Roll Number:

PH107(August 27, 2014) QUIZ 1 a. Please write your Name, Roll Number, Division and Tutorial Batch on answer

sheets and Roll Number on the Question Paper. b. All steps must be shown. All explanations must be clearly given for getting credit.

Just the correct final answer does not guarantee the full credit.

c. Possession of mobile phone and exchange of calculators during the examination are

strictly prohibited.

Weightage: 15% Time: 50 minutes

1.

(a) Using equipartition law, find the molar specific heat of a gas (at constant volume),

which contains tri-atomic molecules with the atoms bonded (non rigidly) to each

other in a triangular form.

[2 marks]

Triatomic molecule has 3 atoms and hence total no. of degrees of freedom=3x3=9 Out of these, 3 are translational, 3 are rotational and the remaining 3 are vibrational. According to the equipartition theorem, the average energy per degree of freedom

for translation and rotation is TkB2

1each and that for vibrational freedom is kBT.

………………(1 mark)

Therefore, the total energy per mole is ABB NTkTk

3

2

133 =6NAkBT

And 6 6V A B

dEC N k R

dT ………………(1 mark)

(b) In a dispersive medium, the frequency- wavelength relationship of certain type of

waves is such that phase velocity is ov for a wavelengtho , but is

2o

v for the

wavelength 2o . Find the group velocity of the waves at the wavelengths o and 2

o ,

in terms of o

v . Also find the angular frequency of the wave for these two

wavelengths, in terms of ov and o .

[4 marks]

……………………………………………………..(1 mark)

2

1v

. ., v ( is a constant)

p

p

or kk k

i e Ak k A

0 0

00 0

00 0 0

0 0

0 00 0

0 0

v 2 2v

v ( ) 2v and

vv ( 2 ) 2 v

2

2 v2( ) v v

v 2 v2(

(1 mark)

(2 m2 ) v 2

arks)2 4

g p

g

g

dAk

dk

k

k

2. A photon with energy equal to 23o

m c is scattered by a particle of rest mass om ,

initially at rest. After the scattering, both the photon and the particle move in different

directions making equal angles with the direction of the incident photon. Find (a)

the value of (b) the energy of the scattered photon in terms of 2

om c and (c) de

Broglie wave length of the scattered particle. [5 marks]

h’, h’/c h, h/c

Ee, pe

e

0

'sin sin .............(1)

'cos ............(2)

'p

Putting this in eqn.(2),

'2 cos 2 cos

2cos .'

'2cos

Using Compton effect equat

(1 mark)

(1 mar

ion, '- (1 os )

k)

c

e

e

e

hp

c

h hp

c c

h

c

h hp

c c

or

h

m c

0

2

0

0

. ., (2cos 1) (1 cos )...............(3)

Also, 33

hi e

m c

hc hm c

m c

1

2

0

e 0

3

0

(1 mark)

(1 mark)

(1 ma

putting in equation 3, cos (4 / 5)

8'

' 5

15'

8

' 15p

8

81.3 1 rk

1)0

5dB

e

h

h

h m c

hm c

c

h hnm

p m c

3. A particle of mass m is confined to a region where a potential a

xVV 0

x exists. It is given that V0 has the dimensions of potential and a has the

dimension of length. Both V0 and a are constants. Using the uncertainty principle with

xpx= h, find out the ground state energy of the particle. [4 marks]

0

222

2

2 2

02

2

0

3

1/32 2

003

0

2 2 2

00 02

0

(1 mark)

(1 mark)

(1 mar

2

2

since 0,4

( )2 8

04

4 4

8 8 4

k)

x

x x

xV V

a

x x

hp

x

hp p p

x

p h xE V x V

m mx a

VdE h

dx mx a

Vh h ax

mx a mV

xh h h aE V

mx a m mV

2/3 1/32

0

0 0

1/32 2 2

0

2

1/32 2

0

2

4

4 1

4 8

2(1 m

7

3ark

2)

V h a

a mV

h aV h m

m m h a a

h V

ma

Even if a student has taken x=x, full credit should be give, provided that all

other steps are correct.

Useful data

2 2 2 2 4

3

1 cos

2.43 10 nm

o

o

o

E p c m c

h

m c

h

m c