Ratio of Longest Wavelengths in Lyman and Balmer Series in Hydrogen Atom Spectrum

The hydrogen atom spectrum consists of a series of spectral lines that are produced when an electron transitions between different energy levels within the atom. These spectral lines are characterized by their wavelengths, and they can be grouped into different series based on the final energy level of the electron.

The Lyman series corresponds to transitions where the electron's final energy level is n = 1, while the Balmer series corresponds to transitions where the final energy level is n = 2. The longest wavelength in each series corresponds to the lowest energy transition within that series.

Lyman Series:
In the Lyman series, the electron transitions from higher energy levels (n > 1) to the lowest energy level (n = 1). The formula for calculating the wavelength of a spectral line in the Lyman series is given by:

1/λ = R_H * (1 - 1/n^2)

Where λ is the wavelength, R_H is the Rydberg constant for hydrogen (approximately 1.097 x 10^7 m^-1), and n is the energy level of the electron.

For the longest wavelength in the Lyman series, we need to find the transition with the smallest change in energy level. This occurs when n = 2. Plugging in n = 2 into the formula, we get:

1/λ_Lyman = R_H * (1 - 1/2^2)
1/λ_Lyman = R_H * (1 - 1/4)
1/λ_Lyman = R_H * (3/4)

Balmer Series:
In the Balmer series, the electron transitions from higher energy levels (n > 2) to the second energy level (n = 2). The formula for calculating the wavelength of a spectral line in the Balmer series is similar to the Lyman series:

1/λ = R_H * (1 - 1/n^2)

For the longest wavelength in the Balmer series, we need to find the transition with the smallest change in energy level. This occurs when n = 3. Plugging in n = 3 into the formula, we get:

1/λ_Balmer = R_H * (1 - 1/3^2)
1/λ_Balmer = R_H * (1 - 1/9)
1/λ_Balmer = R_H * (8/9)

Ratio of Longest Wavelengths:
To find the ratio of the longest wavelengths in the Lyman and Balmer series, we divide the equation for the Lyman series by the equation for the Balmer series:

(1/λ_Lyman) / (1/λ_Balmer) = (R_H * (3/4)) / (R_H * (8/9))
(1/λ_Lyman) / (1/λ_Balmer) = (3/4) / (8/9)
(1/λ_Lyman) / (1/λ_Balmer) = (3/4) * (9/8)
(1/λ_Lyman) / (1/λ_Balmer) = 27/32

So, the ratio of the longest wavelength in the Lyman series to