BOHR MODEL for Hydrogen ATOM
The main postulates are:

In an atom electron revolve around the nucleus in certain definite circular paths known as orbit or energy shells.

Each orbit is associated with definite energy hence known as energy levels or shells. These are numbered as 1, 2, 3,…. or K, L, M ….These integral numbers are known as Principal Quantum numbers

Only these orbits are permitted for electron which angular momentum (L = m vr) is whole no. multiple of $ \frac {h}{2 \pi}$ (where n = 1, 2, 3, …)
$L=n \frac {h}{2 \pi}$
$m v r = n \frac {h}{2 \pi}$
where, m = Mass of e = 9.1 X 10^{31 }Kg
v =revolving electron velocity
r = radius of orbit

As long as electron present in a particular orbit, it neither absorbs nor looses energy. Therefore remains constant.

When energy is supplied to electron it absorbs the energy & jumps to higher level & when energy is released it jumps back to the lower level. In doing so it emits energy

The radii of the stationary state orbit is given by
R_{n} =n^{2} a_{0} Where n is the orbit number and a_{0} =52.9 pm
vii. The energy of the stationary state is given by
$E_{n}=2.18 X 10^{18}(\frac {1}{n^2}) Joule $
Here negative sign states that energy of the electron in a atom is less that energy of a free electron.A free electron at rest is infinitely away from the nuclues and is assigned a energy level 0. We can get this value by putting infinity in the above equation
viii. Bohr’s theory can also be applied to the ions containing only one electron, similar to that present in hydrogen atom. For example, He+ ,Li2+, Be3+ and so on.
The energies of the stationary states associated with these kinds of ions (also known as hydrogen like species) are given by the expression.
$E_{n}=2.18 X 10^{18}(\frac {Z^2}{n^2}) Joule $
And the radius of the orbit is given by
$r_n= \frac {52.9 n^2}{Z^2} pm $
Achievements of Bohr Theory:

He explained atomic spectrum of hydrogen atom.

He explained stability of atom.

Bohr theory helped in calculating energy of electron in hydrogen atom & hydrogen like atoms. (i.e. species)
$E =\frac {2r^{2} me^{4}Z^{2}}{n^2r^2}$
$E =\frac {1312 Z^2}{n^2}$ unit – Kila joule/ mole
$E_{n}=2.18 X 10^{18}(\frac {1}{n^2}) $ unit – Joule/ atom
Limitations of Bohr Theory

It could not explain the spectrum of atoms containing more than 1 electron or multi electrons

Bohr Theory failed to explain fine structure of spectral lines.

It could not explain Zeeman effect & stark effect.

This theory failed to explain ability of atoms to form molecule by chemical bonds.

It was not in accordance with Heisenberg uncertainty principle.
Solved Example
Question 1
Calculate the energy associated with with first orbit in He^{+} ion.Also what is the radius of the orbit?
Solution
$E_{n}=2.18 X 10^{18}(\frac {Z^2}{n^2}) Joule $
Here n=1 and Z=2
E=8.72 X10^{18} Joule
And the radius of the orbit is given by
$r_n= \frac {52.9 n^2}{Z^2} pm $
So
r=13.225 pm
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