To explain line spectra, Neils Bohr proposed that the angular momentum of the electrons orbiting the atom is quantized:

mvr = nh/2p

where m is the mass of the electron, r is the radius of the orbit, and v is the orbital speed of the electron.

The electron is experiencing uniform circular motion. The only force on it is the attractive force between the negative electron and the positive nucleus:

kZe²/r² = mv²/r

Therefore mv² = kZe²/r

If you rearrange Bohr's angular momentum equation to solve for the velocity:

v = nh/2pmr

and then plug that into the equation:

mv² = kZe²/r you get:

mn²h²/4p²m²r² = kZe²/r

Solving this for the radius of the nth orbit gives:

r_n = [h²/4p²mkZe²] n²

r_n = [5.29 x 10^-11 m] n²

5.29 x 10^-11 m is known as the Bohr radius.

So, Bohr's assumption that the angular momentum is quantized produces the result that the radii of the electron's allowed orbits are also quantized.