Bohr's contribution to Rutherford's model of atom :
Rutherford's model of atom failed to explain the stability of atom as well as spectrum of radiations emitted and absorbed by an atom. To explain the concept of stability and the spectrum of an atom, Niels Henrik David Bohr applied Planck’s quantum theory of radiation to Rutherford’s atomic model. He used classical as well as quantum concepts to form his theory.
Postulates of Bohr’s Atom Model
Bohr made the following assumptions popularly known as Postulates of Bohr’s atom model while proposing a new model for the atom:
Postulate 1.
Rutherford’s model of an atom is acceptable to the extent that an atom has a small positively charged core called nucleus where whole mass and positive charge of the atom are supposed to be concentrated.
Postulate 2.
Bohr’s quantization condition for angular momentum states that electrons can revolve only in those energy levels, in which its angular momentum is an integral multiple of $h/2\pi$.
$L = mvr = \frac{nh}{2\pi} \quad \cdots (1)$
Equation (1) is known as Bohr’s quantization of angular momentum. It restricts the number of allowed orbits for electrons.
Here, $m =$ mass of an electron, $v =$ velocity of an electron, $r =$ radius of the orbit, $h =$ Planck’s constant and $n = 1,2,3,\ldots$ (an integer called principal quantum number). In fact, this postulate defines a fixed stable orbit and tells that angular momentum of an electron in an atom is quantized.
Postulate 3.
An electron revolves around the nucleus with a definite fixed energy in a circular path of fixed radius known as stationary state without gaining or losing the energy. The stationary state of electron is also known as energy level.
Postulate 4.
An electron emits energy in the form of a photon or radiation, when it jumps from higher energy level to lower energy.
If $E_i$ and $E_f$ are the energies associated with the orbits of principal quantum numbers $n_i$ and $n_f$ respectively ($n_i > n_f$), then the amount of energy radiated in the form of a photon is given by,
Bohr’s frequency condition.
$h\nu = E_i - E_f$
where, $\nu$ is the frequency of the emitted radiation and $h$ is Planck’s constant.
Eqn. (2) is called Bohr’s frequency condition.
Numerical Example :
Solution: According to Bohr’s postulate,
$L_n = n \frac{h}{2\pi}
Here $n=2$
$L_2 = 2 \times \frac{h}{2\pi} = \frac{h}{\pi} \text{J s}$