Physics 221: Optical and Thermal Physics Final Exam, Sec. 500, 9 May Please fill in your Student ID number (UIN): IMPORTANT

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1 Physics 221: Optical and Thermal Physics Final Exam, Sec. 500, 9 May 2005 Instructor: Dr. George R. Welch, 415 Engineering-Physics, Print your name neatly: Last name: First name: Sign your name: Please fill in your Student ID number (UIN): IMPORTANT Read these directions carefully: There are 8 problems totaling 200 points. Check your exam to make sure you have all the pages. Work each problem on the page the problem is on. You may use the back. If you need extra pages, I have plenty up front. Indicate what you are doing! I cannot give full credit for merely writing down the answer. Neatness counts! I will give generous partial credit if I can tell that you are on the right track. This means you must be neat and organized. Each problem with its associated figure is self explanatory. If you must ask a question, then come to the front, being as discrete as possible so as not to disturb others. Put your name on each page it is asked for. You will lose credit if you fail to print your name on each page it is asked for.

2 Problem points. (a) A material has linear coefficient of thermal expansion α. Assume this is the same in all directions. Denote the mass per unit volume of the material by ρ. Calculate the change in density ρ of the substance when the temperature changes by T. Express the answer in terms of α, ρ, T, and numerical factors. Assume T 1/α. (b) Use the ideal gas law to find the coefficient of volume expansion for an ideal gas at temperature T. Assume the pressure is constant. If you work neatly I will find more partial credit for you!

3 Problem points. An ideal gas has pressure p and mass-density ρ. Calculate the root-mean square velocity v 2 of the atoms in this gas. Be neat. Neatness helps. Work neatly.

4 Problem points. A block of ice with mass 1.00 kg and temperature -10 C is placed in a perfectly insulating container. A small amount, kg, of liquid water at 20 C is poured on the ice and the container is sealed. What is the final temperature and phase of the contents of the sealed container? Use the following data for water: Specific heat of ice is 2100 J/(kg C ). Specific heat of liquid water is 4186 J/(kg C ). Latent heat of fusion of water is 333 kj/kg. You must be neat. Neatness will count as part of the grade on this problem.

5 Problem 4. (25 points) A phase diagram for carbon dioxide (CO 2 ) is shown in your book on page 474. convenience, that diagram is reproduced below. For your P (atm) SOLID 73 Critical point LIQUID 5.11 Triple point GAS T ( oc) Suppose a sample of CO 2 has pressure of 50 atmospheres and has temperature 10 C. This sample is in a sealed container, and there is nothing else in the container. The entire volume of the container is occupied by CO 2 at 10 C. (a) What is the phase of this sample of CO 2? (b) The sample is now heated to 50 C, and the pressure is raised to 100 atmospheres. Suppose this is done in two ways: 1) The initial pressure is held constant, and the temperature is slowly increased to 50 C. Then, the temperature is held constant, and the pressure is slowly increased to 100 atmospheres. 2) The initial temperature is held constant, and the pressure is slowly increased to 100 atmospheres. Then, the pressure is held constant, and the temperature is slowly increased to 50 C. Explain the important differences between the two cases. State what phase transitions occur in each case. Also, as the sample is heated and pressurized, its volume may change. In which case, and at what point, will the volume of the container reach its maximum value? You may write on the next page. Please write neatly, and tersely. Do not write a book. For full credit you must keep your answer well below 100 words. Think about what you want to say before you start writing. Please write neatly. I have to be able to read it to grade it.

6 Problem 4, continued... Remember to write neatly. I have to be able to read it to grade it. Think about what you want to say before you start writing. Remember to write neatly. I have to be able to read it to grade it.

7 Problem 5. (25 points) A sealed container contains an ideal monatomic gas. The pressure and volume are both known, and denoted as P 0 and V 0. Heat is allowed to flow out of the gas at constant volume until its pressure drops to 2/3 of its original value. Then heat is added to the gas so that it expands at constant pressure until its volume is doubled. (a) Calculate the total work done by the gas. (b) Calculate the change in internal energy of the gas. (c) Calculate the total heat flow into or out of the gas. Express all answers in terms of P 0, V 0, and numerical factors. Underline each of your three answers. Hint: You are given that the gas is monatomic. This is important. You need to work Neatly! Don t forget to be neat.

8 Problem 6. (25 points) At very low temperatures, the molar specific heat of many substances varies as the cube of the absolute temperature: C v = 12π4 5 R T 3 T 3 D where R is the ideal gas constant, and T D depends on the material. This is sometimes called Debye s law. For silicon, it is found experimentally that T D = 645 K. Suppose a small block of silicon with mass grams is inside a cryostat (device for maintaining low temperatures) at T = 1 K. If 1 µj (one micro-joule) of heat is added to this piece of silicon, what will be the final temperature? (The atomic mass of silicon is grams per mole.) Don t forget to be neat. Neatness helps!

9 Problem 7. (25 points) The figure below is a PV diagram for a reversible heat engine in which an ideal gas is the working fluid. Starting at point 1, heat is added to the gas in such a way that it expands isobarically to point 2. Then, heat is removed from the gas so that the pressure drops isochorically to point 3. Finally, the gas is compressed adiabatically back to point 1. P P1 1 2 P3 3 V 1 V 2 V Calculate the efficiency of this heat engine. Express your answer only in terms of the compression ratio V 2 /V 1 and the ratio of specific heats γ. Feel free to use the next page, but be neat! Present your work neatly and clearly.

10 Problem 7, continued... Remember to work neatly. Being neat and organized will help you through the algebra. Present your work neatly and clearly.

11 Problem 8. (25 points) 1 kg of water at temperature 10 C, is mixed in an insulated container with 1 kg of water at temperature 70 C. Calculate the change in entropy. Important: Show your work. The specific heat of water is given in problem 3. Work NEATLY.