Bohr's Theory of Hydrogen atom

The first Bohr's postulate  states that  the  angular momentum  of the orbital electron is some integral multiples of h/2π.    mvn rn = nh/ 2π

                                            or    2πrn  = n[h/mvn] = nλ

i.e., the circumference of all the  allowed electronic orbits in the hydrogen atom must accommodate some integral number of a kind of wave characterizing the orbital electron - matter wave associated with the electron. It implies that                                                                                  

                                               h/2πm =  vnrn /n = constant                                                       

The wave characteristics of the electron make the electronic orbits to be discrete.

     For stability of the orbital electron in the Hydrogen atom, the electro-static force of attraction between the nucleus and the electron in the circular orbit  must be equal to the centrifugal force at every instant of its motion.  

                             electrostatic force of attraction         =  centrifugal force

                                               e^2/[4πεo] rn^2                          =  mvn^2/rn                                      

                                                e^2/[4πεo] m     = constant    =  vn^2 rn

These two conditions explain the dependence of one variable with the other used in Bohr's theory of hydrogen atom.  

(1) Orbital velocity (vn) Vs Orbital radius (rn) 

     The equation predicts  vn^2rn  = constant i.e, vn is inversely proportional to rn^(1/2)  i.e., as rn increases the orbital velocity decreases. By substituting rn = n^2 ao  we get  vn^2 n^2 ao  = constant , where ao is Bohr radius, the radius of the innermost orbit in hydrogen atom.  

(2) Orbital velocity (vs) Vs Quantum number(n)

    From the above equations one can derive  vn^2 rn =  e^2/ m[4πεo] = constant and  vnrn/n  =  [h/2πm] = constant. Dividing one by the other we have   vn  = [e^2/2n εo h] ,  or nvn = constant ,which indicates that vn is inversely proportional to n.i.e, as we go away from the center ,the orbital electron moves slower.The very same conclusion can be obtained from the equation  vnrn /n =  vn^2 rn / vnn  = constant. since  vn2rn  = constant , vn n must be a constant. Again from the  relation  mvnrn   = nh/2π,   vn rn /n  = constant.  vn rn /n  =  vn^2rn /nvn  or nvn  = constant  as vn2 rn  = constant . 

    One can establish the same relationship between vn and n from the Bohr's condition on the circumference of the electronic orbits. All orbits irrespective of its quantum number accommodate some integral number of matter waves associated with the orbital electron 

                                                              2πrn = nλ = nh/m vn  

                                                            2πm/h = n/vnrn = constant

Since vn^2rn  = constant  nvn = constant

(3) Orbital readius (rn) Vs Quantum number (n)

     The relation  mvnrn   = nh/2π   gives  vnrn/n  = constant.  vn rn /n  =  vn^2 rn /n vn  or n vn  = constant  as vn^2rn  = constant . On Squaring the relation  [vn rn /n]2  = vn^2 rn  /[n^2 /rn] or  n^2 /rn  = constant as vn^2 rn  = constant. It shows that rn  is directly proportional to n^2. By substituting the value of rn  , we get   vn^2 n^2 ao = constant

 (4) Rate of variation of orbital velocity with respect to radius (dvn/drn)

          Differentiating  v^2 r = constant with respect to r, v^2 + 2vr (dv/dr) = 0, which gives  dvn/drn = -vn/2rn. By substituting the values of rn  and the corresponding vn  one can find out the rate with which the orbital velocity falls with radius of the orbit.

rn =  n^2 h^2 εo /mπ e^2   = n^2  ao

mvnrn  =  mvn(n^2)ao =  nh/2π  or  vn  =   h/2π mnao  which states nvn = constant 

Substituting these values in dvn/drn  we arrive at   dvn/drn = - vn/2rn   = -  h/4πmao^2 n^3. i.e.,  dvn/drn  is negative and varies inversely proportional to n3.