Name Cyclic Processes Cyclic Processes A fixed quantity of ideal gas is contained within a metal cylinder that is sealed with a movable, frictionless, insulating piston. (The piston can move up or down without the slightest resistance from friction, but no gas can enter or leave the cylinder. The piston is heavy, but there can be no heat transfer to or from the piston itself.) The cylinder is surrounded by a large container of water with high walls as shown. We are going to describe two separate processes, Process #1 and Process #2. movable piston Time A ideal gas water Entire system at room temperature. At initial time A, the gas, cylinder, and water have all been sitting in a room for a long period of time, and all of them are at room temperature. Step 1. We now begin Process #1: The water container is gradually heated, and the piston very slowly moves upward. At time B the heating of the water stops, and the piston stops moving when it is in the position shown in the diagram below: Time B Piston in new position. Temperature of system has changed. Question #1: During the process that occurs from time A to time B, which of the following is true: (a) positive work is done on the gas by the environment, (b) positive work is done by the gas on the environment, (c) no net work is done on or by the gas. Question #2: During the process that occurs from time A to time B, the gas absorbs x joules of energy from the water. Which of the following is true: The total kinetic energy of all of the gas molecules (a) increases by more than x joules; (b) increases by x joules; (c) increases, but by less than x joules; (d) remains unchanged; (e) decreases by less than x joules; (f) decreases by x joules; (g) decreases by more than x joules. Explain your answer: 1
Step 2. Now, empty containers are placed on top of the piston as shown. Small lead weights are gradually placed in the containers, one by one, and the piston is observed to move down slowly. While this happens, the temperature of the water is nearly unchanged, and the gas temperature remains practically constant. (That is, it remains at the temperature it reached at time B, after the water had been heated up.) containers lead weight weights being added Piston moves down slowly. Temperature remains same as at time B. Step 3. At time C we stop adding lead weights to the container and the piston stops moving. (The weights that we have already added up until now are still in the containers.) The piston is now found to be at exactly the same position it was at time A. Time C Weights in containers. Piston in same position as at time A. Temperature same as at time B. Question #3: During the process that occurs from time B to time C, does the total kinetic energy of all the gas molecules increase, decrease, or remain unchanged? Question #4: During the process that occurs from time B to time C, is there any net energy flow between the gas and the water? If no, explain why not. If yes, is there a net flow of energy from gas to water, or from water to gas? Explain your answer: 2
Step 4. Now, the piston is locked into place so it cannot move; the weights are removed from the piston. The system is left to sit in the room for many hours, and eventually the entire system cools back down to the same room temperature it had at time A. When this finally happens, it is time D. Time D Piston in same position as at time A. Temperature same as at time A. Question #5: During the process that occurs from time C to time D, the water absorbs y joules of energy from the gas. Which of the following is true: The total kinetic energy of all of the gas molecules (a) increases by more than y joules; (b) increases by y joules; (c) increases, but by less than y joules; (d) remains unchanged; (e) decreases, by less than y joules; (f) decreases by y joules; (g) decreases by more than y joules. Question #6: Consider the entire process from time A to time D. (i) Is the net work done by the gas on the environment during that process (a) greater than zero, (b) equal to zero, or (c) less than zero? Explain your answer: (ii) Is the total heat transfer to the gas during that process (a) greater than zero, (b) equal to zero, or (c) less than zero? Explain your answer: 3
Now let us begin Process #2. The piston is unlocked so it is again free to move. We start from the same initial situation as shown at time A and D (i.e., same temperature and position of the piston). Just as before, we heat the water and watch as the piston rises. However, this time, we will heat the water for a longer period of time. As a result, the piston ends up higher than it was at time B. Step 5. Step 6. Now, weights are added to the piston and it begins to move down. (Temperature does not change during this process.) However, this time, more weights than before must be added to get the piston back to the position it had at time C. Again, the piston is locked and the weights are removed. After many hours, the system returns to the same temperature that it had at time A and time D (and the piston is in the same position as it was at those times). This final state occurs at time E. Question #7: Consider the total kinetic energy of all of the gas molecules at times A, D, and E; call those E A, E D, and E E. Rank these in order of magnitude (greatest to least, using > or < signs). If two or more of these are equal, indicate that with an = sign. Explain your answer: Question #8: Consider the following positive quantities: Q 1, Q2, W1, W2. These represent the absolute values of the total heat transfer to the gas during Process #1 and Process #2, and of the net work done by the gas during Processes #1 and #2. Rank these four quantities from largest to smallest. If two or more are equal, indicate with = sign. Explain your answer: Question #9: Sketch a P-V diagram of Process #1 and Process #2 on the same set of axes; label (with the appropriate letter) the states that occur at times A, B, C, D, and E. (Use the other side of this sheet if necessary.) Explain how you can use the diagram to justify your answer to Question #8. 4
Question #10: A. For the process A B: i. Is the work done by the system (W AB ) positive, negative, or zero? ii. Is heat transferred to the system, away from the system, or is there no heat transfer? iii. Does the internal energy increase, decrease, or remain the same? Explain. B. For the process B C: i. Is the work done by the system (W BC ) positive, negative, or zero? ii. Is heat transferred to the system, away from the system, or is there no heat transfer? iii. Does the internal energy increase, decrease, or remain the same? Explain. C. For the process C D: i. Is the work done by the system (W CD ) positive, negative, or zero? ii. Is heat transferred to the system, away from the system, or is there no heat transfer? iii. Does the internal energy increase, decrease, or remain the same? Explain. 5
Question #11: Rank the absolute values W AB, W BC, and W CD from largest to smallest; if two or more are equal, indicate with an = sign. largest smallest Explain your reasoning. Question #12: Rank the absolute values of internal energy change U AB, U BC, and U CD from largest to smallest; if two or more are equal, indicate with an = sign. largest smallest Explain your reasoning. Question #13: Rank the absolute values of heat transfer Q AB, Q BC, and Q CD from largest to smallest; if two or more are equal, indicate with an = sign. Assume that the low temperature is 300 K, and the high temperature is 600 K. largest smallest Explain your reasoning. 6
Question #14: Are your answers to Questions #11, 12, and 13 consistent with your answers to parts (i) and (ii) of Question #6? Explain. Question #15: Is the net change in volume from time A to time D (V D V A ) greater than zero, equal to zero, or less than zero? Is this consistent with your answers to Questions #6(i) and #14? Explain. Question #16: Is the net change in temperature from time A to time D (T D T A ) greater than zero, equal to zero, or less than zero? Is this consistent with your answers to Questions #6(ii) and #14? Explain. 7
A Sample Cyclic Process A thermodynamic system undergoes a three-step process. An adiabatic expansion takes it from state 1 to state 2; heat is added at constant pressure to move the system to state 3; and an isothermal compression returns the system to state 1. The system consists of a diatomic ideal gas with C V = 5R/2. The number of moles is chosen so nr = 100 J/K. The following information is known about states 2 and 3. Pressure: P 2 = P 3 = 100 kpa Volume: V 3 = 0.5 m 3 What is the temperature of the system in state 3? The system does 20000 J of work in the constant pressure process that takes it from state 2 to state 3. What is the volume and temperature of the system in state 2? V 2 = T 2 = Complete the table below, filling in the missing numerical values, with units of joules. The total work done by the system in the cycle is 19400 J. Note that there is a way to do this without integrating or using logarithms. Process Q E W int 1 2 2 3 20000 3 1 Complete cycle -19400 Step 1: First fill in all the terms that are zero. Remember that each row in the table satisfies the First Law of Thermodynamics. Also remember that Eint = ncv T. Chapter 15 The Laws of Thermodynamics Page 9