STATISTICAL MECHANICS & THERMODYNAMICS

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1 UVA PHYSICS DEPARTMENT PHD QUALIFYING EXAM PROBLEM FILE STATISTICAL MECHANICS & THERMODYNAMICS UPDATED: NOVEMBER 14, a. Explain what is meant by the density of states, and give an expression for the density of states of a free partile per unit interval of wavevetor k = k in three dimensions for a ube of length L. In speial relativity the energy of a partile is related to its momentum by E( p) = ( p) + ( m ), where m is the mass and the speed of light. Highly relativisti partiles with energy greatly in exess of their rest mass therefore have a linear relation E = p between momentum and energy: their veloity is (almost) the speed of light. b. Find the one-partile partition funtion Z 1 for suh partiles.. Find the N partile partition funtion in an approximation that ignores the effet of quantum statistis. d. Find the internal energy, the heat apaities C V and γ = C / C P V C P, and the adiabati index 2. A simple model for a binary alloy (e.g. β-brass, whih is a mixture of Zn and Cu) is to onsider a simple ubi lattie in whih eah site is oupied by either an A atom or a B atom. Take the hemial potentials for the A and B atoms to be µ A and µ B. In addition, eah pair of neighboring atoms will interat with eah other and ontribute an energy E = φ AA, φ BB or φ AB, depending on whether the pair is both A, both B or one of eah. a) Define a variable t i whih is when the site has an A atom and 1 when the site has a B atom. Write down the energy as a funtion of t i. b) Show how this model may be mapped on to the lassial Ising model, and identify J and H in terms of the parameters speified above. (In the Ising model, ) E = J S S H S + E i J i o < i, j> i=1 1

2 3. An Einstein solid onsists of a lattie in 3-dimensional spae with N lattie sites. Eah lattie site ontains three harmoni osillators, one for eah diretion in spae, eah of whih has frequeny. The harmoni osillators are not oupled together. Use the anonial ensemble to ompute the following assuming the Hamiltonian is given by 3N ˆ 1 ˆ H = ω ni + i= 1 2 where ˆn i is the number operator for energy quanta of the i th harmoni osillator. The number operators have eigenvetors n i >, so that ˆn i n i >= n i n i >. Note that the temperature is arbitrary. (a) The entropy of the solid. To get full redit, you will need to show expliitly how you evaluated the partition funtion. (b) The internal energy () The heat apaity of the Einstein solid. 4. Consider the following approximate equation of state for a non-ideal gas, whih takes the form of a trunated expansion in density b p nkbt n n where n N / V is the density, and b and are positive onstants. (a) Sketh the isotherms pn ( ) for different values of the temperature (it s easier to work in terms of density here rather than volume). (b) 1 V By finding the isothermal ompressibility T as a funtion of n, V p T onlude that there is a ritial temperature T below whih this equation of state must break down. Find T and the orresponding density n and pressure p. () Desribe in as muh detail as possible (without doing any alulations) what atually happens below the ritial point. UVa Physis Department PhD Qualifying Exam Problem File 2

3 5. A three-dimensional gas of noninterating bosoni partiles obeys the dispersion relation ε(k) = A k 1/2. (a) Obtain an expression for the density n(t, z) where z = e βµ is the fugaity. Simplify your expression as the best you an. You may find onvenient to define ν 1 k 1 t z ζ ν ( z) = dt = f 1 e t Γ ν 1 k ν ( ) (b) Find the ritial temperature for Bose ondensation, T (n). Your answer should only inlude the density n, the onstant A, the physial onstants and numerial fators (whih may be expressed in terms of integrals or infinite sums). () What is the ondensate density n o when T= ½ T? k = 1 6. (a) Derive the following expression, ln Z E = β, 1 (where β = ) for the average energy of a system desribed by the kt B Boltzmann distribution, with the anonial partition funtion Z. (b) Calulate Z and use this formula to find the average energy of 1) a free partile in one dimension onfined to a box of size 2L; ) a lassial 1D harmoni osillator with energy K + V = mv + kx. 2 2 () Now suppose that the potential is V = x α, with α >. Find the average energy per partile in this ase, and explain what happens as α. UVa Physis Department PhD Qualifying Exam Problem File 3

4 7. A simple model for a linear moleule onsists of a one-dimensional system of N partiles (eah of mass m) onneted in a line by idential springs (eah with onstant κ). Classial mehanis tells us that the natural vibrational frequenies are given by nπ ωn = 2ωsin 2 N for ω = κ / m and n = 1, 2, (N-1). You an assume N >> 1, and that the moleule is in thermal equilibrium at temperature T suh that assume NkT ω. Note that you an t kt ω, so the system is not neessarily in the lassial limit. (a) Derive an integral expression for the heat apaity of the moleule under these onditions. If the integral you obtain is non-trivial, you don t need to evaluate it. (b) Evaluate the heat apaity in the low- and high-temperature limits. Verify that your answer agrees with expetations in the high-temperature limit. In the low temperature limit, find the leading-order dependene on temperature as T. 8. Consider a two-dimensional gas of N non-interating eletrons (spin ½) in a box of area A. (a) What is the density of single partile states in k-spae? (b) Calulate the density of single partile states with respet to energy, D(ε). () Find the Fermi wavevetor k F and the Fermi energy, µ(t=). (d) Find the internal energy E at T= as a funtion of N and A. (e) Find the surfae tension σ at T= as a funtion of N and A. 9. (a) For a gas of free eletrons in d dimensions, ompute the isothermal ompressibility, κt() at zero temperature, in terms of the mean number of partiles per unit volume, n=<n>/v and the Fermi energy, ε F. Give the answer to an overall numerial onstant C d. (b) Estimate numerially the Fermi temperature of metalli opper by treating the eletrons as a gas of free partiles in 3-dimensions. The density of opper is UVa Physis Department PhD Qualifying Exam Problem File 4

5 892 kg.m -3, its atomi weight is 63.5 and it may be assumed that there is one ondution eletron per atom. 1 P (hint: κt ( ) = V ). V NT, = 1. N atoms of spin ½ and mass m are onfined to a volume V. The atoms an form bound pairs of spin zero and binding energy u per pair, where u >. The atoms and bound pairs are otherwise non-interating partiles. The hemial potential of the atom is µ 1 and that of the bound pair is µ 2. (a) Using the free energy approah, derive a relation that expresses µ 2 in terms of µ 1 when the system is in thermal equilibrium. (Note: No redit will be awarded if you fail to present a derivation based on free energy onsideration.) (b) What is the number of bound pairs at T =? Why? (You may assume that N is very large.) () What must µ 1 be if the pairs undergo a Bose-Einstein ondensation (BEC)? Why? (d) Find the atomi density N/V for the pairs to undergo a BEC at the transition temperature T, assuming k B T << u. [If the expression you obtain ontains an integral that does not depend on any physial onstant or parameter, you may let the integral be equal to a number alled γ.] (e) Identify the phases (i.e., atoms, pairs, et) that exist at (i) T = ; (ii) T > T > ; and (iii) T > T. Justify your answer. 11. A vessel of volume V 1 ontains N moleules of a lassial ideal gas held at temperature T and pressure P 1. The energy of a moleule may be written in the form E j p 2m px y pz ( px, py, pz ) = ε j 2m where ε j denotes the energy levels orresponding to the internal states j of the moleules of the gas. (a) Evaluate the free energy, F. Expliitly display the dependene on the volume V 1, leaving your answer in terms of the internal free energy 2m UVa Physis Department PhD Qualifying Exam Problem File 5

6 Now onsider another vessel of volume V 2, also at temperature T, ontaining the same number of moleules of idential gas held at pressure P 2. (b) Give an expression for the total entropy of the two gases in terms of P 1, P 2, T and N. () The vessels are then onneted to permit the gases to mix without doing work. Evaluate expliitly the hange in entropy of the system. Does your answer make sense? Chek it by onsidering the speial ase V 1 = V 2 (P 1 = P 2 ). 12. Consider a large regular lattie of magneti atoms in an external field, B. The spindependent part of the Hamiltonian is approximately 1 J Si Sj µ Si B 2, i, j i where the double sum is restrited to the q nearest neighbors of eah atom. For simpliity you an assume spin ½ and that the external magneti field is along the z-diretion. (a) Whih sign of J leads to ferromagnetism? Choose this sign for the rest of the problem. (b) Using mean field theory (the Weiss moleular-field approximation), find the self-onsistent equation for the mean value of eah spin, or equivalently for the magnetization, when the atomi density is n = N/V atoms per unit volume. () Find the Curie temperature T in terms of J and q. (d) For T < T, the onset of magnetization is proportional to (T - T) β. Calulate the value of β in the mean field theory. 13. (a) For a lassial system with the Hamiltonian (, ) H q p at a temperature τ, show that N 2 N pi ki = + q 2m 2 i= 1 i i= 1 2 i 2 pi 2m i τ 2 k i 2 2 = qi τ = 2 (b) Consider a system of a large number of lassial partiles and assume a UVa Physis Department PhD Qualifying Exam Problem File 6

7 general dependene of the energy of eah partile on the generalized oordinate or momentum omponent q given by ε (q), where q ± ( q) = + lim ε. Show that in thermal equilibrium, the generalized equipartition theorem holds: ε ( q) q = τ. q 14. In a uniform magneti field, B, applied along the z -diretion, the orbital motion of the eletron projeted on the xy -plane makes a irular motion with angular frequeny, ω = eb / m (also known as the ylotron frequeny) due to the Lorentz fore. The irular motion an be regarded as a quantized harmoni osillation. The energy levels of this eletron are given by the following expression: 1 2 ( ) E = ω n+ 2 + pz /2m (a) Provide an expression for the grand partition funtion. Ignore the eletron spin ontribution. Assume that eah Landau level has degeneray, and onsider it to be g ( eb / ) L x L y where L is the length. Leave your answer in the form of an integral. (b) Evaluate the grand partition funtion in the high temperature limit. Assume the fugaity f e /kt satisfies f 1 in this limit. () Evaluate the magnetization, M, in this high temperature limit. Simply stating the definition of M is not suffiient. 15. An eletron in a magneti field H has energy ε μ B H, depending on whether the spin magneti moment is aligned, respetively, parallel or anti-parallel to the field. Here ε is the eletron kineti energy and μ B is the Bohr magneton. Assume that an eletron gas with a fixed number N of eletrons in a volume V is nearly degenerate; N and V are large enough so that the thermodynami limit applies. The gas is neutral due to a bakground harge in this problem we ll be onerned only with the eletrons. Ignoring any eletron orbital motion, evaluate the following quantities at low temperatures: (a) The density of states, g(ε), assoiated with translational motion of the eletrons in the ontainer. UVa Physis Department PhD Qualifying Exam Problem File 7

8 (b) The average number of partiles having their spins parallel or anti-parallel to the field. (Use the grand anonial ensemble). M N N, where án + ñ is the number () The average magnetization µ B ( + ) of partiles with their spins parallel to the magneti field and án - ñ is the number of partiles with their spins anti-parallel. Evaluate this quantity at low temperatures, by whih we mean that you an assume (here the quantity μ is the hemial potential) that (μ ± μ B H)/ 1, where τ kt. If we all x (μ ± μ B H)/, you may therefore want the following large-x approximation: if 2 u 2 3/2 π 1/2 I ( x) du, then I ( x) x + x /2 u x 1/2 e + (d) The zero-field spin paramagneti suseptibility, χ, of the eletron gas at low temperatures. (This quantity is defined in the limit as H of M / H). Your answer should inlude the first orretion to the low-temperature leading term. You an leave your answer without expliit evaluation of µ. 16. Problem #16 The rotational motion of a lassial two-atom moleule about its enter of mass is desribed by two angular variables φ and θ (see the figure above) and the orresponding anonial momenta pφ and pθ. The kineti energy of the lassial rotational motion has the form E rot = pθ + 2 φ 2I 2I sin θ p Consider an ideal gas of suh two-atom moleules and UVa Physis Department PhD Qualifying Exam Problem File 8

9 1. Calulate the partition funtion of the lassial rotational motion. 2. Find the entropy and the heat apaity of the rotational motion. Can the result for the heat apaity be understood without alulations? The energy of a gas of photons at temperature T in a volume V is known to be U = αvt, where α is a onstant that an ultimately be expressed in terms of fundamental onstants. a. Determine the entropy S of this gas. b. Determine the equation of state of the photon gas. A ertain engine uses the photon gas as its working material. The engine yle shown as a (reversible) pv diagram in the figure, following the path , where paths 12 and 34 are isobari (onstant pressure) and paths 23 and 41 are adiabati.. Draw the yle shown as an ST diagram, identifying the (S, T) oordinates that orrespond to the (p, V) points 1, 2, 3, and 4. Use an arrow to indiate the diretion of the yle as you have drawn it on the ST diagram, and desribe the shape of the yle on the ST diagram. d. Use your ST plot (NOT the pv plot) to determine the effiieny of the engine. As always, explain your reasoning. Problem # Here are two fats about a substane: The entropy is the following funtion of temperature T and volume V: α 1 V T S = R V T At the partiular temperature T, the work the substane does on its surroundings as it expands from V to V is UVa Physis Department PhD Qualifying Exam Problem File 9

10 W T V = RT log V Here R is a onstant with units of J/K and α is a dimensionless onstant. (a) Find the Helmholtz free energy A, assuming that it is zero at T =T, V = V. (b) Find the equation of state for this substane. Is there a point in parameter spae where the substane behaves as an ideal gas? () Take α = 1 and define V 1 through log(v 1 /V ) = 1. At what temperature T would the system do twie as muh work in going from V to V 1 as it does at T = T? 19. The partition funtion for an ideal gas of N bukyball moleules (C 6, i.e., eah moleule is made up of 6 arbon atoms) in a volume V at temperature T an be written in the form N z Z =. N! (a) If z in turn an be written in the form z = V θ ζ, on what variables (i.e., p, V, N, µ, T, ) does ζ depend? What is θ? Explain your answers! (b) Derive the equation of state for the C 6 -bukyball gas. When N l bukyballs ondense to form a liquid, the rudest approximation one an make is to treat these moleules as if they still form a gas, with the additional provisos that (i) eah C 6 moleule is assumed to have a potential energy ε due to interations with the rest of the moleules, and (ii) eah moleule is free to move in a volume of N l v, where v is a onstant. () Based on the above information, find the partition funtion for a bukyball liquid onsisting of N l bukyballs in terms of the funtion ζ and relevant parameters given in this problem. (You may want to use the approximation lnn! N lnn N.) (d) Find an expression for the vapor pressure of an ideal C 6 gas in equilibrium with its liquid phase. UVa Physis Department PhD Qualifying Exam Problem File 1

11 2. The entropy-temperature (S-T) yle of a nameless reversible engine is shown in the figure. The highest and lowest temperatures T h and T l as well as the highest and lowest entropies S h and S l are marked on the figure. The area of the loop in the ST-plane is A in some units. Clearly S(T) as well as the inverse T(S) are double-valued funtions exept at the extremes T h, T l, S h, and S l. Aordingly, these funtions are more properly written as S(T) = f ± (T) and T(S) = g ± (S), where the two signs + and - denote the upper and lower branhes of these double-valued funtions, respetively. Problem #2 For parts (b)-(d) of this problem express your answers in terms of the above-defined variables and funtions. (a) Using an arrow, indiate on the figure the diretion of the S-T path traversed by the engine yle when it does work in eah yle. Explain this result. (b) What is (i) the work done by the engine in one yle and (ii) the heat input to the engine over one yle? () Suppose now that you have a Carnot engine that operates between the same temperature extremes T h and T l as our nameless engine. Draw a representation on the ST-plot above, or on your own reprodution of it, of this Carnot yle, inluding the diretion of the path, assuming that the Carnot engine does the same amount of work per yle as the nameless engine. Desribe and quantify your plot. (d) Is the nameless engine more effiient than, equal in effiieny to, or less effiient than the Carnot yle you desribed in part ()? Prove this result. 21. Here are two fats about a substane: The entropy is the following funtion of T and V: α 1 V T S = R V T UVa Physis Department PhD Qualifying Exam Problem File 11

12 At onstant temperature T the work the substane does on its surroundings as it expands from V to V is V W = RT ln. T V (a) Find the Helmholtz free energy F, assuming that it is zero at the state values speified by the subsript. (b) Find the equation of state of this stuff. Is there a point in parameter spae where it is ideal? () For this part we ll simplify the algebra by assuming that ln(v/v ) = 1 and also α = 1. At what temperature T would the system do twie as muh work in going from V to V as it does at T = T? 22. You are interested in learning about the p-v-t (pressure-temperature-volume) relation as well as the energy, U, of a non-interating Bose system. [Feel free to use β = 1/kT if you like]. The grand anonial partition funtion of the system is given by Q. (a) How an you express the pressure and the energy in terms of Q? (b) What is Q for a gas of non-interating bosons, expressed as an appropriate produt (i.e. your system is in a small box and the energies are disrete). As a seond part to this, express ln Q as a disrete sum. () Your system has some peuliar features: the box is a d-dimensional ube with sides of length, L, and the energy-momentum relation is ε = αp ν, where α is a proportionality onstant. Using these fats, allow the box to get large and onvert the sum over allowed energies to an integral over a ontinuum of energies; express ln Q as an integral. (d) Show that pv ν = U. d 23. (a) Find the Bose-Einstein ondensation temperature T BEC for a large number, N, of non-interating atoms of mass M onfined to a volume V. Assume the atoms have spin, or have integer spin but the spin degeneray is lifted by an applied magneti field. A derivation of the full result is required (8 points), but you an get partial redit by just answering suh questions as: What is the momentum of an atom in the ondensate? What is the value of the hemial potential for UVa Physis Department PhD Qualifying Exam Problem File 12

13 T T BEC? What an be said about the value of T BEC by dimensional analysis alone? (b) How an one reate, in pratie, suh a BE ondensate? (1 point) () Estimate the number density N/V neessary to obtain T BEC = 1 µk if the atoms are 23 Na (1 point). Useful integral: dx e x x π = ζ ( 3/ 2) A set of N eletrons are onfined to a restrited area A of a plane but an move freely within those onstraints. When a magneti field B is applied perpendiular to the plane, the Landau quantization of the eletron orbits produes energy levels 1 E = ω +, where =,1,2,... 2 and ω = eb / m is the ylotron frequeny. The degeneray of the ξ = 2 eba /( h) =Φ/ Φ (independent of ), th quantum level is where Φ= BA is the magneti flux passing through the area A and Φ = h /2e is the magneti flux quantum. In what follows ignore (i) interations between the eletrons, and (ii) interations of eletron spin with the magneti field, i.e., imagine that the eletrons have no magneti dipole moment. (a) Calulate the magneti field dependene of the Fermi energy E F. (You an assume that the temperature is at absolute zero.) (b) Sketh the Fermi energy as a funtion of magneti field in the range.8b < B < 1.2B, where B = Φ N / A. () Explain the physial origin of the Fermi energy hange when the field is lose to B. (d) Eletrons of density ρ = 1 19 m 3 are onfined to a 1 nm-thik layer. Estimate the strength of the magneti field that will produe the shift in Fermi energy near B. (e) Estimate the temperature at whih the Fermi energy effet of part () an be observed in the laboratory. Some possibly useful quantities: UVa Physis Department PhD Qualifying Exam Problem File 13

14 11 2 Φ = h / 2e = Tesla-m e 4 = ev/tesla m The Boltzmann onstant k B = ev/k 25. Consider the Stirling yle, onsisting of the following: (a) Isothermal ompression from volume V a to volume V b at temperature T 1, (b) Heating from temperature T 1 to temperature T 2 at fixed volume V b, () Isothermal expansion from V b to V a at temperature T 2, and (d) Cooling bak to temperature T 1 at fixed volume V a. What is the effiieny of the Stirling yle? (η Stirling work done divided by total heat intake in steps (ii) and (iii)). Assume that the yle is arried out with an ideal lassial gas with a temperature independent heat apaity C V at fixed volume. Compare this result with the effiieny of a Carnot engine operating between the same two temperatures. By writing an expliit formula for the ratio η Carnot /η Stirling, show that this ratio is always > Distinguishable partiles of spin 1/2 with magneti moment µ B are plaed in an external magneti field H (i.e. the energy of eah partile is S z µ B H and its magnetization is S µ ). The number of partiles is not fixed. z B (a) Find the grand anonial partition funtion at hemial potential µ and temperature T (with µ < ( kt + µ H )). (b) Find the magnetization from the grand anonial partition funtion. B 27. Consider an ideal gas of massless bosons in thermal equilibrium, where the number N of partiles is a onserved quantity. (a) Derive an expression for the number of thermally exited partiles N ex (i.e. partiles with momenta p ). UVa Physis Department PhD Qualifying Exam Problem File 14

15 N (b) If the number-density is ρ = = 1 2 m 3, find the ritial temperature, below V whih Nex < N. Why does this ritial temperature not our in a photon gas? b g b g b g Useful information: ζ 2 = , ζ 3 = , ζ 4 = Consider a system onsisting of a liquid and a gas in phase equilibrium. Assume that the gas phase an be approximated by an ideal gas and that the volume oupied by the liquid phase is negligible. (a) Derive the Clausius-Clapeyron equation, the equation whih determines the slope of the phase boundary in the pv-plane. The equation is dp dt q = T v, where q is the latent heat of vaporization per moleule and v is the hange in volume when one moleule is transferred from the liquid to the gas. (b) Determine dv/dt, the hange in volume per partile of the gas per unit hange of temperature along the phase equilibrium urve. Express your answer in terms of p, q, and kt. () Determine the speifi heat per moleule urve of the gas along the phase equilibrium urve. This quantity is defined as urve q g = T urve, where q g is the amount of heat per moleule added to the gas per unit hange in the temperature, with the restrition that as the temperature is varied, the pressure and volume are varied to keep the system along the oexistene urve. (Thus, do not inlude any latent heat of transformation). Express your answer in terms of q and p. (d) Find an expression for the temperature dependene of the phase oexistene boundary p(t) at low temperatures, assuming q is independent of temperature in that region. 29. (a) A gas does work when it expands adiabatially. What is the energy soure for this work? Consider an adiabati urve plotted for a gas on a P-V diagram. Give a general argument, either physial or mathematial, to prove that the UVa Physis Department PhD Qualifying Exam Problem File 15

16 adiabati urve must always be steeper than the isothermal one where they ross. (b) Forty-four grams of CO 2 taken to be an ideal gas, are used to operate a Carnot heat engine between the isotherms T 1 = 273K and T 2 = 373K, and the adiabatis S 1 = 198 Joules/K and S 2 = 222 Joules/K. In one yle, how muh work does the engine do, and what is its effiieny? By what fator does the volume hange during either isothermal segment of the yle. 3. Consider a system of N distinguishable non-interating spins in a magneti field B. Eah spin has a magneti moment of size µ, and eah an point either parallel or antiparallel to the field. Thus, the energy of a partiular state is N niµ B n i = ± 1 i= 1 b g (a) Determine the thermodynamially defined internal energy U of this system as a funtion of β = 1/kT, B, and N by employing an ensemble haraterized by these variables. (b) Determine the entropy S of this system as a funtion of β, B, and N. () Determine the behavior of the energy U and the entropy S for this system in the limit T. 31. (a) Corret the following Kelvin statement of the Seond Law: "No proess exists whih onverts all of a given amount of heat into work". (b) The following proess appears to violate the above inorret statement: A ylinder of gas is at a pressure p, a temperature T and volume V. Heat Q is added at onstant volume thus inreasing the pressure to p + p and the temperature to T + T. The gas is now allowed to expand adiabatially and reversibly doing an amount of work equal to Q and thus onverting the heat added into useful work. Plot the p-v phase diagram. Integrate the ombined First and Seond Laws and show that S > for the proess starting at the initial point and ending at the final point. UVa Physis Department PhD Qualifying Exam Problem File 16

17 32. An ideal monatomi gas is ontained in a thermally insulated ylinder. The gas is ompressed by a piston moving with a uniform veloity u, whih is small ompared to the average moleular speed. Using kineti theory, ompute the rate of hange of the pressure with time. Hene ompute dp/dv and show that the result is onsistent with the 5 3 adiabati equation of state PV = onstant. Use the following standard symbols: P, V, T = pressure, volume, temperature A = area of piston m = mass of moleule n = number of moleules per unit volume v z = omponent of moleular veloity normal to the piston 33. (a) What is the molar speifi heat at onstant volume of a perfet diatomi gas at room temperature? A rough estimate is suffiient. The presene of moleular vibrations an usually be negleted. Explain why this is orret. (b) What is the speifi heat at onstant pressure of the same gas? 34. The surfae temperature of the sun is T (=55K); its radius is R (= 7 x 1 1 m); the radius of the moon is R; the mean distane between sun and moon is L (= 1.5 x 1 13 m). Assume that both the sun and the moon absorb all eletromagneti radiation inident upon them. (a) What is the total flux of energy from the sun? It is suffiient to indiate how this depends on R o and T. (b) What is the inident flux of energy on an element of the lunar surfae, as a funtion of a properly defined angle θ? What is the flux of energy radiated by the same element? () Negleting the thermal ondutivity of the lunar soil, ompute the temperature distribution on the lunar surfae. 35. A thin-walled vessel of volume V, kept at onstant temperature, ontains a gas whih slowly leaks out through a small hole of area A. The outside pressure is very low ompared to the pressure in the vessel. (a) Relate the flow of moleules through the hole to the density and some average UVa Physis Department PhD Qualifying Exam Problem File 17

18 veloity of the gas moleules in the vessel. It is not neessary to ompute expliitly numerial fators of order unity. (b) Find the time required for the pressure (or density) in the vessel to derease to 1/e of its initial value. 36. A neutron star onsists mainly of neutrons, but it also ontains protons and eletrons in thermal equilibrium through the reations n p+ e + ν and p+ e n+ ν (The antineutrino or neutrino on the right side of these equations esapes the star, and hene do not reah equilibrium with the other partiles; their hemial potentials an be set to zero.) In the model of the neutron star to be onsidered here, the neutrons, protons, and eletrons are all taken to be ultrarelativisti, Ek ( ) = k. (Real life is more ompliated.) The star is eletrially neutral. (a) Obtain an expression for the Fermi energy E F as a funtion of the number density N/V for an ultrarelativisti gas of idential fermions. n (b) Find the thermal equilibrium relation between E F, star, where the supersripts on E F label partile types. p E F, and e E F for the neutron () Using the results of the first part of the problem, find in our model the relative numbers of neutrons, protons, and eletrons. 37. A sample of weight W hangs from an elasti thread of ross-setion A o and length L o (at equilibrium). For a small hange of the weight, W, the length of the thread hanges by an amount L proportional to W, L = C W. (a) How do you expet C to depend on A and L? (b) Write a free energy whih is a funtion of W and a free energy whih is a funtion of L. Make an expliit analogy to the usual F and G funtions. () Write an expression for the root mean square flutuation in the position of the sample. Exhibit the dependene on temperature and on the geometrial fators A and L? UVa Physis Department PhD Qualifying Exam Problem File 18

19 38. A solution of heavy moleules in water is put in a entrifuge (a ylinder of radius R) and spun about its symmetry axis with onstant angular veloity ω. Let the mass of a moleule be M and the mass of an equivalent volume of water be m, with M > m. Suppose that the initial onentration of the solution is. Find the equilibrium onentration in the spinning entrifuge as a funtion of the radial oordinate r (r < R). Assume that the moleules in the solution do not interat with eah other. 39. Consider a dilute gas of idential moleules, eah having (in addition to very tightly bound eletrons) one loosely bound eletron with binding energy -E b. As a funtion of temperature and of the volume per moleule what fration of the moleules are ionized? 4. A losed ylinder is divided by partitions into 3 equal ompartments of volume V whih eah ontain one mole of a different inert ideal gas. The gases are at the same temperature. Calulate the hange in entropy whih ours when the partitions are removed allowing the three gases to diffuse isothermally to a uniform mixture. Use either marosopi thermodynamis or statistial thermodynamis. UVa Physis Department PhD Qualifying Exam Problem File 19