The Bohr Model & Hydrogen Spectra

Electrons in an atom can only occupy specific fixed energy levels, like rungs on a ladder — they can never sit in between. When an electron jumps down from a higher rung to a lower one, it emits a single photon carrying exactly that energy difference, which is why hydrogen glows in sharp, distinct colors instead of a smooth rainbow.

1/λ = R · (1/n₁² − 1/n₂²)

λ — wavelength (m): the wavelength of light given off by the electron's jump
R — Rydberg constant (1/m): a fixed constant that sets the scale of hydrogen's energy levels
n₁ — lower level (—): the energy level the electron lands on, closer to the nucleus
n₂ — upper level (—): the energy level the electron starts from, farther from the nucleus

An electron in a hydrogen atom drops from energy level n = 3 to energy level n = 2 — the first line of the Balmer series. Find 1/λ in terms of the Rydberg constant R.

n₁ = 2
n₂ = 3

1/λ = R · (1/n₁² − 1/n₂²)
1/λ = R · (1/4 − 1/9) = R · (9/36 − 4/36)

1/λ = 5R/36

Which photon carries more energy: an electron dropping from n = 2 to n = 1 (Lyman), or from n = 3 to n = 2 (Balmer)?

Lyman (n = 2 → 1)
Balmer (n = 3 → 2)
They carry the same energy

For the transition n = 2 to n = 1 (the first Lyman line), what is 1/λ in terms of R?

R/4
3R/4
3R/2