Chapter 4, Classical statistical mechanics Video Solutions, Statistical Physics of Particles

Problem 1

Classical harmonic oscillators: consider $N$ harmonic oscillators with coordinates and momenta $\left\{q_i, p_i\right\}$, and subject to a Hamiltonian
$$\mathcal{H}\left(\left\{q_i, p_i\right\}\right)=\sum_{i=1}^N\left[\frac{p_i^2}{2 m}+\frac{m \omega^2 q_i^2}{2}\right] .$$
(a) Calculate the entropy $S$, as a function of the total energy $E$.
(Hint. By appropriate change of scale, the surface of constant energy can be deformed into a sphere. You may then ignore the difference between the surface area and volume for $N \gg 1$. A more elegant method is to implement this deformation through a canonical transformation.)
(b) Calculate the energy $E$, and heat capacity $C$, as functions of temperature $T$, and $N$.
(c) Find the joint probability density $P(p, q)$ for a single oscillator. Hence calculate the mean kinetic energy, and mean potential energy for each oscillator.

Jack Hou

Numerade Educator

06:38

Problem 2

Quantum harmonic oscillators: consider $N$ independent quantum oscillators subject to a Hamiltonian
$$\mathcal{H}\left(\left\{n_i\right\}\right)=\sum_{i=1}^N \hbar \omega\left(n_i+\frac{1}{2}\right),$$
where $n_i=0,1,2, \cdots$ is the quantum occupation number for the ith oscillator.
(a) Calculate the entropy $S$, as a function of the total energy $E$.
(Hint. $\Omega(E)$ can be regarded as the number of ways of rearranging $M=\sum_i n_i$ balls, and $N-1$ partitions along a line.)
(b) Calculate the energy $E$, and heat capacity $C$, as functions of temperature $T$, and $N$.
(c) Find the probability $p(n)$ that a particular oscillator is in its $n$th quantum level.
(d) Comment on the difference between heat capacities for classical and quantum oscillators.

Amit Srivastava

Numerade Educator

08:46

Problem 3

Relativistic particles: $N$ indistinguishable relativistic particles move in one dimension subject to a Hamiltonian
$$\mathcal{H}\left(\left\{p_i, q_i\right\}\right)=\sum_{i=1}^N\left[c\left|p_i\right|+U\left(q_i\right)\right],$$
with $U\left(q_i\right)=0$ for $0 \leq q_i \leq L$, and $U\left(q_i\right)=\infty$ otherwise. Consider a microcanonical ensemble of total energy $E$.
(a) Compute the contribution of the coordinates $q_i$ to the available volume in phase space $\Omega(E, L, N)$.
(b) Compute the contribution of the momenta $p_i$ to $\Omega(E, L, N)$.
(Hint. The volume of the hyperpyramid defined by $\sum_{i=1}^d x_i \leq R$, and $x_i \geq 0$, in $d$ dimensions is $R^d / d$ !.)
(c) Compute the entropy $S(E, L, N)$.
(d) Calculate the one-dimensional pressure $P$.
(e) Obtain the heat capacities $C_L$ and $C_P$.
(f) What is the probability $p\left(p_1\right)$ of finding a particle with momentum $p_1$ ?
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Eduard Sanchez

Numerade Educator

01:42

Problem 4

Hard sphere gas: consider a gas of $N$ hard spheres in a box. A single sphere excludes a volume $\omega$ around it, while its center of mass can explore a volume $V$ (if the box is otherwise empty). There are no other interactions between the spheres, except for the constraints of hard-core exclusion.
(a) Calculate the entropy $S$, as a function of the total energy $E$.
(Hint. $(V-a \omega)(V-(N-a) \omega) \approx(V-N \omega / 2)^2$.)
(b) Calculate the equation of state of this gas.
(c) Show that the isothermal compressibility, $\kappa_T=-V^{-1} \partial V /\left.\partial P\right|_T$, is always positive.
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Penny Riley

Numerade Educator

02:02

Problem 5

Non-harmonic gas: let us re-examine the generalized ideal gas introduced in the previous section, using statistical mechanics rather than kinetic theory. Consider a gas of $N$ non-interacting atoms in a $d$-dimensional box of "volume" $V$, with a kinetic energy
$$\mathcal{H}=\sum_{i=1}^N A\left|\vec{p}_i\right|^s$$
where $\vec{p}_i$ is the momentum of the $i$ th particle.
(a) Calculate the classical partition function $Z(N, T)$ at a temperature $T$. (You don't have to keep track of numerical constants in the integration.)
(b) Calculate the pressure and the internal energy of this gas. (Note how the usual equipartition theorem is modified for non-quadratic degrees of freedom.)
(c) Now consider a diatomic gas of $N$ molecules, each with energy
$$\mathcal{H}_i=A\left(\left|\vec{p}_i^{(1)}\right|^s+\left|\vec{p}_i^{(2)}\right|^s\right)+K\left|\vec{q}_i^{(1)}-\vec{q}_i^{(2)}\right|^t$$
where the superscripts refer to the two particles in the molecule. (Note that this unrealistic potential allows the two atoms to occupy the same point.) Calculate the expectation value $\left\langle\left|\vec{q}_i^{(1)}-\vec{q}_i^{(2)}\right|^t\right\rangle$, at temperature $T$.
(d) Calculate the heat capacity ratio $\gamma=C_P / C_V$, for the above diatomic gas.

Stanley Enemuo

Numerade Educator

01:18

Problem 6

Surfactant adsorption: a dilute solution of surfactants can be regarded as an ideal threedimensional gas. As surfactant molecules can reduce their energy by contact with air, a fraction of them migrate to the surface where they can be treated as a two-dimensional ideal gas. Surfactants are similarly adsorbed by other porous media such as polymers and gels with an affinity for them.
(a) Consider an ideal gas of classical particles of mass $m$ in $d$ dimensions, moving in a uniform attractive potential of strength $\varepsilon_d$. By calculating the partition function, or otherwise, show that the chemical potential at a temperature $T$ and particle density $n_d$ is given by
$$\mu_d=-\varepsilon_d+k_B T \ln \left[n_d \lambda(T)^d\right], \quad \text { where } \quad \lambda(T)=\frac{h}{\sqrt{2 \pi m k_B T}} .$$
(b) If a surfactant lowers its energy by $\varepsilon_0$ in moving from the solution to the surface, calculate the concentration of floating surfactants as a function of the solution concentration $n\left(=n_3\right)$, at a temperature $T$.
(c) Gels are formed by cross-linking linear polymers. It has been suggested that the porous gel should be regarded as fractal, and the surfactants adsorbed on its surface treated as a gas in $d_f$-dimensional space, with a non-integer $d_f$. Can this assertion be tested by comparing the relative adsorption of surfactants to a gel, and to the individual polymers (presumably one-dimensional) before cross-linking, as a function of temperature?

Lottie Adams

Numerade Educator

03:57

Problem 7

Molecular adsorption: $N$ diatomic molecules are stuck on a metal surface of square symmetry. Each molecule can either lie flat on the surface, in which case it must be aligned to one of two directions, $x$ and $y$, or it can stand up along the $z$ direction. There is an energy $\operatorname{cost}$ of $\varepsilon>0$ associated with a molecule standing up, and zero energy for molecules lying flat along $x$ or $y$ directions.
(a) How many microstates have the smallest value of energy? What is the largest microstate energy?
(b) For microcanonical macrostates of energy $E$, calculate the number of states $\Omega(E, N)$, and the entropy $S(E, N)$.
(c) Calculate the heat capacity $C(T)$ and sketch it.
(d) What is the probability that a specific molecule is standing up?
(e) What is the largest possible value of the internal energy at any positive temperature?

Stanley Enemuo

Numerade Educator

26:30

Problem 8

Curie susceptibility: consider $N$ non-interacting quantized spins in a magnetic field $\vec{B}=B \hat{z}$, and at a temperature $T$. The work done by the field is given by $B M_z$, with a magnetization $M_z=\mu \sum_{i=1}^N m_i$. For each spin, $m_i$ takes only the $2 s+1$ values $-s,-s+$ $1, \cdots, s-1, s$.
(a) Calculate the Gibbs partition function $Z(T, B)$. (Note that the ensemble corresponding to the macrostate $(T, B)$ includes magnetic work.)
(b) Calculate the Gibbs free energy $G(T, B)$, and show that for small $B$,
$$G(B)=G(0)-\frac{N \mu^2 s(s+1) B^2}{6 k_B T}+\mathcal{O}\left(B^4\right) .$$
(c) Calculate the zero field susceptibility $\chi=\partial M_z /\left.\partial B\right|_{B=0}$, and show that it satisfies Curie's law
$$\chi=c / T$$
(d) Show that $C_B-C_M=c B^2 / T^2$, where $C_B$ and $C_M$ are heat capacities at constant $B$ and $M$, respectively.

Dr. Rajveer Singh

Numerade Educator

Problem 9

Langmuir isotherms: an ideal gas of particles is in contact with the surface of a catalyst.
(a) Show that the chemical potential of the gas particles is related to their temperature and pressure via $\mu=k_B T\left[\ln \left(P / T^{5 / 2}\right)+A_0\right]$, where $A_0$ is a constant.
(b) If there are $\mathcal{N}$ distinct adsorption sites on the surface, and each adsorbed particle gains an energy $\epsilon$ upon adsorption, calculate the grand partition function for the two-dimensional gas with a chemical potential $\mu$.
(c) In equilibrium, the gas and surface particles are at the same temperature and chemical potential. Show that the fraction of occupied surface sites is then given by $f(T, P)=$ $P /\left(P+P_0(T)\right)$. Find $P_0(T)$.
(d) In the grand canonical ensemble, the particle number $N$ is a random variable. Calculate its characteristic function $\langle\exp (-i k N)\rangle$ in terms of $Q(\beta \mu)$, and hence show that
$$\left\langle N^m\right\rangle_c=-\left.\left(k_B T\right)^{m-1} \frac{\partial^m \mathcal{G}}{\partial \mu^m}\right|_T,$$
where $\mathcal{G}$ is the grand potential.
(e) Using the characteristic function, show that
$$\left\langle N^2\right\rangle_c=\left.k_B T \frac{\partial\langle N\rangle}{\partial \mu}\right|_T .$$
(f) Show that fluctuations in the number of adsorbed particles satisfy
$$\frac{\left\langle N^2\right\rangle_c}{\langle N\rangle_c^2}=\frac{1-f}{\mathcal{N} f}$$

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01:09

Problem 10

Molecular oxygen has a net magnetic spin $\vec{S}$ of unity, that is, $S^z$ is quantized to $-1,0$, or +1 . The Hamiltonian for an ideal gas of $N$ such molecules in a magnetic field $\vec{B} \| \hat{z}$ is
$$\mathcal{H}=\sum_{i=1}^N\left[\frac{\vec{p}_i^2}{2 m}-\mu B S_i^z\right],$$
where $\left\{\vec{p}_i\right\}$ are the center of mass momenta of the molecules. The corresponding coordinates $\left\{\vec{q}_i\right\}$ are confined to a volume $V$. (Ignore all other degrees of freedom.)
(a) Treating $\left\{\vec{p}_i, \vec{q}_i\right\}$ classically, but the spin degrees of freedom as quantized, calculate the partition function, $\tilde{Z}(T, N, V, B)$.
(b) What are the probabilities for $S_i^z$ of a specific molecule to take on values of $-1,0$, +1 at a temperature $T$ ?
(c) Find the average magnetic dipole moment, $\langle M\rangle / V$, where $M=\mu \sum_{i=1}^N S_i^z$.
(d) Calculate the zero field susceptibility $\chi=\partial\langle M\rangle /\left.\partial B\right|_{B=0}$.

Raj Bala

Numerade Educator

01:14

Problem 11

One-dimensional polymer: consider a polymer formed by connecting $N$ disc-shaped molecules into a one-dimensional chain. Each molecule can align along either its long axis (of length $2 a$ ) or short axis (length $a$ ). The energy of the monomer aligned along its shorter axis is higher by $\varepsilon$, that is, the total energy is $\mathcal{H}=\varepsilon U$, where $U$ is the number of monomers standing up.
(a) Calculate the partition function, $Z(T, N)$, of the polymer.
(b) Find the relative probabilities for a monomer to be aligned along its short or long axis.
(c) Calculate the average length, $\langle L(T, N)\rangle$, of the polymer.
(d) Obtain the variance, $\left\langle L(T, N)^2\right\rangle_c$.
(e) What does the central limit theorem say about the probability distribution for the length $L(T, N)$ ?

Manik Pulyani

Numerade Educator

02:15

Problem 12

Polar rods: consider rod-shaped molecules with moment of inertia $I$, and a dipole moment $\mu$. The contribution of the rotational degrees of freedom to the Hamiltonian is given by
$$\mathcal{H}_{\text {rot. }}=\frac{1}{2 I}\left(p_\theta^2+\frac{p_\phi^2}{\sin ^2 \theta}\right)-\mu E \cos \theta,$$
where $E$ is an external electric field. ( $\phi \in[0,2 \pi], \theta \in[0, \pi]$ are the azimuthal and polar angles, and $p_\phi, p_\theta$ are their conjugate momenta.)
(a) Calculate the contribution of the rotational degrees of freedom of each dipole to the classical partition function.
(b) Obtain the mean polarization $P=\langle\mu \cos \theta\rangle$ of each dipole.
(c) Find the zero-field polarizability
$$\chi_T=\left.\frac{\partial P}{\partial E}\right|_{E=0}$$
(d) Calculate the rotational energy per particle (at finite $E$ ), and comment on its highand low-temperature limits.
(e) Sketch the rotational heat capacity per dipole.

Penny Riley

Numerade Educator