What quantum number of the hydrogen atom comes closest to giving a 24-nm-diameter electron orbit?

n = 15. A Bohr orbit with n = 15 comes closest to having a 24 nm diameter.

The formula for the radius r of the nth orbital of a hydrogen atom is

r = n^2·a

where

a = the Bohr radius = 0.0529 nm

We can solve this equation to get

n = √ (r/a)

If d = 24 nm, r = 12 nm.

∴ n = √ (12 nm/0.0529 nm) = √227 = 15.1

n must be an integer, so n = 15.