Thermodynamics The goal of thermodynamics is to understand how heat can be converted to work Main lesson: Not all the heat energy can be converted to mechanical energy This is because heat energy comes with disorder (entropy), and overall disorder cannot decrease Temperature v T Random directions of velocity Higher temperature means higher velocities v Add up energy of all molecules: Internal Energy of gas U Mechanical energy: all atoms move in the same direction 1 Mv2 2 Statistical mechanics For one atom 1 1 1 1 1 1 3 E = mv2 + mv2 + mv2 = kT + kT + kT = kT h i h2 xi h2 yi h2 z i 2 2 2 2 Ideal gas: No Potential energy from attraction between atoms 3 U = NkT h i 2 v T Pressure v Pressure is caused because atoms T bounce off the wall Lx p − x px ∆px =2px 2L ∆t = x vx ∆p 2mv2 mv2 F = x = x = x ∆t 2Lx Lx ∆p 2mv2 mv2 F = x = x = x ∆t 2Lx Lx 1 mv2 =2 kT = kT h xi ⇥ 2 kT F = Lx Pressure v F kT 1 kT A = LyLz P = = = A Lx LyLz V NkT Many particles P = V Lx PV = NkT Volume V = LxLyLz Work Lx ∆V = A ∆Lx v A = LyLz Work done BY the gas ∆W = F ∆Lx We can write this as ∆W =(PA) ∆Lx = P ∆V This is useful because the body could have a generic shape Internal energy of gas decreases U U ∆W ! − Gas expands, work is done BY the gas P dW = P dV Work done is Area under curve V P Gas is pushed in, work is done ON the gas dW = P dV − Work done is negative of Area under curve V By convention, we use POSITIVE sign for work done BY the gas Getting work from Heat Gas expands, work is done BY the gas P dW = P dV V Volume in increases Internal energy decreases ... this means Temperature decreases Pressure decreases But how do we get the engine back to its initial state ? P P dW = P dV dW = P dV − V V We will have to put in exactly as much work as we got out ... so this does not help The heat engine P Gas expands, does work Add heat, so pressure goes back up Cool gas down, so pressure drops further Compress gas so original volume restored V Two relations P Work done by the gas Work done on the gas V (1) Net work done equal Area inside cycle (2) Net work done equals net heat added (since engine returns to original state) 36. Consider the quasi-static adiabatic expansion of an ideal gas from an initial state i to a final state f. Which of the following statements is NOT true? (A) No heat flows into or out of the gas. (B) The entropy of state i equals the entropy of state f. (C) The change of internal energy of the gas is - z PdV. 38. An AC circuit consists of the elements shown (D) The mechanical work done by the gas is above, with R = 10,000 ohms, L = 25 millihenries, and C an adjustable capacitance. The AC voltage PdV. z generator supplies a signal with an amplitude of (E) The temperature of the gas remains constant. 40 volts and angular frequency of 1,000 radians per second. For what value of C is the amplitude of the current maximized? (A) 4 nF (B) 40 nF GRE0177 (C) 4 mF (D) 40 mF (E) 400 mF 39. Which two of the following circuits are high-pass filters? 37. A constant amount of an ideal gas undergoes the cyclic process ABCA in the PV diagram shown above. The path BC is isothermal. The work done by the gas during one complete cycle, beginning and ending at A, is most nearly (A) 600 kJ (B) 300 kJ (C) 0 (D) -300 kJ (E) -600 kJ (A) I and II (B) I and III (C) I and IV (D) II and III (E) II and IV Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 30 36. Consider the quasi-static adiabatic expansion of an ideal gas from an initial state i to a final state f. Which of the following statements is NOT true? (A) No heat flows into or out of the gas. (B) The entropy of state i equals the entropy of state f. (C) The change of internal energy of the gas is - z PdV. 38. An AC circuit consists of the elements shown (D) The mechanical work done by the gas is above, with R = 10,000 ohms, L = 25 millihenries, and C an adjustable capacitance. The AC voltage PdV. z generator supplies a signal with an amplitude of (E) The temperature of the gas remains constant. 40 volts and angular frequency of 1,000 radians per second. For what value of C is the amplitude of the current maximized? (A) 4 nF (B) 40 nF GRE0177 (C) 4 mF (D) 40 mF (E) 400 mF 39. Which two of the following circuits are high-pass filters? 37. A constant amount of an ideal gas undergoes the cyclic process ABCA in the PV diagram shown above. The path BC is isothermal. The work done by the gas during one complete cycle, beginning and ending at A, is most nearly (A) 600 kJ (B) 300 kJ (C) 0 (D) -300 kJ (E) -600 kJ (A) I and II (B) I and III (C) I and IV (D) II and III (E) II and IV Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 30 27. Electromagnetic radiation emitted from a nucleus is most likely to be in the form of 27. Electromagnetic radiation emitted from a nucleus is(A) most gamma likely rays to be in the form of (B) microwaves (A) gamma rays (C) ultraviolet radiation (B) microwaves (D) visible light (C) ultraviolet radiation (E) infrared radiation (D) visible light (E) infrared radiation GREPracticeBook 28. A sample of nitrogen gas undergoes the cyclic thermodynamic process shown above. Which of the 28. A sample of nitrogen gas undergoes the cyclic following gives the net heat transferred to the system thermodynamic process shown above. Which of the followingin one complete gives thecycle net 1 heat 2 transferred 3 1 ?to the system (A)in one 80 complete J cycle 1 2 3 1 ? (A)(B) 8040 J (C) 40 J (B) 40 J (D) 80 J (C) 40 J (E) 180 J (D) 80 J (E) 180 J GO ON TO THE NEXT PAGE. -14- 20 GO ON TO THE NEXT PAGE. -14- 27. Electromagnetic radiation emitted from a nucleus is most likely to be in the form of 27. Electromagnetic radiation emitted from a nucleus is(A) most gamma likely rays to be in the form of (B) microwaves (A) gamma rays (C) ultraviolet radiation (B) microwaves (D) visible light (C) ultraviolet radiation (E) infrared radiation (D) visible light (E) infrared radiation GREPracticeBook 28. A sample of nitrogen gas undergoes the cyclic thermodynamic process shown above. Which of the 28. A sample of nitrogen gas undergoes the cyclic following gives the net heat transferred to the system thermodynamic process shown above. Which of the followingin one complete gives thecycle net 1 heat 2 transferred 3 1 ?to the system (A)in one 80 complete J cycle 1 2 3 1 ? (A)(B) 8040 J (C) 40 J (B) 40 J (D) 80 J (C) 40 J (E) 180 J (D) 80 J (E) 180 J GO ON TO THE NEXT PAGE. -14- 20 GO ON TO THE NEXT PAGE. -14- P Volume in increases dW = P dV Temperature decreases (atoms slower) Pressure decreases (atoms slower and also more distance to travel) V P To keep temperature constant, add heat Pressure still drops, since volume rises To keep pressure constant, need to add even more heat V Terminology Isothermal: constant temperature Isobaric: constant pressure Adiabatic: no heat added or removed GREPracticeBook 29. For an ideal gas, consider the three thermodynamic processes—labeled 1, 2, and 3 —shown in the PV diagram 29. For an ideal gas, consider the three thermodynamic above. Each process has the same initial state and the same processes—labeled 1, 2, and 3 —shown in the PV diagram final volume. One process is adiabatic, one is isobaric, and above. Each process has the same initial state and the same one is isothermal. Which of the following correctly identifies final volume. One process is adiabatic, one is isobaric, and the three processes? one is isothermal. Which of the following correctly identifies the threeAdiabatic processes? Isobaric Isothermal (A) 1 2 3 Adiabatic Isobaric Isothermal (B) 2 1 3 (A) 1 2 3 (C) 2 3 1 (B) 2 1 3 (D) 3 1 2 (C) 2 3 1 (E) 3 2 1 (D) 3 1 2 (E) 3 2 1 GO ON TO THE NEXT PAGE. -16- 22 GO ON TO THE NEXT PAGE. -16- GREPracticeBook 29. For an ideal gas, consider the three thermodynamic processes—labeled 1, 2, and 3 —shown in the PV diagram 29. For an ideal gas, consider the three thermodynamic above. Each process has the same initial state and the same processes—labeled 1, 2, and 3 —shown in the PV diagram final volume. One process is adiabatic, one is isobaric, and above. Each process has the same initial state and the same one is isothermal. Which of the following correctly identifies final volume.