Chapter 22

Chapter 22Chapter 22

Heat Engines, Entropy and the Heat Engines, Entropy and the

Second Law of ThermodynamicsSecond Law of Thermodynamics

First Law of Thermodynamics – First Law of Thermodynamics – ReviewReview

• The first law states that a change in internal The first law states that a change in internal energy in a system can occur as a result of energy in a system can occur as a result of energy transfer by heat, by work, or by both.energy transfer by heat, by work, or by both.

• The law makes no distinction between the The law makes no distinction between the results of heat and the results of work.results of heat and the results of work.

First Law of Thermodynamics – First Law of Thermodynamics – ReviewReview

• In according with first law of thermodynamics the In according with first law of thermodynamics the energy is always conservedenergy is always conserved

• What does it mean to conserve energy if the What does it mean to conserve energy if the total amount of energy in the universe does not total amount of energy in the universe does not change regardless of what we do?change regardless of what we do?

• The first law of thermodynamic does not tell the The first law of thermodynamic does not tell the whole story: whole story: Energy is always conserved, but Energy is always conserved, but some forms of energy are more useful than some forms of energy are more useful than others.others.

First Law – Missing PiecesFirst Law – Missing Pieces

• There is an important distinction between There is an important distinction between heatheat and and workwork that is not evident from the that is not evident from the first lawfirst law

• The first law makes no distinction between The first law makes no distinction between processes that occur spontaneously and processes that occur spontaneously and those that do notthose that do not– An example is that it is impossible to design a An example is that it is impossible to design a

device that takes in energy and converts it all device that takes in energy and converts it all to workto work

The Second Law of ThermodynamicsThe Second Law of Thermodynamics

• The possibility or impossibility of putting energy to The possibility or impossibility of putting energy to use is the subject of the use is the subject of the Second Law of Second Law of Thermodynamics.Thermodynamics.

• For example, it is easy to convert work into thermal For example, it is easy to convert work into thermal energy, but it is impossible to remove energy as energy, but it is impossible to remove energy as heat from a single reservoir and convert it internally heat from a single reservoir and convert it internally into work with no other changes.into work with no other changes.

• This experimental fact is one statement of the This experimental fact is one statement of the

second law of thermodynamics.second law of thermodynamics.


• Kelvin’s statementKelvin’s statement: : No system can take energy No system can take energy as heat from a single reservoir and convert it as heat from a single reservoir and convert it entirely into work without additional net entirely into work without additional net changes in the system or its surroundings.changes in the system or its surroundings.

• Clausus statementClausus statement:: A process whose only net A process whose only net result is to transfer energy as heat from a result is to transfer energy as heat from a cooler object to a hotter one is impossible. cooler object to a hotter one is impossible.


• A common example of conversion of work into heat A common example of conversion of work into heat is movement with friction.is movement with friction.

• Suppose you spend two minutes pushing a block Suppose you spend two minutes pushing a block along a tabletop in a closed path, leaving the block along a tabletop in a closed path, leaving the block in its initial position. Also, suppose that the block-in its initial position. Also, suppose that the block-table system is initially in thermal equilibrium with table system is initially in thermal equilibrium with the surroundings. the surroundings.


• The work you do on the system is converted into The work you do on the system is converted into internal energy of the system and the block-table internal energy of the system and the block-table system becomes warmer system becomes warmer →→ the system is no the system is no longer in thermal equilibrium with its longer in thermal equilibrium with its surroundings surroundings →→ the system will transfer energy the system will transfer energy as heat to its surroundings until it returns to the as heat to its surroundings until it returns to the thermal equilibriumthermal equilibrium


• Because the final and initial states of the system Because the final and initial states of the system are the same, the are the same, the 1-st Law1-st Law of of TDTD dictates that the dictates that the energy transferred to the environment as heat energy transferred to the environment as heat equals to the work done on the system. equals to the work done on the system.

• The reverse process never occur – a block and The reverse process never occur – a block and table that are warm will never spontaneously cool table that are warm will never spontaneously cool by converting their internal energy into the work by converting their internal energy into the work that causes the block to push your hand around the that causes the block to push your hand around the

table!table!


• Such an amazing occurrence would not violate the Such an amazing occurrence would not violate the first law of thermodynamics or any other physical first law of thermodynamics or any other physical law, it does, however, violate the second law of law, it does, however, violate the second law of thermodynamics.thermodynamics.

• There is a lack of symmetry in the roles played by There is a lack of symmetry in the roles played by heat and work that is not evident from the first law. heat and work that is not evident from the first law. This lack of symmetry is related to the fact that This lack of symmetry is related to the fact that some processes are some processes are irreversibleirreversible..


• Establishes which processes do and which do Establishes which processes do and which do not occurnot occur

• Some processes can occur in either direction Some processes can occur in either direction according to the first lawaccording to the first law

• They are observed to occur only in one direction They are observed to occur only in one direction according to the second law.according to the second law.

Irreversible ProcessesIrreversible Processes

• An irreversible process is one that occurs An irreversible process is one that occurs naturally in one direction onlynaturally in one direction only

• No irreversible process has been observed to No irreversible process has been observed to run backwardsrun backwards

• An important engineering implication is the An important engineering implication is the limited efficiency of heat engineslimited efficiency of heat engines

Heat EngineHeat Engine

• A A heat engineheat engine is a device that takes in energy by is a device that takes in energy by heat and, operating in a cyclic process, expels a heat and, operating in a cyclic process, expels a fractionfraction of that energy by means of work of that energy by means of work

• A heat engine carries some working substance A heat engine carries some working substance through a cyclical processthrough a cyclical process

Heat Engine.Heat Engine.

• The working substance The working substance absorbs energy by heat from absorbs energy by heat from a high temperature energy a high temperature energy reservoir (reservoir (QQhh))

• Work is done by the engine Work is done by the engine ((WWengeng))

• Energy is expelled as heat to Energy is expelled as heat to a lower temperature reservoir a lower temperature reservoir ((QQcc))

Heat Engine.Heat Engine.• Since it is a cyclical process, Since it is a cyclical process,

ΔΔEEintint = 0 = 0– Its initial and final internal Its initial and final internal

energies are the sameenergies are the same

• Therefore, Therefore, QQnetnet = = WWengeng

• The work done by the engine The work done by the engine

equals the net energy equals the net energy absorbed by the engineabsorbed by the engine

• The work is equal to the area The work is equal to the area enclosed by the curve of the enclosed by the curve of the PVPV diagram diagram– The working substance is a The working substance is a

gasgas

Thermal Efficiency of a Heat EngineThermal Efficiency of a Heat Engine

• Thermal efficiencyThermal efficiency is defined as the ratio is defined as the ratio of the net work done by the engine during of the net work done by the engine during one cycle to the energy input at the one cycle to the energy input at the higher temperaturehigher temperature

• We can think of the efficiency as the ratio We can think of the efficiency as the ratio of what you gain to what you giveof what you gain to what you give

eng 1h c c

h h h

W Q Q Qe

Q Q Q

More About EfficiencyMore About Efficiency

• In practice, all heat engines expel only a In practice, all heat engines expel only a fraction of the input energy by mechanical fraction of the input energy by mechanical workwork

• Therefore, their efficiency is always less Therefore, their efficiency is always less than 100%than 100%– To have To have ee = 100% = 100%, energy expelled as heat to , energy expelled as heat to

a lower temperature reservoir, a lower temperature reservoir, QQCC,, must be must be 00

Perfect Heat EnginePerfect Heat Engine

• No energy is expelled to No energy is expelled to the cold reservoirthe cold reservoir

• It takes in some amount It takes in some amount of energy and does an of energy and does an equal amount of workequal amount of work

• ee = 100% = 100%

• It is an impossible It is an impossible engineengine

During each cycle a heat engine absorbs During each cycle a heat engine absorbs 200 J200 J of heat of heat from a hot reservoir, does work, and exhausts from a hot reservoir, does work, and exhausts 160 J160 J to a to a cold reservoir. What is the efficiency of the engine?cold reservoir. What is the efficiency of the engine?

During each cycle a heat engine absorbs During each cycle a heat engine absorbs 200 J200 J of heat of heat from a hot reservoir, does work, and exhausts from a hot reservoir, does work, and exhausts 160 J160 J to a to a cold reservoir. What is the efficiency of the engine?cold reservoir. What is the efficiency of the engine?

hQ

W

Qh= 200J

W = Qh – Qc = 200J – 160J = 40J

%2020.0200

40

J

J

Q

W

h

Heat Pumps and RefrigeratorsHeat Pumps and Refrigerators

• Heat engines can run in reverseHeat engines can run in reverse– This is not a natural direction of energy transferThis is not a natural direction of energy transfer

– Must put some energy into a device to do thisMust put some energy into a device to do this

– Devices that do this are called heat pumps or Devices that do this are called heat pumps or refrigeratorsrefrigerators

• ExamplesExamples– A refrigerator is a common type of heat pumpA refrigerator is a common type of heat pump

– An air conditioner is another example of a heat An air conditioner is another example of a heat pumppump

Heat Pump ProcessHeat Pump Process

• Energy is extracted from Energy is extracted from the cold reservoir, the cold reservoir, QQCC

• Energy is transferred to Energy is transferred to the hot reservoir, the hot reservoir, QQhh

• Work must be done Work must be done onon the engine, the engine, WW

Coefficient of PerformanceCoefficient of Performance

• The effectiveness of a heat pump is The effectiveness of a heat pump is described by a number called the described by a number called the coefficient of performancecoefficient of performance ( (COPCOP))

• In In heating modeheating mode, the , the COPCOP is the ratio of is the ratio of the heat transferred in to the work requiredthe heat transferred in to the work required

energy transferred at high tempCOP =

work done by heat pumphQ

W

COP, Heating ModeCOP, Heating Mode

• COPCOP is similar to efficiency is similar to efficiency

• QQhh is typically higher than is typically higher than WW

– Values of Values of COPCOP are generally greater than are generally greater than 11

– It is possible for them to be less than It is possible for them to be less than 11

• We would like the We would like the COPCOP to be as high as to be as high as possiblepossible

COP, Cooling ModeCOP, Cooling Mode

• In In cooling modecooling mode, you “gain” energy from , you “gain” energy from a cold temperature reservoira cold temperature reservoir

• A good refrigerator should have a high A good refrigerator should have a high COPCOP– Typical values are Typical values are 55 or or 66

COP cQ

W

A refrigerator has a A refrigerator has a coefficient of performancecoefficient of performance equal to equal to 5.005.00. The refrigerator takes in . The refrigerator takes in 120 J120 J of of energy from a cold reservoir in each cycle. Find (a) energy from a cold reservoir in each cycle. Find (a) the work required in each cycle and (b) the energy the work required in each cycle and (b) the energy expelled to the hot reservoir.expelled to the hot reservoir.

A refrigerator has a A refrigerator has a coefficient of performancecoefficient of performance equal to equal to 5.005.00. The refrigerator takes in . The refrigerator takes in 120 J120 J of of energy from a cold reservoir in each cycle. Find (a) energy from a cold reservoir in each cycle. Find (a) the work required in each cycle and (b) the energy the work required in each cycle and (b) the energy expelled to the hot reservoir.expelled to the hot reservoir.

COP refrigerator cQW

(a)(a) 120 JcQ COP 5.00 24.0 JW

(b) Heat expelled = Heat removed + Work done.

120 J 24 J 144 Jh cQ Q W

Carnot EngineCarnot Engine

• A theoretical engine developed by Sadi A theoretical engine developed by Sadi CarnotCarnot

• A heat engine operating in an ideal, A heat engine operating in an ideal, reversible cycle (now called a reversible cycle (now called a Carnot Carnot cyclecycle) between two reservoirs is the most ) between two reservoirs is the most efficient engine possibleefficient engine possible– This sets an upper limit on the efficiencies of This sets an upper limit on the efficiencies of

all other engines all other engines

Carnot’s TheoremCarnot’s Theorem

• No real heat engine operating between No real heat engine operating between two energy reservoirs can be more two energy reservoirs can be more efficient than a efficient than a Carnot engineCarnot engine operating operating between the same two reservoirsbetween the same two reservoirs– All real engines are less efficient than a All real engines are less efficient than a

Carnot engineCarnot engine because they do not operate because they do not operate through a reversible cyclethrough a reversible cycle

– The efficiency of a real engine is further The efficiency of a real engine is further reduced by friction, energy losses through reduced by friction, energy losses through conduction, etc.conduction, etc.

Carnot CycleCarnot Cycle

Overview of Overview of the the processes in processes in a Carnot a Carnot cyclecycle

Carnot Cycle, Carnot Cycle, AA to to BB• AA →→ BB is an isothermal expansion is an isothermal expansion

• The gas is placed in contact with The gas is placed in contact with the high temperature reservoir, the high temperature reservoir, TThh

• The gas absorbs heat The gas absorbs heat ||QQhh||

• The gas does work The gas does work WWABAB in raising in raising

the pistonthe piston

Carnot Cycle, Carnot Cycle, BB to to CC• BB →→ CC is an adiabatic expansion is an adiabatic expansion

• The base of the cylinder is The base of the cylinder is replaced by a thermally insulated replaced by a thermally insulated wallwall

• No heat enters or leaves the No heat enters or leaves the systemsystem

• The temperature falls from The temperature falls from TThh to to

TTcc . . The gas does work The gas does work WWBCBC

Carnot Cycle, Carnot Cycle, CC to to DD• The gas is placed in contact with The gas is placed in contact with

the cold temperature reservoirthe cold temperature reservoir

• CC →→ DD is an isothermal is an isothermal compressioncompression

• The gas expels energy The gas expels energy QQcc

• Work Work WWCDCD is done on the gas is done on the gas

Carnot Cycle, Carnot Cycle, DD to to AA• D D → → AA is an adiabatic is an adiabatic

compressioncompression

• The gas is again placed against The gas is again placed against a thermally nonconducting walla thermally nonconducting wall– So no heat is exchanged with the So no heat is exchanged with the

surroundingssurroundings

• The temperature of the gas The temperature of the gas increases from increases from TTcc to to TThh

• The work done on the gas is The work done on the gas is WWDADA

Carnot Cycle, Carnot Cycle, PVPV Diagram Diagram

The work done by the The work done by the engine is shown by engine is shown by the area enclosed by the area enclosed by the curve, the curve, WWengeng

The net work is equal The net work is equal to to ||QQhh| – || – |QQcc||

EEintint = 0 = 0 for the entire for the entire

cyclecycle

Efficiency of a Carnot EngineEfficiency of a Carnot Engine

• Carnot showed that the efficiency of the engine Carnot showed that the efficiency of the engine depends on the temperatures of the reservoirsdepends on the temperatures of the reservoirs

• Temperatures must be in KelvinsTemperatures must be in Kelvins

• All Carnot engines operating between the same All Carnot engines operating between the same two temperatures will have the same efficiencytwo temperatures will have the same efficiency

h

CC

h

C

h

C

T

Teand

T

T

Q

Q 1

Notes About Carnot EfficiencyNotes About Carnot Efficiency

• Efficiency is Efficiency is 00 if if TThh = = TTcc

• Efficiency is Efficiency is 100%100% only if only if TTcc = 0 K = 0 K– Such reservoirs are not availableSuch reservoirs are not available– Efficiency is always Efficiency is always less than 100%less than 100%

• The efficiency increases as The efficiency increases as TTcc is lowered is lowered and as and as TThh is raisedis raised

• In most practical cases, In most practical cases, TTcc is near room is near room temperature, temperature, 300 K300 K– So generally So generally TThh is raised to increase efficiencyis raised to increase efficiency

Carnot Heat Pump COPsCarnot Heat Pump COPs

• In heating mode:In heating mode:

• In cooling mode:In cooling mode:

ch

hh

TT

T

W

QCOP

ch

cc

TT

T

W

QCOP

A Carnot heat engine uses a steam boiler at A Carnot heat engine uses a steam boiler at 100100CC as the as the high-temperature reservoir. The low-temperature reservoir high-temperature reservoir. The low-temperature reservoir is the outside environment at is the outside environment at 20.020.0CC. Energy is exhausted . Energy is exhausted to the low-temperature reservoir at the rate of to the low-temperature reservoir at the rate of 15.4 W15.4 W. (a) . (a) Determine the useful power output of the heat engine. (b) Determine the useful power output of the heat engine. (b) How much steam will it cause to condense in the high-How much steam will it cause to condense in the high-temperature reservoir in temperature reservoir in 1.00 h1.00 h??

A Carnot heat engine uses a steam boiler at A Carnot heat engine uses a steam boiler at 100100CC as the as the high-temperature reservoir. The low-temperature reservoir high-temperature reservoir. The low-temperature reservoir is the outside environment at is the outside environment at 20.020.0CC. Energy is exhausted . Energy is exhausted to the low-temperature reservoir at the rate of to the low-temperature reservoir at the rate of 15.4 W15.4 W. (a) . (a) Determine the useful power output of the heat engine. (b) Determine the useful power output of the heat engine. (b) How much steam will it cause to condense in the high-How much steam will it cause to condense in the high-temperature reservoir in temperature reservoir in 1.00 h1.00 h??

The efficiency is: 1 1 ccc

h h

QTe

T Q

c

h

Qc t

Qh

t

TT

273 100 K

15.4 W 19.6 W273 20 K

h c h

c

Q Q Tt t T

A Carnot heat engine uses a steam boiler at A Carnot heat engine uses a steam boiler at 100100CC as the high- as the high-temperature reservoir. The low-temperature reservoir is the temperature reservoir. The low-temperature reservoir is the outside environment at outside environment at 20.020.0CC. Energy is exhausted to the low-. Energy is exhausted to the low-temperature reservoir at the rate of temperature reservoir at the rate of 15.4 W15.4 W. (a) Determine the . (a) Determine the useful power output of the heat engine. (b) How much steam useful power output of the heat engine. (b) How much steam will it cause to condense in the high-temperature reservoir in will it cause to condense in the high-temperature reservoir in 1.00 h1.00 h??

(a)(a) engh cQ W Q The useful power output is:

eng19.6 W 15.4 W 4.20 Wh cW Q Q

t t t

(b)(b) hh V

QQ t mL

t

26

3600 s19.6 J s 3.12 10 kg

2.26 10 J kgh

V

Q tm

t L

Gasoline EngineGasoline Engine

• In a gasoline engine, six processes occur In a gasoline engine, six processes occur during each cycleduring each cycle

• For a given cycle, the piston moves up For a given cycle, the piston moves up and down twiceand down twice

• This represents a four-stroke cycleThis represents a four-stroke cycle

• The processes in the cycle can be The processes in the cycle can be approximated by the Otto cycleapproximated by the Otto cycle

Otto CycleOtto Cycle

• The The PVPV diagram of an diagram of an Otto cycle is shown at Otto cycle is shown at rightright

• The Otto cycle The Otto cycle approximates the approximates the processes occurring in processes occurring in an internal combustion an internal combustion engineengine

The Conventional Gasoline EngineThe Conventional Gasoline Engine

Gasoline Engine – Gasoline Engine – Intake StrokeIntake Stroke

• During the intake stroke, During the intake stroke, the piston moves the piston moves downwarddownward

• A gaseous mixture of air A gaseous mixture of air and fuel is drawn into the and fuel is drawn into the cylindercylinder

• Energy enters the system Energy enters the system as potential energy in the as potential energy in the fuelfuel

• OO →→ AA in the Otto cycle in the Otto cycle

Gasoline Engine – Gasoline Engine – Compression StrokeCompression Stroke• The piston moves upwardThe piston moves upward

• The air-fuel mixture is The air-fuel mixture is compressed adiabaticallycompressed adiabatically

• The temperature increasesThe temperature increases

• The work done on the gas The work done on the gas is positive and equal to the is positive and equal to the negative area under the negative area under the curvecurve

• AA → → BB in the Otto cycle in the Otto cycle

Gasoline Engine Gasoline Engine – Spark– Spark

• Combustion occurs when Combustion occurs when the spark plug firesthe spark plug fires

• This is not one of the strokes This is not one of the strokes of the engineof the engine

• It occurs very quickly while It occurs very quickly while the piston is at its highest the piston is at its highest positionposition

• Conversion from potential Conversion from potential energy of the fuel to internal energy of the fuel to internal energyenergy

• BB →→ CC in the Otto cycle in the Otto cycle

Gasoline Engine Gasoline Engine – Power Stroke– Power Stroke

• In the power stroke, the In the power stroke, the gas expands adiabaticallygas expands adiabatically

• This causes a temperature This causes a temperature dropdrop

• Work is done by the gas Work is done by the gas

• The work is equal to the The work is equal to the area under the curvearea under the curve

• CC →→ DD in the Otto cycle in the Otto cycle

Gasoline Engine Gasoline Engine – Valve Opens– Valve Opens

• This is process This is process DD →→ AA in the in the Otto cycleOtto cycle

• An exhaust valve opens as the An exhaust valve opens as the piston reaches its bottom piston reaches its bottom positionposition

• The pressure drops suddenlyThe pressure drops suddenly• The volume is approximately The volume is approximately

constantconstant– So no work is doneSo no work is done

• Energy begins to be expelled Energy begins to be expelled from the interior of the cylinderfrom the interior of the cylinder

Gasoline Engine – Gasoline Engine – Exhaust StrokeExhaust Stroke

• In the exhaust stroke, In the exhaust stroke, the piston moves the piston moves upward while the upward while the exhaust valve remains exhaust valve remains openopen

• Residual gases are Residual gases are expelled to the expelled to the atmosphereatmosphere

• The volume decreasesThe volume decreases

• AA →→ OO in the Otto cycle in the Otto cycle

Otto Cycle EfficiencyOtto Cycle Efficiency

• If the air-fuel mixture is assumed to be an If the air-fuel mixture is assumed to be an ideal gas, then the efficiency of the Otto ideal gas, then the efficiency of the Otto cycle is cycle is

is the ratio of the molar specific heatsis the ratio of the molar specific heats

• VV11//VV22 is called the is called the compression ratiocompression ratio

1

1 2

11e

V V

Otto Cycle EfficiencyOtto Cycle Efficiency

• Typical values:Typical values:─ Compression ratio of Compression ratio of 88

─ = 1.4= 1.4

─ ee = 56% = 56%

• Efficiencies of real engines are Efficiencies of real engines are 15%15% to to 20%20%– Mainly due to friction, energy transfer by Mainly due to friction, energy transfer by

conduction, incomplete combustion of the air-conduction, incomplete combustion of the air-fuel mixturefuel mixture

Diesel EnginesDiesel Engines• Operate on a cycle similar to the Otto cycle Operate on a cycle similar to the Otto cycle

without a spark plugwithout a spark plug• The compression ratio is much greater and so The compression ratio is much greater and so

the cylinder temperature at the end of the the cylinder temperature at the end of the compression stroke is much highercompression stroke is much higher

• Fuel is injected and the temperature is high Fuel is injected and the temperature is high enough for the mixture to ignite without the enough for the mixture to ignite without the spark plugspark plug

• Diesel engines are more efficient than gasoline Diesel engines are more efficient than gasoline engines engines

A A 1.60-L1.60-L gasoline engine with a compression ratio gasoline engine with a compression ratio of of 6.20 6.20 has a useful power output of has a useful power output of 102 hp102 hp. . Assuming the engine operates in an idealized Otto Assuming the engine operates in an idealized Otto cycle, find the energy taken in and the energy cycle, find the energy taken in and the energy exhausted each second. Assume the fuel-air exhausted each second. Assume the fuel-air mixture behaves like an ideal gas with mixture behaves like an ideal gas with γγ= 1.40= 1.40..

A A 1.60-L1.60-L gasoline engine with a compression ratio of gasoline engine with a compression ratio of 6.20 6.20 has a useful power output of has a useful power output of 102 hp102 hp. Assuming the engine . Assuming the engine operates in an idealized Otto cycle, find the energy taken in operates in an idealized Otto cycle, find the energy taken in and the energy exhausted each second. Assume the fuel-air and the energy exhausted each second. Assume the fuel-air mixture behaves like an ideal gas with mixture behaves like an ideal gas with γγ= 1.40= 1.40..

Otto 1 0.4007 5 11 2

1 1 11 1 1

6.206.20e

V V

Otto 0.518e We have assumed the fuel-air mixture to behave like We have assumed the fuel-air mixture to behave like a a diatomic gas. diatomic gas. NowNow

eng eng

h h

W W te

Q Q t

A A 1.60-L1.60-L gasoline engine with a compression ratio of gasoline engine with a compression ratio of 6.20 6.20 has a useful power output of has a useful power output of 102 hp102 hp. Assuming the engine . Assuming the engine operates in an idealized Otto cycle, find the energy taken in operates in an idealized Otto cycle, find the energy taken in and the energy exhausted each second. Assume the fuel-air and the energy exhausted each second. Assume the fuel-air mixture behaves like an ideal gas with mixture behaves like an ideal gas with γγ= 1.40= 1.40..

eng

eng

eng

3

746 W 1 hp102 hp

0.518

146 kW

746 W146 10 W 102 hp 70.8 kW

1 hp

h

h

h c

c h

c

W tQt eQtQ W Q

WQ Qt t tQ

t

Irreversibility and DisorderIrreversibility and Disorder

There are many irreversible processes that can There are many irreversible processes that can not be described by the heat-engine or not be described by the heat-engine or refrigerator statements of the second law, such refrigerator statements of the second law, such as a glass falling to the floor and breaking or a as a glass falling to the floor and breaking or a balloon popping.balloon popping.

However all irreversible processes have one However all irreversible processes have one thing in common – the system and its thing in common – the system and its surroundings moves towards a less ordered surroundings moves towards a less ordered state. state.

Suppose a box containing a gas of mass Suppose a box containing a gas of mass MM at temperature at temperature T T is is moving along a frictionless table with a velocity moving along a frictionless table with a velocity vvcmcm. The total kinetic . The total kinetic

energy of the gas has two components: that associated with the energy of the gas has two components: that associated with the movement of the center of the mass movement of the center of the mass ½Mv½Mvcmcm

22, and the energy of the , and the energy of the

motion of its molecules relative to its center of the massmotion of its molecules relative to its center of the mass

The center of the mass energy The center of the mass energy ½Mv½Mv22cmcm

is ordered is ordered

mechanical energy that could be converted entirely mechanical energy that could be converted entirely into work.into work.

For example, if a weight were attached to a moving For example, if a weight were attached to a moving box by a string passing over a pulley, this energy box by a string passing over a pulley, this energy could be used to lift the weight.could be used to lift the weight.

The relative energy of the molecules is the The relative energy of the molecules is the internal thermal energy of the gas, which is related internal thermal energy of the gas, which is related to its temperature to its temperature TT. It is random, non-ordered . It is random, non-ordered energy that can not be converted entirely into work. energy that can not be converted entirely into work.

Now, suppose that the block hits a fixed wall and stops. This inelastic Now, suppose that the block hits a fixed wall and stops. This inelastic collision is clearly an irreversible process. The ordered mechanical energy collision is clearly an irreversible process. The ordered mechanical energy of the gas is converted into random internal energy and the temperature of of the gas is converted into random internal energy and the temperature of the gas rises. The gas still has the same total energy, but now all of the the gas rises. The gas still has the same total energy, but now all of the energy is associated with the random motion of the molecules about of energy is associated with the random motion of the molecules about of center of the mass of the gas, which is now at rest. center of the mass of the gas, which is now at rest.

EntropyEntropy

Thus, the gas become less ordered (more disordered), and Thus, the gas become less ordered (more disordered), and gas lost some of its ability to do work. gas lost some of its ability to do work.

There is a thermodynamic function called There is a thermodynamic function called entropy entropy S S that is a that is a measure of the disorder of the system.measure of the disorder of the system.

Entropy Entropy SS, like pressure , like pressure PP, volume , volume VV, temperature , temperature TT, and , and internal energy internal energy EEintint, is a function of the state of the system., is a function of the state of the system.

The change in entropy The change in entropy dSdS of the system as it goes from one of the system as it goes from one state to another is defined asstate to another is defined as

where where dQdQrevrev is the energy that must be transferred to the is the energy that must be transferred to the

system as heat in the reversible process that bring the system as heat in the reversible process that bring the system from the initial state to the final state.system from the initial state to the final state.

T

dQdS rev

Entropy of an Ideal GasEntropy of an Ideal Gas Consider an arbitrary reversible quasi-static process in which a system consisting of an ideal gas adsorbs an amount Consider an arbitrary reversible quasi-static process in which a system consisting of an ideal gas adsorbs an amount

of heat of heat dQdQrevrev. .

According to the first law, According to the first law, dQdQrevrev is related to is related to dEdEintint and and WW by by

or

If we will divide each term by If we will divide each term by TT: :

PdVdQdWdQdE revrev int

V

dVnRTdQdTC revv

V

dVnR

T

dQ

T

dTC revv

Entropy of an Ideal GasEntropy of an Ideal Gas

V

dVnR

T

dTC

T

dQdS v

rev

V

dVnR

T

dQ

T

dTC revv

We will assume We will assume CCvv to be constant and by integrating we will find to be constant and by integrating we will find

the entropy change of an ideal gas that undergoes a reversible the entropy change of an ideal gas that undergoes a reversible expansion from an initial state of volume expansion from an initial state of volume VV11 and temperature and temperature T T11

to a final state of volume to a final state of volume VV22 and temperature and temperature TT22

1

2

1

2 lnlnV

VnR

T

TC

T

dQS v

Entropy and the Second LawEntropy and the Second Law

• Entropy is a measure of disorderEntropy is a measure of disorder

• The entropy of the Universe increases The entropy of the Universe increases in all real processesin all real processes– This is another statement of the second law of This is another statement of the second law of

thermodynamicsthermodynamics

Entropy and HeatEntropy and Heat

• The original formulation of entropy deals The original formulation of entropy deals with the transfer of energy by heat in a with the transfer of energy by heat in a reversible processreversible process

• If If dQdQrevrev is the amount of energy transferred is the amount of energy transferred

by heat when a system follows a reversible by heat when a system follows a reversible path then the change in entropy, path then the change in entropy, SS is is

rdQS

T

Entropy and HeatEntropy and Heat

• The change in entropy depends only on The change in entropy depends only on the endpoints and is independent of the the endpoints and is independent of the path followedpath followed

• The entropy change for an irreversible The entropy change for an irreversible process can be determined by calculating process can be determined by calculating the change in entropy for a reversible the change in entropy for a reversible process that connects the same initial and process that connects the same initial and final pointsfinal points

More About Change in EntropyMore About Change in Entropy

• dQdQrevrev is measured along a reversible path, is measured along a reversible path,

even if the system may have followed an even if the system may have followed an irreversible pathirreversible path

• The meaningful quantity is the The meaningful quantity is the changechange in in entropy and not the entropy itselfentropy and not the entropy itself

• For a finite process,For a finite process,f f

r

i i

dQS dS

T

Change in EntropyChange in Entropy

• The change in entropy of a system going The change in entropy of a system going from one state to another has the same from one state to another has the same value for all paths connecting the two value for all paths connecting the two statesstates

• The finite change in entropy depends only The finite change in entropy depends only on the properties of the initial and final on the properties of the initial and final equilibrium statesequilibrium states

SS for a Reversible Cycle for a Reversible Cycle

• SS = 0 for any reversible cycle = 0 for any reversible cycle

• In general, In general,

– This integral symbol indicates the integral is over a This integral symbol indicates the integral is over a closed pathclosed path

0 T

dQrev

Entropy Changes in Irreversible Entropy Changes in Irreversible ProcessesProcesses

• To calculate the change in entropy in a To calculate the change in entropy in a real system, remember that entropy real system, remember that entropy depends only on the state of the systemdepends only on the state of the system

• Do not use Do not use QQ, the actual energy transfer in , the actual energy transfer in the processthe process– Distinguish this from Distinguish this from QQrev rev , the amount of , the amount of

energy that would have been transferred by energy that would have been transferred by heat along a reversible pathheat along a reversible path

– QQrevrev is the correct value to use for is the correct value to use for SS

• In general, the total entropy and therefore In general, the total entropy and therefore the total disorder always increases in an the total disorder always increases in an irreversible processirreversible process

• The total entropy of an isolated system The total entropy of an isolated system undergoes a change that cannot undergoes a change that cannot decreasedecrease– This is another statement of the second law of This is another statement of the second law of

thermodynamicsthermodynamics



• If the process is irreversible, then the total If the process is irreversible, then the total entropy of an isolated system always entropy of an isolated system always increasesincreases– In a reversible process, the total entropy of an In a reversible process, the total entropy of an

isolated system remains constantisolated system remains constant

• The change in entropy of the Universe The change in entropy of the Universe must be greater than zero for an must be greater than zero for an irreversible process and equal to zero for a irreversible process and equal to zero for a reversible processreversible process

Heat Death of the UniverseHeat Death of the Universe

• Ultimately, the entropy of the Universe should Ultimately, the entropy of the Universe should reach a maximum valuereach a maximum value

• At this value, the Universe will be in a state of At this value, the Universe will be in a state of uniform temperature and densityuniform temperature and density

• All physical, chemical, and biological processes All physical, chemical, and biological processes will ceasewill cease– The state of perfect disorder implies that no energy is The state of perfect disorder implies that no energy is

available for doing workavailable for doing work

– This state is called the This state is called the heat death of the Universeheat death of the Universe

SS in Thermal Conduction in Thermal Conduction

• The cold reservoir absorbs The cold reservoir absorbs QQ and its entropy and its entropy changes by changes by QQ//TTcc

• At the same time, the hot reservoir loses At the same time, the hot reservoir loses QQ and and its entropy changes by its entropy changes by --QQ//TThh

• Since Since TThh > > TTcc , the increase in entropy in the cold , the increase in entropy in the cold reservoir is greater than the decrease in entropy reservoir is greater than the decrease in entropy in the hot reservoirin the hot reservoir

• Therefore, Therefore, SSUU > 0 > 0– For the system and the UniverseFor the system and the Universe

SS in a Free Expansion in a Free Expansion

• Consider an adiabatic free expansionConsider an adiabatic free expansion

• QQ = 0 = 0 and cannot be used since that is for and cannot be used since that is for an irreversible processan irreversible process

1f fr

ri i

dQS dQ

T T

SS in Free Expansion in Free Expansion

• For an isothermal process, this becomesFor an isothermal process, this becomes

• Since Since VVff > > VVii , , SS is positive is positive• This indicates that both the entropy and the This indicates that both the entropy and the

disorder of the gas increase as a result of the disorder of the gas increase as a result of the irreversible adiabatic expansion irreversible adiabatic expansion

i

f

V

VnRS ln

SS in Calorimetric Processes in Calorimetric Processes

• The process is irreversible because the system The process is irreversible because the system goes through a series of nonequilibrium statesgoes through a series of nonequilibrium states

• Assuming the specific heats remain constant Assuming the specific heats remain constant and no mixing takes place:and no mixing takes place:

– If mixing takes place, this result applies only to If mixing takes place, this result applies only to identical substancesidentical substances

SS will be positive and the entropy of the Universe will be positive and the entropy of the Universe increasesincreases

1 1 2 2ln lnf f

c h

T TS m c m c

T T

Entropy on a Microscopic ScaleEntropy on a Microscopic Scale

• We can treat entropy from a microscopic We can treat entropy from a microscopic viewpoint through statistical analysis of viewpoint through statistical analysis of molecular motionsmolecular motions

• A connection between entropy and the number A connection between entropy and the number of microstates (of microstates (WW) for a given macrostate is ) for a given macrostate is

SS = = kkBB ln ln WW– The more microstates that correspond to a given The more microstates that correspond to a given

macrostate, the greater the entropy of that macrostatemacrostate, the greater the entropy of that macrostate

• This shows that entropy is a measure of disorderThis shows that entropy is a measure of disorder

Entropy, Molecule ExampleEntropy, Molecule Example

• One molecule in a two-sided container has a 1-in-2 One molecule in a two-sided container has a 1-in-2 chance of being on the left sidechance of being on the left side

• Two molecules have a 1-in-4 chance of being on the left Two molecules have a 1-in-4 chance of being on the left side at the same timeside at the same time

• Three molecules have a 1-in-8 chance of being on the left Three molecules have a 1-in-8 chance of being on the left side at the same timeside at the same time

Entropy, Molecule Example ExtendedEntropy, Molecule Example Extended

• Consider Consider 100 100 molecules in the containermolecules in the container

• The probability of separating The probability of separating 5050 fast fast molecules on one side and molecules on one side and 5050 slow slow molecules on the other side is (molecules on the other side is (½½))100100

• If we have one mole of gas, this is found to If we have one mole of gas, this is found to be extremely improbablebe extremely improbable

Entropy, Marble ExampleEntropy, Marble Example

• Suppose you have a bag with Suppose you have a bag with 5050 red red marbles and marbles and 5050 green marbles green marbles

• You draw a marble, record its color, return You draw a marble, record its color, return it to the bag, and draw another it to the bag, and draw another

• Continue until four marbles have been Continue until four marbles have been drawndrawn

• What are possible macrostates and what What are possible macrostates and what are their probabilities?are their probabilities?

Entropy, Marble Example, ResultsEntropy, Marble Example, Results

• The most ordered are the least likelyThe most ordered are the least likely

• The most disorder is the most likelyThe most disorder is the most likely

What change in entropy occurs when a What change in entropy occurs when a 27.9-g27.9-g ice cube at ice cube at –12–12CC is transformed into steam at is transformed into steam at 115115CC??


273 K 273 K273 K1ice

ice ice 261 K261 K 261 K

ln

2730.0270 kg 2090 J kg C ln273 K ln261 K 0.0270 kg 2090 J kg C ln

261

2.54 J K

f

i

mc dTdQS mc T dT mc T

T T

S

S

As the ice melts its entropy change is

50.0270 kg 3.33 10 J kg32.9 J K

273 KfmLQ

ST T


As liquid water warms from 273 K to 373 K,

liquidliquid

373ln 0.0270 kg 4186 J kg C ln 35.3 J K

273

ff

ii

mc dT TS mc

T T

As the water boils and the steam warms,

steam

6

ln

0.0270 kg 2.26 10 J kg 3880.0270 kg 2010 J kg C ln 164 J K 2.14 J K

373 K 373

fv

i

TmLS mc

T T

S

The total entropy change is

2.54 32.9 35.3 164 2.14 J K 236 J K

Chapter 22

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Transcript of Chapter 22