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QUESTION: 1

If light was of purely wave nature, the total energy radiated by a black-body would be (T is the temperature of the blackbody)

Solution:

If light was of purely wave nature, then electromagnetic wave theory predicts that the total energy radiated by a black body over all wavelength is infinite. This is known as ultraviolet catastrophe.

QUESTION: 2

Two particles are said to be distinguishable when

Solution:

In Quantum mechanical description of particles, the de-Broglie wavelength of a particles is a measure of the wave character of the particle and hence, in a way, measures the inherent imprecision in locating a particle. It means that a particle is speed over a length and not localized at precisely fixed spot. Hence, the average distance between two particles (mean free path) should be large as compared to their de-Broglie wavelength for them to be ‘distinguishable’.

Options (a) and (d) are properties that are associated with ‘indistingunishable’ particles.

QUESTION: 3

A reversible heat engine (Carnot engine) with a monatomic gas molecules as the working substance. If a diatomic molecules is used as the working substance, the efficiency of the same engine is

Solution:

The efficiency of a Cannot engine does not depend on the working substance.

*Answer can only contain numeric values

QUESTION: 4

Due to outside heat and green house effect in a glass-sealed room in a building, the energy inside increases at the rate of 0.5 kJ hour (which leads to increase in the temperature of the room). To make the room comfortable. an air conditioner is fitted. The room is to be kept at a cozy 20^{0}C against 40^{0}C outside. Assuming the air-conditioner is able to work at its maximum performance. How many units of electricity is consumed by the air-conditioner (in units of kJ hour)? (upto 2 decimal places)

Solution:

QUESTION: 5

The plot that describes an engine underdoing a thermodynamic process with maximum efficiency is

Solution:

A Cannot engine lias the m axim um efficiency. In the cyclic process of a cannot engine, two path s are isothermals and the other two are adiabatics (or isentropics).

QUESTION: 6

The density fluctuations in a pure substance are maximum at its

Solution:

The number density fluctuations in a grand canonical ensemble is given by

Now, at the critical point of a substance, the isothermal compresibility becomes large (maximum).

As a result, number density fluctuations are Large at critical point.

Hence mass density fluctuations are also large (or maximum) at the critical point.

QUESTION: 7

The density of states for a system of 4 electrons in a 2-D space having total energy in the range (E, E + dE) is proportional to

Solution:

The number of allowed microstates for N particles are

QUESTION: 8

A collection of 20 non-rigid diatomic molecules can be described by a phase space of dimension ________

Solution:

Degree of freedom of one non rigid diatomic molecule

QUESTION: 9

A particle of mass m starts from rest a height H and moves under the gravitational force mg. It strikes a horizontal surface with a coefficient of restitution equal to a perfect one. The phase space diagram of the particle is ('h' is height from the surface)

Solution:

*Answer can only contain numeric values

QUESTION: 10

3 identical spin 1/2 fermions are to be filled in 2 non-degenerate energy states. If a magnetic field is switched on in the direction, how many distinct configurations of fermions in the given energy states arise?

Solution:

3 identical fermions are filled in 2 non degenerate energy states.

These 2 states are possible.

But in presence of a magnetic field, there arises two configuration of each of the above two splits in presence Df magnetic field. Therefore 4 distinct configurations arise for the given system.

*Answer can only contain numeric values

QUESTION: 11

A rigid linear diatomic ideal gas at room temperatrue is expanded at constant presure. What percentage of the heat supplied is used to increase the internal energy of the gas? (Answer must be given upto 2 decimal places)

Solution:

For Linear diatomic molecule (ligid).

QUESTION: 12

A classical ideal gas of atoms with masses 'm' is confined in a 2-D potential given by

at a temperature T. If k_{B }is the Boltzmann constant, the value of

Solution:

*Answer can only contain numeric values

QUESTION: 13

If a body radiates heat to the outside such that the entropy of the universe increases propotional to T^{2}. and the heat radiated is propotional to T^{α}, then α is __________

Solution:

If entropy increases as T^{2}, we have

*Answer can only contain numeric values

QUESTION: 14

Consider a random walker on a square lattice. Walking to each neighbouring site is equally probable. The

probability that the walker is at a site which is only 2 steps away if he takes 4 steps in total is _____ (upto 2

decimal places)

Solution:

To effectively make only 2 steps, 1 out of the four steps lias to be in a direction opposite to one of the other 3.

Case-1 The walker takes a step back just after the 1 st step.

*Answer can only contain numeric values

QUESTION: 15

In S-dimensions, two metals A and B have the ratio of Fermi energies as If the ratio of the number densities of free electrons in the metals is 4 : 9, then the value of S' is _______

Solution:

In S-dimensions. the particle density varies with Fermi energy as

QUESTION: 16

A vertically suspended square plate of side 'a' acts as adsorbent molecules of particular gas having N identical molecules out of a mixture once adsorbed, the gas molecules are free to move in the square plate but do not possess the energy to escape. The average energy of the gas at low temperatures (mga » k_{B}T) is

Solution:

*Answer can only contain numeric values

QUESTION: 17

3- particles are to be distributed in 5 non degenerate energy levels. Let 'S_{D}' be the entropy of such a system if the particles are distinguishable and 'SB be the entropy if they are indistinguishable particles of integral spin. Then S_{D} - S_{B} = ________ k_{B} (upto 2 decimal places)

Solution:

If the particles are distinguishable, then

If the integral spin particles are indistinguishable, then

*Answer can only contain numeric values

QUESTION: 18

3- particles are to be distributed in 5 non degenerate energy levels. Let 'S_{D}' be the entropy of such a system if the particles are distinguishable and 'S_{B}' be the entropy if they are indistinguishable particles of integral spin. Then S_{D}-S_{B} = _____ k_{B} (upto 2 decimal places)

Solution:

The single particle partition function is

QUESTION: 19

Photons can undergo Bose-Einstein condensation because

Solution:

Photon can undergo Bose-Einstein condensation due to the following properties

(i) They have symmetric wavefiuictions with respect to particle exchange

(if) They can have overlapping wavefiuictions.

(iii) The chemical potential of photons is zero.

QUESTION: 20

N non-interacting particles with magnetic moments are subjected to a magnetic field along the direction. Each magnetic moment lias a doubly degenerate energy level zero and non-degenerate energy levels of energies -μH and μH. The collection is in thermal equilibrium with a heat reservoir at temperature T. The magnetisation per particle is

Solution:

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