Chapter 6- Answers to selected exercises

1. A system of $N$ classical ultrarelativistic particles, in a container of volume $V$, at temperature $T$, is given by the Hamiltonian

$$\mathcal{H}=\dsum\limits_{i=1}^{N}c\left\vert \vec{p}_{i}\right\vert ,$$

where $c$ is a positive constant. Obtain an expression for the canonical partition function. Calculate the entropy per particle as a function of temperature and specific volume. What is the expression of the specific heat at constant volume? All the thermodynamic quantities are easily obtained from the classical canonical partition function, given by

$$Z=\frac{1}{h^{3N}N!}V^{N}\left[ \dint d^{3}p\exp \left( -\beta c\... ...rac{1}{h^{3N}N!}\left[ \frac{8\pi V}{\left( \beta c\right) ^{3}}\right] ^{N}.$$

2. Consider a set of $N$ one-dimensional harmonic oscillators, described by the Hamiltonian

$$\mathcal{H}=\dsum\limits_{i=1}^{N}\left[ \frac{1}{2m}\vec{p}_{i}^{\,2}+\frac{ 1}{2}m\omega ^{2}x_{i}^{n}\right] ,$$

where $n$ is a positive and even integer. Use the canonical formalism to obtain an expression for the classical specific heat of this system. Note that the canonical partition function may be written as $Z=Z_{1}^{N}$, where

$$Z_{1}=\left( \frac{2\pi m}{\beta }\right) ^{1/2}\left( \frac{2}{\... ...ega ^{2}}\right) ^{n/2}\int_{-\infty }^{+\infty }dy\exp \left( -y^{n}\right) .$$

Thus,

$$u=-\frac{\partial }{\partial \beta }\ln Z_{1}=\frac{n+2}{2n}k_{B}T.$$

3. Consider a classical system of $N$ very weakly interacting diatomic molecules, in a container of volume $V$, at a given temperature $T$. The Hamiltonian of a single molecule is given by

$$\mathcal{H}_{m}=\frac{1}{2m}\left( \vec{p}_{1}^{\,2}+\vec{p} _{2... ...\right) +\frac{1}{2}\kappa \left\vert \vec{r}_{1}-\vec{r}_{2}\right\vert ^{2},$$

where $\kappa >0$ is an elastic constant. Obtain an expression for the Helmholtz free energy of this system. Calculate the specific heat at constant volume. Calculate the mean molecular diameter,

$$D=\left\{ \left\langle \left\vert \vec{r}_{1}-\vec{r}_{2}\right\vert ^{2}\right\rangle \right\} ^{1/2}.$$

        Now consider another Hamiltonian, of the form $\epsilon$

$$\mathcal{H}_{m}=\frac{1}{2m}\left( \vec{p}_{1}^{\,2}+\vec{p} _{2}^{\,2}\right) +\epsilon \left\vert r_{12}-r_{o}\right\vert ,$$

where $\epsilon$ and $r_{o}$ are positive constants, and $r_{12}=\left\vert \vec{r}_{1}-\vec{r}_{2}\right\vert$. What are the changes in your previous answers? Taking into account the thermodynamic limit ( $V\rightarrow \infty$), the first Hamiltonian is associated with the partition function

$$Z_{1}=\left( \frac{2\pi m}{\beta }\right) ^{3}\int d^{3}r_{1}\int... ...2\pi m}{\beta } \right) ^{3}V\left( \frac{2\pi }{\beta \kappa }\right) ^{3/2},$$

from which we obtain the Helmholtz free energy (and the value of $D$). It interesting to check the constant value of the molecular specific heat, $c=\left( 9/2\right) k_{B}$. The second Hamiltonian represents a more complicated model of a diatomic molecule. The partition function is given by

$$Z_{1}=\left( \frac{2\pi m}{\beta }\right) ^{3}\int d^{3}r_{1}\int... ...o}\right) \right\vert \right] \sim \left( \frac{2\pi m}{\beta }\right) ^{3}VI,$$

where

$$I=4\pi \dint\limits_{0}^{r_{o}}r^{2}\exp \left[ -\beta \epsilon \... ...}}^{\infty }r^{2}\exp \left[ -\beta \epsilon \left( r-r_{o}\right) \right] dr=$$

$$=\frac{2r_{o}^{2}}{\beta \epsilon }+\frac{4}{\left( \beta \epsilo... ...{\left( \beta \epsilon \right) ^{4}}\exp \left( -\beta \epsilon r_{o}\right) .$$

Now it is interesting to obtain the specific heat as a function of temperature.

4. Neglecting the vibrational motion, a diatomic molecule may be treated as a three-dimensional rigid rotator. The Hamiltonian $\mathcal{H}_{m}$ of the molecule is written as a sum of a translational, $\mathcal{H}_{tr}$, plus a rotational, $\mathcal{H}_{rot}$, term (that is, $\mathcal{H}_{m}=\mathcal{H} _{tr}+\mathcal{H}_{rot}$). Consider a system of $N$ very weakly interacting molecules of this kind, in a container of volume $V$, at a given temperature $T$. (a) Obtain an expression for $\mathcal{H}_{rot}$ in spherical coordinates. Show that there is a factorization of the canonical partition function of this system. Obtain an expression for the specific heat at constant volume. (b) Now suppose that each molecule has a permanent electric dipole moment $\vec{\mu}$ and that the system is in the presence of an external electric field $\vec{E}$ (with the dipole $\vec{\mu}$ along the axis of the rotor). What is the form of the new rotational part of the Hamiltonian? Obtain an expression for the polarization of the molecule as a function of field and temperature. Calculate the electric susceptibility of this system. The Lagrangian of a free rotator (two atoms of mass $m$ and a fixed interatomic distance $a$) is given by

$$L_{rot}=\frac{ma^{2}}{4}\overset{}{\left[ \left( \frac{d\theta }{dt}\right) ^{2}+\left( \frac{d\varphi }{dt}\right) ^{2}\sin ^{2}\theta \right] },$$

from which we have the Hamiltonian

$$H_{rot}=\frac{p_{\theta }^{2}}{ma^{2}}+\frac{p_{\varphi }^{2}}{ma^{2}\sin ^{2}\theta }.$$

Therefore,

$$u_{rot}=\left\langle H_{rot}\right\rangle =\frac{1}{2}k_{B}T+\frac{1}{2} k_{B}T=k_{B}T.$$

In the presence on an electric field (taken along the $z$ direction), and with the magnetic moment along the axis of the rotator, we have

$$H=\frac{p_{\theta }^{2}}{ma^{2}}+\frac{p_{\varphi }^{2}}{ma^{2}\sin ^{2}\theta }-E\mu \cos \theta .$$

The associated partition function is given by

$$Z_{1}=\dint\limits_{0}^{\pi }d\theta \dint\limits_{0}^{2\pi }d\va... ...ta p_{\varphi }^{2}}{ma^{2}\sin ^{2}\theta }+\beta E\mu \cos \theta .\right] =$$

$$=\frac{4\pi ^{2}ma^{2}}{\beta }\frac{2\sinh \left( \beta \mu E\right) }{ \beta \mu E}.$$

Thus, we have

$$\left\langle \mu \cos \theta \right\rangle =\mu \left[ \coth \lef... ...a \mu E\right) -\frac{1}{\beta \mu E}\right] =\mu L\left( \beta \mu E\right) ,$$

where $L\left( x\right)$ is known as the Langevin function.

5. Consider a classical gas of $N$ weakly interacting molecules, at temperature $T$, in an applied electric field $\vec{E}$. Since there is no permanent electric dipole moment, the polarization of this system comes from the induction by the field. We then suppose that the Hamiltonian of each molecule will be given by the sum of a standard translational term plus an ``internal term.'' This internal term involves an isotropic elastic energy, which tends to preserve the shape of the molecule, and a term of interaction with the electric field. The configurational part of the internal Hamiltonian is be given by

$$\mathcal{H}=\frac{1}{2}m\omega _{o}^{2}r^{2}-q\vec{E}\cdot \vec{r}.$$

Obtain the polarization per molecule as a function of field and temperature. Obtain the electric susceptibility. Compare with the results of the last problem. Make some comments about the main differences between these results. Do you know any physical examples corresponding to these models? First, we calculate the configurational partition function

$$Z_{1}=\dint\limits_{0}^{\infty }r^{2}dr\dint\limits_{0}^{\pi }\si... ...p \left[ -\frac{\beta m\omega _{o}^{2}}{2}r^{2}+\beta qEr\cos \theta \right] =$$

$$=\beta \left( \frac{2\pi }{\beta m\omega _{o}^{2}}\right) ^{3/2}\exp \left( \frac{\beta q^{2}E^{2}}{2m\omega _{o}^{2}}\right) .$$

The polarization is given by

$$\left\langle qr\cos \theta \right\rangle =\frac{1}{\beta }\frac{\partial }{ \partial E}\ln Z_{1}=\frac{q^{2}E}{m\omega _{o}^{2}},$$

so the dielectric susceptibility is just a constant (in sharp contrast to the previous result for permanent electric dipoles!)

6. The equation of state of gaseous nitrogen at low densities may be written as

$$\frac{pv}{RT}=1+\frac{B\left( T\right) }{v},$$

where $v$ is a molar volume, $R$ is the universal gas constant, and $B\left( T\right)$ is a function of temperature only. In the following table we give some experimental data for the second virial coefficient, $B\left( T\right)$ , as a function of temperature.

$$\begin{tabular}{\vert l\vert l\vert} \hline $T\left( \mathrm{K}... ... \hline $373$\ & $6.2$\ \\ \hline $600$\ & $21.7$\ \\ \hline \end{tabular}$$

Suppose that the intermolecular potential of gaseous nitrogen is given by

$$V\left( r\right) =\left\{ \begin{array}{ll} \infty , & 0<r<a, \\ -V_{0}, & a<r<b, \\ 0, & r>b. \end{array} \right.$$

Use the experimental data of this table to determine the best values of the parameters $a$, $b$, and $V_{0}$. According to Section 6.4 (although nitrogen is a gas of diatomic molecules), the virial coefficient is given by

$$B=-2\pi \dint\limits_{0}^{\infty }r^{2}\left\{ \exp \left[ -\beta... ...i \dint\limits_{a}^{b}r^{2}\left[ \exp \left( \beta V_{o}\right) -1\right] dr.$$

Now it is straightforward to fit the parameters $a$, $b$ and $V_{o}$.