Bohr's Theory of Hydrogen Atom

Niels Bohr suggested that the problem about hydrogen spectrum can be solved if we can make some assumptions. According to classical theory, the frequency of the electromagnetic waves emitted by a revolving electron is equal to the frequency of revolution. As the electrons radiate energy, their angular velocities would change continuously and they would emit a continuous spectrum against line spectrum actually observed. So, Bohr concluded that even if electromagnetic theory successfully explained the macroscopic phenomenon, it could not be applied to explain microscopic phenomenon, that in atomic scale. So he made bold suggestions called as Bohr's postulates.

Bohr's Theory of Hydrogen Atom

Postulates

(i) Every atom consists of nucleus and suitable number of electrons revolved around the nucleus in circular orbits.

(ii) Electrons revolved only in certain non-radiating orbits called stationery orbits for which the total angular momentum is an integral multiple of h/2p where h is plank's constant.

L is the Angular momentum of the revolving electrons

(iii) Radiation occurs when an electron jumps from one permitted orbit to another. It is emitted when electron jumps from higher orbit to a lower orbit

i.e., E₂ - E₁ = hf, where f is frequency of radiation.

Derivation of Radius

Radii of orbits

According to Bohr's second postulate

Since

Where m is mass of electron, v is linear velocity, r is radius of orbit in which e revolves around the nucleus.

Now

[Because necessary centripetal force is provided by the electrostatic force of attraction between electron and nucleus] whose charge is Ze where Z is the atomic number of the atom.

Substituting for v,

For Hydrogen atom Z = 1

i.e., r a n² the stationary orbits are not equally spaced

On substituting the value h = 6.6x10^-34 J-sec

n = 1

K = 9 x 10⁹Nm²/c²

m = 9.1 x 10^-31kg

e = 1.6 x 10^-19c, we get

r = 5.29 x 10^-11m

This is called the Bohr radius.

Velocity of electron in a stationary orbit substituting the expression for r in the equation.

We get

The resulting expression is

Calculation shows that when n=1, velocity v of the electron is 1/137 time velocity of light is vacuum i.e.,

The total energy (T.E) of the electron in stationary orbit

The energy of electron revolving in a stationery orbit is of two types. Kinetic energy due to velocity and potential energy due to the position of the electron.

Now

(-ve is for charge of an electron)

Now T.E = K.E + P.E

Spectral series of hydrogenations.

On putting the value m , k , e, h, we get

For hydrogen

The -ve sign implies that electron is bound to the nucleus. As n increases, the total energy of electron is more than that in the inner orbits.