**Bohr’s atomic Theory**

It is based on the application of quantum theory of radiation.

**Postulates:-**

1. The electrons revolve around the nucleus with definite velocity in certain fixed closed paths called orbits. There are numbered as 1, 2, 3, 4…… (Or) K, L, M, N from the Nucleus.

2. The angular momentum of electron is an integral multiple of the factor \(\frac{h}{2\pi }\).

\(mvr=n\frac{h}{2\pi }\)

n – Principle quantum number

m – Mass of electron

v – Velocity of electron

h – Planck’s constant

3. Each stationary state is associated with definite amount of energy.

4. As long as an electron is revolving in an orbit neither loses nor gains energy. Hence these orbits are called stationary states.

5. The energy of an electron changes only when it moves from one orbit to another.

6. Outer orbits have higher energy and Inner orbits have lower energy.

7. The energy is absorbed when electron moves from inner orbit to outer orbit.

The energy is releases when electron jumps from outer orbit to inner orbit.

8. ΔE = E₂ – E₁ = hυ.

**According to Bohr’s theory for hydrogen atom:**

a) Radius of nth orbit Hydrogen atom.

r_{n} = 0.529 x 10⁻⁸ n² cm

If n = 1, r = 0.529 ⁰A

b) For Hydrogen like ions (He⁺, Li²⁺, Be³⁺).

The radii may be given as \({{r}_{n}}=\frac{0.529\times {{n}^{2}}}{Z}\overset{0}{\mathop{A}}\,\)

Z – Atomic number of species

\(r=\frac{{{n}^{2}}{{h}^{2}}}{4{{\pi }^{2}}mkZ{{e}^{2}}}\)

Where, k = \(\frac{1}{4\pi {{\varepsilon }_{0}}}\) = 9 x 10⁹ Nm²/C²

e = charge of electron

c) Energy of electron in hydrogen atom.

\({{E}_{n}}=\frac{-13.6}{{{n}^{2}}}ev/atom\) = \(\frac{-2.18X{{10}^{-18}}}{{{n}^{2}}}\)J

Kinetic energy in n^{th} shell = \(\frac{13.6\times {{Z}^{2}}}{{{n}^{2}}}eV\)

Potential energy in n^{th} shell = \(\frac{-27.2\times {{Z}^{2}}}{{{n}^{2}}}eV\)

d) Energy expression for hydrogen like ions (He⁺, Li²⁺, Be³⁺, is E_{n} = E_{H} x Z².

e) Velocity of electron in the first orbit of hydrogen atom is 2.18 x 10⁸ cms⁻¹.

f) Velocity of electron in the one electron species He⁺, Li²⁺, Be³⁺ is given as V_{n} = \(\frac{2.18\times {{10}^{8}}}{n}\) x Z cm s⁻¹

g) Time period of revolution of electron in n^{th} orbit T_{n} = \(\frac{2\pi r}{{{v}_{n}}}=\frac{{{n}^{3}}}{{{Z}^{2}}}\) x 1.5 x 10⁻¹⁶ sec

h) Orbital frequency: Number of revolutions per second of an electron in a shell. It is given as: \(\frac{{{Z}^{3}}}{{{n}^{3}}}\) x 6.66 x 10¹⁵

i) When an electron returns from n_{2} to n_{1} state, the number of lines in the spectrum will be \(\frac{\left( {{n}_{2}}-{{n}_{1}} \right)\left( {{n}_{2}}-{{n}_{1}}+1 \right)}{2}\)

**Advantages of Bohr’s model:-**

- It explains the stability of the atom.
- It explains the atomic spectrum of hydrogen.
- It also explains hydrogen like species spectrum

Ex: He⁺, Li²⁺and Be³⁺. - Experimentally determined frequencies of spectral line are in close agreement with those calculated by Bohr’s theory.

**Limitations:-**

- It fails to explain spectra of multi electron atoms.
- It could not explain fine structure or atomic spectrum.
- Bohr’s theory is not in agreement with Heisenberg’s uncertainty principle.
- It could not explain the ability of atoms to form molecules by chemical bonding.
- It is unable to explain the splitting of spectral lines in the presence of magnetic field (Zeeman Effect) or an electric field (Stark effect).
- Ignores dual behavior of matter.