Thermodynamics the Goal of Thermodynamics Is to Understand How Heat Can Be Converted to Work

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

P Gas is pushed in, work is done ON the gas dW = P dV Work done is negative of Area under curve

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

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

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

(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

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(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?

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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

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GREPracticeBook

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

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GREPracticeBook

(E) 3 2 1

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5. A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is 1 (A) kT 2

(B) kT 3 7. Two long, identical bar magnets are placed under (C) kT 2 a horizontal piece of paper, as shown in the figure above. The paper is covered with iron filings. (D) 3kT When the two north poles are a small distance apart and touching the paper, the iron filings (E) 6kT move into a pattern that shows the magnetic field lines. Which of the following best illustrates the 6. An ideal monatomic gas expands quasi-statically pattern thatGRE0177 results? to twice its volume. If the process is isothermal, (A) the work done by the gas is Wi . If the process is adiabatic, the work done by the gas is Wa . Which of the following is true? (A) W = W i a (B) (B) 0 = Wi < Wa (C) 0 < Wi < Wa (D) 0 = Wa < Wi (E) 0 < W < W a i (C)

(D)

(E)

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5. A three-dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is 1 (A) kT 2

(D)

(E) V

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3. Two simple pendulums A and B consist 6. The electric field of a plane electromagnetic wave

3. Twoof identical simple massespendulums suspended A and from B consis stringst 6. ofThe wave electric number field kof anda plane angular electromagnetic frequency w wave is of identicallength L Amasse and sL suspendedB , respectively. from strings The two ofgiven wave by number E = E0 (ke x and+ ey angular) sin(kz frequency wt). Which w of is pendulums oscillate in equal gravitational fields. the following gives the direction of the associated of length LA and LB , respectively. The two given by E = E0(ex + ey) sin(kz wt). Which of If the period of pendulum B is twice the period pendulums oscillate in equal gravitational fields. magneticthe following field gives B ? the direction of the associated Ifof thependulum period ofA ,pendulum which of theB is following twice the is period true magnetic(A) ez field B ? of the lengths of the two pendulums? of pendulum A, which of the following is true (B) e + e (A) ez x y of the lengths1 of the two pendulums? (A) L = L (B)(C) ex + ey B 4 A x y 1 (D) e e (A) L = L (C) xex zey B 14 A (B) L = L (D)(E) ey ez B 2 A x z 1 (E) e e (B) LB = LA y z 2 (C) LB = LA 7. Which of the following is true about any system

that undergoes a reversible thermodynamic (C) LB = LA 7. Which of the following is true about any system process? (D) LB = 2LA that undergoes a reversible thermodynamic (D) L = 2L (A)process? There are no changes in the internal energy B A of the system. (E) L = 4L (A) There are no changes in the internal energy B A (B) The temperature of the system remains of the system. (E) L = 4L constant during the process. B A (B) The temperature of the system remains (C) The entropy of the system and its constant during the process. environment remains unchanged. (C) The entropy of the system and its (D) The entropy of the system and its environment remains unchanged. environment must increase. (D) The entropy of the system and its (E) The net work done by the system is zero. environment must increase.

(E) The net work done by the system is zero. 8. For which of the following thermodynamic GREPracticeBook processes is the increase in the internal energy 8. For which of the following thermodynamic 4. For the circuit shown in the figure above, what is of an ideal gas equal to the heat added to the gas? the current i through the 2 W resistor? processes is the increase in the internal energy 4. For the circuit shown in the figure above, what is (A)of an Constant ideal gas temperature equal to the heat added to the gas? (A)the current 2 A i through the 2 W resistor? (B) Constant volume (B) 4 A (A) Constant temperature (A) 2 A (C) Constant pressure (C) 5 A (B) Constant volume (B) 4 A (D) Adiabatic (D) 10 A (C) Constant pressure (C) 5 A (E) Cyclic (E) 20 A (D) Adiabatic (D) 10 A (E) Cyclic (E) 20 A 5. By definition, the electric displacement current through a surface S is proportional to the 5. By definition, the electric displacement current through(A) magnetic a surface flux Sthrough is proportional S to the (B) rate of change of the magnetic flux through S (A) magnetic flux through S (C) time integral of the magnetic flux through S (B) rate of change of the magnetic flux through S (D) electric flux through S (C) time integral of the magnetic flux through S (E) rate of change of the electric flux through S (D) electric flux through S

S (E) rate of change of the electric flux through

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3. Two simple pendulums A and B consist 6. The electric field of a plane electromagnetic wave

(E) The net work done by the system is zero. 8. For which of the following thermodynamic GREPracticeBook processes is the increase in the internal energy 8. For which of the following thermodynamic 4. For the circuit shown in the figure above, what is of an ideal gas equal to the heat added to the gas? the current i through the 2 W resistor? processes is the increase in the internal energy 4. For the circuit shown in the figure above, what is (A)of an Constant ideal gas temperature equal to the heat added to the gas? (A)the current 2 A i through the 2 W resistor? (B) Constant volume (B) 4 A (A) Constant temperature (A) 2 A (C) Constant pressure (C) 5 A (B) Constant volume (B) 4 A (D) Adiabatic (D) 10 A (C) Constant pressure (C) 5 A (E) Cyclic (E) 20 A (D) Adiabatic (D) 10 A (E) Cyclic (E) 20 A 5. By definition, the electric displacement current through a surface S is proportional to the 5. By definition, the electric displacement current through(A) magnetic a surface flux Sthrough is proportional S to the (B) rate of change of the magnetic flux through S Increase in internal energy equals heat added (A) magnetic flux through S (C) time integral of the magnetic flux through S minus Work done by gas (B) rate of change of the magnetic flux through S (D) electric flux through S (C) time integral of the magnetic flux through S (E) rate of change of the electric flux through S (D) electric flux through S

S (E) rate of change of the electric flux through

GO ON TO THE NEXT PAGE. -6- 12 GO ON TO THE NEXT PAGE. -6- Entropy P Volume in increases dW = P dV Temperature decreases (atoms slower)

Pressure decreases (atoms slower and also more distance to travel) V

Does anything stay the SAME?

Why do the atoms slow down? v Stationary wall v

So atoms slow down when we expand the box .... (A) V (B) v v V

In frame of wall, Lab frame velocity is lower

V (C) (D) v V v 2V

In frame of wall, In lab frame, is lower reflects with speed of reflected atom same speed is even lower 1-dimension

Disorder in position x = L v Disorder in velocity v =2v L

velocity

position v L v v0

L L0

velocity old disorder region new disorder region

position

The Area of the two regions is the same Entropy Three notational modifications

(A) velocity momentum

position position

A = x p

(B) Consider Log of Area instead of Area

ln A =ln(x p)

S = k ln A = k ln(x p) Note: If there are N atoms then multiply entropy by N

S = Nkln A = Nkln(x p)

36. Consider the quasi-static adiabatic expansion GRE0177 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 (C) 4 mF (D) 40 mF (E) 400 mF

39. Which two of the following circuits are high-pass filters?

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36. Consider the quasi-static adiabatic expansion GRE0177 of an ideal gas from an initial state i to a final state f. Which of the following statements is NOT true? P (A) No heat flows into or out of the gas. dW = P dV (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 adjustableV 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 (C) 4 mF (D) 40 mF (E) 400 mF

39. Which two of the following circuits are high-pass filters?

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If entropy does not change in such processes, then how DOES it change ?

Two ways: (A) Add heat

(B) Change things quickly (irreversible process) (A) Add heat

momentum momentum

position position

Q S / (Definition of temperature Q = T S in thermodynamics)

93. An inertial reference frame S moves at constant GREPracticeBook 93. Anspeed inertial with respectreference to framea second S inertialmoves atreference constant speedframe withS. An respect observer to a in second S measures inertial thereference energy frameE, momentum S. An observer p, and in position S measures x of athe moving energy Eparticle, momentum at time pt , for and a positionparticular x event. of a moving An particleobserver at in time S measurest for a particular energy Eevent. , momentum An p , and position x at time t for the same observer in S measures energy E , momentum movingp , and particleposition at x the at same time event.t for Which the same of the movingfollowing particle is an expression at the same of event. a relativistic Which of the 91. The diagram above shows a Carnot cycle for an followinginvariant for is anthis expression event? of a relativistic 91. Theideal diagram air conditioner, above shows which a isCarnot to cool cycle a house for an on a invariant(A) xx for this event? hotideal summer air conditioner, day. The whichair conditioner is to cool absorbs a house heat on a (B)(A) xp xp athot the summer lower temperatureday. The air insideconditioner and pumps absorbs it toheat (B)(C) ttp p theat the environment lower temperature at the higher inside temperature and pumps outside.it to (C)(D) tE tE 2 2 2 2 Whichthe environment of the following at the higher gives thetemperature ratio of the outside. heat (E)(D) E Epc E p c Q absorbed in the house (i.e., between points b 2 2 2 2 Whichbc of the following gives the ratio of the heat (E) E pc Epc 94. Consider three identical, ideal capacitors. The first Qandbc absorbedc on the cycle)in the houseto the work(i.e., betweendone during points the b cycle? capacitor is charged to a voltage V and then and c on the cycle) to the work done during the 94. Consider three identical, ideal capac0 itors. The first disconnected from the battery. The other two cycle?(A) 0 capacitor is charged to a voltage V0 and then (B) 0.033 capacitors, initially uncharged and connected in (A) 0 disconnected from the battery. The other two (C) 0.97 (B) 0.033 capacitors,series, are then initially connected uncharged across and the connected first in (D) 1.0 (C) 0.97 capacitor. What is the final voltage on the first (E) 30. series, are then connected across the first (D) 1.0 capacitor.capacitor? What is the final voltage on the first (E) 30. 92. A particle in an infinite square well with walls at capacitor?V (A) 0 922 92. xA = particle 0 and inx =an L infinite has energy square E well with walls. The at V5 2 (A) 0 92mL22 V x = 0 and x = L has energy E . The 50 probability that the particle is between 2x = 0 and (B) 2mL V3 probabilityx = L/6 is that the particle is between x = 0 and (B) 0 V3 (C) 0 x(A) = L 1/36/6 is V2 (B) 1/6 (C) 0 (A) 1/36 22V (C) 1/3 (D) 0 (B) 1/6 (D) 1/2 2V3 (C) 1/3 0 (E) 1 (D) (E) V03 (D) 1/2 (E) 1 (E) V 0

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93. An inertial reference frame S moves at constant S GREPracticeBook 93. Anspeed inertial with respectreference to framea second S inertialmoves atreference constant speedframe withS. An respect observer to a in second S measures inertial thereference energy frameE, momentum S. An observer p, and in position S measures x of athe moving energy TotalEparticle, momentumwork at time done pt , for and= a Total positionparticular heat x event. of aabsorbed moving An particleobserver at in time S measurest for a particular energy Eevent. , momentum An p , and position x at time t for the same observer in S measures energy E , momentum movingp , and particleposition Qat x the =at same timeT event.tSfor Which the same of the movingfollowing particle is an expression at the same of event. a relativistic Which of the 91. The diagram above shows a Carnot cycle for an followinginvariant for is anthis expression event? of a relativistic ideal air conditioner, which is to cool a house on a invariant(A) xx for this event? 91. The diagram above shows a Carnot cycle for an Qemitted = 305S hotideal summer air conditioner, day. The whichair conditioner is to cool absorbs a house heat on a (B)(A) xp xp athot the summer lower temperatureday. The air insideconditioner and pumps absorbs it toheat (B)(C) ttp p Qabsorbed = 295S theat the environment lower temperature at the higher inside temperature and pumps outside.it to (C)(D) tE tE 2 2 2 2 Whichthe environment of the following at the higher gives thetemperature ratio of the outside. heat (E)(D) E Epc E p c Q absorbed in the house (i.e., between points b 2 2 2 2 Whichbc of the following gives the ratio of the heat (E)W E= 10pc S Epc 94. Consider three identical, ideal capacitors. The first Qandbc absorbedc on the cycle)in the houseto the work(i.e., betweendone during points the b cycle? capacitor is charged to a voltage V and then and c on the cycle) to the work done during the 94. Consider three identical, ideal capac0 itors. The first disconnected from the battery. The other two cycle?(A) 0 capacitor is charged to a voltage V0 and then (B) 0.033 Qabsorbedcapacitors, initially295 uncharged and connected in (A) 0 disconnected from the battery. The other two (C) 0.97 = 30 (B) 0.033 capacitors,series, are then initially connected uncharged across and the connected first in (D) 1.0 W 10 ⇡ (C) 0.97 capacitor. What is the final voltage on the first (E) 30. series, are then connected across the first (D) 1.0 capacitor.capacitor? What is the final voltage on the first (E) 30. 92. A particle in an infinite square well with walls at capacitor?V (A) 0 922 92. xA = particle 0 and inx =an L infinite has energy square E well with walls. The at V5 2 (A) 0 92mL22 V x = 0 and x = L has energy E . The 50 probability that the particle is between 2x = 0 and (B) 2mL V3 probabilityx = L/6 is that the particle is between x = 0 and (B) 0 V3 (C) 0 x(A) = L 1/36/6 is V2 (B) 1/6 (C) 0 (A) 1/36 22V (C) 1/3 (D) 0 (B) 1/6 (D) 1/2 2V3 (C) 1/3 0 (E) 1 (D) (E) V03 (D) 1/2 (E) 1 (E) V 0

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57. Which of the following statements is (are) true for a Maxwell-Boltzmann description of an ideal gas 57. Which of the following statements is (are) true for of atoms in equilibrium at temperature T ? a Maxwell-Boltzmann description of an ideal gas of atomsI. The inaverage equilibrium velocity at temperatureof the atoms Tis ?zero. II. The distribution of the speeds of the atoms I. The average velocity of the atoms is zero. has a maximum at u = 0. II. The distribution of the speeds of the atoms III. The probability of finding an atom with zero has a maximum at u = 0. kinetic energy is zero. III. The probability of finding an atom with zero (A) Ikinetic only energy is zero. 56. A sample of N molecules has the distribution of (B) II only (A) I only 56. Aspeeds sample shown of N in moleculesthe figure hasabove. the distributionPduu is an of (C) I and II (B) II only estimate of the number of molecules with speeds (D) I and III speeds shown in the figure above. P u du is an (C) I and II (E) II and III estimatebetween ofu theand number u + du of, and molecules this number with isspeeds (D) I and III nonzero only up to 3u , where u is constant. (E) II and III between u and u + du0 , and this number0 is 58. A monatomic ideal gas changes from an initial

Whichnonzero of only the followingup to 3u ,gives where the uvalue is constant. of a ? GREPracticeBook 0 0 58. Astate monatomic (Pi , Vi , idealTi, ngasi) tochanges a final fromstate an (P finitial , Vf , Which of theN following gives the value of a ? Tf , nf ), where Pi < Pf , Vi = Vf , Ti < Tf and (A) a state (Pi , Vi , Ti, ni) to a final state (Pf , Vf , 3u0 N nTi ,= nnf ),. Which where ofP the < Pfollowing, V = V gives, T < the T change and (A) a N f f i f i f i f (B) a 3u 0 nin =entropy n . Which of the of gas? the following gives the change 2u i f N0 in entropy3 of theTf gas? (B) a N (A) nR ln (C) a 2 u0 2 T u 3 Tif N0 (A) nR ln (C) a 2 T u3 N 3 Ti (D) a 0 (B) nR ln 2u 2 Tf 3 N0 3 Ti (D) a (B) nR ln 2 u 2 Tf (E) aN 0 5 f (C) nR ln 2 TTi (E) a N 5 f (C) nR ln 25 Ti (D) nR ln 2 Tf 5 Ti (D) nR ln 2 T (E) 0 f

(E) 0

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Whichnonzero of only the followingup to 3u ,gives where the uvalue is constant. of a ? GREPracticeBook 0 0 58. Astate monatomic (Pi , Vi , idealTi, ngasi) tochanges a final fromstate an (P finitial , Vf , Which of theN following gives the value of a ? Tf , nf ), where Pi < Pf , Vi = Vf , Ti < Tf and (A) a state (Pi , Vi , Ti, ni) to a final state (Pf , Vf , 3u0 N nTi ,= nnf ),. Which where ofP the < Pfollowing, V = V gives, T < the T change and (A) a N f f i f i f i f (B) a 3u 0 nin =entropy n . Which of the of gas? the following gives the change 2u i f N0 in entropy3 of theTf gas? (B) a N (A) nR ln Q = T S (C) a 2 u0 2 T u 3 Tif N0 (A) nR ln (C) a 2 T u3 N 3 Ti (D) a 0 (B) nR ln Q 2u 2 Tf S = 3 N0 3 Ti (D) a (B) nR ln T 2 u 2 Tf (E) aN 0 5 f (C) nR ln 2 TTi Q No work Q = U (E) a N 5 f (C) nR ln 25 Ti (D) nR ln 3 3 2 Tf 5 Ti U = NkT = nRT (D) nR ln 2 2 2 Tf (E) 0 3 U = nRT (E) 0 2

3 Tf dT 3 T S S = nR S = nR f i 2 T 2 T ZTi 3 T = nR ln f 2 Ti

GO ON TO THE NEXT PAGE. -40- 46 GO ON TO THE NEXT PAGE. -40- Irreversibility How do we increase entropy ?

Two ways: (A) Add heat

(B) Change things quickly (irreversible process)

What is we move the piston at half the speed?

In each bounce, half as much slowdown V v 2V But there will be twice as many bounces ... So total slowdown will be the same ...

So we don't have to specify the speed of the piston ... But what happens if we move the piston VERY fast?

velocities remain same Atoms don't touch the piston, so they don't slow down !

momentum momentum

position position

Entropy increases !

GRE0177 47. A sealed and thermally insulated container of total 50. At 20∞C, a pipe open at both ends resonates at a volume V is divided into two equal volumes by frequency of 440 hertz. At what frequency does an impermeable wall. The left half of the con- the same pipe resonate on a particularly cold day tainer is initially occupied by n moles of an ideal when the speed of sound is 3 percent lower than gas at temperature T. Which of the following it would be at 20∞C ? gives the change in entropy of the system when the wall is suddenly removed and the gas expands (A) 414 Hz to fill the entire volume? (B) 427 Hz (C) 433 Hz (A) 2nR ln2 (D) 440 Hz (B) nR ln2 (E) 453 Hz 1 (C) nR ln2 2 51. Unpolarized light of intensity I0 is incident on a (D) -nR ln2 series of three polarizing filters. The axis of the second filter is oriented at 45° to that of the first (E) -2nR ln2 filter, while the axis of the third filter is oriented

48. A gaseous mixture of O (molecular mass 32 u) at 90° to that of the first filter. What is the intensity 2 of the light transmitted through the third filter? and N2 (molecular mass 28 u) is maintained at (A) 0 urms ()N2 constant temperature. What is the ratio (B) I0/8 urms()O2 of the root-mean-square speeds of the molecules? (C) I0/4 7 (D) I0/2 (A) 8 (E) I / 2 0 7 (B) 8

8 (C) 7

8 2 (D) F I 7 H K 8 (E) lnF I H 7K

49. In a Maxwell-Boltzmann system with two states of energies ⑀ and 2⑀, respectively, and a degeneracy of 2 for each state, the partition function is 52. The conventional unit cell of a body-centered cubic Bravais lattice is shown in the figure above. (A) e-⑀ /kT The conventional cell has volume a3 . What is (B) 2e-2⑀ /kT the volume of the primitive unit cell? -3⑀ /kT (C) 2e (A) a3/8 (D) e-⑀ /kT + e-2⑀ /kT (B) a3/4 -⑀ /kT -2⑀ /kT (C) a3/2 (E) 2[e + e ] (D) a3 (E) 2a3

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47. A sealed and thermally insulated container of total 50. At 20∞C, GRE0177 a pipe open at both ends resonates at a volume V is divided into two equal volumes by frequency of 440 hertz. At what frequency does an impermeable wall. The left half of the con- the same pipe resonate on a particularly cold day tainer is initially occupied by n moles of an ideal when the speed of sound is 3 percent lower than gas at temperature T. Which of the following it would be at 20∞C ? gives the change in entropy of the system when the wall is suddenly removed and the gas expands (A) 414 Hz to fill the entire volume? (B) 427 Hz (C) 433 Hz (A) 2nR ln2 L 2L v v (D) 440 Hz (B) nR ln2 ! ! (E) 453 Hz 1 (C) nR ln2 momentum momentum 2 51. Unpolarized light of intensity I0 is incident on a (D) -nR ln2 series of three polarizing filters. The axis of the second filter is oriented at 45° to that of the first (E) -2nR ln2 filter, while the axis of the third filter is oriented

position at 90° to that of theposition first filter. What is the intensity 48. A gaseous mixture of O2 (molecular mass 32 u) A 2A of the light transmitted through the third filter? and! N2 (molecular mass 28 u) is maintained at (A) 0 S = k ln A k ln(2A)=k ln A + kulnrms 2()N2 constant temperature. What is the ratio (B) I0/8 ! urms()O2 (C) I0/4 ofS thetotal root-mean-square= Nkln 2 = nR speedsln 2 of the molecules? 7 (D) I0/2 (A) 8 (E) I / 2 0 7 (B) 8

8 (C) 7

8 2 (D) F I 7 H K 8 (E) lnF I H 7K

Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. 38 Irreversibility

velocities remain same

Can we reverse this process ? NO !

If we try to move piston back, it WILL hit the atoms and speed them up ...

Slow process are called QUASISTATIC. They are REVERSIBLE. There is NO net increase in entropy

Sudden changes are IRREVERSIBLE. There IS a net increase in entropy (disorder)

Entropy can increase but never decrease

3. Two simple pendulums A and B consist 6. The electric field of a plane electromagnetic wave

that undergoes a reversible thermodynamic (C) LB = LA 7. Which of the following is true about any system GREPracticeBook process? (D) LB = 2LA that undergoes a reversible thermodynamic (D) L = 2L (A)process? There are no changes in the internal energy B A of the system. (E) L = 4L (A) There are no changes in the internal energy B A (B) The temperature of the system remains of the system. (E) L = 4L constant during the process. B A (B) The temperature of the system remains (C) The entropy of the system and its constant during the process. environment remains unchanged. (C) The entropy of the system and its (D) The entropy of the system and its environment remains unchanged. environment must increase. (D) The entropy of the system and its (E) The net work done by the system is zero. environment must increase.

(E) The net work done by the system is zero. 8. For which of the following thermodynamic

processes is the increase in the internal energy 8. For which of the following thermodynamic 4. For the circuit shown in the figure above, what is of an ideal gas equal to the heat added to the gas? the current i through the 2 W resistor? processes is the increase in the internal energy 4. For the circuit shown in the figure above, what is (A)of an Constant ideal gas temperature equal to the heat added to the gas? (A)the current 2 A i through the 2 W resistor? (B) Constant volume (B) 4 A (A) Constant temperature (A) 2 A (C) Constant pressure (C) 5 A (B) Constant volume (B) 4 A (D) Adiabatic (D) 10 A (C) Constant pressure (C) 5 A (E) Cyclic (E) 20 A (D) Adiabatic (D) 10 A (E) Cyclic (E) 20 A 5. By definition, the electric displacement current through a surface S is proportional to the 5. By definition, the electric displacement current through(A) magnetic a surface flux Sthrough is proportional S to the (B) rate of change of the magnetic flux through S (A) magnetic flux through S (C) time integral of the magnetic flux through S (B) rate of change of the magnetic flux through S (D) electric flux through S (C) time integral of the magnetic flux through S (E) rate of change of the electric flux through S (D) electric flux through S

S (E) rate of change of the electric flux through

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3. Two simple pendulums A and B consist 6. The electric field of a plane electromagnetic wave

(E) The net work done by the system is zero. 8. For which of the following thermodynamic

S (E) rate of change of the electric flux through

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