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*Answer can only contain numeric values

QUESTION: 1

For black body radiation, entropy varies with temperature as Find the value of α?

Solution:

For black body radiation,

(Stefan-Boltzman's lane)

The correct answer is: 3

*Answer can only contain numeric values

QUESTION: 2

For a fermi gas (in 8 dimension) value of the ratio C_{p}/C_{V} is?

Solution:

Average energy of Fermious is given by

pV^{5/3}= constant

The correct answer is: 1.667

*Answer can only contain numeric values

QUESTION: 3

A quantity of heat ΔH is transferred from a large heat reservoir at temperature T to another large heat reservoir at temperature T/2. The heat reservoir have such large capacities that there is no observable change in true temperature. The ratio of the entropy change of the entire system to the entropy change in heat reservoir at temperature T/2 is?

Solution:

Now,

The correct answer is: 0.5

*Answer can only contain numeric values

QUESTION: 4

At high temperature, entropy of a spin half system is equal to Nk_{B} time the natural logarithm of what number?

Solution:

zcosh(βμH) = 1 and μHβ[tanh(βμH)] = 0

∴ Entropy for spin 1/2 system which given by

S = Nk[ln{2cosh(βμH)} – βμHtanh(βμH)]

Reduces to S = Nk_{B}ln 2

The correct answer is: 2

*Answer can only contain numeric values

QUESTION: 5

Virial of a system is defined as V = Σq_{i}p_{i} , where q_{i} is a generalized coordinate and p_{i} conjugate momentum. Value of V for a system consisting of N particles at temperature T is equal to aNk_{B}T. Find the value of constant a?

Solution:

Virial theorem states that

V = –2k·E

Since

⇒ V = –3NkT

⇒ a = –3

The correct answer is: -3

*Answer can only contain numeric values

QUESTION: 6

Entropy of a system of N classical harmonic oscillator having frequency ω and at temperature T is given by Find the value of constant x?

Solution:

For Harmonic oscillator

∴ Single Oscillator partition function

∴ N-oscillator partition function

Z = (Z_{1} )N

∴ Helmholtz free energy function

F = –kTlnZ

∴ Entropy

∴ x = 1

The correct answer is: 1

*Answer can only contain numeric values

QUESTION: 7

The temperature of a cavity of fixed volume is doubled. Its energy increases by α times whereas number of photons increase by β times. Then calculate α/β?

Solution:

Energy of blackbody radiation αT^{4} and equilibrium number of photons in cavity αT^{3
}

The correct answer is: 2

*Answer can only contain numeric values

QUESTION: 8

For non interacting fermions entropy value with temperature for T → 0 as . Find the value of α.

Solution:

For T → 0 internal energy

⇒ α = 1

The correct answer is: 1

*Answer can only contain numeric values

QUESTION: 9

The mean internal energy of a one-dimensional harmonic oscillator in equilibrium with a heat bath of temperature T is in the units of k_{B}T.

Solution:

The correct answer is: 1

*Answer can only contain numeric values

QUESTION: 10

A system of N particles is enclosed in a volume V at a temperature T. The logarithm of the partition function is given by ln Z = N ln{V – bN}(k_{B}T)^{3/2} , where b is a constant with approximate dimensions. The internal energy of the gas in the units of Nk_{B}T is given as?

Solution:

The correct answer is: 1.5

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