Thermal & Statistical Physics Study Questions for the Spring 018 Department Exam December 6, 017 1. a. Define the chemical potential. Show that two systems are in diffusive equilibrium if 1. You may start with F = F1 + F (free energy) and use the fact that 1 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 in terms of T and other parameters of the problem. c. Now find the free energy F and the specific heat at constant volume CV.. Consider a three-dimensional Fermi-Dirac gas of particles within some volume V. a. Write down the distribution function for a Fermi-Dirac gas (ave. # of particles in a state). Sketch it at T = 0, labeling the axes and indicating the Fermi energy. b. Write down an expression for the total number of spin 1/ particles, N, and then evaluate this expression at T=0 to find the zero-temperature 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. The Diesel engine cycle consists of (see figure) processes: Suppose the cycle is applied to n moles of an ideal diatomic gas. In terms of the given pressures and volumes and the gas constant R, give expressions for a. the heat QH given to the gas during combustion. b. the heat QC removed from the gas during the cooling process. c. the entropy change of the gas in process BC. d. the entropy change of the gas in process CD. AB: Adiabatic compression from volume V0 to volume V1, where pressure = p1. The compression ratio rc = V0/V1 is high enough that it causes ignition of the air-fuel mixture without needing a spark. BC: Expansion at constant pressure p1 from volume V1 to V < V0, during the burning of the fuel. CD: Adiabatic expansion from V to V0, with expansion ratio re = V0/V. DA: Constant volume (V0) cooling back to the original temperature at point A. Treat the gas as an ideal gas, and suppose that the constant pressure (Cp) and constant volume (Cv) heat capacities are known. a. Calculate the efficiency of the cycle, defined as the ratio of the work output to the heat adsorbed per cycle, W/QH. b. Express as a function of C p / C v, re, and r c. c. Calculate the entropy changes in each process of the cycle, and sketch the cycle in a S-T diagram. 4. A system consists of three particles, each of which has three non-degenerate energy states--0, and 3. Determine the canonical partition function if the particles are a. classical particles (note: classical particles are distinguishable) b. fermions c. bosons
5. Consider a system of N non-degenerate distinguishable particles each with a magnetic moment which may be antiparallel, perpendicular, or parallel to an applied magnetic field. The corresponding energy W under magnetic field B at temperature T=To. B is µb, 0, and -µb, respectively. The system is in equilibrium a. Calculate the average energy E and average entropy S of the system. b. Calculate the average number of particles in each level. c. Calculate the average energy E and the entropy S of the system right after turning over the magnetic field ( B to B ). What is the temperature T of the system at this moment? 6. 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. 7. Consider a 1-dimensional ideal electron gas, length L( V) and temperature T. a. Show that the energy for 1 electron may be written 1 / m n n 1,,3,... L b. Find the partition function for N electrons and hence show that PV = NRT. 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. 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. n(x) / s. Show that c. Consider a dilute gas in a steady state with a concentration gradient within the relaxation time and to first order, the particle flux may be written n(x) kt Jx nvx D ; D x m 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. kt 10. 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. 11. A system of N distinguishable non-interacting particles has one-particle energy levels of En where the degeneracy of the nth level is equal to n, and n=1,, 3,... a. Show that the partition function of the system is k T. where 1 B e Z 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. N n,
1. Consider a particle in a 1-dimensional box of side L with available energy levels (in the presence of a magnetic field B along Z) n n m B; / L m n is an integer greater than zero and m m k. z ˆ a. What is the partition function for this particle? b. What is the partition function for N classical, identical, noninteracting particles in the same box in the presence of a magnetic field. Hence find an expression for the internal energy U, and Helmholtz free energy F. c. What is the equilibrium pressure and magnetization of this gas? 13. Consider an extreme relativistic classical gas (ignore spin) with single particle energy ε = p c + m c 4 pc where p is momentum and c is the velocity of light. a. If this gas is confined to 1-dimension (length L) show that = nhc/l (n = 1,, 3,...) and then find the partition function Z1 for a single particle. b. Now find F and show that PV = NkT 14. Consider an ideal gas of diatomic molecules. The rotational motion is quantized according to j j ( j1) o, j = 0, 1,,... The multiplicity of each rotational level is g j j 1. a. Find the rotational part of the partition function Z for one molecule. b. Convert the sum (for Z) to an integral and evaluate the integral for kt o. From this find an expression for the specific heat capacity at constant volume at low temperatures. 15. 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). 4 3/ 4 1/ 1/ 4
16. a. From the condition that two coexisting phases of a given chemical substance must have the same chemical potential, 1, 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 = 0. 17. In a temperature range near absolute temperature T, the tension force F of a stretched plastic rod is related to its length L by the expression F at (L L o) where a and Lo are positive constants, Lo being the unstretched length of the rod. When L = Lo, the heat capacity CL of the rod (measured at constant length) is given by the relation CL = bt, where b is a constant. a. Write down the fundamental thermodynamic relation for this system, expressing ds in terms of de and dl. S/ L. b. The entropy S(T,L) of the rod is a function of T and L. Compute T c. Knowing S(To,Lo), find S(T,L) at any other temperature T and length L. (It is most convenient to calculate first the change of entropy with temperature at the length Lo where the heat capacity is known.) d. If the rod is thermally insulated but stretched a small distance L from equilibrium, find the change in T. e. Calculate the heat capacity CL (L,T) of the rod when its length is L instead of Lo. 18. A lead bullet with mass m=10 grams, leaves a gun with speed v=500m/s and a temperature of 150 o C. It is shot into a large body of water at 5 o C. The specific heat of lead is 18 J/kg K, and for water it is 4190 J/kg K. a. Describe briefly what time-dependent temperature changes might occur in the bullet and in the water near it. b. Estimate the energy transferred to the water. c. Estimate the total entropy change after the bullet has cooled to the water temperature.
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. 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 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 temperature 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? 1. 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 Cv. c. Why iscv 3/ Nk?. 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 / I m, quantum number m (m=0, ±1, ±, ±3 ) and the rotational energy is given by m where I is the moment of inertia of the 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=.. c. Find the probability that m=1 given that / I d. Find the rotational contribution to the internal energy of the gas (N molecules) in the high temperature limit, where kt / I. 3. Similar to the van der Waals equation of state is the Dieterici equation of state. p(v b) = RTe a/rtv Find the critical constants pc, Vc and Tc in this model of a weakly interacting gas. This equation of state was proposed to account for the interaction of gas atoms with walls.
4. 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 max (Debye frequency) is given by 1/ 3 c 6N m. L b. Show that at high temperatures the internal energy U = 3NkBT 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. 5. For a photon gas in a cavity, a. Show that p = ( U ) = V j n j ħ ( dω j ), where n j is the number of photons in mode j. S dv b. Also show that ( dω j dv ) = ω j 3V This leads us to the equation of state for the photon gas. c. Show that p = U 3V This pressure is called radiation pressure. d. Find an expression for the radiation pressure in terms of the temperature, and calculate its value at the center of the sun, where the temperature is ~10 7 K.
6. 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. A sample of helium gas inside a cylinder terminated with a piston doubles its volume from Vi = 1m 3 to Vf = 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?
8. A 1.00 kg block of ice at -0.0 o C is left to melt in the ocean, which is at +30.0 o C. The specific heat of ice is 0 J/kg K, of water is 4186 J/kg K, and the heat of fusion of water is 333 kj/kg. a. Calculate the entropy change of the ocean. b. Calculate the total entropy change of (ocean plus block of ice). 9. Consider a volume of gas at pressure P and temperature T. 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. 30. Consider a system made out of one single classical oscillator in contact with a heat reservoir at a n temperature T. The potential energy of this one-dimensional oscillator is V(x) 0 x / a where 0 and a are constants. What is the thermal energy of the oscillator? (Note: Don't get scared by the integral that you will face; only powers of kt in the partition function would contribute to the thermal energy.) 31. Equipartition. A classical harmonic oscillator p Kq H m is in thermal contact with a heat bath at temperature T. Calculate the partition function for the oscillator in the canonical ensemble and show explicitly that E k T, and B B E E k T. 3. For a gas of molecules with diameter d, number density n and at a temperature T, find a. the mean free path, b. their average speed, and c. the pressure of the gas using kinetic arguments.
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. Q W E a b b c c a