Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121, April 29, 2008.The Second Law of Thermodynamics.

http://www.horizons.uc.edu/MasterJuly1998/oncampus.htm

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121.April 29, 2008.

• Course Information

• Topics to be discussed today:

• The Second Law of Thermodynamics

• Applications of the Second Law - the Carnot Engine

• Entropy

• The End!

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121.April 29, 2008.

• Homework set # 10 is now available and is due onWednesday evening, April 30, at 11.30 pm.

• Exam # 3 will be returned in workshops.

• Any complaints about Exam # 3 must be addressed by me;please write down why you feel the exam was not gradedproperly and hand it to me before next Monday, May 5. Iwill do the re-grading on Tuesday morning, May 6.

• I have distributed information about the score you need toobtain on the final exam to get a C-, a B-, and an A- in thiscourse.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121.Grade Requirements.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121.Final Exam - Details.

• The final exam will be held on Thursday May 8 between 4pm and 7 pm in Hubbell. The final exam will cover all thematerial discussed in the course; there will be NO particularfocus on thermodynamics. NOTE: no error analysis.

• Extra office hours of the TAs and me will be announced viaemail later this week.

• The formula sheet to be attached to the final exam will bedistributed via email by the end of the weekend, and willalso be available on the WEB.

• Please use all the resources at your disposal to prepare forthe exam (e.g. the detailed solutions of previous exams, thehomework assignments, the study guide, etc.).

• The final exam grades will be distributed via email onMonday May 12.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second law of thermodynamics.

• There are several different formsof the second law ofthermodynamics:

• It is not possible to completelychange heat into work with noother change taking place.

• Heat flows naturally from a hotobject to a cold object; heat willnot flow spontaneously from acold object to a hot object.

• Many naturally processes do notviolate conservation of energywhen executed in reverse, butwould violate the second law.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second law of thermodynamics.Heat engines.

• Most engines rely on atemperature difference to operate.

• Let’s understand why:

• The steam pushes the piston to theright and does work on the piston:

Win = nRTin(1-Vin/Vout)

• To remove the steam, the pistonhas to do work on the steam:

Wout = nRTout(1-Vout/Vin)

• If Tin = Tout: Win + Wout = 0 (nonet work is done).

• In order to do work Tin > Tout andwe must thus cool the steambefore compression starts.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second law of thermodynamics.Heat engines.

• The efficiency of an engine is defined asthe ratio of the heat extracted from the hotreservoir and the work done:

Efficiency = | W | / | QH |

• The work done and the heat extracted areusually measured per engine cycle.

• Because of the second law, no engine canhave a 100% efficiency!

• Note: the cost of operation does not onlydepend on the cost of maintaining thehigh temperature reservoir, but may alsoinclude the cost of maintaining the coldtemperature reservoir.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second law of thermodynamics.Heat pumps.

• In many cases (heat engines), theconversion of flow of heat towork is the primary purpose ofthe engine (e.g. the car engine).

• In many other applications (heatpumps), work is converted to aflow of heat (e.g. airconditioning).

• The performance of a heat pumpis usually specified by providingthe coefficient of performance K:

K = | QL | / | W |

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second law of thermodynamics.Heat pumps.

Note: you can not cool your house by opening the door ofyour refrigerator!

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second law of thermodynamics.Heat pumps.

Note: You usually pay for the work done but not for the heat extracted from the outside. You can thus get more energythan what you pay for!

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second law of thermodynamics.Heat pumps.

• Heat pumps:

• The heat capacity increases withincreasing outside temperature.

• Additional heaters may berequired in colder climates.

• The heat capacity can also beincreased by changing the sourceof heat from the air to the ground.

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Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second law of thermodynamics.Heat pumps.

• Heat pumps:

• Heaters in the winter: take heatfrom the outside to the inside.

• Air conditioners in the summer:take heat from the inside to theoutside.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

A “perfect” engineused to determine the limits on efficiency.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

• Step 1: a to b.

• The gas is in contact with a heatbath at temperature TH and weightis removed from the piston.

• The gas expands, whilemaintaining a constanttemperature (the change in theinternal energy is thus equal to 0J).

• Using the first law ofthermodynamics we see that

Q

H= W

H= n R T

Hln

Vb

Va

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

• Step 2: b to c.

• The gas is isolated from theenvironment and some moreweight is removed from thepiston.

• The gas expands and during theadiabatic expansion, thetemperature of the gas willdecrease.

• For adiabatic expansion pVγ isconstant, and we can thus relatedstate b to state c:

THVbγ-1 = TCVc

γ-1

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

• Step 3: c to d.

• The gas is in contact with a heatbath at temperature TC and weightis added to the piston.

• The gas is compressed, whilemaintaining a constanttemperature (the change in theinternal energy is thus equal to 0J).

• Using the first law ofthermodynamics we see that

Q

c= W

c= n R T

Cln

Vc

Vd

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

• Step 4: d to a.

• The gas is isolated from theenvironment and some moreweight is added to the piston.

• The gas is compressed and duringthe adiabatic compression, thetemperature of the gas willincrease.

• For adiabatic expansion pVγ isconstant, and we can thus relatedstate b to state c:

THVaγ-1 = TCVd

γ-1

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

• The adiabatic expansion andcompression steps can be used toshow that Vb/Va = Vc/Vd.

• This relation between thevolumes is very useful since itallows us to determine the ratioof the heat flows:

QH

QC

=

TH

lnV

b

Va

TC

lnV

c

Vd

=T

H

TC

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

• The efficiency of the Carnotcycle can now be determined.

• Note that the work done by theCarnot engine is the differencebetween the heat extracted fromthe hot reservoir and the heatdumped in the cold reservoir:

e =Q

H- Q

C

QH

= 1 -T

C

TH

=T

H-T

C

TH

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

• If we look at the efficiency of the Carnot cycle:

you see that the efficiency improves when the temperaturedifference between the hot and the cold bath increases. Thisis why it sometimes pays to increase the cooling of yourengine!

• Carnot’s theorem tells us that no real engine can have anefficiency more than that of the Carnot engine.

e =Q

H- Q

C

QH

= 1 -T

C

TH

=T

H-T

C

TH

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Second Law of Thermodynamics.The Carnot Cycle.

• The Carnot cycle is an exampleof a reversible process, which is aprocess that can be done inreverse.

• A reversible process requires thatany changes are made infinitelyslowly.

• Real processes are not reversibledue to for example friction ,turbulence in the gas, etc.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Entropy.

• Our study of the Carnot cycle showed thatQH/TH + QL/TL = 0.

• Since any reversible cycle can beapproximated by a series of Carnot cycles,we expect that the integral of dQ/T along theclosed path of the cycle is 0 J.

• One consequence is that the integral of dQ/Tbetween a and b is independent of the path.

• The quantity dQ/T is called the entropy dS.• The entropy difference between a and b is

thus path independent, and entropy is a statevariable.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Entropy.

• Entropy is a state variable, but Q and W arenot state variables since they depend on thepath used to get from a to b.

• For a reversible process, the change in theentropy of the engine is opposite to thechange in entropy of the environment thatprovides the heat required (or absorbs theheat generated). The net change in thetotal entropy is thus 0 J/K.

• For an irreversible process, the net changein the total entropy will be larger than 0J/K.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Entropy and the Second Law ofThermodynamics.

• We can express the second law ofthermodynamic can be expressedin terms of entropy:

• The entropy of an isolated systemwill never decrease. It either staysthe same (reversible process) orincreases (irreversible process).

• Natural processes tend to movetoward a state of greater disorder.

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121.Do you violate the second law?

• During the past 4 months, your brain hopefully has absorbedmuch of what I have covered, and concepts associated withmechanics should be in a much more ordered state in yourbrain on May 8 compared to their order on January 17.

• Do you violate the second law by going from disorder toorder?

• Not if you include the disorder you dumped into yourenvironment due to sweating over the exams and homeworkassignments. If you include that disorder, this course has aresulted in a greater disorder in our Universe (since clearlythe impact of Physics 121 is irreversible).

Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester

Physics 121. The End.You are free (and so am I)!