Free solutions & answers for Statistical Mechanics Chapter 3 - (Page 1) [step by step] (2024)

Verify that the quantity \((k / \mathcal{N}) \ln \Gamma\), where $$ \Gamma(\mathcal{N}, U)=\sum_{\left\langle n_{r}\right\rangle}^{\prime}W\left\\{n_{r}\right\\}, $$ is equal to the (mean) entropy of the given system. Show that this leads toessentially the same result for \(\ln \Gamma\) if we take, in the foregoingsummation, only the largest term of the sum, namely the term\(W\left\\{n_{r}^{*}\right\\}\) that corresponds to the most probabledistribution set. [Surprised? Well, note the following example: For all \(N\), the summation over the binomial coefficients \({ }^{N} C_{r}=N !/[r !(N-r !)]\) gives $$ \sum_{r=0}^{N}{ }^{N} C_{r}=2^{N_{N}} $$ therefore, $$ \ln \left\\{\sum_{r=0}^{N}{ }^{N} C_{r}\right\\}=N \ln 2 . $$ Now, the largest term in this sum corresponds to \(r \simeq N / 2\); so, forlarge \(N\), the logarithm of the largest term is very nearly equal to $$ \begin{aligned} & \ln \\{N !\\}-2 \ln \\{(N / 2) !\\} \\ \approx & N \ln N-2 \frac{N}{2} \ln \frac{N}{2}=N \ln 2 \end{aligned} $$

Making use of the fact that the Helmholtz free energy \(A(N, V, T)\) of athermodynamic system is an extensive property of the system, show that $$ N\left(\frac{\partial A}{\partial N}\right)_{V, T}+V\left(\frac{\partialA}{\partial V}\right)_{N, T}=A . $$ [Note that this result implies the well-known relationship: \(N \mu=A+PV(\equiv G)\).]

Prove that, quite generally, $$ C_{P}-C_{V}=-k \frac{\left[\frac{\partial}{\partialT}\left\\{T\left(\frac{\partial \ln Q}{\partialV}\right)_{T}\right\\}\right]_{V}^{2}}{\left(\frac{\partial^{2} \lnQ}{\partial V^{2}}\right)_{T}}>0 . $$ Verify that the value of this quantity for a classical ideal classical gas is\(N k\).

If an ideal monatomic gas is expanded adiabatically to twice its initialvolume, what will the ratio of the final pressure to the initial pressure be?If during the process some heat is added to the system, will the finalpressure be higher or lower than in the preceding case? Support your answer byderiving the relevant formula for the ratio \(P_{f} / P_{i}\).

(a) The volume of a sample of helium gas is increased by withdrawing thepiston of the containing cylinder. The final pressure \(P_{f}\) is found to beequal to the initial pressure \(P_{i}\) times \(\left(V_{i} / V_{f}\right)^{1.2},V_{i}\) and \(V_{f}\) being the initial and final volumes. Assuming that theproduct \(P V\) is always equal to \(\frac{2}{3} U\), will (i) the energy and (ii)the entropy of the gas increase, remain constant, or decrease during theprocess? (b) If the process were reversible, how much work would be done and how muchheat would be added in doubling the volume of the gas? Take \(P_{i}=1\mathrm{~atm}\) and \(V_{i}=1 \mathrm{~m}^{3}\).

(a) Evaluate the partition function and the major thermodynamic properties ofan ideal gas consisting of \(N_{1}\) molecules of mass \(m_{1}\) and \(N_{2}\)molecules of mass \(m_{2}\), confined to a space of volume \(V\) at temperature\(T\). Assume that the molecules of a given kind are mutually indistinguishable,while those of one kind are distinguishable from those of the other kind. (b) Compare your results with the ones pertaining to an ideal gas consistingof \(\left(N_{1}+N_{2}\right)\) molecules, all of one kind, of mass \(m\), suchthat \(m\left(N_{1}+N_{2}\right)=m_{1} N_{1}+m_{2} N_{2}\).

Consider a system of \(N\) classical particles with mass \(m\) moving in a cubicbox with volume \(V=L^{3}\). The particles interact via a short-ranged pairpotential \(u\left(r_{i j}\right)\) and each particle interacts with each wallwith a short-ranged interaction \(u_{\text {wall }}(z)\), where \(z\) is theperpendicular distance of a particle from the wall. Write down the Lagrangianfor this model and use a Legendre transformation to determine the Hamiltonian\(H\). (a) Show that the quantity \(P=-\left(\frac{3 H}{\partial V}\right)=\frac{-1}{3L^{2}}\left(\frac{\partial H}{\partial L}\right)\) can clearly be identified asthe instantaneous pressure - that is, the force per unit area on the walls. (b) Reconstruct the Lagrangian in terms of the relative locations of theparticles inside the box \(\boldsymbol{r}_{i}=L s_{i}\), where the variables\(s_{i}\) all lie inside a unit cube. Use a Legendre transformation to determinethe Hamiltonian with this set of variables. (c) Recalculate the pressure using the second version of the Hamiltonian. Showthat the pressure now includes three contributions: (1) a contribution proportional to the kinetic energy, (2) a contribution related to the forces between pairs of particles, and (3) a contribution related to the force on the wall. Show that in thethermodynamic limit the third contribution is negligible compared to the othertwo. Interpret contributions 1 and 2 and compare to the virial equation ofstate (3.7.15).

Show that the partition function \(Q_{N}(V, T)\) of an extreme relativistic gasconsisting of \(N\) monatomic molecules with energy-momentum relationship\(\varepsilon=p c, c\) being the speed of light, is given by $$ Q_{N}(V, T)=\frac{1}{N !}\left\\{8 \pi V\left(\frac{k T}{hc}\right)^{3}\right\\}^{N} $$ Study the thermodynamics of this system, checking in particular that $$ P V=\frac{1}{3} U, \quad U / N=3 k T, \quad \text { and } \quad\gamma=\frac{4}{3} $$ Next, using the inversion formula (3.4.7), derive an expression for thedensity of states \(g(E)\) of this system.