Radius of the permitted electronic orbits 

       It is governed by the relation  mvnrn = n[h/2π] where  h is  Planck's constant . Its value is 6.626 ×10−34 joule-seconds. Squaring both sides, m^2 vn^2 rn^2 = n^2 h^2/4 π^2  or vn^2  =   n^2h^2 /4π^2m^2  rn^2 .   Substituting this value in 

                     e^2/[4πεo] m    =  vn^2 rn =  n^2 h^2/4 π^2m^2  rn

                                         rn = [n^2 h^2/4 π^2 m^2] [4πεo m/ e^2 ] =   n^2 εoh^2/mπ e^2 = n^2 ao              It shows that the radius of the orbital electron is directly proportional to n^2 , that is why n is called as principle orbital quantum number. As we go away from the center the orbits are widely separated. For the innermost orbit  n=1, and r1 = ao , a constant called Bohr radius. The Bohr radius  is a fundamental constant in atomic physics, representing the most probable distance between the electron and the nucleus in the hydrogen atom in its ground state. The permitted electronic orbits in the hydrogen atoms have radii rn = n^2 ao. It shows that the radius of higher orbits   in hydrogen atom are 4 ao , 9 ao , 16 ao .....     n^2 ao .and  the spacing between the n th and (n+1) th orbits is (2n+1)ao .  

      The radius of the first orbit in the hydrogen atom is called Bohr radius and is given by   

                                  ao  =  h^2 εo /mπ e^2                                                                                                                                                                      

=[7 x (6.626 x 10^-34)2 x 8.85 x 10^-12]/[22 x (1.602 x 10^-19)2 x 9.108 x 10^-31]                              = 5.289 x 10^-11 m 

 Total energy of the orbital electron          

      The orbital electron has both kinetic energy due to its orbital motion and potential energy by virtue of its position in the electrostatic field. Kinetic energy of the single electron in the hydrogen atom = (1/2) m v^2. Since  e^2/[4πεo]r = mv^2 , K.E = (1/2)[e^2/[(4πεo)r] = e^2/8 πεor.  Potential energy of the electron = - e^2/4 πεor =  - 2 e^2/8 πεor .Total energy is

                                        TE= K.E + P.E = -   e^2/8πεor.                                                         

  It is negative and is responsible for its binding with the nucleus 

     Knowing the practical value of atomic binding energy of the electron in the innermost orbit of the hydrogen atom from the knowledge on its ionization energy, one can estimate Bohr radius. The validity of the theory can be verified by comparing the practical with the theoretical values. 

Binding energy =  13.6 eV  = 13.6 x 1.602 x 10^-19  = 2.1787 x 10^-18 j  ; [1 eV = 1.602 x 10^-19 joule]                e^2/8 πεor = 13.6 eV = 2.1787 x 10^-18 j

or         r = [1.602 x 10^-19]^2 / 8 x (22/7) x 8.85 x 10^-12 x 2.1787 x 10^-18                                                                                                                                                                                                              = [7 x (1.602)^2  x 10^-38] / 8 x 22 x 8.85 x 2.1787 x 10^-30                                                                   = [17.964828/3393.54312] x 10^-8                                                                                                               = 5.293 x 10^-11 m                                                                                                                                              It is very same as that of Bohr radius derived earlier.