Lecture 9 Overview (Ch. 1-3)

1. 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!

2. V 3 V2 2 V1 1 T1 T2 T 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?

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)

4. Problem 1 (cont.) Sackur-Tetrode equation: (b) V 3 V2 2 V1 1 1 – 2 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.

5. Problem 1 (cont.) (b) - for quasi-static processes 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)

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

7. P 2 1 P1 3 V1 V2 V 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. Problem 2

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

9. P 2 1 P1 3 V1 V2 V Problem 2 (cont.) (c) 1 – 2 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.

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

11. 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. Problem 3 P 10 Start with the definition: P0 we need to find the equation of this process V=V(T) 20 30 0 V0 V 40

12. P S=const adiabat P0 T=const isotherm 0 V0/2 5V0/8 V0 V Problem 3 (cont.) 50 Does it make sense? C/R the line touches an isotherm 2.5 1.5 0 1/2 5/8 1 V/ V0 the line touches an adiabat

13. 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

14. Problem 5 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 UAand 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.

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