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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Atoms Molecules and Nuclei Page 2

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2.

Bohr Model (for Hydrogen Atom)

 

First Postulate: An electron in an atom revolves around the nucleus in a circular orbit. The centripetal force is provided by the electrostatic force exerted by the nucleus on the electron.

mv2

=

1

 

e2

 

 

 

r

4pe0

r2

  Postulate of stationary orbit

Second Postulate: The electron can revolve around the nucleus only in certain allowed orbits those for which the angular momentum of electron is an integral multiple of . These orbits are called stationary orbits.

mvr

=

nh

 

2p

   Postulate of quantisation of angular momentum

 

where h

= Planck's constant

n

= principle quantum number

 

= 1, 2, 3, 4, .

 

Third Postulate: Energy is radiated by the electron only when it jumps from an outer orbit of higher energy to an inner orbit of lower energy. The radiated energy is in the form of photons of energy hu given by

En2 - En1 = hu

  Postulate of radiation

 

where h

= Planck's constant

u

= frequency of radiation.

 

 

3.

Energy Quantization

 


Main features of Max Planck's quantum theory:

  1. Energy is radiated by a source not continuously but in packets or bundles of energy. Each packet is known as 'photon'.
  2. Each photon has energy hu.

3.      Energy is absorbed or emitted by a body in integral multiples of hu. (Energy is said to be quantised).

 

 

4.

Radius and Energy of Bohr Orbit

 

 

(i) Radius of Bohr Orbit

  1. Find expression for v2 using first postulate

mv2

=

1

 

e2

 

 

 

r

4pe0

r2

\ v2 =

1

 

e2

 

 

4pe0

rm

(1)

  1.  
  2. Find expression for v2 using 2nd postulate

mvr

=

nh

 

2p

\ v2 =

n2h2

 

4p2m2r2

(2)

  1.  
  2. Equating the two values of v2, find expression for 'r'

    R.H.S. of (1) = R.H.S. of (2)

\ r =

e0 n2 h2

 

p m e2

(3)

From equation (3),  rn a n2

 

(ii) Energy of the Electron

 

  1. Write expression for P.E. of an electron
    P.E. = Electrostatic potential due to nucleus
    charge on the electron

P.E. = (- e)

1

 

e

=

- e2

 

 

 

4pe0

r

4pe0r

  1.  
  2. Write expression for kinetic energy of an electron using postulate 1.

K.E.

=

1

m v2

=

e2

 

 

2

8pe0r

 

 

 

 

3.

 

Total energy = P.E. + K.E. = E =

- m e4

 

8e02n2 h2

 

 

 

 

Thus, we see that 

En

a  

1

 

n2

 



 

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