2018 PUSO Thermodynamics Written Exam

Transcription

1 Team Number: Team Name: Participant Names: For Office Use Only: / PUSO Thermodynamics Written Exam Instructions: You will have 25 minutes to complete the attached exam. There are a total of 87.5 possible points on this exam. There are no tiebreakers ties will be broken according to the official rules, based on the lab portion of the competition. Please show your work and include units where applicable.

2 For calculation problems, please show your work and include units! Problem 1 (1pt). The entropy S of an ideal gas is related to the quantity W, the number of real microstates corresponding to the gas macrostate, via the formula S = k log W for some constant k. The formula is named after a German physicist. What is his name? (Hint: the constant k J/K is also named after him.) Problem 2 (5pt). Calculate the work performed by 0.50 moles of oxygen expanding isothermally at 25 C from 1.0 to.25 atm of pressure. 1

3 Problem 3 (5pt). Sketch the Carnot cycle of a heat engine, on a graph of pressure vs volume. You need not have specific numerical values on the graph. Be sure to identify the type of process occurring for each leg of the cycle, and to draw arrows on each leg to indicate direction. Problem 4 (5pt). Find the minimum amount of work needed to extract 1.4J of heat from a body at the temperature of 50. C when the temperature of the environment is 125 C. 2

4 Problem 5 (2pt). What is the thermal efficiency of the associated Carnot cycle for Problem 4? Problem 6 (5pt). An ideal monotomic gas expands adiabatically to a volume 1.5 times larger than its initial volume. The initial temperature is 15 C. Find the final temperature. (Hint: The heat capacity ratio (a.k.a. adiabatic index) for an ideal monotomic gas is γ = 5/3.) 3

5 Problem 7 (2pt). Derive the heat capacity ratio γ (a.k.a. adiabatic index) for an ideal diatomic gas by a degree of freedom count. If a molecule has f degrees of freedom, the two are related by the formula γ = f. You must explain your answer to explain full credit - a correct value for γ without explanation will be worth half credit. Problem 8 (1pt). With a diagram like the one in Problem 3, how does one graphically interpret the work done by the heat engine? 4

6 Problem 9 (2pt). State the first law of thermodynamics (in the context of a closed system) in terms of the following quantities: change in internal energy U, heat supplied to the system Q, and work done by the system W. Problem 10 (1pt). is a well-known thought experiment in thermodynamics that postulates how the second law of thermodynamics might be violated. In this thought experiment, the namesake agent controls a small door between two chambers of gas A and B, and sorts the molecules by opening and shutting the door quickly as gas molecules approach, so that fast molecules pass from chamber A to B, while slow molecules remain in chamber A. Thus, chamber B heats up while chamber A cools down. Problem 11 (1pt). Which French physicist is credited for the first formulation of the second law of thermodynamics in 1824? (He showed that there are limitations in transforming heat into work.) Problem 12 (2pt). There are many equivalent statements of the second law of thermodynamics. Give any statement of the second law of thermodynamics. Problem 13 (1pt). Two such statements are attributed to Lord Kelvin and Rudolph Clausius. State either. If one of these statements was the one you gave in Problem 12, then you must state the other. You need not identify who the statement is attributed to. (Hint: One statement concerns a heat engine, and the other concerns heat transfer between two bodies.) 5

7 Problem 14 (1pt). Why might the thought experiment as explained in Problem 10 violate the second law of thermodynamics? Problem 15 (1pt). Why might the thought experiment from Problem 10 not actually violate the second law of thermodynamics? Give a brief explanation. Note: multiple acceptable answers possible. Problem 16 (1pt). Describe James Joule s paddle wheel experiment, which led him to an estimate for the mechanical equivalent of heat (which by his calculation was J/cal), i.e. the amount of work needed to raise the temperature of a pound of water by one degree Fahrenheit Problem 17 (1pt). This temperature scale is defined with absolute zero at 0 and with the triple point of water at approximately 492. a. Celsius b. Rømer c. Rankine d. Kelvin 6

8 Problem 18 (1pt). Which of the following is a cgs unit of energy? a. Joule b. Erg c. British Thermal Unit, BTU d. Dyne Problem 19 (1pt). In the Fahrenheit scale of temperature, 0 degrees is defined as which of the following? a. The point at which pure water freezes. b. The point at which ethyl alcohol freezes. c. Approximately the point at which brine water freezes. d. Approximately the point at which it is too cold to go skiing. Problem 20 (1pt). Which of the following is a true statement about Temperature scales. a. From a theoretical standpoint, only differences in temperatures are important; Temperature scales are equivalent to an additive constant. b. Originally degrees were defined as the height of an expanding column of mercury; Now scales like the Kelvin scale are defined completely independently of any physical phenomenon. c. Originally degrees were inaccurately defined by the expansion of physical materials; Modern scales are defined via electronic sensors. d. The modern interpretation of temperature is related to the average kinetic energy of a material s molecules; The modern interpretation has nothing to do with heat engines. 7

9 Problem 21 (1pt). Which of the following is true about the Kelvin and Celsius Scales. I. 0 on the Kelvin scale represents the freezing point of water II. The Kelvin scale is defined with as the freezing point of water. III. The Celsius scale is defined with 0.01 as the triple point of water. a. Choice I only. b. Choice II only. c. Choice III only. d. Choice II and III only. e. None of the choices are correct. Problem 22 (2.5pts). Do the following conversions: a Joules to Kilojoules b. 100 calories to Joules c Joules to BTU d degrees Celsius to degrees Fahrenheit (or, degrees Freedom) e degrees Celsius to Rankine Problem 23 (1pt). In the Debye model: a. The heat capacity of many solids is accurately predicted to be constant at high temperatures, and proportional to T 3 at low temperatures, where T represents temperature. b. The heat capacity of many solids is accurately predicted to be constant at high temperatures, but inaccurately predicts the heat capacity to be proportional to T 3 at low temperatures due to Quantum considerations. c. The heat capacity of many solids is inaccurately predicted to be proportional to T 3 at all temperatures. d. The heat capacity of many solids is inaccurately predicted to be constant at all temperatures. 8

10 Problem 24 (1pt). Which of the following is true at low temperatures: a. The heat capacity of many solids is constant and equal to 3R. b. Entropy becomes poorly defined. c. The ideal gas law breaks down because gas molecules become closer together. d. The ideal gas law breaks down because interactions between gas molecules becomes significant. Problem 25 (1pt). Which of the following statements are true: a. The distribution of speeds of gas molecules always follows a Boltzman distribution. b. The distribution of speeds of gas molecules always follows a Boltzman distribution when the pressure is high enough and the temperature is low enough. c. The distribution of speeds of gas molecules always follows a Boltzman distribution when the pressure is low enough and the temperature is high enough. d. The distribution of speeds of gas molecules can never follow a Boltzman distribution. Problem 26 (1pt). Alice, Lawrence, and Derek spent too much time outside and are cold. To heat up, Alice shines a really bright light on Lawrence and Derek until their palms are sweaty. How is heat transferred to Lawrence and Derek? a. Conduction. b. Convection. c. Radiation. d. Both choices a and b. 9

11 Problem 27 (1pt). In her quest to heat things up, Alice goes to a pond, which has frozen over, and shines her humongous light on it. After a while, the ice melts and the pond water begins to heat up. How is heat transferred from the top layer of water to the rest of the pond? a. Conduction. b. Convection. c. Radiation. d. Both choices a and b. Problem 28 (1pt). For an ideal gas, how many variables are needed to define its thermodynamic state? a. Infinite b. 1 c. 2 d. 3 Problem 29 (1pt). Which is true of Entropy? a. During some process, the entropy of a system can never decrease. b. Given the temperature and heat, the entropy change of a system during some process can be calculated. c. Entropy can only be defined up to some additive constant. d. Because the entropy change during a process depends on the heat, entropy is not a state function. 10

12 Problem 30 (1pt). For some mixture of phases and substances, call the degrees of freedom of the system to be the amount of state variables that can be freely chosen. A mixture of two non-reacting gases has how many degrees of freedom? a. 1 b. 2 c. 3 d. 4 Problem 31 (1pt). A system consists of a single substance, present in 3 phases. How many degrees of freedom are there in the system? a. 1 b. 2 c. 3 d. 4 Problem 32 (1pt). Which of the following is true for a system at constant temperature and volume: a. During some process, the magnitude of the change in Gibb s Free Energy represents the minimum amount of work that can be done by the system. b. During some process, the magnitude of the change in the Hemholtz Free Energy represents the maximum amount of work that can be done by the system. c. The Gibb s Free Energy is minimized at equilibrium. d. The Hemholtz Free Energy is maximized at equilibrium. 11

13 Problem 33 (5pt). Sketch a P V graph of a substance in the liquid-gas states. Include and label on your graph: a. The critical point b. The critical isotherm passing through the liquid-gas phases c. An isotherm purely in the gas phase d. The saturated vapor curve e. The phases represented by different regions on the graph Problem 34 (3pt). This question has two parts: a. If water has a heat of vaporization of kj mol, what is its vapor pressure at 65 C? b. If hexane has a vapor pressure of 213torr at 40 C and a vapor pressure of 836torr at 80 C, what is its heat of vaporization in kj mol? 12

14 Problem 35 (5pt). An ice cube with mass 50g, initially of temperature 5 C is placed into a cup of 200g of warm water, initially at temperature 40 C. What is the final temperature and state of the system after thermal equilibrium is achieved, assuming no heat is lost to the environment? The specific heat of water is 4186 J kg C and the specific heat of fusion is J. kg, the specific heat of ice is 2060 J kg C, 13

15 Problem 36 (8pt). A ring of mass M and specific heat C and inner radius R is originally at 0 C. A sphere of mass m and radius r which is originally at 100 C is placed on top of the ring. The two objects come to thermal equilibrium, and no heat is lost to the surroundings. The sphere just barely passes through the ring. What is the specific heat of the sphere? Assume that L = Lα T where L is the change in length of a solid from its original length L due to a temperature increase T. The constant of proportionality is given by α r for the ring and α s for the sphere. 14

16 Problem 37 (5pt). In the anime Kill la Kill, the character Mako is positioned precariously over a vat of boiling oil. While Mako is eventually saved, later on, other characters fall into the vat of oil, frying to a crisp immediately. Derek, being the weeb he is (but also a science guy) wants to assess the physical validity of this scenario. Assume that Mako can be modeled as a 2-dimensional circle of radius r, mass m, specific heat c, and originally at a temperature 0 C. Now assume that Mako is dunked into a vat of oil at a fixed temperature of 300 C. How long will it take for thermal equilibrium to be reached? How long will it take for Mako to reach 150 C? Assume that heat transfer occurs at a rate per unit area q = k T, where T represents the difference in temperature across the boundary and k is the heat transfer coefficient. Also assume that Mako is small enough so that she heats up uniformly. 15

17 Problem 38 (8pt). Seeking to refine his model, Derek now assumes that Mako expands while heating up. How long will it take for Mako to reach a temperature of 150 C? Assume that A = Aα T, where A is the change in area of a 2-dimensional Mako from its original area A due to a temperature increase T and α is the thermal expansion coefficient. 16

18 Thermodynamics - Written Test - ANSWER KEY February 10, 2018 Unless specified otherwise, questions are to be graded all-or-nothing. Maximum points possible per question are indicated on the test. 1. Boltzmann 2. (Sample Solution) Subscript i and subscript f indicate initial and final respectively. The work W done by the gas is given by the following formula W = Vf V i = nrt P dv Vf = nrt log dv V ) V i ( Vf V i by ideal gas law by log I mean the natural logarithm, students might also write ln = (0.5mol)(8.314J/molK)(298K) log 4 since P f /P i = 1/4 and V 1/P when isothermal = 1717J or 1700J Some students may give the rounded answer to follow significant figures. Accept either. Alternately, students might directly write down the formula W = nrt log(p i /P f ) which is also an acceptable form of work. In the unlikely event that students use different letters in their formulas, just match what they wrote to the key, as closely as possible. Three points for the correct answer. figures is fine.) (Anything correct within two significant Two points for writing down either W = nrt log(v f /V i ) or W = nrt log(p i /P f ), or if the student explicitly includes a step where the numbers are substituted in, as in the sample solution. If they did not write down either W = nrt log(v f /V i ) or W = nrt log(p i /P f ), award one point if the student either wrote down W = V f V i P dv or noted V 1/P for an isothermal process. (Award a maximum total of one point for this, i.e. if the student wrote both, still give them one point.) 1

19 Thermodynamics - Written Test - ANSWER KEY February 10, Example: Figure 1: Carnot Cycle heat engine) Two points for correct shape. That is, a closed plane figure with four sides curved as in the example. The sides should also slope in the same direction as in the example above, though parameters like steepness and length may vary. Half a point for each correctly labeled leg of the cycle (i.e., isothermal or adiabatic). Two points possible in total. This must be the same as the diagram. One point for one complete set of correct arrows (all or nothing). This must be the same as the diagram. 4. (Sample Solution) The minimum amount work needed is achieved by a Carnot cycle. Write W for the work needed, Q for heat, and T H and T C for the temperatures of the hot and cold reservoirs, respectively. The formula is W = Q T H T C = (1.4J) =.325J Three points for the correct answer T C 398K 323K 323K Two points for an equivalent symbolic expression as in the sample solution, or if the student explicitly includes a step where the numbers are substituted in, as in the sample solution. 5. (Sample Solution) The Carnot efficiency is given by η = 1 T C T H = T H T C T H = =.188 2

20 Thermodynamics - Written Test - ANSWER KEY February 10, 2018 One point for correct answer One point for any equivalent (correct) symbolic expression as in the sample solution, or if the student explicitly includes a step where the numbers are substituted in, as in the sample solution. 6. (Sample Solution) By ideal gas law and for an adiabatic process P i V i = nrt i P f V f = nrt f P i V γ i = P f V γ f. Dividing the equations given by ideal gas law implies (after some rearranging) T f /T i = (V f /V i )(P f /P i ) = (V f /V i )(V f /V i ) γ = (V f /V i ) 1 γ = (1/5) 2/ Substituting T i = 288K gives 220K or 53.3 C. Three points for the correct answer. Accept anything within two significant figures in Kelvin (if they give an answer in Celsius, add 273 to convert their answer into Kelvin to check whether it is within two significant figures.) Do not accept answers in other units. Two points for an expression like T f /T i = (V f /V i ) 1 γ, or at least some manipulations like that given in the sample solution. If they did not derive or write down a correct formula, award one point if the noted that P V γ is constant for an adiabatic process. 7. γ = 7/5, 5 degrees of freedom. (Three translational degrees of freedom, two rotational degrees of freedom.) One point for correct γ value. Half point for correct number of degrees of freedom. Half point for correct explanation of degrees of freedom. 8. area inside the enclosed region 9. U = Q W 10. Maxwell s Demon 3

21 Thermodynamics - Written Test - ANSWER KEY February 10, Nicolas Léonard Sadi Carnot (last name only acceptable) 12. (Close statements perfectly acceptable.) Total entropy can never decrease over time for an isolated system. or A Carnot cycle s efficiency depends only on the temperatures of the two reservoirs, and is the most efficient heat engine possible. (Carnot) or Heat can never pass from a colder to a warmer body without some other change occurring at the same time. (Clausius). or It is impossible to create a cyclically operating device, whose sole purpose is to absorb heat energy from a single thermal reservoir and deliver an equivalent amount of work (no heat engine has 100% thermal efficiency). (Kelvin-Planck). or It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects. (Kelvin) 13. See the previous problem. Accept either of the last two statements for the statement attributed to Lord Kelvin. 14. The entropy is decreasing in the described thought experiment, the cooler chamber gets cooler and the warmer chamber gets warmer. 15. The demon may generate entropy in separating the molecules or making measurements. (Accept anything similar-sounding, or which you judge is reasonable.) 16. A paddle wheel was put in water and spun by a weight under the influence of gravity - Joule calculated the work done by the paddle wheel, and the corresponding increase in temperature of the wheel (i.e. the transformation of work into heat.) 17. c 18. b 19. c 20. e 21. c 22. Conversion Solutions: Half a point is awarded for each part: (a) 1.6 kilojoule (b) joules (c) BTU (d) degrees Fahrenheit (e) Rankine 4

22 Thermodynamics - Written Test - ANSWER KEY February 10, a 24. d 25. e 26. c 27. a 28. d 29. e 30. c 31. e 32. b 33. Example: Figure 2: P-V Graph of Liquid-Gas Phases 34. Two parts: Two points are awarded for solving part a and one point is awarded for solving part b. (a) 195.8torr or mmhg or 26.1kP a (b) 31.4 kj mol 35. Three points are awarded for stating that the final temperature is C, and two points are awarded for stating that the final state is liquid 5

23 Thermodynamics - Written Test - ANSWER KEY February 10, C s = C M r R 100rα s m R r+100rα r. One point of partial credit may be awarded for finding the following constraint 100mC s = T f (mc s + MC). Two points each may be awarded for writing down the correct system of equations R f R = Rα r T f and R f r = rα s (T f 100). The remaining three points are awarded for the correct final answer. 37. This question has two parts: (a) For the first part, 1 point is awarded for stating that thermal equilibrium will never be reached (b) The other 4 points are awarded for finding t = mc ln(2). For this part, two 2πr 2 k dt points may be awarded for writing down the correct differential equation, 300 T = 2πr 2 k dt. One point may be awarded for finding the correct expression of temperature as a function of time, T (t) = 300(1 exp 2πr2 kt mc ( ) ). Equivalent expressions mc may also be accepted for partial credit. mc 38. t = ln( α). Four points of partial credit may be awarded for writing 2πr 2 k(1+300α) dt down the correct differential equation, = 2πr2 k dt. Three more points may (1+αT )(300 T ) mc be awarded for finding an expression involving the temperature as a function of time such as one of the following: (a) ln 300(1+αT ) = ( α) 2πr2 k t 300 T mc ( ) (b) 300(1 + αt ) = (300 T ) exp ( α) 2πr2 k t mc ( ) (c) T (t) = 300(γ 1) where γ = exp ( α) 2πr2 k t 300α+γ mc Of course, any of the above expressions (or something equivalent) is acceptable for partial credit. 6

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