27. None of the given options match this result. There may be an error in the provided options.