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Unformatted text preview: First 176 Midterm Exam: Answers Professor Greenside Wednesday, February 18 A reminder for future quizzes and exams: work on those problems first that are worth the most points. The true false and multiple choice questions can be interesting to think about but reward you with few points compared to the problems that require writing out some answer. True or False Questions (2 points each) For each of the following statements, please circle T or F to indicate whether a given statement is true or false respectively. 1. T / F For an ideal gas, the ratio of the specific heats C P /C V is equal to the adiabatic exponent γ . Answer: T This problem required a short calculation. From the ideal gas law in the form V = NkT/P and from the formulas U = Nf ( kT/ 2), C v = ( ∂U/∂T ) V = Nfk/ 2, and C P = ( ∂U/∂T ) P + P ( ∂V/∂T ) P given at the beginning of the midterm, you can calculate that: C p = ( ∂U/∂T ) P + P ( ∂V/∂T ) P (1) = ( Nfk/ 2) + P ( Nk/P ) (2) = ( f/ 2 + 1) Nk (3) = ( f + 2) Nk/ 2 (4) = ( f + 2) f fN k 2 ¶ (5) = γC V , (6) so the statement is true. 2. T / F It is possible for the heat capacity C P of a substance to be infinite at a finite temperature. Answer: T This was mentioned briefly in Schroeder in the discussion of specific heats. From the definition Eq. (1.4) on page 28, C = Q/ Δ T , we can see that the specific heat could be infinite if it is possible to add heat to a system (positive numerator) without changing its temperature (zero denominator). But this is exactly what happens during some phase transitions such as the melting of ice or the boiling of water 1 . Looking for a sharp peak in the specific heat (experimentally or computationally in a simulation) is also a way to discover a phase transition in some novel substance or in some novel regime: a careful measurement (say carried out during a computer-controlled experiment) of the substance’s specific heat as a function of temperature might reveal a large peak, suggesting a phase transition. 3. T / F When a closed path is traced out in the P V-plane of an ideal gas, starting at some point A on the path and returning back to A , the total amount of heat absorbed by the gas will depend on the position of the starting point A on the path. Note: for this problem, assume that the total work done on the gas after following the path from A back to A is nonzero. 1 I say “some” phase transitions because there are phase transitions of a different kind than the melting of a solid or the vaporization of a liquid, say the loss of ferromagnetism as a magnet is heated above a certain temperature, for which the temperature does not stay constant during the transition. 1 Answer: F This is a qualitative question that you can settle by drawing some representative closed loop, e.g., a circle, in the P V-plane of an ideal gas. If you start at different points on the closed curve and trace all the way around back to the same points, you should be able to convince yourself that...
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