Thermal and Statistical Physics Department Exam Last updated November 4, L π

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1 Thermal and Statistical Physics Department Exam Last updated November 4, a. Define the chemical potential µ. Show that two systems are in diffusive equilibrium if µ 1 =µ. You may start with F = F 1 + F (free energy) and use the fact that µ 1 =µ should minimize F. b. Now consider a 1-d gas of length L and number of distinguishable and non-interacting particles N. Assume that temperature T is high enough that we are in the classical regime. Find the chemical potential µ in terms of T and other parameters of the problem. c. Now find the free energy F and the specific heat at constant volume C V.. a. Write down the distribution function for a Fermi-Dirac gas (ave. # of particles in a state) and sketch it at T = 0 labeling the axes and indicating the Fermi energy. b. Using the above, write down an expression for the total number of spin 1/ particles, N, and then evaluate this expression to find the Fermi energy for an extreme relativistic gas of electrons with energy - momentum relation E = pc. c. Find the total energy of this system at T = 0 in terms of the Fermi energy and the total number of particles. 3. Consider 3-d lattice of N atoms arranged in a box of side L. a. Briefly explain why there are only 3N independent phonon modes and show that ω max (Debye frequency) is given by 1/3 πc 6N ω m =. L π b. Show that at high temperatures the internal energy U = 3Nk B T and that at low temperatures it is proportional to T 4. You can use the Bose-Einstein distribution to determine the occupation number of the states. 4. Consider two single-particle states and two particles. Find the entropy of this system when the following statistics apply: a. Maxwell-Boltzmann (classical) b. Fermi-Dirac c. Bose-Einstein d. Derive an approximate condition relating temperature and density to determine when or when not quantum statistics must be used. 5. a. Write down the partition function for a spin (s = ± ½) in a magnetic field. Use this to find an expression for the magnetization of N spins in a magnetic field B at temperature T. The magnetization of an individual spin is ± m. b. Show that for high enough temperatures the susceptibility dm/db is Nm /T.

2 6. a. Write down the Grand Partition function (Gibbs sum), Ω. Define all quantities used and explain what the summations are over. b. Show that the average number of particles in a Grand Canonical ensemble is given by log N = τ Ω µ where τ= kt and µ is the chemical potential. c. Now consider a dilute gas of hydrogen atoms. Assume each atom may have the following states: State No. of Electrons Energy Ground 1 0 Positive ion 0 Negative ion +δ Find the condition that the average number of electrons per atom is The partition function, Q for a photon gas at temperature T is given by ln(q) = ln 1 exp( β ω) k where ω= ck (c is velocity of light, k is wavevector), β= 1/kBT (kb is Boltzmann's constant), and the summation k is over all available states. 3 4 x π a. Show that total energy U = 3 PV. Hint: dx = o exp( x) b. Show that the specific heat at constant volume T. It is easiest to find U as a function of temperature first by changing the sum to an integral. 8. In a mono-atomic crystalline solid each atom can occupy either a regular lattice site or an interstitial site. The energy of an atom at an interstitial site exceeds the energy of an atom at a lattice site by ε. Assume that the number of atoms, lattice sites, and interstitial sites are all equal in number. a. Calculate the entropy of the crystal in the state where exactly n of the N atoms are at interstitial sites. b. What is the temperature of the crystal in this state, if the crystal is at thermal equilibrium? c. If ε = 0.5 ev and the temperature of the crystal is 73 K, what is the fraction of atoms at interstitial sites? Hint: You have two choices to make in this problem which atoms to put in interstitial sites, and which interstitial sites to put them in. 9. A system contains N sites. At a given site there is only one accessible orbital state, but this orbital can be occupied by 0, 1 or electrons of (opposite spin). The site energy is ε if occupied by two electrons, and zero otherwise. In addition, there is an externally applied magnetic field H that acts on the electron spin magnetic moment m. a. Find a formula for the Gibbs sum (Grand Partition function) on one site, as a function of the inverse temperatureβ= 1/koT, chemical potential µ, and the above given quantities. b. What is the expected number of electrons per site <N>? c. What is the probability that a site is unoccupied? d. If the average number of electrons per site is found to be <N>=1, what is the chemical potential µ?

3 10. The surface temperature of the sun is To (= 5800 o K); its radius is R (= m) while the radius of the earth is r (= m). The mean distance between the sun and the earth is L (= m). In first approximation one can assume that both the sun and the earth absorb all electromagnetic radiation incident upon them. Assume the earth has reached a steady state at some temperature T. a. Find an approximate expression for the temperature T of the earth in terms of the astronomical parameters mentioned above. b. Calculate the temperature T numerically. 11. A system of N distinguishable noninteracting particles has one-particle energy levels of En degeneracy of the nth level is equal to n, and n=1,, 3,... a. Show that the partition function of the system is β= k T. where ( ) 1 B Z N βε e = βε ( e 1) b. Calculate the entropy of the system of particles. c. Determine the behavior of the entropy at high temperature. d. Determine the behavior of the specific heat at high temperatures. 1. Consider a large reservoir energy Uo ε in thermal contact with a system with energy ε. = nε, where the a. Give an argument based on the multiplicity (number of distinguishable ways a state with given energy may be obtained) which leads to the result P( ε) ~ exp( ε / kt) where P( ε ) is the probability of finding the system in a particular state with energy ε. Hint: recall that the entropy S = k log (multiplicity). b. A system of N magnetic moments ( µ ) may orient themselves parallel or antiparallel to an applied field. Deduce from (a) the total magnetization M 1µ i as a function of H and T. 13. Consider a two-state system with energies 0 and ε. a. Calculate the partition function b. Find the average energy at a temperature T. c. Find the heat capacity and graph it versus temperature. d. Sketch the heat capacity as a function of the temperature.

4 h 14. An ideal diatomic gas has rotational energy levels given by ε j = j( j + 1), with degeneracies 8π I gj = j+ 1. h a. For oxygen, with θ rot = = K, what fraction of the molecules is in the lowest rotational energy 8π IkB state at T=50K? b. Repeat for hydrogen, with θ rot = 85K. 15. A paramagnetic salt contains 1 mole of noninteracting ions, each with a magnetic moment of one Bohr magnet on ( µ B =9.74 x 10-4 J/T). A magnetic field H is applied along a particular direction; the permissible states of the ionic moments are either parallel or antiparallel to the field. a. Suppose the salt is maintained at a constant temperature of 4.0K, and H is increased slowly from 1 T to 10 T. What is the magnitude of the heat transfer from the thermal reservoir? b. Now the system is thermally isolated, and H is decreased from 10 T to 1 T (adiabatic demagnetization). What is the final temperature of the salt? 16. a. From the condition that two coexisting phases of a given chemical substance must have the same chemical potential, µ 1 =µ, derive the Claussius-Clayperon equation dp S S = 1 dt V V 1 b. Drawn below is the phase diagram for He 4. Carefully explain why the slope of the liquid-solid phase boundary is zero at T = A classical monatomic ideal gas of N particles (each of mass m) is confined to a cylinder of radius r and infinite height. A gravitation field points along the axis of the cylinder downwards. a. Determine the Helmholtz energy A. b. Find the internal energy U and the specific heat C V. c. Why is Cv 3/ Nk? 18. A thin-walled vessel of volume V is kept at a constant temperature T. A gas leaks slowly out of the vessel through a small hole of area A into surrounding vacuum. Find the time required for the pressure in the vessel to drop to half of its original value.

5 19. Consider a paramagnetic substance with the equation of state M = AH/(T-T 0 ). Here M is the magnetization, H is the applied magnetic field, A and T 0 are constants, and T is the temperature. The equation of state is valid only for T>T 0. Show that C M, the heat capacity at constant magnetization, is independent of M. 0. The nuclei of certain crystalline solids have spin one. According to quantum theory, each nucleus can therefore be in any one of three quantum states labeled by the quantum number m where m = 1, 0, -1. A nucleus has the same energy ε in the state m = 1 and m = - 1, and an energy of zero in the state m = 0. i) Find an expression for the partition function for N such nuclei kept in contact with a heat bath at an absolute temperature T. ii) From the above expression for the partition function, compute the entropy of the system. Show that one gets the correct limit of the entropy for both T and T A sample of helium gas inside a cylinder terminated with a piston doubles its volume from V i = 1m 3 to V f = m 3. During this process the pressure and volume are related by PV 6/5 = A = constant. Assume that the product PV always equals U 3, where U is the internal energy. a. What is the change in energy of the gas? b. What is the change in entropy of the gas? c. How much heat was added to or removed from the gas?. Consider diatomic molecules adsorbed on a flat surface at temperature T. The molecules are free to move on the surface and to rotate within the plane of the surface this is a non-interacting D gas with rotational motion about one axis. The rotational state of the molecules is given by a single quantum number m (m=0, ε = / I m, where I is the moment of inertia of the ±1, ±, ±3 ) and the rotational energy is given by m ( ) molecule. a. Find an expression for the rotational partition function of a single molecule. You need not evaluate the infinite series. b. Find the ratio of the probabilities of finding a molecule in states m=3 and m=. ε / I. c. Find the probability that m=1 given that ( ) d. Find the rotational contribution to the internal energy of the gas (N molecules) in the high temperature limit, where kt >> / I. 3. a. Write down the Boltzmann equation for a gas. Explain the meaning of each of the quantities appearing in it. Make sure you write the right-hand-side (collision integral) in its general form as an integration over distribution functions. b. What assumptions have gone into this equation? Quantify them if you can. When would you expect these assumptions to be valid? Indicate where these assumptions are used in the derivation of the Boltzmann equation. c. Consider a dilute gas in a steady state with a concentration gradient( n(x)/ s). Show that within the relaxation time τ and to first order, the particle flux may be written n(x) Jx = nvx = D ; D = x ktτ m

6 where V x is the average value of vx. Assume that locally the particles follow a Maxwell-Boltzmann distribution m f o(v,x) = n(x) exp 3/ ( mv / kt ). πkt ( ) 4. Suppose a particular type of rubber band has the equation of state f/l = A o T where A o is a constant, f is the tension and L is the length per unit mass. a. Write down the fundamental thermodynamic relation (nd law of thermodynamics) for this system, expressing ds in terms of de and dl. C b. Compute L where C L is the constant-length heat capacity per unit mass. L T c. Compute the quantity in b) if the equation of state is given by f/l=a o T. 5. a. Compare the 4 level systems below. More than one particle may occupy a level. Which system has highest temperature lowest temperature lowest specific heat highest entropy b. Consider the system below. Explain why it can be thought of as having a negative temperature. (Hint: Consider the Boltzmann factor e -E/kT ). c. If this system (of part (b)) is brought into contact with a large reservoir at temperature TR (positive) draw a graph indicating how its temperature will change as a function of time as it comes to equilibrium with the reservoir. (Take the initial temperature of the system to be TS a negative number). d. Explain why no problems with the third law are encountered in part (c).

7 6. A system has the following equations of state: 3 u = pv p = A v T where A is a constant, u=u/n, v=v/n, and N is the number of particles. a. Show that the entropy of this system can be written as 1/4 7 A S(U, V, N) = U V N 3 3 b. Calculate the Helmholtz free energy, F(T,V,N). 7. Consider a volume of gas at pressure P and temperature T. 4 3/4 1/ 1/4 a. Write down the multiplicity of states in this volume as a function of entropy S. b. Let So be the equilibrium entropy and Eo be the equilibrium energy. Expand the entropy as a function of energy S(E) about So to determine the multiplicity of states as a function of energy. c. Find an expression for the mean square fluctuation in energy in terms of the heat capacity of the gas. 8. A dilute system consisting of 50 independent, indistinguishable particles is in thermal contact with a reservoir at a constant temperature T = ε/k B, where k B is the Boltzmann constant. Each particle has three energy levels of energy 0, ε, and ε and degeneracy 300, 600 and 100, respectively. a. Calculate the canonical partition function for this system. b. How many particles are in each energy level? c. Determine the entropy of this system. 9. a. Obtain the Van der Waals equation of state ( )( ) p + N a / V V Nb = N kt by making the following two corrections to the ideal gas: (i) use an effective volume, instead of the usual volume, (ii) include interactions in a mean field form. State clearly where the mean field assumption comes in. b. What is the work done by the gas if it expands isothermally from V 1 to V at some temperature T. n = + ω0; n = 0, 1,,... Treat this single oscillator to be a small system coupled to a heat bath at temperature T. What is the probability of finding the oscillator in its nth quantum state? 31. A one-dimensional harmonic oscillator has evenly spaced quantum energy levels, ε n = n ω. (We choose the zero of energy at the n=0 level.) 30. A quantum harmonic oscillator has energy levels E ( n 1/) a. Find an expression for the Helmholtz free enrgy, F, as a function of temperature. b. Find the limiting behavior of the entropy for T 0. Comment on its physical significance. c. Repeat b) for kbt>> ω.

8 3. A 1.00 gram drop of water is supercooled to o C (remains in liquid state). Then suddenly and irreversibly it freezes and becomes solid ice at the temperature of the surrounding air, o C. The specific heat of ice is.0 J/g K, of water is J/g K, and the heat of fusion of water is 333 J/g. a. How much heat leaves the drop as it freezes? b. What is the change in entropy of the drop as it freezes? 33. A sample of ideal gas is taken through the cyclic process abca shown in the figure. At point a, T = 300 K. a. What are the temperatures of the gas at points b and c? b. Complete the table by inserting the correct numerical value in each indicated cell. Note that Q is positive when heat is absorbed by the gas and W is positive when work is done by the gas. E is the change in internal energy of the gas. a b b c c a Q W E

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