Bohr's Theory of Hydrogen Atom

 

Bohr's Theory of Hydrogen Atom

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Postulates

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(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

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Radii of orbits

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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.

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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

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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.