Thermodynamics And Statistical Physics MCQ

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• Which one of the following is a first order phase transition?
  ক

  Vaporization of a liquid at its boiling point

  খ

  Ferromagnetic to paramagnetic

  গ

  Normal liquid He to superfluid He

  ঘ

  Superconducting to normal state
• The probability that an energy level E at a temperature T is unoccupied by a fermion of chemical potential μ is given by
  ক

  $$\frac{1}{{{e^{\left( {\varepsilon - \mu } \right){k_B}T}} + 1}}$$

  খ

  $$\frac{1}{{{e^{\left( {\varepsilon - \mu } \right){k_B}T}} - 1}}$$

  গ

  $$\frac{1}{{{e^{\left( {\mu - \varepsilon } \right){k_B}T}} + 1}}$$

  ঘ

  $$\frac{1}{{{e^{\left( {\mu - \varepsilon } \right){k_B}T}} - 1}}$$
• The vapour pressure p (in mm of Hg) of a solid, at temperature T, is expressed by In p = 23 - 3863/T and that of its liquid phase by In p = 19 - 3063/T. The triple point (in kelvin) of the material is
  ক

  185

  খ

  190

  গ

  195

  ঘ

  200
• The wavefunctions of two identical particles in stated n and s are given by $${\phi _n}\left( {{r_1}} \right)$$  and $${\phi _s}\left( {{r_2}} \right)$$ , respectively. The particles obey Maxwell-Boltzmann statistics. The state of the combined two particles system is expressed as
  ক

  $${\phi _n}\left( {{r_1}} \right) + {\phi _s}\left( {{r_2}} \right)$$

  খ

  $$\frac{1}{{\sqrt 2 }}\left$$

  গ

  $$\frac{1}{{\sqrt 2 }}\left$$

  ঘ

  $${\phi _n}\left( {{r_1}} \right){\phi _s}\left( {{r_2}} \right)$$
• The dimension of phase space of ten rigid diatomic molecules is
  ক

  5

  খ

  10

  গ

  50

  ঘ

  100
• A system of N non-interacting classical point particles constrained to move on the two-dimensional surface of a sphere. The internal energy of the system is
  ক

  $$\frac{3}{2}$$ Nk_BT

  খ

  $$\frac{1}{2}$$ Nk_BT

  গ

  Nk_BT

  ঘ

  $$\frac{5}{2}$$ Nk_BT
• The phase diagram of a free particle of mass m and kinetic energy E, moving in one-dimensional box with, perfectly elastic walls at x = 0 and x = L, is given by
  ক

  <img src="/images/option-image/engineering-physics/thermodynamics-and-statistical-physics/1689597303-54-1.png" title="Thermodynamics and Statistical Physics mcq question image" alt="Thermodynamics and Statistical Physics mcq question image">

  খ

  <img src="/images/option-image/engineering-physics/thermodynamics-and-statistical-physics/1689597332-54-2.png" title="Thermodynamics and Statistical Physics mcq question image" alt="Thermodynamics and Statistical Physics mcq question image">

  গ

  <img src="/images/option-image/engineering-physics/thermodynamics-and-statistical-physics/1689597353-54-3.png" title="Thermodynamics and Statistical Physics mcq question image" alt="Thermodynamics and Statistical Physics mcq question image">

  ঘ

  <img src="/images/option-image/engineering-physics/thermodynamics-and-statistical-physics/1689597369-54-4.png" title="Thermodynamics and Statistical Physics mcq question image" alt="Thermodynamics and Statistical Physics mcq question image">
• The mean free path of the particles of a gas at a temperature T₀ and pressure p₀ has a value λ₀. If the pressure is increased to 1.5 p₀ and the temperature is reduced to 0.75 T₀, the mean free path
  ক

  remains unchanged

  খ

  is reduced to half

  গ

  is doubled

  ঘ

  is equal to 1.125 λ₀
• For any process, the second law of thermodynamics requires that the change of entropy of the universe be
  ক

  positive only

  খ

  positive or zero

  গ

  zero only

  ঘ

  negative or zero
• The mean internal of a one-dimensional classical harmonic oscillator in equilibrium with a heat bath of temperature T is
  ক

  $$\frac{1}{2}$$k_BT

  খ

  k_BT

  গ

  $$\frac{3}{2}$$k_BT

  ঘ

  3k_BT
• A Carnot cycle operates on a working substance between two reservoirs at temperatures T₁ and T₂, where, T₁ &gt; T₂. During each cycle an amount of heat Q₁ is extracted from the reservoir at T₁ and an amount Q₂ is delivered to the reservoir at T₂. Which of the following statements is incorrect?
  ক

  Work done in one cycle is Q₁ - Q₂

  খ

  $$\frac{{{Q_1}}}{{{T_1}}} = \frac{{{Q_2}}}{{{T_2}}}$$

  গ

  Entropy of the hotter reservoir decreases

  ঘ

  Entropy for the universe (consisting of the working substance and the two reservoirs) increases
• The pressure for a non-interacting Fermi gas with internal energy U at temperature T is
  ক

  $$p = \frac{3}{2}\frac{U}{V}$$

  খ

  $$p = \frac{2}{3}\frac{U}{V}$$

  গ

  $$p = \frac{3}{5}\frac{U}{V}$$

  ঘ

  $$p = \frac{1}{2}\frac{U}{V}$$
• The pressure versus temperature diagram of a given system at certain low temperature range is found to be parallel to the temperature axis in the liquid to solid transition region. The change in the specific volume remains constant in this region. The conclusion one can get from the above is
  ক

  the entropy of solid is zero in this temperature region

  খ

  the entropy increases when the system goes from liquid to solid phase in this temperature region

  গ

  the entropy decreases when the system transforms from liquid to solid phase in this region

  ঘ

  the change in entropy is zero in the liquid to solid transition region
• The total number of accessible states of N non-interacting particles of spin $$\frac{1}{2}$$ is
  ক

  2^N

  খ

  N²

  গ

  $${2^{\frac{{\text{N}}}{2}}}$$

  ঘ

  N
• Each of the two isolated vessels, A and B of fixed volumes, contains N molecules of a perfect monatomic gas at pressure p. The temperatures of A and B are T₁and T₂ respectively. The two vessels are brought into thermal contact. At equilibrium, the change in entropy is
  ক

  $$\frac{3}{2}N{k_B}\ln \left$$

  খ

  $$N{k_B}\ln \left( {\frac{{{T_2}}}{{{T_1}}}} \right)$$

  গ

  $$\frac{3}{2}N{k_B}\ln \left$$

  ঘ

  $$2N{k_B}$$
• A second order phase transition is one in which
  ক

  the plot of entropy as a function of temperature shows a discontinuity

  খ

  the plot of specific heat as a function of temperature shows a discontinuity

  গ

  the plot of volume as a function of pressure shows a discontinuity

  ঘ

  the plot of comprehensibility as a function of temperature is continuous
• A sample of ideal gas with initial pressure p and volume V is taken through an isothermal expansion proceed during which the change in entropy is found to be ΔS. The universal gas constant is R. Then the work done by the gas is given by
  ক

  $$\frac{{pV\Delta S}}{{nR}}$$

  খ

  $$nR\Delta S$$

  গ

  $$pV$$

  ঘ

  $$\frac{{p\Delta S}}{{nRV}}$$
• The partition function of a single gas molecule is $${{Z_\alpha }}$$ . The partition function of N such non-interacting gas molecules is then given by
  ক

  $$\frac{{{{\left( {{Z_\alpha }} \right)}^N}}}{{N!}}$$

  খ

  $${\left( {{Z_\alpha }} \right)^N}$$

  গ

  $$N\left( {{Z_\alpha }} \right)$$

  ঘ

  $$\frac{{{{\left( {{Z_\alpha }} \right)}^N}}}{N}$$
• The specific heat of an ideal Fermi gas in three-dimensions at very low temperature (T) varies as
  ক

  T

  খ

  $${{\text{T}}^{\frac{3}{2}}}$$

  গ

  T²

  ঘ

  T³
• Identify which one is a first order phase transition?
  ক

  A liquid to gas transition at its critical temperature

  খ

  A liquid to gas transition close to its triple point

  গ

  A paramagnetic to ferromagnetic transition in the absence of a magnetic field

  ঘ

  A metal to superconductor transition in the absence of a magnetic field
• Two particles are said to be distinguishable when
  ক

  the average distance between them is large compared to their de-Broglie wavelengths

  খ

  the average distance between them is small compared to their de-Broglie wavelengths

  গ

  they have overlapping wavepackets

  ঘ

  their total wavefunction is symmetric under particle exchange
• The free energy of a photon gas enclosed in a volume V is given by $$F = - \frac{1}{3}aV{T^{ - 4}},$$    where a is a constant and T is the temperature of the gas. The chemical potential of the photon gas is
  ক

  zero

  খ

  $$\frac{4}{3}aV{T^3}$$

  গ

  $$\frac{1}{3}a{T^{ - 4}}$$

  ঘ

  $$aV{T^{ - 4}}$$
• Thermodynamic variables of a system can be volume V, pressure p, temperature T, number of particles N, internal energy E and chemical potential μ, etc. For a system to be specified by Microcanonical (MC), Canonical Ensemble (CE) and Grand Canonical (GC) ensembles, the parameters required for the respective ensembles are
  ক

  MC : (N, V, T); CE : (E, V, N); GC : (V, T, μ)

  খ

  MC : (E, V, N); CE : (N, V, T); GC : (V, T, μ)

  গ

  MC : (V, T, μ); CE : (N, V, T); GC : (E, V, N)

  ঘ

  MC : (E, V, N); CE : (V, T, μ): GC : (N, V, T)
• Which among the following sets for Maxwell relation is correct? (U-internal energy, H-enthalpy, A-Helmholtz free energy and G-Gibbs free energy)
  ক

  $$T = {\left( {\frac{{\partial U}}{{\partial V}}} \right)_S}{\text{ and }}p = {\left( {\frac{{\partial U}}{{\partial S}}} \right)_V}$$

  খ

  $$V = {\left( {\frac{{\partial H}}{{\partial p}}} \right)_S}{\text{ and }}T = {\left( {\frac{{\partial H}}{{\partial S}}} \right)_p}$$

  গ

  $$p = - {\left( {\frac{{\partial G}}{{\partial V}}} \right)_T}{\text{ and }}V = {\left( {\frac{{\partial G}}{{\partial p}}} \right)_S}$$

  ঘ

  $$p = - {\left( {\frac{{\partial A}}{{\partial S}}} \right)_T}{\text{ and }}S = - {\left( {\frac{{\partial A}}{{\partial p}}} \right)_V}$$
• Two identical particles have to be distributed among three energy levels. Let r_B, r_F and r_C represent the ratios of probability of finding two particles to that of finding one particle in a given energy state. The subscripts B, F and C correspond to whether the particles are Bosons, Fermions and classical particles, respectively. The r_B : r_F : r_C is equal to
  ক

  $$\frac{1}{2}$$ : 0 : 1

  খ

  1 : $$\frac{1}{2}$$ : 1

  গ

  1 : $$\frac{1}{2}$$ : $$\frac{1}{2}$$

  ঘ

  1 : 0 : $$\frac{1}{2}$$
• The number of states for a system of N identical free particles in a three-dimensional space having total energy between E and E + δE (δE ≪ E), is proportional to the
  ক

  (E^{3N/2 - 1})δE

  খ

  E^N/2 δE

  গ

  NE^1/2 δE

  ঘ

  E^N δE
• Consider a system of two non-interacting classical particles which can occupy any of the three energy levels with energy values E = 0, ε and 2ε having degeneracies g(E) = 1, 2 and 4 respectively, The mean energy of the system is
  ক

  $$\varepsilon \left$$

  খ

  $$\varepsilon \left$$

  গ

  $$\varepsilon {\left^2}$$

  ঘ

  $$\varepsilon \left$$
• The internal energy of n moles of a gas is given $$E = \frac{3}{2}nRT - \frac{a}{V},$$    where V is the volume of the gas at temperature T and a is a positive constant. One mole of the gas in state (T₁, V₁) is allowed to expand adiabatically into vacuum to a final state (T₂, V₂). The temperature T₂ is
  ক

  $${T_1} + \frac{a}{R}\left( {\frac{1}{{{V_2}}} + \frac{1}{{{V_1}}}} \right)$$

  খ

  $${T_1} - \frac{2}{3}\frac{a}{R}\left( {\frac{1}{{{V_2}}} - \frac{1}{{{V_1}}}} \right)$$

  গ

  $${T_1} + \frac{2}{3}\frac{a}{R}\left( {\frac{1}{{{V_2}}} - \frac{1}{{{V_1}}}} \right)$$

  ঘ

  $${T_1} - \frac{1}{3}\frac{a}{R}\left( {\frac{1}{{{V_2}}} - \frac{1}{{{V_1}}}} \right)$$
• Consider a system of N atoms of an ideal gas of type A at temperature T and volume V. It is kept in diffusive contact with another system of N atoms of another ideal gas of type B at the same temperature T and volume V. Once the combined system reaches equilibrium,
  ক

  the total entropy of the final system is the same as the sum of the entropy of the individual system always

  খ

  the entropy of mixing is 2Nk_B In 2

  গ

  the entropy of the final system is less than that of sum of the initial entropies of the two gases

  ঘ

  the entropy of mixing is non-zero when the atoms A and B are of the same type
• For a two-dimensional free electron gas, the electronic density n, and the Fermi energy E_F, are related by
  ক

  $$n = \frac{{{{\left( {2m{E_F}} \right)}^{\frac{3}{2}}}}}{{3{\pi ^2}{\hbar ^3}}}$$

  খ

  $$n = \frac{{m{E_F}}}{{\pi {\hbar ^2}}}$$

  গ

  $$n = \frac{{m{E_F}}}{{2\pi {\hbar ^2}}}$$

  ঘ

  $$n = \frac{{{2^{\frac{3}{2}}}{{\left( {m{E_F}} \right)}^{\frac{1}{2}}}}}{{\pi \hbar }}$$