Q.No:1 GATE-2012
A classical gas of molecules, each of mass \(m\), is in thermal equilibrium at the absolute temperature, \(T\). The velocity components of the molecules along the Cartesian axes are \(\nu_x, \nu_y\) and \(\nu_z\). Then mean value of \((\nu_x+\nu_y)^2\) is
(A)
\(\frac{k_B T}{m}\)
(B)
\(\frac{3}{2} \frac{k_B T}{m}\)
(C)
\(\frac{1}{2} \frac{k_B T}{m}\)
(D)
\(\frac{2k_B T}{m}\)
Check Answer
Option D
Q.No:2 GATE-2012
Consider a system whose three energy levels are given by \(0, \varepsilon\) and \(2\varepsilon\). The energy level \(\varepsilon\) is two-fold degenerate and the other two are non-degenerate. The partition function of the system with \(\beta=\frac{1}{k_B T}\) is given by
(A)
\(1+2e^{-\beta \varepsilon}\)
(B)
\(2e^{-\beta \varepsilon}+e^{-2\beta \varepsilon}\)
(C)
\((1+e^{-\beta \varepsilon})^2\)
(D)
\(1+e^{-\beta \varepsilon}+e^{-2\beta \varepsilon}\)
Check Answer
Option C
Q.No:3 GATE-2012
A paramagnetic system consisting of \(N\) spin-half particles, is placed in an external magnetic field. It is found that \(N/2\) spins are aligned parallel and the remaining \(N/2\) spins are aligned antiparallel to the magnetic field. The statistical entropy of the system is,
(A)
\(2Nk_B \ln{2}\)
(C)
\(\frac{N}{2}k_B \ln{2}\)
(C)
\(\frac{2N}{2}k_B \ln{2}\)
(D)
\(Nk_B \ln{2}\)
Check Answer
Option D
Q.No:4 GATE-2012
At a certain temperature \(T\), the average speed of nitrogen molecules in air is found to be \(400 m/s\). The most probable and the root mean square speeds of the molecules are, respectively,
(A)
\(355 m/s, 434 m/s\)
(B)
\(820 m/s, 917 m/s\)
(C)
\(152 m/s, 301 m/s\)
(D)
\(422 m/s, 600 m/s\)
Check Answer
Option A
Q.No:5 GATE-2013
Two gases separated by an impermeable but movable partition are allowed to freely exchange energy. At equilibrium, the two sides will have the same
(A)
pressure and temperature
(B)
volume and temperature
(C)
pressure and volume
(D)
volume and energy
Check Answer
Option A
Q.No:6 GATE-2013
The entropy function of a system is given by \(S(E)=aE(E_0-E)\) where \(a\) and \(E_0\) are positive constants. The temperature of the system is
(A)
negative for some energies
(B)
increases monotonically with energy
(C)
decreases monotonically with energy
(D)
Zero
Check Answer
Option A
Q.No:7 GATE-2013
Consider a linear collection of \(N\) independent spin \(1/2\) particles, each at a fixed location. The entropy of this system is (\(k\) is the Boltzmann constant)
(A)
Zero
(B)
\(Nk\)
(C)
\(\frac{1}{2}Nk\)
(D)
\(Nk\ln{(2)}\)
Check Answer
Option D
Q.No:8 GATE-2013
Consider a gas of atoms obeying Maxwell-Boltzmann statistics. The average value of \(e^{i\vec{a}.\vec{p}}\) over all the momenta \(\vec{p}\) of each of the particles (where \(\vec{a}\) is a constant vector and \(a\) is its magnitude, \(m\) is the mass of each atom, \(T\) is temperature and \(k\) is Boltzmann's constant) is,
(A)
One
(B)
Zero
(C)
\(e^{-\frac{1}{2}a^2 mkT}\)
(D)
\(e^{-\frac{3}{2}a^2 mkT}\)
Check Answer
Option C
Q.No:9 GATE-2013
There are four energy levels \(E, 2E, 3E\) and \(4E\) (where \(E>0\)). The canonical partition function of two particles is, if these particles are
two identical fermions
(A)
\(e^{-2\beta E}+e^{-4\beta E}+e^{-6\beta E}+e^{-8\beta E}\)
(B)
\(e^{-3\beta E}+e^{-4\beta E}+2e^{-5\beta E}+e^{-6\beta E}+e^{-7\beta E}\)
(C)
\((e^{-\beta E}+e^{-2\beta E}+e^{-3\beta E}+e^{-4\beta E})^2\)
(D)
\(e^{-2\beta E}-e^{-4\beta E}+e^{-6\beta E}-e^{-8\beta E}\)
Check Answer
Option B
Q.No:10 GATE-2013
There are four energy levels \(E, 2E, 3E\) and \(4E\) (where \(E>0\)). The canonical partition function of two particles is, if these particles are
two distinguishable particles
(A)
\(e^{-2\beta E}+e^{-4\beta E}+e^{-6\beta E}+e^{-8\beta E}\)
(B)
\(e^{-3\beta E}+e^{-4\beta E}+2e^{-5\beta E}+e^{-6\beta E}+e^{-7\beta E}\)
(C)
\((e^{-\beta E}+e^{-2\beta E}+e^{-3\beta E}+e^{-4\beta E})^2\)
(D)
\(e^{-2\beta E}-e^{-4\beta E}+e^{-6\beta E}-e^{-8\beta E}\)
Check Answer
Option C
Q.No:11 GATE-2014
At a given temperature, \(T\), the average energy per particle of a non-interacting gas of two-dimensional classical harmonic oscillators is ___________.\(k_B T\) (\(k_B\) is the Boltzmann constant).
Check Answer
Ans 1.99-2.01
Q.No:12 GATE-2014
For a system of two bosons, each of which can occupy any of the two energy levels \(0\) and \(\varepsilon\), the mean energy of the system at a temperature \(T\) with \(\beta=\frac{1}{k_B T}\) is given by
(A)
\(\frac{\varepsilon e^{-\beta \varepsilon}+2\varepsilon e^{-2\beta \varepsilon}}{1+2e^{-\beta \varepsilon}+e^{-2\beta \varepsilon}}\)
(B)
\(\frac{1+\varepsilon e^{-\beta \varepsilon}}{2e^{-\beta \varepsilon}+e^{-2\beta \varepsilon}}\)
(C)
\(\frac{2\varepsilon e^{-\beta \varepsilon}+\varepsilon e^{-2\beta \varepsilon}}{2+e^{-\beta \varepsilon}+e^{-2\beta \varepsilon}}\)
(D)
\(\frac{\varepsilon e^{-\beta \varepsilon}+2\varepsilon e^{-2\beta \varepsilon}}{2+e^{-\beta \varepsilon}+e^{-2\beta \varepsilon}}\)
Check Answer
Option A
Q.No:13 GATE-2015
Consider a system of \(N\) non-interacting spin-\(1/2\) particles, each having a magnetic moment \(\mu\), is in a magnetic field \(\vec{B}=B\hat{z}\). If \(E\) is the total energy of the system, the number of accessible microstates \(\Omega\) is given by
(A)
\(\Omega=\frac{N!}{\frac{1}{2}\left(N-\frac{E}{\mu B}\right)! \frac{1}{2}\left(N+\frac{E}{\mu B}\right)!}\)
(B)
\(\Omega=\frac{\left(N-\frac{E}{\mu B}\right)!}{\left(N+\frac{E}{\mu B}\right)!}\)
(C)
\(\Omega=\frac{1}{2}\left(N-\frac{E}{\mu B}\right)! \frac{1}{2}\left(N+\frac{E}{\mu B}\right)!\)
(D)
\(\Omega=\frac{N!}{\left(N+\frac{E}{\mu B}\right)!}\)
Check Answer
Option A
Q.No:14 GATE-2015
The entropy of a gas containing \(N\) particles enclosed in a volume \(V\) is given by \(S=Nk_B \ln{\left(\frac{aVE^{3/2}}{N^{5/2}}\right)}\), where \(E\) is the total energy, \(a\) is a constant and \(k_B\) is the Boltzmann constant. The chemical potential \(\mu\) of the system at a temperature \(T\) is given by
(A)
\(\mu=-k_B T\left[\ln{\left(\frac{aVE^{3/2}}{N^{5/2}}\right)}-\frac{5}{2}\right]\)
(B)
\(\mu=-k_B T\left[\ln{\left(\frac{aVE^{3/2}}{N^{5/2}}\right)}-\frac{3}{2}\right]\)
(C)
\(\mu=-k_B T\left[\ln{\left(\frac{aVE^{3/2}}{N^{3/2}}\right)}-\frac{5}{2}\right]\)
(D)
\(\mu=-k_B T\left[\ln{\left(\frac{aVE^{3/2}}{N^{3/2}}\right)}-\frac{3}{2}\right]\)
Check Answer
Option A
Q.No:15 GATE-2015
The average energy \(U\) of a one dimensional quantum oscillator of frequency \(\omega\) and in contact with a heat bath at temperature \(T\) is given by
(A)
\(U=\frac{1}{2}\hbar\omega\coth{\left(\frac{1}{2}\beta\hbar\omega\right)}\)
(B)
\(U=\frac{1}{2}\hbar\omega\sinh{\left(\frac{1}{2}\beta\hbar\omega\right)}\)
(C)
\(U=\frac{1}{2}\hbar\omega\tanh{\left(\frac{1}{2}\beta\hbar\omega\right)}\)
(D)
\(U=\frac{1}{2}\hbar\omega\cosh{\left(\frac{1}{2}\beta\hbar\omega\right)}\)
Check Answer
Option A
Q.No:16 GATE-2016
The entropy \(S\) of a system of \(N\) spins, which may align either in the upward or in the downward direction, is given by \(S=-k_B N[p\ln{p}+(1-p)\ln{(1-p)}]\). Here \(k_B\) is the Boltzmann constant. The probability of alignment in the upward direction is \(p\). The value of \(p\), at which the entropy is maximum, is ___________. (Give your answer upto one decimal place)
Check Answer
Ans 0.5
Q.No:17 GATE-2016
\(N\) atoms of an ideal gas are enclosed in a container of volume \(V\). The volume of the container is changed to \(4 V\), while keeping the total energy constant. The change in the entropy of the gas, in units of \(Nk_B\ln{2}\), is ___________, where \(k_B\) is the Boltzmann constant.
Check Answer
Ans 2
Q.No:18 GATE-2016
A two-level system has energies zero and \(E\). The level with zero energy is non-degenerate, while the level with energy \(E\) is triply degenerate. The mean energy of a classical particle in this system at a temperature \(T\) is
(A)
\(\frac{Ee^{-E/k_B T}}{1+3e^{-E/k_B T}}\)
(B)
\(\frac{Ee^{-E/k_B T}}{1+e^{-E/k_B T}}\)
(C)
\(\frac{3Ee^{-E/k_B T}}{1+e^{-E/k_B T}}\)
(D)
\(\frac{3Ee^{-E/k_B T}}{1+3e^{-E/k_B T}}\)
Check Answer
Option D
Q.No:19 GATE-2017
Consider \(N\) non-interacting, distinguishable particles in a two-level system at temperature \(T\). The energies of the levels are \(0\) and \(\varepsilon\), where \(\varepsilon>0\). In the high temperature limit (\(k_B T\gg \varepsilon\)), what is the population of particles in the level with energy \(\varepsilon\)?
(A)
\(\frac{N}{2}\)
(B)
\(N\)
(C)
\(\frac{N}{4}\)
(D)
\(\frac{3N}{4}\)
Check Answer
Option A
Q.No:20 GATE-2018
A microcanonical ensemble consists of \(12\) atoms with each taking either energy \(0\) state, or energy \(\epsilon\) state. Both states are non-degenerate. If the total energy of this ensemble is \(4\epsilon\), its entropy will be _____________. \(k_B\) (up to one decimal place), where \(k_B\) is the Boltzmann constant.
Check Answer
Ans 6.1-6.3
Q.No:21 GATE-2018
The partition function of an ensemble at a temperature \(T\) is
\[
Z=\left(2\cosh{\frac{\varepsilon}{k_B T}}\right)^N,
\]
where \(k_B\) is the Boltzmann constant. The heat capacity of this ensemble at \(T=\frac{\varepsilon}{k_B}\) is \(X Nk_B\), where the value of \(X\) is ____________. (up to two decimal places).
Check Answer
Ans 0.41-0.43
Q.No:22 GATE-2018
The quantum effects in an ideal gas become important below a certain temperature \(T_Q\) when de Broglie wavelength corresponding to the root mean square thermal speed becomes equal to the inter-atomic separation. For such a gas of atoms of mass \(2\times 10^{−26} kg\) and number density \(6.4\times 10^{25} m^{-3}\), \(T_Q=\) ___________. \(\times 10^{-3} K\) (up to one decimal place).
(\(k_B=1.38\times 10^{-23} J/K, h=6.6\times 10^{-34} J-s\))
Check Answer
Ans 78-90
Q.No:23 GATE-2019
Consider a one-dimensional gas of \(N\) non-interacting particles of mass \(m\) with the Hamiltonian for a single particle given by,
\[
H=\frac{p^2}{2m}+\frac{1}{2}m\omega^2(x^2+2x)
\]
The high temperature specific heat in units of \(R=Nk_B\) (\(k_B\) is the Boltzmann constant) is
(A)
\(1\)
(B)
\(1.5\)
(C)
\(2\)
(D)
\(2.5\)
Check Answer
Option A
Q.No:24 GATE-2019
Consider two systems A and B each having two distinguishable particles. In both the systems, each particle can exist in states with energies \(0, 1, 2\) and \(3\) units with equal probability. The total energy of the combined system is \(5\) units. Assuming that the system A has energy \(3\) units and the system B has energy \(2\) units, the entropy of the combined system is \(k_B \ln{\lambda}\). The value of \(\lambda\) is __________________.
Check Answer
Ans 12
Q.No:25 GATE-2021
Consider a system of three distinguishable particles, each having spin \(S=1/2\) such that \(S_z=\pm 1/2\) with corresponding magnetic moments \(\mu_z=\pm \mu\). When the system is placed in an external magnetic field \(H\) pointing along the \(z\)-axis, the total energy of the system is \(\mu H\). Let \(x\) be the state where the first spin has \(S_z=1/2\). The probability of having the state \(x\) and the mean magnetic moment (in the \(+z\) direction) of the system in state \(x\) are
(A)
\(\frac{1}{3}, \frac{-1}{3}\mu\)
(B)
\(\frac{1}{3}, \frac{2}{3}\mu\)
(C)
\(\frac{2}{3}, \frac{-2}{3}\mu\)
(D)
\(\frac{2}{3}, \frac{1}{3}\mu\)
Check Answer
Option A
Q.No:26 GATE-2021
Consider a single one-dimensional harmonic oscillator of angular frequency \(\omega\), in equilibrium at temperature \(T=(k_B \beta)^{-1}\). The states of the harmonic oscillator are all non-degenerate having energy \(E_n=\left(n+\frac{1}{2}\right)\hbar \omega\) with equal probability, where \(n\) is the quantum number. The Helmholtz free energy of the oscillator is
(A)
\(\frac{\hbar \omega}{2}+\beta^{-1} \ln{[1-\exp{(\beta \hbar \omega)}]}\)
(B)
\(\frac{\hbar \omega}{2}+\beta^{-1} \ln{[1-\exp{(-\beta \hbar \omega)}]}\)
(C)
\(\frac{\hbar \omega}{2}+\beta^{-1} \ln{[1+\exp{(-\beta \hbar \omega)}]}\)
(D)
\(\beta^{-1} \ln{[1-\exp{(-\beta \hbar \omega)}]}\)
Check Answer
Option B
Q.No:27 GATE-2022
Consider a non-interacting gas of spin \(1\) particles, each with magnetic moment \(\mu\), placed in a weak magnetic field \(B\), such that \(\frac{\mu B}{k_B T}\ll 1\). The average magnetic moment of a particle is
(a)
\(\frac{2\mu}{3}\left(\frac{\mu B}{k_B T}\right)\)
(b)
\(\frac{\mu}{2}\left(\frac{\mu B}{k_B T}\right)\)
(c)
\(\frac{\mu}{3}\left(\frac{\mu B}{k_B T}\right)\)
(d)
\(\frac{3\mu}{4}\left(\frac{\mu B}{k_B T}\right)\)
Check Answer
Option a
Q.No:28 GATE-2017
Consider a triatomic molecule of the shape shown in the figure below in three dimensions. The heat capacity of this molecule at high temperature (temperature much higher than the vibrational and rotational energy scales of the molecule but lower than its bond dissociation energies) is:
(A)
\(\frac{3}{2}k_B\)
(B)
\(3k_B\)
(C)
\(\frac{9}{2}k_B\)
(D)
\(6k_B\)
Check Answer
Option D
Q.No:29 GATE-2023
A simple harmonic oscillator with an angular frequency \(\omega\) is in thermal equilibrium with a reservoir at absolute temperature \(T\), with \(\omega=\frac{2k_B T}{\hbar}\). Which one of the following is the partition function of the system?
(A)
\(\frac{e}{e^2-1}\)
(B)
\(\frac{e}{e^2+1}\)
(C)
\(\frac{e}{e-1}\)
(D)
\(\frac{e}{e+1}\)
Check Answer
Option A
Q.No:30 GATE-2023
Consider 6 identical, non-interacting, spin \(\frac{1}{2}\) atoms arranged on a crystal lattice at absolute temperature \(T\). The z-component of the magnetic moment of each of these atoms can be \(\pm \mu_B\). If \(P\) and \(Q\) are the probabilities of the net magnetic
moment of the solid being \(2\mu_B\) and \(6\mu_B\) respectively, what is the value of \(\frac{P}{Q}\) (in integer)?
Check Answer
Ans 15
Q.No:31 GATE-2016
\(N\) atoms of an ideal gas are enclosed in a container of volume \(V\). The volume of the container is changed to \(4 V \), while keeping the total energy constant. The change in the entropy of the gas, in units of \(Nk_B\ln{2}\), is ___________, where \(k_B\) is the Boltzmann constant.
Check Answer
Ans 2
Q.No:32 GATE-2024
The Hamiltonian of a system of \( N \) particles in volume \( V \) at temperature \( T \) is
\[
H = \sum_{i=1}^{2N} a_i q_i^2 + \sum_{i=1}^{2N} b_i p_i^2
\]
where \( a_i \) and \( b_i \) are positive constants. The ensemble average of the Hamiltonian is \( aNk_B T \), where \( k_B \) is the Boltzmann constant. The value of \( \alpha \) is _____ (in integer).
Check Answer
Ans 2
Q.No:33 GATE-2025
A paramagnetic material containing paramagnetic ions with total angular
momentum \(J = \frac{1}{2}\) is kept at absolute temperature \(T\).
The ratio of the magnetic field required for 80% of the ions to be in the
lowest energy state to that required for having 60% of the ions in the
lowest energy state at the same temperature is
A) \(\frac{2 \ln 2}{\ln( \frac{3}{2} )}\)
B) \(\frac{\ln 2}{\ln( \frac{3}{2} )}\)
C) \(\frac{3 \ln 2}{\ln( \frac{3}{2} )}\)
D) \(\frac{\ln 3}{\ln( \frac{3}{2} )}\)
Check Answer
Option A
Q.No:34 GATE-2025
Consider a two-level system with energy states \(+\varepsilon\) and \(-\varepsilon\).
The number of particles at \(+\varepsilon\) level is \(N_{+}\) and the number of particles
at \(-\varepsilon\) level is \(N_{-}\). The total energy of the system is \(E\) and the total
number of particles is \(N = N_{+} + N_{-}\). In the thermodynamic limit, the inverse of
the absolute temperature of the system is
(Given: \(\ln N! \simeq N \ln N - N\))
A) \(\frac{k_{B}}{2\varepsilon} \ln \left[ \frac{N - \frac{E}{\varepsilon}}{N + \frac{E}{\varepsilon}} \right]\)
B) \(\frac{k_{B}}{\varepsilon} \ln N\)
C) \(\frac{k_{B}}{2\varepsilon} \ln N\)
D) \(\frac{k_{B}}{\varepsilon} \ln \left[ \frac{N - \frac{E}{\varepsilon}}{\,N + \frac{E}{\varepsilon}} \right]\)
Q.No.1 Discussion :
Q.No.2 Discussion :
option c
Q.No.3 Discussion :
Answer
Q.No.4 Discussion :
Correct Answer: Option A
Q.No.5 Discussion :
Answer
Q.No.6 Discussion :
Ans: A. Negative for some energies
Q.No.7 Discussion :
Answer
Q.No.8 Discussion :
Q.No.9 Discussion :
Ans
Q.No.10 Discussion :
Ans
Q.No.11 Discussion :
My solution
Q.No.12 Discussion :
ans option A
Q.No.13 Discussion :
QNO13 solution
Q.No.14 Discussion :
Ans
Q.No.15 Discussion :
Option A
Second page of the answer
Q.No.16 Discussion :
Answer
Q.No.17 Discussion :
Answer
Q.No.18 Discussion :
Q18 solution
Q.No.19 Discussion :
Q.No.20 Discussion :
solution
Q.No.21 Discussion :
Q.No.22 Discussion :
84.2
Q.No.23 Discussion :
Ans
Q.No.24 Discussion :
Question 24
Q.No.25 Discussion :
Option A
Q.No.26 Discussion :
ans
Q.No.27 Discussion :
q no. 27 solution
Q.No.28 Discussion :
Solution
Q.No.29 Discussion :
Q.No.30 Discussion :
Answer
Q.No.31 Discussion :