Lecture 9 Overview (Ch. 1-3) Format of the first midterm: three problems with multiple questions. Total: 100 points. The Ideal Gas Law, calculation of W, Q and dS for various ideal gas processes. Einstein solid and two-state paramagnet, multiplicity and entropy, the stat. phys. definition of T, how to get from the multiplicity to the equation of state. Only textbook and cheat-sheets (handwritten!) are allowed. No homeworks and lecture notes. DO NOT forget to bring your calculator!

Problem 1 One mole of a monatomic ideal gas goes through a quasistatic three-stage cycle (1-2, 2-3, 3-1) shown in the Figure. T1 and T2 are given. (a) (10) Calculate the work done by the gas. Is it positive or negative? (b) (20) Using two methods (Sackur-Tetrode eq. and dQ/T), calculate the entropy change for each stage and for the whole cycle, Stotal. Did you get the expected result for Stotal? Explain. (c) (5) What is the heat capacity (in units R) for each stage? T V V1 V2 T1 T2 1 2 3

Problem 1 (cont.) (a) 1 – 2 V  T  P = const (isobaric process) 2 – 3 V = const (isochoric process) 3 – 1 T = const (isothermal process)

Problem 1 (cont.) (b) 1 – 2 2 – 3 3 – 1 Sackur-Tetrode equation: V 3 V  T  P = const (isobaric process) T1 T2 T 2 – 3 V = const (isochoric process) 3 – 1 T = const (isothermal process) as it should be for a quasistatic cyclic process (quasistatic – reversible), because S is a state function.

Problem 1 (cont.) (b) 1 – 2 2 – 3 3 – 1 - for quasi-static processes V V  T  P = const (isobaric process) V1 1 T1 T2 T 2 – 3 V = const (isochoric process) 3 – 1 T = const (isothermal process)

Problem 1 (cont) (c) Let’s express both Q and dT in terms of dV : V 3 V2 2 1 – 2 V  T  P = const (isobaric process) V1 1 T1 T2 T 2 – 3 V = const (isochoric process) 3 – 1 T = const (isothermal process), dT = 0 while Q  0 At home: recall how these results would be modified for diatomic and polyatomic gases.

Problem 2 One mole of a monatomic ideal gas goes through a quasistatic three-stage cycle (1-2, 2-3, 3-1) shown in the Figure. Process 3-1 is adiabatic; P1 , V1 , and V2 are given. (a) (10) For each stage and for the whole cycle, express the work W done on the gas in terms of P1, V1, and V2. Comment on the sign of W. (b) (5) What is the heat capacity (in units R) for each stage? (c) (15) Calculate Q transferred to the gas in the cycle; the same for the reverse cycle; what would be the result if Q were an exact differential? (d) (15) Using the Sackur-Tetrode equation, calculate the entropy change for each stage and for the whole cycle, Stotal. Did you get the expected result for Stotal? Explain. P 1 2 3 V V1 V2 P1

Problem 2 (cont.) (a) P V V1 V2 P1 1 – 2 2 – 3 3 – 1 P = const (isobaric process) 2 – 3 V = const (isochoric process) 3 – 1 adiabatic process

Problem 2 (cont.) (c) P V V1 V2 P1 1 – 2 2 – 3 3 – 1 P = const (isobaric process) 2 – 3 V = const (isochoric process) 3 – 1 adiabatic process For the reverse cycle: If Q were an exact differential, for a cycle Q should be zero.

Problem 2 (cont.) (d) P P1 V1 V2 1 – 2 V 2 – 3 3 – 1 Sackur-Tetrode equation: P (d) 1 2 P1 3 V1 V2 1 – 2 V  T  P = const (isobaric process) V 2 – 3 V = const (isochoric process) 3 – 1 Q = 0 (quasistatic adiabatic = isentropic process) as it should be for a quasistatic cyclic process (quasistatic – reversible), because S is a state function.

we need to find the equation of this process V=V(T) Problem 3 Calculate the heat capacity of one mole of an ideal monatomic gas C(V) in the quasi-static process shown in the Figure. P0 and V0 are given. P 10 Start with the definition: P0 we need to find the equation of this process V=V(T) 20 30 V0 V 40

Problem 3 (cont.) P V P0 V0 T=const isotherm S=const adiabat V0/2 T=const isotherm S=const adiabat V0/2 5V0/8 Problem 3 (cont.) 50 Does it make sense? C/R the line touches an isotherm 2.5 1.5 1/2 5/8 1 V/ V0 the line touches an adiabat

Problem 4 (10) The ESR (electron spin resonance) set-up can detect the minimum difference in the number of “spin-up” and “spin-down” electrons in a two-state paramagnet N-N =1010. The paramagnetic sample is placed at 300K in an external magnetic field B = 1T. The component of the electron’s magnetic moment along B is  B =  9.3x10-24 J/T. Find the minimum total number of electrons in the sample that is required to make this detection possible. - the high-T limit

Problem 5 (a) Consider a system whose multiplicity is described by the equation: where U is the internal energy, V is the volume, N is the number of particles in the system, Nf is the total number of degrees of freedom, f(N) is some function of N. (10) Find the system’s entropy and temperature as functions of U. Are these results in agreement with the equipartition theorem? Does the expression for the entropy makes sense when T  0? (5) Find the heat capacity of the system at fixed volume. (15) Assume that the system is divided into two sub-systems, A and B; sub-system A holds energy UA and volume VA, while the sub-system B holds UB=U-UA and VB=V-VA. Show that for an equilibrium macropartition, the energy per molecule is the same for both sub-systems. (a) - in agreement with the equipartition theorem When T  0, U  0, and S  -  - doesn’t make sense. This means that the expression for  holds in the “classical” limit of high temperatures, it should be modified at low T.

Problem 5 (cont.) (b) (c)