Radius and Energy Levels of Hydrogen Atom:

Consider an electron of mass ‘m’ and charge ‘e’ revolving around a nucleus of charge Ze (where, Z = atomic number and e is the charge of the proton) with a tangential velocity v. r is the radius of the orbit in which electron is revolving.

By Coulomb’s Law, the electrostatic force of attraction between the moving electron and nucleus is Coulombic force = KZe²/r²

K = 9 x10⁹ Nm² C^–2

In C.G.S. Units, value of K = 1 dyne cm² (esu)^–2

The centrifugal force acting on the electron is mv²/r

mv²/r= KZe²/r²                                         … (1)

or     v² =   KZe²/mr                                          … (2)

According to Bohr’s postulate of angular momentum quantization, we have

mvr = nh/2π

v = nh/2πmr

Equating (2) and (3) 

KZe2/mr = n²h²/4π²m2r²

Where n = 1, 2, 3 - - - - - ∞ 

Hence only certain orbits whose radii are given by the above equation are available for the electron. The greater the value of n, i.e., farther the energy level from the nucleus the greater is the radius.

The radius of the smallest orbit (n=1) for hydrogen atom (Z=1) is r_o.

r_o = n²h²/4π²me²K = 1² x (6.626 x 10^-34)² / 4x(3.14)² x9x10^-31 x (1.6x10^-19)²x9x10⁹ =5.29 x 10^–11 m=0.529 Å

Radius of n^th orbit for an atom with atomic number Z is simply written as

r_n = 0.529 x n²/z Å