Statistical Mechanics: Solutions (Sekerka)

These solutions come with absolutely no guarantees. A couple of the solutions have errors, I just want to archive my solutions that is why I'm posting them here.

Problem 1

Solution 1
Consider an idealized crystal (Kossel Crystal) made up of atoms that are rigid cubes of edge length a. Assume that the crystal is in the shape of a cube having edge na where n greater than or equal to 2 is an integer. Thus the total number of atoms in the crystal is Ntot = n3.

a) Calculate the following as a function of n:
• Nbulk = the number of atoms interior to the crystal
• Nfaces = the number of atoms on the faces of the crystal that are not at corners or edges
• Nedges = the number of atoms at the edges of the crystal that are not at corner
• Ncorners = the number of atoms at the corners (vertices) of the crystal

Note from the Euler theorem that faces + verticies = edges + 2, in this case 6 + 8 = 12 + 2. Check your results to be sure that Ntot = Nbulk + Nfaces + Nedges + Ncorners.

b) Next, assume that these atoms only have nearest neighbor forces such that the bonds between neighbors have energy −b where b > 0. An atom in the bulk will have energy ebulk = −6b/2 = −3b, since each bond is shared by two atoms. Determine efaces, eedges, and ecorners for the other types of atoms.

c) Calculate the total energy, Etotal, of the crystal due to all atoms and the total energy, Ebulkonly, due only to bulk atoms, both as functions of n. The fraction of energy that is not associated with bulk atoms is

f := (Etotal − Ebulkonly)/Etotal.

Determine f as a function of n and make a plot of f versus n for n = 2 to 10, 000. What do you conclude from your graph about the energy of this crystal as a function of its size? How big would the crystal have to be to have 99.9% of its energy accounted for by bulk atoms?

Problem 2
Solution 2
a) Use the method of Jacobians to express (∂T/∂p)S,N in terms of measurable quantities such as V , T, α := V−1(∂V/∂T)p,N, ;κT := −V−1(∂V/∂p)T,N, and CP = T(∂S/∂T)p,N. You will also need a Maxwell relation that you can find from dG. The quantity (∂T/∂p)S,N is called the isentropic (reversible adiabatic) change in temperature with applied pressure.

b) Evaluate your result for an ideal monatomic gas, for which pV = NRT in terms of the variables T and p. Integrate the resulting equation to show that T/p2/5 = constant. Substitute for T to obtain a relationship obeyed in the V, p plane. This relationship describes the reversible adiabatic segments of the Carnot cycle.

Problem 3
Solution 3
a) Use the method of Jacobians to express ratio of the isentropic coefficient of expansion
αs := (1/V)( ∂V/∂T)S,N
to the isobaric coefficient of thermal expansion
α:= (1/V) (∂V/∂T)p,N
in terms of only the ratio = Cp/Cv . You will need to use a Maxwell relation that you can find from dG and you can use relations for Cp − CV derived in class or in Appendix C to simplify your answer.

b) Evaluate your result for S for a monotonic ideal gas. Also evaluate S directly from the isentropic curve for an ideal gas derived in Problem 2b. c) Give a physical explanation of why S is negative.

Problem 4
Solution 4
A single component material can exist in two phases, the phase and the phase. In the
phase its equation of state is
p/kT = A + Bμα/kT
and in the phase it is
p/kT = C + D(μγ/kT)2

where A, B, C and D are positive functions of the absolute temperature T, D > B and C < A. μα is the chemical potential of α, μγ , is the chemical potential of γ, p is the pressure and k is Boltzmann’s constant.

Note: To simplify the notation, define P := p/(kT) and M := μ/(kT), so that
Mα= (P − A)/B;(Mγ)2/ = (P − C)/D.
Be careful when taking square roots to guarantee that physical quantities such as molar volumes are positive.

Problem 5
Solution 5
A monocomponent system has equations of state (in intensive form) as follows:
T = 2(θ/R)s; p = 2(Rθ/v02)v
where s is the entropy per mole, v is the volume per mole, and the other symbols are

a) Find the fundamental equation in extensive form (U as a function of S, V and N) in the energy representation

b) From U find the chemical potential μ

c) Show explicity that the Euler equation is satisfied.

Problem 6
Solution 6
Consider particles (e.g., spin one) each of which have three quantum states with equally spaced non-degenerate energy levels separated by energy ε, as depicted in Figure 1. Consider a system consisting of N = 4 particles.

Figure 1: Energy level diagram of a three-state quantum subsystem (particle).

a) Assume that the particles have negligible interaction energy and are distinguishable by virtue of their position in a solid. Make a table of the microstates (configurations of the particles) of this system that correspond to a total energy of E = 5". Determine the total number of microstates Ω(E,N) for the given values of E and N. For example, one microstate corresponds to the configuration: particle #1 in ε2 , particle #2 in ε2 , particle #3 in ε1 , and particle #4 in ε0.

b) If the particles in part a were indistinguishable Bozons, how many distinct microstates would there be?

Problem 7
Solution 7
We will show in class that the natural logarithm of a sum of positive terms of the form
s = ΣNm=0 Tm
is given approximately by the logarithm of the largest term Tm* in the sum, i.e.,
ln s ≈ ln Tm*,
provided that Tm* = O(eN) for large N. The purpose of this problem is to have you work out a specific example for which the exact value is known.

a) Consider the function s = (a+b)N where a and b are positive constants. By means of the binomial theorem,

s = ΣNm=0 (N!/m!(N − m))ambN−m.

Find the maximum term in the sum, take its logarithm and compare to ln s. Hint: ln x is a monotonically increasing function of x so the maximum of x and the maximum of ln x occur at the same place. Use Stirling’s approximation in the form lnN! ≈ N lnN − N and then do the required differentiation.

b) How can you explain the precision of your result, given the fact that other terms in the sum were neglected?

c) For Ndistinguishable particles each having three equally spaced states with energies 0, ε and 2ε, the multiplicity function is

g(N,m) = Σp [ N!/(N − p)!(m − p)!(2p − m)! ]

where m = E/ε and E is the total energy. The sum is restricted to values of p for which
all of the factors in the denominator are positive or zero (remember 0! = 1). Find the value of p that determines the maximum term in the sum. This will hopefully convince you that the microcanonical ensemble is not a very tractable way to solve problems.

Problem 8
Solution 8

In Appendix C page 504, Eq(7,a,b), Pathria gives the volume and area of a hypersphere of radius R and dimensionality n:

Vn = (πn/2/(n/2)! )Rn ; Sn = (2πn/2/Γ(n/2) )Rn-1 = (nπn/2/(n/2)! )Rn-1.

Here, the gamma function Γ(n/2+1) = (n/2)Γ(n/2) = (n/2)! suffices to define the factorial whenever n is an odd integer. We also have Γ(1/2) = π(1/2). For large n, practically all of the volume of a hypersphere lies near its surface.

a) Calculate exactly the fraction of volume of a hypersphere that lies within ΔR of its surface. Equate this fraction to 1−ε where ε is very small and solve the resulting equation exactly for ΔR/R in terms of ε and n.

b) Find an approximate solution to this equation for ΔR/R very much less than 1 but for n possibly extremely large. If you are not careful to note that nΔR/R might be large, you will get the wrong answer! Evaluate your answer for ε = 10−10 and n = 1023.

c) Estimate the fraction of volume within ΔR of the surface of a hypersphere by means of the derivative dV = ΔR(dV/dR), equate again to 1 − ε and solve for ΔR/R. Note that for ε = 10−10 this result is practically independent of ε for ε very much less than 1. Why does this result differ from the result of part (b)? What went wrong in this analysis?

d) Based on what you have learned, go back to part (a) and show that a good approximation is

ε = exp(−nΔR/R)

provided that ΔR/R is small, even when nΔR/R is not small.

Problem 9
Solution 9
Pathria Problem 1.7 page 27 which we repeat here with a slight change of wording and notation:

Study the statistical mechanics of an extreme relativistic gas characterized by the single particle energy states

ε(nx, ny, nz, ) = (hc/2L)(nx2+ ny2+ nz2)1/2 = hc/2V1/3 (nx2+ ny2+ nz2)

along the lines followed in Pathria section 1.4 and Sekerka 14.5.1 under the heading Scaling Analysis. Show that the ratio Cp/CV = 4/3 in this case. Show also that this gas satisfies the ideal gas law and relate its pressure to its energy density. Hint: You don’t need to worry about computing the dependence of on N; all you need to know is the functional form of in regard to its dependence on V and E.

Problem 10
Solution 10
We will show in class that the entropy of N atoms of an ideal gas at temperature T and volume V is given by

S = NkB[ln(VnQ/N) + 5/2]

where nQ = (mkBT/2πhbar2)3/2 is the quantum concentration.

Consider a rigid box of volume V consisting of two chambers having volumes V' and V" separated by a partition. Assume that the box and its contents are maintained at temperature T throughout any processes that take place within it. Let the volume V' be filled with N' molecules of an ideal gas and V" be filled with N" molecules of the same gas. If the partition separating V' from V" is removed and the gases are allowed to mix, we will show in class that there will be no entropy change provided that the initial pressures of the two gases are equal. Such will be the case if V'/N' = V"/" = V/N, where V = V' + V" and N = N'+ N". However, if V'/N'≠ V"/N" but still V = V'+ V" and N = N'+ N", the initial pressures of the two gases will not be equal and there will be an entropy change ΔS ≠ 0.

a) Calculate the dimensionless entropy change ΔS/NkB for a process at constant T when atoms of the same gas are mixed as discussed above under conditions for which V'/N' and V"/N" are arbitrary but still V = V'+ V" and N = N'+ N". Show that your answer can be written entirely as a function of two variables, x = N"/N and y = V"/V.

b) Consider your answer as a function of y at fixed x and then as a function of x at fixed y. What is the minimum value of ΔS/NkB and for what conditions does it occur? Sketch curves of ΔS/NkB as a function of y for a few values of x.

c) How much heat Q is absorbed by the system during this process? Is this process reversible or irreversible, and under what conditions? Note: Be careful, this is not an isolated system!

Problem 11
Solution 11
The purpose of this problem is to allow you to solve a problem for a three state system by using the microcanonical ensemble for N = 100 particles and to compare the result with the result you would obtain by specifying the temperature and using the canonical ensemble. We consider N = 100 identical but distinguishable subsystems, each of which has three equally spaced energy levels, ε0 = 0, ε1 = ε, and ε2 = 2ε. Consider a quantum state of the whole system having energy E = mε. The multiplicity g(N , m) of this state will be the coefficient of tm in the expansion of (1 + t + t2 )100 which we can evaluate by using Mathematica. We are interested in the state for m = 30 so we dump all output except for m = 30 and neighboring states m = 29 and m = 31. The resulting values are:
• g29 := g(100, 29) = 13382209963495358875422764000
• g30 := g(100, 30) = 51118572167520610863352020240
• g31 := g(100, 31) = 189247030822069326713933356800

(a) What is the entropy S of this system (in units of kB ) for energy E = 30ε?

(b) Calculate approximately the value of t := kB T /ε for E = 30ε.
Hint: Evaluate the required derivative f'm by using the finite difference formula

where f is some appropriate function of m.

(c) Use the canonical ensemble to solve this same problem and obtain an expression for U/ε as a function of T . Evaluate this expression for the value of T calculated in part b and compare your result with E/ε = 30. What do you conclude from this comparison?

Problem 12
Solution 12
This problem is motivated by Pathria problem 2.7, page 41.
Consider N identical but distinguishable and very weakly interacting one-dimensional simple harmonic oscillators at fixed positions in a solid, each with energy levels separated by hbarω. We know that the multiplicity of a state having total energy E = mhbarω is
and that the entropy S = kB ln Ω. According to classical statistical mechanics, all we know is that

S = kB ln (∆ω/ωo)

where ∆ω is the volume of phase space between E − ∆E and E. Classically, the constant ωo remains unknown, but we can determine it by comparison with a quantum mechanical result in an approriate limit.

(a) Find an approximate expression for the above expression for Ω in the limit of high temperatures and express your results in terms of E/hbarω and N .
Hint: Write out the expression for Ω in more detail and think about the relative values of N and m at high temperatures.

(b) The classical Hamiltonian for N simple harmonic oscillators is
where ω = k/m. Calculate ∆ω to a reasonable approximation.
Hint: For large N we know that ∆ω is practially equal to the entire volume of phase space corresponding to energies ≤ E. Calculate that volume by making an appropriate change of variables to map it onto the volume of a hypersphere.

(c) Compare the results of parts (a) and (b) and deduce the value of ωo needed to get agreement between classical and quantum mechanical results. Explain why this value of ωo differs from that given by Sekerka Eq. (12.19)?

Problem 13
Solution 13
Suppose that the partition function Z(β) for some system is known as a function of
β = 1/kB T .

(a) Deduce a general formula for the heat capacity at constant volume, CV , in terms of derivatives of Z with respect to β.

(b) Apply your formula to calculate the heat capacity of two very weakly interacting one-dimensional simple harmonic oscillators having angular frequencies ω1 and ω2 . How would our answer change of there were N1 oscillators having ω1 and N2 having ω2?

(c) Sketch the heat capacities of a system composed of these two oscillators as a function of
temperature for ω1 < < ω2 .

(d) Suppose that you have a system consisting of many very weakly interacting harmonic oscillators with angular frequencies 0 ≤ ω ≤ ω and that they are distibuted such that there are G(ω)dω oscillators with angular frequencies between ω and ω + dω. Apply your general result to write an expression for the heat capacity of such a system.

Problem 14
Solution 14
For the canonical ensemble, the temperature, T , is specified and the average internal energy U of a system is calculated. However, the members of the canonical ensemble have a distribution of energies. This problem is designed to illustrate their dispersion of energy.

(a) Show for any linear averaging process, <> , that
(b) Show that
where E is the energy of a member of the ensemble and τ = 1/β and CV is the heat capacity at constant volume.

(c) Use the result of part b to calculate (E−U)2 for N identical but distinguishable harmonic oscillators having angular frequency ω. Give approximate forms of your result for very high and very low temperatures and discuss briefly their physical significance.

(d) Evaluate the result of part (b) for a monatomic ideal gas and give numerical values of the root mean square energy dispersion per particle in units of kB T , namely the quantity √(E − U)2 /(NkB T ), for N = 100 and N = 1020 .
Note: ∂U/∂τ = (1/k)CV where CV is the heat capacity of the system. U, E and CV pertain to the whole system, so each is of order N . If we introduce the heat capacity per particle cv := CV /N and take the square root of the result of part b, we obtain
which shows that the dispersion in energy per particle is proportional to 1/√N . Therefore, as N → ∞, one approaches the thermodynamic limit in which the energy fluctuations go to zero.

Problem 15
Solution 15
Consider N identical but distinguishable very weakly interacting four-level quantum sub-systems (particles) with degenerate energy levels and energy gaps as shown in Figure 1. Each system has seven quantum states, with degeneracies as shown in the figure.
Figure 1: Energy level diagram of a four-level quantum subsystem having seven quantum states, some of which are degenerate. The ground level is a doublet, the first excited level is a singlet, the next excited level is a triplet, and the highest level is a singlet.

(a) Write an explicit expression for the internal energy, U, for the entire system of particles
at general temperature T .

(b) For kB T = ε/2, what is the ratio of the the fraction of ALL subsystems (particles) in quantum states having energy 5ε to those in state ε3 having having energy ε? Evaluate your result numerically.

(c)For kBT = ε/2, what is the approximate value of U/(N ε)? Evaluate your result numerically and compare with the exact numerical result.

(d) For a very high temperature, say T > 1000ε/kB, what is the entropy S of the entire system of particles?

(e) Plot the heat capacity CV at constant volume of this system of particles as a function of temperature for temperatures up to 20ε/kB . Explain the shape of your graph.

Problem 16-20
Solution 16
Solution 17
Solution 18
Solution 19
Solution 20

Problem 21-25
Problem 21
Problem 22

Problem 23
Problem 24
Problem 25

Problem 26-30
Solution 26
Solution 27
Solution 28
Solution 29
Solution 30

Problem 31-35
Solution 31
Solution 32
Solution 33
Solution 34
Solution 35

Problem 36-40
Solution 36
Solution 37
Solution 38
Solution 39
Solution 40




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