1. IV. Kinetic theory (continued – see previous lecture) 5. Heat capacitance a) Monoatomic gas } b) Equipartition principle. Degrees of freedom Diatomic gas, including rotations: Diatomic gas, including rotations and vibrations:

2. Molar Specific Heats of Gases at Constant Volume C V At Room Temperature C V (J/mol·K) MonatomicHe12.47 Ar12.47 DiatomicH 2 20.42 N 2 20.76 O 2 21.10 PolyatomicCO 2 28.46 SO 2 31.39 3R/2 = 12.47 J/mol·K 5R/2 = 20.79 J/mol·K 7R/2 = 29.10 J/mol·K

3. How many rotational degrees of freedom does the water molecule have? Question 1.1 2.2 3.3 4.4

4. 5. Distribution of molecular speeds (Maxwell distribution) c) A hint of quantum theory b) Heat capacitance of solids

5. Molar Specific Heats of Elemental Solids at Constant Volume due to Lattice Vibrations The rule of Dulong and Petit only holds at high temperatures. At low T, quantum mechanical effects reduce C V.

6. V. The first low of Thermodynamics (conservation of energy) dU = dQ - dW 1. Macro- a micro- parameters Equation of state for monatomic ideal gas: Thermodynamic equilibrium 2. Internal energy (kinetic plus potential energy of particles) dU = dQ + dW ext dW ext = -dW dW = P dV dQ = T dS dU = dQ - dW = T dS - P dV ΔU = ΔQ - ΔW

7. 3. Work V P V1V1 V2V2 12 Δ W = P(V 2 - V 1 )>0 V P V2V2 V1V1 21 Δ W = P(V 2 - V 1 )<0 ΔW 12 = 0 ΔW 23 > 0 ΔW 34 > 0 ΔW 42 < 0 Work is path dependent Heat is path dependent Internal energy is path independent V P V1V1 V2V2 12 ΔWΔW V P V1V1 V2V2 1 2 3 4 ΔWΔW

8. Example1: A quantity of air is taken from state a to state b along a path that is a straight line in the PV-diagram, where V a =0.070 m 3, V b =0.110 m 3, P a =1.00*10 5 Pa, P b =1.40*10 5 Pa. Assume that the gas is ideal. a) What is the work, W, done by the gas in this process? V P a b VaVa VbVb W = ½ (P a + P b )(V b - V a ) W = ½ (1.00 + 1.40)* 10 5 Pa*(0.1100 - 0.0700 ) m 3 W = 4.8* 10 3 J b) What happen with temperature and internal energy of this gas? For ideal gas:

9. Example2: This P-V diagram represents a system consisting of a fixed amount of ideal gas that undergoes three different processes in going from state A to state B. Rank work, heat transfer, change in internal, kinetic, and potential energy for each process. V State A I ΔW 1 < ΔW 2 < ΔW 3 ΔU 1 = ΔU 2 = ΔU 3 = U B - U A ΔT 1 = ΔT 2 = ΔT 3 = T B - T A ΔU = ΔQ - ΔW ΔQ 1 <ΔQ 2 <Δ Q 3 For ideal gas: PE=0 and Δ PE=0 T~ K=U ΔK 1 = ΔK 2 = ΔK 3 = K B - K A P State B 2 3

10. Example3: A system consisting of a quantity of ideal gas is in equilibrium state A. It is slowly heated and as it expands, its pressure varies. It ends up in equilibrium state B. Now suppose that the same quantity of ideal gas again starts in state A, but undergoes a different thermodynamic process (i.e., follows a different path on a P-V diagram), only to end up again in the same state B as before. Consider the net work done by the system and the net heat absorbed by the system during these two different processes. Which of these statements is true? 1. The work done may be different in the two processes, but the heat absorbed must be the same. 2. The work done must be the same in the two processes, but the heat absorbed may be different. 3. The work done may be different in the two processes, and the heat absorbed may be different in the two processes. 4. Both the work done and the heat absorbed must be the same in the two processes, but are not equal to zero. 5. Both the work done and the heat absorbed by the system must be equal to zero in both processes. Note: See example 2

11. ~1/3 of questions test understanding of concepts/principles ~2/3 of questions test numerical application of concepts/principles Bring soft (#2) pencils, erasers, and your scientific calculator Studying: Review during lecture: Friday, Dec. 12 Practice problems and solutions for Ch. 20 are posted Compare your homework solutions with the posted solutions Review the lecture questions and the text supplements Review and solve the example problems in the text Solve problems on the practice exams -- Seven Phys 221 practice Final Exams are posted plus practice problems on waves and thermodynamics (Ch. 15-20) The formula sheet that will be provided with the exam will be posted Meet with me before the exam to clear up any serious problems you are having with the course material Final Exam: Tues, Dec. 16, 2008, 7:00-9:00 p.m.

12. Physics 221 Final Exam Conflicts If you are enrolled in any of the three following courses you are entitled to take a Physics 221 Final Exam Make-up at a time and place to be arranged. Math 195, Math 196, Acct 215 Prior to final exam week, students must request to take the make-up exam. Physics 221 Final Exam Tuesday, Dec. 16, 7:00-9:00 p.m. 20 2-point problems + 20 4-point problems (multiple choice) Total: 40 problems worth 120 course points ~ 1/3 Ch 17-20 ~ 1/2 Comprehensive (prior to Ch. 17) ~ 1/5 Laboratory Final Exam

13. One cm 3 of liquid water at 100 ˚C is boiled off at 1 atm pressure and is converted to 1670 cm 3 of steam at 100 ˚C. The “system” is the water. The work done by the system when it is converted to steam is __ L·atm. (1 L = 1000 cm 3 ) Question 1 1. 0.17 2. 1.7 3. 17 4. 1700 1.  1 2. 2 3.  3 4. 3 Question 2: The work done by the system in going from point 1 to point 2 is ___ L·atm.

14. 4. Thermodynamic processes Adiabatic process: ΔQ = 0 ΔU = - ΔW ΔU = ΔQ - ΔW Isochoric process: ΔV = 0 ΔW = 0 ΔU = ΔQ ΔW = P ΔV Isobaric process: ΔP = 0 ΔW = PΔV Isothermal process: ΔT = 0 For ideal gas (!) only: ΔT = 0 ΔU = 0 ΔQ = ΔW Closed cycle process: ΔT = ΔP = ΔV = ΔU = 0 ΔQ = ΔW

15. The processes on a PV diagram (ideal gas)

17. The work done by the gas during this process is ___ L·atm. Question 1.1.5 2.2.7 3.3.9 4.5.4 An ideal gas undergoes an isothermal expansion from V 1 = 1.00 L to V 2 = 2.72 L and p 2 = 1.00 atm as shown in the figure.

18. The net heat absorbed by the system during one cycle is ___ L·atm Question 1. 2 2.-2 3. 4 4.-4

19. Example: A monatomic ideal gas undergoes an increase in pressure from p 1 = 1.00 atm to p 2 = 3.00 atm at V = 24.0 L as shown in the figure. The heat absorbed by the gas during this process is ??? L·atm. For Ideal monatomic gas:

20. Heat capacitance of an ideal gas at constant volumeat constant pressure For an ideal gas:

21. Relating heat capacities at constant volume and pressure

22. Adiabatic processes for an ideal gas For an adiabatic process: dQ = 0 dU = dQ -PdV For an ideal gas: PV=nRT

23. Example: V V1V1 P V2V2 1 2 (a) (b)

24. The final pressure of the air is ___ atm. (  = 1.40 for air) Example The volume of the air inside the cylinder of an engine decreases from 1.00 L to 0.100 L during adiabatic compression. The initial pressure of the air is 1.00 atm and initial temperature is 27 ˚C. The final temperature of the air is ___ ˚C.

25. Adiabatic processes for an ideal gas (2) PV=nRT Work

26. The work done by the gas during this process is ___ L·atm. (  = 1.40 for air) Question 1.-8 2.-4 3. 2 4. 6 The volume of the air inside the cylinder of an engine decreases from 1.00 L to 0.100 L during adiabatic compression. The initial pressure of the air is 1.00 atm and the final pressure is 25.1 atm.

27. The heat absorbed by the gas during this process is ___ L·atm. Question Some air undergoes an isobaric expansion at 1.00 atm pressure from V 1 = 5.00 L to V 2 = 9.00 L. (C V = 5R/2)