Neil Bohr proposed the Bohr atomic model in 1915. It came about through the transformation of Rutherford's atomic model. Rutherford's model introduces the nuclear atom model, explaining that negative charge electrons surround the nucleus (positively charged).
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 a proton) with a velocity v and r is the radius of the orbit in which the electron is revolving.
For an electron orbiting in a hydrogen atom the necessary centripetal force is the electrostatic force of attraction between the negative electron and the massive, positively charged proton that is the nucleus.
F = k = (where ∈o is permittivity of free space) k = 9 ×10^{9 }Nm2C^{2 }in SI unit. In C.G.S. units, value of k = 1 dyne cm2 (esu)–2 The centrifugal force acting on the electron = 

Since the electrostatic force balances the centrifugal force, for the stable electron orbit. i.e., electrostatic force = centrifugal force. Thus,
∴ = …(1)
or v2 = …(2)
According to Bohr’s postulate of angular momentum quantization, we have
or mvr =
or v =
or v2 = …(3)
Equating equation (2) and (3) we get
=
∴r =
where n = 1,2,3      ∞
The greater the value of n, farther is the energy level from the nucleus, and greater is the radius of the orbit.
The radius of the smallest orbit (n=1) for hydrogen atom (Z=1).
= 5.29 ×10^{11 }m = 0.529 Å
Radius of nth orbit for an atom with atomic number Z is simply written as
rn = 0.529 × n^{2}/ZÅ
Velocity of electron, v = …(4)
Radius of the nth orbit, r =.
Substituting value of r in equation (4)
Velocity of the electron in the nth orbit is given as
Velocity of electron in the first orbit of hydrogen atom is
v1 = = 2.18 ×10^{8} cm/s
∴ velocity of an electron in the nth orbit of an atom is given as
vn = 2.18 ×10^{8 }xZ/n cm/s
The total energy E of an electron is the sum of kinetic energy and potential energy.
Kinetic energy of an electron = [Putting the value from equation (1)]
Potential energy =
The −ve sign of the potential energy of the electron is due to the fact that electron is attracted towards the nucleus and energy is released (at infinity the potential energy of the electron is zero).
Total energy,
E = K.E. + P. E.
where n = 1,2,3……….∞
This expression shows that only certain energies are allowed to the electron.
Since this energy expression consist of so many fundamental constant, we are giving you the following simplified expressions.
E = –21.8 ×10^{12}×Z^{2}/n^{2} erg per atom
= –21.8 ×10^{19}×Z^{2}/n^{2}J per atom
= –13.6 × Z^{2}/n^{2} eV per atom
1 electron volt is defined as the amount of energy gained by an electron when it accelerates through a field of 1V.
∴ 1 eV = 1.6 ×10^{19} J
1eV = 3.83 ×10^{23}kcal
1eV = 1.602 ×10^{12}erg
E = –313.6 ×Z^{2}/n^{2 }kcal / mole (1 cal = 4.18 J)
If any students need to take the online test to check their concepts or undertstanding then they can visit Online Test Atomic Structure.