Thermodynamics And Statistical Physics — Part 1 MCQs with Explanations

$$\frac{1}{{1 + {e^{ - \varepsilon /{k_B}T}}}}$$ is the correct answer to “A system has two energy levels with energies E and 2E. The lower level is four-fold degenerate while the upper level is doubly degenerate. If there are N non-interacting classical particles in the system, which is in thermodynamic equilibrium at temperature T, the fraction of particles in the upper level is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$\frac{3}{5}{E_F}$$ is the correct answer to “A system of non-interacting Fermi particles with Fermi energy E F has the density of states proportional to √E, where E is the energy of a particle. The average energy per particle at temperature T = 0 is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

clockwise is the correct answer to “A piston containing an ideal gas is originally in the state X (see figure). The gas is taken through a thermal cycle X → Y → X as shown The work done by the gas is positive, if the direction of the thermal cycle is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$${\left[ {\exp \left( {\frac{{\hbar \omega }}{{{k_B}T}}} \right) - 1} \right]^{ - 1}}$$ is the correct answer to “A photon gas is at thermal equilibrium at temperature T. The mean number of photons in an energy state $$\varepsilon = h\omega $$ is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

h3N is the correct answer to “In a classical micro-canonical ensemble for a system of N non-interacting particles, the fundamental volume in phase space which is regarded as equivalent to one micro-state is (where, h is the Planck's constant.)”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

Cp, β and K are all infinite is the correct answer to “In the region of co-existence of a liquid and vapour phases of a material”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$${N^{\frac{2}{3}}}$$ is the correct answer to “For a Fermi gas of N particles in three-dimensions at T = 0 K, the Fermi energy E F is proportional to”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$2 + {e^{\frac{{ - 2\varepsilon }}{{kT}}}} + {e^{\frac{{ - \varepsilon }}{{kT}}}}$$ is the correct answer to “The partition function of two Bose particles, each of which can occupy any of the two energy levels 0 and $$\varepsilon $$ is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

the particles are interacting is the correct answer to “It is necessary to apply quantum statistics to a system of particles, if”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

f(EF) has a value of $$\frac{1}{2}$$ is the correct answer to “Consider the Fermi-Dirac distribution function f(E) at room temperature (300 K), where E refers tb energy. If E F is the Fermi energy, which of the following is true”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

NkB ln 2 is the correct answer to “A system of N non-interacting and distinguishable particles of spin 1 is in thermodynamic equilibrium. The entropy of the system is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$T < \frac{{{\lambda ^2}}}{{2\pi m{k_B}}}{\left[ {\frac{N}{{V\xi \left( {\frac{3}{2}} \right)}}} \right]^{\frac{3}{2}}}$$ is the correct answer to “Which of the following conditions should be satisfied by the temperature T of a system of N non-interacting particles occupying a volume V, for Bose-Einstein condensation to take place? (where, m is the mass of each particle of the system, k B is the Boltzmann constant, h is the Planck's constant and $$\xi $$ is the well known zeta function.)”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

104 Pa is the correct answer to “If the isothermal compressibility of a solid is K T = 10 -10 (Pa) -1 , the pressure required to increase its density by 1% is approximately”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

a van der Waal's gas is the correct answer to “The equation of state of a dilute gas at very high temperature is described by $$\frac{{pV}}{{{k_B}T}} \approx 1 + \frac{{B\left( T \right)}}{V},$$ where V is the volume per particle and B(T) is a negative quantity. One can conclude that this is a property of”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

1H1 is the correct answer to “Which of the following atoms cannot exhibit Bose-Einstein condensation, even in principle”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$\frac{n}{{{T^{\frac{3}{2}}}}} \ll 1$$ is the correct answer to “Which of the following relations between the particle number density n and temperature T must hold good for a gas consisting of non-interacting particles to be described by quantum statistics”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

diverges and its entropy has finite discontinuity is the correct answer to “In a first order phase transition, at the transition temperature, specific heat of the system”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$\frac{{N!}}{{n!\left( {N' - n} \right)!}}\frac{{N!}}{{n!\left( {N' - n} \right)!}}$$ is the correct answer to “A monatomic crystalline solid comprises of N atoms, out of which n atoms are in interstitial positions. If the available interstitial sites are N', then number of possible micro-states is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

Its energy increases 16 times and the number of photons increase 8 times is the correct answer to “The temperature of a cavity of fixed volume is doubled. Which of the following is true for the black-body radiation inside the cavity”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

417 is the correct answer to “A heat pump working on the Carnot cycle maintains the inside temperature of a house at 22°C by supplying 450 kJ/s. If the outside temperature is 0°C, the heat taken, in kJ/s, from the outside air is approximately”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$NkT\ln \frac{{\hbar \omega }}{{kT}}$$ is the correct answer to “If the partition function of a harmonic oscillator with frequency $$\omega $$ at a temperature T is $$\frac{{kT}}{{\hbar \omega }},$$ then the free energy of N such independent oscillators is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$S = \frac{4}{3}aV{T^3},\,p = \frac{a}{3}{T^4}$$ is the correct answer to “The free energy for a photon gas is given by $$F = - \left( {\frac{a}{3}} \right)V{T^{ - 4}},$$ where a is a constant. The entropy S and the pressure p of the photon gas are”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

√E is the correct answer to “For an energy state E of a photon gas, the density of states is proportional to”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

3.5% is the correct answer to “The probability that a state which is 0.2 eV above the Fermi energy in a metal atom at 700 K is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

500 K is the correct answer to “Consider black body radiation in a cavity maintained at 2000 K. If the volume of the cavity is reversibly and adiabatically increased from 10 cm 3 to 640 cm 3 , the temperature of the cavity changes to”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$n = \frac{{m{E_F}}}{{2\pi {\hbar ^2}}}$$ is the correct answer to “For a two-dimensional free electron gas, the electronic density n, and the Fermi energy E F , are related by”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

the entropy of mixing is non-zero when the atoms A and B are of the same type is the correct answer to “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,”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$${T_1} + \frac{2}{3}\frac{a}{R}\left( {\frac{1}{{{V_2}}} - \frac{1}{{{V_1}}}} \right)$$ is the correct answer to “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 (T 1 , V 1 ) is allowed to expand adiabatically into vacuum to a final state (T 2 , V 2 ). The temperature T 2 is”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$\varepsilon \left[ {\frac{{\exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + 2\exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}{{1 + \exp \left( {\frac{{ - \varepsilon }}{{{k_B}T}}} \right) + \exp \left( {\frac{{ - 2\varepsilon }}{{{k_B}T}}} \right)}}} \right]$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

NE1/2 δE is the correct answer to “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”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

1 : 0 : $$\frac{1}{2}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$V = {\left( {\frac{{\partial H}}{{\partial p}}} \right)_S}{\text{ and }}T = {\left( {\frac{{\partial H}}{{\partial S}}} \right)_p}$$ is the correct answer to “Which among the following sets for Maxwell relation is correct? (U-internal energy, H-enthalpy, A-Helmholtz free energy and G-Gibbs free energy)”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

MC : (E, V, N); CE : (N, V, T); GC : (V, T, μ) is the correct answer to “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”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

zero is the correct answer to “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”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

the average distance between them is large compared to their de-Broglie wavelengths is the correct answer to “Two particles are said to be distinguishable when”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

A liquid to gas transition close to its triple point is the correct answer to “Identify which one is a first order phase transition”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

T is the correct answer to “The specific heat of an ideal Fermi gas in three-dimensions at very low temperature (T) varies as”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$${\left( {{Z_\alpha }} \right)^N}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

$$\frac{{pV\Delta S}}{{nR}}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.

the plot of specific heat as a function of temperature shows a discontinuity is the correct answer to “A second order phase transition is one in which”. This question is standard in competitive examination syllabi for Thermodynamics And Statistical Physics.