Team Quanta presents all possible short questions of Thermal & Statistical Physics’ Chapter#03: Formulation of Statistical Methods.
Q.1: How thermodynamic quantities can be calculated from partition function. Give expressions of pressure, entropy, isothermal compressibility, specific heat and internal energy.
Answer. Different thermodynamics quantities can be calculated from partition function $Z$ .
Free energy is $\mathbf{F=-k_{B}T\lnZ}$.
Pressure is $\mathbf{p=-(\frac{\partial F}{\partial V})_{T}}$
Entropy is $\mathbf{S=-(\frac{ \partial F}{ \partial T})_{V}}$
Isothermal compressibility is,
$\mathbf{\frac{1}{K}=V(\frac{ \partial ^{2}F}{ \partial V^{2}})_{T}}$
Specific heat is $\mathbf{C_{V}=-T(\frac{ \partial ^{2 }F}{ \partial T^{2}})_{V}}$
Internal energy is $\mathbf{\overline{U}=k_{B}T^{2}(\frac{\partial {\lnZ }}{\partial T})_{V}}$
Q.2: Ideal mono atomic gas is enclosed in cylinder of radius and length . The cylinder rotates with angular velocity about its symmetry axis and ideal gas is in equilibrium at temperature in coordinate system rotating with the cylinder. Assume that the gas atoms have mass , have no internal degrees of freedom and obey classical statistics. What is partition function for system?
Answer: Partition function is,
$$\mathbf{Z=\int \int d^3p^\prime d^3re -\left(\frac{\frac{{p^\prime}^2}{2m}+\phi-\frac{1}{2}m\omega^2r^2}{k_BT}\right)=\frac{L}{m^2\omega^2}\left(2\pi mk_BT\right)^\frac{5}{2}\left(e^\frac{m\omega^2a^2}{2k_BT\ }-1\right)}$$
Q.3: What is justification in applying Maxwell-Boltzmann statistics to an ideal gas?
Answer: A system consisting of molecules of an ideal gas under ordinary conditions of temperature and pressure is governed by laws of classical Maxwell-Boltzmann statistics i.e. the gas molecules are treated as distinguishable particles.
Q.4: The partition function of a system is given by, $\mathbf{Z=e^{\alpha T^3V}}$ Where α is constant. Calculate pressure of system. Also find expression for entropy and internal energy of system.
Answer: Pressure is defined as,
$$\mathbf{p=k_BT\left(\frac{\partial \ln{Z}}{\partial V}\right)_T=k_BT\ \left(\frac{\partial \left(\alpha T^3V\right)}{\partial V}\right)_T=k_BT \times \alpha T^3-\alpha k_BT^4}$$
Entropy is,
$$\mathbf{S=k_B\left(\frac{\partial\left(T\ln{Z}\right)}{\partial T}\right)_T=k_B\left(\frac{\partial \left(\alpha T^4V\right)}{\partial T}\right)_T=k_B \times 4 \alpha T^3V=4\alpha k_BT^3V}$$
Internal energy is,
$$\mathbf{U=F+TS=-k_BT\ln{Z}+TS=3\alpha k_BT^4V}$$
Q.5: Consider a rigid lattice of distinguishable spin 1/2 atoms in a magnetic field. The spins have two states, with energies -µoH & µoH for spins up and down respectively, relative to H. The system is at temperature T. Determine canonical partition function for this system.
Answer: Partition function is,
$$\mathbf{Z=e^\frac{\mu_\circ H}{k_BT}+e^{-\frac{\mu_\circ H}{k_BT}}}$$
Q.6: Consider a system of two atoms, each having only three quantum states of energies 0, ∈, 2∈. The system is in contact with a heat reservoir at temperature T. Write down partition function for system if particles obey classical statistics and are distinguishable.
Answer: Partition function for one atom is,
$$\mathbf{z=e^{-\frac{0}{k_BT}}+e^{-\frac{\epsilon}{k_BT}}+e^{-\frac{2\epsilon}{k_BT}}}$$
For two atoms, when atoms are distinguishable and obey classical statistics, then required partition function is,
$$\mathbf{Z=z^2=\left(1+e^{-\frac{\epsilon}{k_BT}}+e^{-\frac{2\epsilon}{k_BT\ }}\right)^2}$$
Q.7: Consider a system of two atoms, each having only three quantum states of energies 0, ∈, 2∈. The system is in contact with a heat reservoir at temperature T. Write down partition function for system if particles obey classical statistics and are indistinguishable.
Answer: Partition function for one atom is,
$$\mathbf{z=e^{-\frac{0}{k_BT}}+e^{-\frac{\epsilon}{k_bT}}+e^{-\frac{2\epsilon}{k_BT}}}$$
For two atoms, when atoms are indistinguishable and obey classical statistics, then required partition function is,
$$\mathbf{Z=\frac{z^N}{N!}=\frac{z^2}{2!}=\frac{\left(1+e^{-\frac{\epsilon}{k_BT}}+e^{-\frac{2\epsilon}{k_BT}}\right)^2}{2!}}$$
Q.8: Consider a system of two particles, each having only three quantum states of energies 0, ∈, 2∈ . The system is in contact with a heat reservoir at temperature T. Write down partition function for system, if particles obey classical statistics and are distinguishable and lowest energy state is doubly degenerate.
Answer: Partition function for one atom is,
$$\mathbf{z=e^{-\frac{0}{k_BT}}+e^{-\frac{\epsilon}{k_bT}}+e^{-\frac{2\epsilon}{k_BT}}}$$
For two atoms, when atoms are distinguishable and obey classical statistics, then required partition function is,
$$\mathbf{Z=z^2=\left(2+e^{-\frac{\epsilon}{k_BT}}+e^{-\frac{2\epsilon}{k_BT}}\right)^2}$$
$$\mathbf{\Rightarrow Z=4+4e^{-\frac{\epsilon}{k_BT}}+5e^{-\frac{2\epsilon}{k_BT}}+2e^{-\frac{3\epsilon}{k_BT}}+e^{-\frac{4\epsilon}{k_BT}}}$$
Q.9: Give definition of entropy in statistical physics.
Answer: In statistical physics, entropy is
$$\mathbf{S=k_{B\ }\ln{W}}$$
KB is Boltzmann constant and W is total number of microscopic states of given macroscopic state.
Q.10: Distinguish between micro canonical and canonical ensembles.
Answer: In micro canonical ensemble, number of particles, volume, energy are fixed and no contact with surrounding. Probability distribution is,
$$\mathbf{P\left(E\right)-\delta(E-E_\circ)}$$
Entropy is related to total number of microscopic states of given macroscopic state by relation,
$$\mathbf{S=k_B\ln{W}}$$
In canonical ensemble, number of particles, volume and temperature are fixed and there is a thermal contact with heat reservoir. Probability function is,
$$\mathbf{P_i=Ce^\frac{E_i}{k_BT}}$$
Canonical partition function
$$\mathbf{Z=\sum_{j}e^{-\frac{E_i}{k_BT}}}$$
Helmholtz free energy
$$\mathbf{F=-k_BT\ln{Z}}$$
Q.11: The volume of a perfect gas of N atoms is doubled, the energy being held constant. What is change of entropy?
Answer: Using
$$\mathbf{\sigma=\ln{\Delta T}}$$
Change in entropy is,
$$\mathbf{\Delta \sigma=\log{\Delta T_2}-\log{\Delta T_1}=\log{\left(\frac{\Delta T_2}{\Delta T_1}\right)}=\log{\left(\frac{V_2}{V_1}\right)^N}=\log{\left(\frac{2V_1}{V_1}\right)^N}=N\log{2}}$$
Q.12: State which of the ensembles would be most appropriate for the systems listed below. Give brief, to the point explanation where required.
- A bottle of mineral water immersed in a swimming pool.
- A thermo flask containing hot tea.
- Photon gas enclosed in a constant temperature enclosure.
Answer: A bottle of mineral water immersed in a swimming pool is an example of canonical ensemble due to heat transfer between bottle and pool. A thermo flask containing hot tea is example of micro canonical ensemble because number of particles, volume and energy are fixed and there is no heat contact with surrounding. Photon gas enclosed in a constant temperature enclosure is an example of grand canonical ensemble.
Q.13: Suppose varies as where r is a constant. How does temperature vary as function of ? For what value of energy is temperature zero?
Answer: Entropy is,
$$\mathbf{S=k_B\ln{W}=k_B{\ln{A}+\gamma\sqrt{UV}}}$$
Temperature is defined as,
$$\mathbf{\frac{1}{T}=\left(\frac{\partial S}{\partial U}\right)_{V,N}=\frac{\gamma k_B}{2}\sqrt{\frac{V}{U}}\ \ \ \ \ \Rightarrow T=\frac{2U^\frac{1}{2}}{\gamma V^\frac{1}{2}k_B}}$$
When temperature is zero, internal energy is zero.
Q.14: The entropy of two dimensional gas of particles in an area is given by expression, $\mathbf{S=nk_B\left{\ln{\left(\frac{A}{N}\right)+\ln{\left(\frac{mU}{2\pi n\hbar^2}\right)}+2}\right}}$ Where is number of particles and is energy of gas. Calculate temperature of gas.
Answer: Temperature is defined as,
$$\mathbf{\frac{1}{T}=\left(\frac{\partial S}{\partial U}\right)_{V,N}=\frac{Nk_B}{U}\ \ \ \Rightarrow T=\frac{U}{Nk_B}}$$
Chemical potential is,
$$\boldsymbol{\mu=-T\left(\frac{\partial S}{\partial N}\right)_{V,N}=k_BT\left{\ln{\left(\frac{N}{A}\right)+\ln{\left(\frac{h^2}{2\pi mk_BT\ }\right)}}\right}}$$
Q.15: Regard atomic motion in a gas as random walk due to collisions, given an order of magnitude estimate of time it would take an air molecule in a room to traverse a distance of 1cm. What about 1m?
Answer. If $\lambda$ is mean free path and $\pi$ is relaxation time, then required time is $\tau(\frac{L}{\lambda})^{3}$ . When $L = 1cm$ then time is 1s and for $L = 1m$ , time is ${10}^{4}$s.
Q.16: State assumptions made in derivation of Maxwell-Boltzmann velocity distribution of gas molecules.
Answer: The main assumptions made in derivation of Maxwell-Boltzmann law of distribution of velocities are:
- The particles of system are distinguishable and identical.
- There is no restriction on number of particles which can occupy a phase space cell.
- The available volume of phase space cell can be as small as we please and may even approach zero.
- The state of each particle is specified by giving its instantaneous position and momentum coordinates or by giving its cell number in phase space.
- The phase space can be divided into a very large number of cells.
Q.17: State Maxwell-Boltzmann law of distribution of momentum by explaining all the terms used.
Answer. Maxwell-Boltzmann law of distribution of momentum is given by,
$$\mathbf{n(p)dp=4 \pi m(\frac{1}{2 \pi mk_{B}T})^\frac{3}{2}\ \ e^{\frac{-p^{2}}{2mk_{B}T}}p^{2}}dp$$
This relation gives number of molecules having momenta between $p$ & $p + dp.k_{B}$ is Boltzmann constant and $T$ is absolute temperature.
Q.18: The average kinetic energy of hydrogen atoms in a certain stellar atmosphere (assumed to be in thermal equilibrium) is 1.0eV. What is temperature of atmosphere in kelvin?
Answer. Temperature is,
$$\mathbf{T=\frac{2E}{3k_{B }}=\frac{2 \times 1.6 \times10^{-19}J}{1.38 \times10^{-23}J/K}=7.7 \times 10^{3}K}$$
Q.19: The average kinetic energy of hydrogen atoms in a certain stellar atmosphere (assumed to be in thermal equilibrium) is 1.0eV.What is ratio of number of atoms in second excited state to number in ground state?
Answer. The energy levels for hydrogen atom are,
$$\mathbf{E_{n}=-\frac{13.6}{n^{2}}eV}$$
From Boltzmann distribution,
$$\mathbf{\frac{N_{3}}{N_{1}}= e^ \frac{{E_{1}-E_{3}}}{{k_{B}T}} = e^\frac{-13.6ev+(\frac{13.6}{9}(\frac{2}{3})}{(\frac{2}{3}ev)}=1.33 \times 10^{-8}}$$
Q.20: Consider a system which has two orbital (single particle) state both of same energy. When orbitals are unoccupied, then energy of system is zero; when one orbital is occupied by one particle, the energy is E. We suppose that the energy of system is much higher, say infinitely high, when both orbitals are occupied. What is ensemble average number of particles in the level?
Answer. The probability that a microscopic state is occupied is proportional to
$$\mathbf{e^\frac{\mu -E }{k_{B}T}}$$
Thus ensemble average number of particles in level is,
$$\mathbf{\overline{N} =\frac{1 \times e^\frac{\mu -E}{k_{B}T}+1 \times e^\frac{\mu -E}{(k_{B}T)}}{1+e^\frac{\mu -E}{k_{B}T}+e^\frac{\mu -E}{k_{B}T}}=\frac{2e^\frac{\mu -E}{k_{B} T}}{1+2e\frac{ \mu -E}{k_{B}T}}=\frac{2}{e^\frac{ \mu -E}{k_{B}T}+2}}$$
Q.21: Discuss an example in which Maxwell-Boltzmann law fails.
Answer: Maxwell-Boltzmann law is very general law. It is valid for localized systems, classical systems and non-degenerate quantum systems. It does not hold for degenerate non-localized quantum systems for example, a system of electrons of spin 1/2 at a low temperature and of high density.
Q.22: Ideal mono atomic gas is enclosed in cylinder of radius and length The cylinder rotates with angular velocity about its symmetry axis and Ideal gas is in quilibrIum at temperature T in coordinate system rotating with the cylinder. Assume that the gas atoms have mass m, have no Internal degrees of freedom and obey classical statistics. What Is Hamiltonian in rotating coordinate system?
Answer: The Hamiltonian for a single atom is,
$$\mathbf{h_i^\prime=\frac{{p^\prime}^2}{2m}+\phi-\frac{1}{2}m\omega^2r^2\ \ \ \ \ where\ \phi=\left{\begin{matrix}
0 & r\leq \left | z \right |< \frac{L}{2}\ \infty
& otherwise
\end{matrix}\right.}$$
Hamiltonian of system is,
$$\mathbf{H^\prime=\sum_{i}h_i^\prime}$$
Q.23: Ideal mono atomic gas is enclosed in cylinder of radius and length . The cylinder rotates with angular velocity about its axis and idealgas in equilibrium at temperature in coordinate system rotating with the cylinder. Assume that the gas atoms have mass , have no internal degrees of freedom and obey classical statistics. Find average particle number density as function of .
Answer: Average particle number density is,
$$\frac{\Delta N}{\Delta V}=N\int{d^3p^\prime e^{-\frac{1}{Z}\left(\frac{\frac{{p^\prime}^2}{2m}+\phi-\frac{1}{2}m\omega^2r^2}{k_BT}\right)}=\frac{N}{\pi L}}\frac{m\omega^2}{2k_BT}\frac{e^\frac{m\omega^2r^2}{2k_BT}}{e^\frac{m\omega^2a^2}{2k_BT}-1}\ \ \ \ for\ r<a$$
Q.24: Consider a rigid lattice of distinguishable spin 1/2atoms in a magnetic field. The spins have two states, with energies -µoH & µoH forspins up and down respectively, relative to H . The system is at temperature T . What is entropy of system?
Answer: Partition function is,
$$\mathbf{Z=e^\frac{\mu_\circ H}{k_BT}+e^{-\frac{\mu_\circ H}{k_BT}}=e^{\beta\mu_\circ H}+e^{-\beta\mu_\circ H}\ \ \ \ ;\beta=\frac{1}{k_BT}}$$
Entropy of system is,
$$\mathbf{S=Nk_B\left{\ln{Z}-\beta\frac{\partial \ln{Z}}{\partial \beta}\right}=Nk_B\left{\ln{2}+\ln{\cos{h}\left(\frac{\mu_\circ H}{k_BT}\right)}-\left(\frac{\mu_\circ H}{k_BT}\right)\tan{h\left(\frac{\mu_\circ H}{k_BT}\right)}\right}}$$
Q.26: A paramagnetic system of magnetic dipoles. Each dipole carries a magnetic moment which can be treated classically, If the system at a finite temperature is in a uniform magnetic field , find induced magnetization in system.
Answer: Average magnetic moment is,
$$\mathbf{\bar{\mu}=\frac{\int{\mu\cos{\theta}e^{x\cos{\theta}}d\Omega}}{\int e^{x\cos{\theta}}d\Omega}=\frac{\mu\int_{0}^{\pi}{\cos{\theta}e^{x\cos{\theta}}\sin{\theta}d\theta}}{\int_{0}^{\pi}{e^{x\cos{\theta}}\sin{\theta}d\theta}}=\mu\left(\cot{hx}-\frac{1}{x}\right)\ ;x=\frac{\mu H}{k_BT}}$$
Magnetization is,
$$\mathbf{M=N\ \bar{\mu}=N\mu\left(\cot{hx}-\frac{1}{x}\right)}$$
Heat capacity is,
$$\mathbf{C_H=-H\frac{\partial M}{\partial T}=Nk_B\left(1-x^2\csc{h^2}x^2\right)}$$
Q.27: A material consists of particles and is in weak external magnetic field H. Each particle can have a magnetic moment mµ along the magnetic field, where m=J,……..,-J where, J being an integer and µ is constant. The system is at temperature T . Find the partition function for this system.
Answer: Partition function is,
$$\mathbf{Z=\sum_{m=-J}^{J}{e^\frac{m\mu H}{k_BT}=\frac{\sin{h}\left{\left(J+\frac{1}{2}\right)\frac{\mu H}{k_BT}\right}}{\sin{h\left(\frac{\mu H}{2k_BT}\right)}}}}$$
Q.28: Consider a hetro nuclear diatomic molecule with moment of inertia . In this case only rotational motion of molecule should be considered. Using classical statistical mechanics, find specific heat of this system at temperature T.
Answer: For classical rotator,
$$\mathbf{E=\frac{1}{2I}\left{p_{\theta\ }^2+\frac{p_\phi^2}{\sin^2{\theta}}\right}}$$
Partition function is,
$$Z=\int{e^{-\beta E}dp_\theta dp_\phi d\theta d\phi=\frac{8 \pi ^2I}{\beta}}$$
Mean energy is,
$$\mathbf{\bar{E}=-\frac{\partial}{\partial\beta}\ln{Z}=\frac{1}{\beta}=k_BT}$$
Heat capacity is,
$$\mathbf{C_V=\left(\frac{\partial \bar{E}}{\partial T}\right)_T=k_B}$$
Q.29: Distinguish between three kinds of ensembles stating their respective distribution functions.
Answer. For macro-canonical ensemble, distribution function is described by,
$P(E)= \delta (E-E_{∘})$
For canonical ensemble, it is
$P_{i}=Ce^\frac{{E_i}}{k_{B}T}$
For grand canonical, we have
$P_{N_{i}}=Ce^-(\frac{{E_{N_{i}}-\mu N}}{k_{B}T})$
Q.30: Explain partition function. Give importance of partition function.
Answer. Partition function:- Partition function is denoted by $Z$ and is defined by relations,
$Z=\sum_{j}\ \ e^- \beta E_{j}=\sum_{j} \ \ e^{\frac{E_{J}}{k_{B}T}}$ quantum partition function
$Z=\int\ e^{\frac{E(p,q)}{k_{B}T}}dT$ classical partition function
Sum is taken over all different quantum states of system.
Importance of partition function:- We have found the partition function by normalization condition needed to get the sum of probabilities equal to one. Its importance lies in fact that it enables us to make a direct connection between quantum states of system and its thermodynamic properties of a system from knowledge of partition function.
Q.31: Name three different types of ensembles. Define grand canonical ensemble and canonical ensemble.
Answer: There are three types of ensembles;
- Micro canonical ensemble
- Canonical ensemble
- Grand canonical ensemble
Micro canonical ensemble:- The micro canonical ensemble describes a system with a precisely given energy and fixed composition (precise number of particles). The micro canonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.
Canonical ensemble:- The canonical ensemble describes a system of fixed composition that is in thermal equilibrium with a heat reservoir of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.
Grand canonical ensemble:- The grand canonical ensemble describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particles. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.
Q.32: Define assembly, classical assembly, quantum assembly. What is micro canonical ensemble?
Answer: A physical body under statistical analysis which is composed of systems is called assembly. If systems in assembly obey classical statistics laws, then assembly is called classical assembly.
If systems in assembly obey quantum statistics laws, then assembly is called quantum assembly.
Micro canonical ensemble:- The micro canonical ensemble describes a system with a precisely given energy and fixed composition (precise number of particles). The micro canonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.
Q.33: Consider two equal volumes of same gas having the same temperature, pressure and containing the same number of molecules. These two systems are separated by a barrier and each has entropy . The two systems are combined by removing the barrier. The entropy of combined system is calculated and comes out as . Comment on the reason for this paradox and state how it can be removed?
Answer: Entropy of an ideal gas in micro-canonical ensemble is given by.
$$\mathbf{\sigma=NlnV\left(\frac{4\pi ME}{3Nh^2}\right)^\frac{3}{2}+\frac{3N}{2}}$$
Let we have two samples of gases each with volume V, energy E and number of moleculesN repectively. We make one system of combining all these terms, then the volume, energy and number of molecules of new combined system become 2V, 2E, & 2N and its entropy will be,
$$\mathbf{\sigma^\prime=2\sigma=2Nln2V\left(\frac{4\pi M\times2E}{3\times2Nh^2}\right)^\frac{3}{2}+\frac{3\times2N}{2}}$$
$$\mathbf{\Rightarrow\sigma^\prime=2Nln2+2NlnV\left(\frac{4\pi ME}{3Nh^2}\right)^\frac{3}{2}+\frac{3\times2N}{2}\ }$$
$$\mathbf{\Rightarrow\sigma^\prime=2Nln2+2\left{NlnV\left(\frac{4\pi ME}{3Nh^2}\right)^\frac{3}{2}+\frac{3N}{2}\right}=2Nln2+2\sigma}$$
$$\mathbf{\Rightarrow\sigma^\prime=2\sigma+2Nln2\ }$$
This does not satisfy additive property of entropy, there is some defect called Gibb’s paradox. Consider a system of N particles. When two particles exchange their places, this formula Counts it a new microstate but actually no new microstate is produced, Then according to this formula, total number of microstates become mN if there are m total number of particles i. e. it overestimates N! times the microstates. To overcome this paradox, we divide entropy by N! .
