Chapter 18

Heat Engines,

Entropy

and the

Second Law of Thermodynamics

First Law of Thermodynamics – Review Review: The First Law is a statement of

Conservation of Energy The Law places no restrictions on the

types of energy conversions that can occur In reality, only certain types of energy conversions

are observed to take place Some events (processes) in one direction are

more probable than those in the opposite direction

William Thomson, Lord Kelvin 1824 – 1907 Physicist and

mathematician Proposed the use of

absolute temperature scale Now named for him

Work in thermodynamics led to idea that energy cannot spontaneously pass from colder to hotter objects

Heat Engine A heat engine is a device that takes in

energy by heat, and, operating in a cycle, expels a fraction of that energy by means of work

A heat engine carries some working substance through a cyclical process

Heat Engine, cont. The working substance

absorbs energy by heat from a high temperature energy reservoir (Qh)

Work is done by the engine (Weng)

Energy is expelled by heat to a lower temperature reservoir (Qc)

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

ΔEint = 0 Its initial and final internal energies

are the same Therefore, Qnet = Weng The work done by the engine

equals the net energy absorbed by the engine

The work is equal to the area enclosed by the curve of the PV diagram

If the working substance is a gas

Thermal Efficiency of a Heat Engine Thermal efficiency is defined as the

ratio of the net work done by the engine during one cycle to the energy input at the higher temperature

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

1eng h c c

h h h

W Q Q Q

Q Q Q

e

More About Efficiency In practice, all heat engines expel only

a fraction of the input energy by mechanical work

Therefore, their efficiency is always less than 100% To have e = 100%, QC must be 0

Second Law: Kelvin-Planck Form

It is impossible to construct a heat engine that, operating in a cycle, produces no other effect than the absorption of energy from a reservoir and the performance of an equal amount of work Means that Qc cannot equal 0

Some Qc must be expelled to the environment

Means that e cannot equal 100%

Perfect Heat Engine No energy is

expelled to the cold reservoir

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

e = 100% It is an impossible

engine

Reversible and Irreversible Processes

A reversible process is one for which the system can be returned to its initial state along the same path And one for which every point along some path is

an equilibrium state An irreversible process does not meet these

requirements Most natural processes are known to be

irreversible Reversible process are an idealization, but some

real processes are good approximations

Reversible and Irreversible Processes, cont A real process that is a

good approximation of a reversible one will occur very slowly

The system is always very nearly in an equilibrium state

Adding and removing the grains of sand is an example of a real process that can occur slowly enough to be reversible

Reversible and Irreversible Processes, Summary The reversible process is an idealization All real processes on Earth are

irreversible

Sadi Carnot 1796 – 1832 First to show the

quantitative relationship between work and heat

Book reviewed importance of the steam engine

Carnot Engine A theoretical engine developed by

Sadi Carnot A heat engine operating in an ideal,

reversible cycle (now called a Carnot Cycle) between two reservoirs is the most efficient engine possible This sets an upper limit on the efficiencies

of all other engines

Work by a Carnot Cycle The net work done by a working

substance taken through the Carnot cycle is the greatest amount of work possible for a given amount of energy supplied to the substance at the upper temperature

Carnot Cycle

Overview of the processes in a Carnot Cycle

Carnot Cycle, A to B A to B is an isothermal

expansion The gas is placed in

contact with the high temperature reservoir, Th

The gas absorbs heat |Qh|

The gas does work WAB in raising the piston

Carnot Cycle, B to C B to C is an adiabatic

expansion The base of the cylinder

is replaced by a thermally nonconducting wall

No heat enters or leaves the system

The temperature falls from Th to Tc

The gas does work WBC

Carnot Cycle, C to D The gas is placed in

contact with the cold temperature reservoir

C to D is an isothermal compression

The gas expels energy QC

Work WCD is done on the gas

Carnot Cycle, D to A D to A is an adiabatic

compression The gas is again placed

against a thermally nonconducting wall

So no heat is exchanged with the surroundings

The temperature of the gas increases from TC to Th

The work done on the gas is WDA

Carnot Cycle, PV Diagram The work done by

the engine is shown by the area enclosed by the curve, Weng

The net work is equal to |Qh| – |Qc|

Eint = 0 for the entire cycle

Efficiency of a Carnot Engine Carnot showed that the efficiency of the

engine depends on the temperatures of the reservoirs

Temperatures must be in Kelvins All Carnot engines operating between the

same two temperatures will have the same efficiency

1C C CCarnot

h h h

Q T Tand e

Q T T

Notes About Carnot Efficiency Efficiency is 0 if Th = Tc

Efficiency is 100% only if Tc = 0 K Such reservoirs are not available Efficiency is always less than 100%

The efficiency increases as Tc is lowered and as Th is raised

In most practical cases, Tc is near room temperature, 300 K So generally Th is raised to increase efficiency

Real Engine vs. Carnot Engine All real engines are less efficient than

the Carnot engine because they all operate irreversibly so as to complete a cycle in a brief time interval

In addition, real engines are subject to practical difficulties that decrease efficiency Friction is an example

Heat Pumps and Refrigerators Heat engines can run in reverse

This is not a natural direction of energy transfer Must put some energy into a device to do this Devices that do this are called heat pumps or

refrigerators Examples

A refrigerator is a common type of heat pump An air conditioner is another example of a heat

pump

Heat Pump Process Energy is extracted

from the cold reservoir, QC

Energy is transferred to the hot reservoir, Qh

Work must be done on the engine, W

Coefficient of Performance The effectiveness of a heat pump is

described by a number called the coefficient of performance (COP)

In heating mode, the COP is the ratio of the heat transferred in to the work required

hheat pump

Qenergy transferred to hot resCOP

work done on pump W

COP, Heating Mode COP is similar to efficiency Qh is typically higher than W

Values of COP are generally greater than 1 It is possible for them to be less than 1

We would like the COP to be as high as possible

COP, Carnot in Heating Mode A Carnot cycle heat engine operating in

reverse constitutes an ideal heat pump It will have the highest possible COP for

the temperatures between which it operates

,h

Carnot heat pumph c

TCOP

T T

COP, Cooling Mode In cooling mode, you “gain” energy

from a cold temperature reservoir

A good refrigerator should have a high COP Typical values are 5 or 6

crefrigerator

QCOP

W

COP, Carnot in Cooling Mode The highest possible COP is again the Carnot

engine running in reverse

As the difference in temperature approaches zero, the theoretical COP approaches infinity In practice, COP’s are limited to values below 10

,c c

Carnot refrigeratorh c h c

Q TCOP

Q Q T T

Second Law – Clausius Form Energy does not flow spontaneously by

heat from a cold object to a hot object The Second Law can be stated in multiple

ways, all can be shown to be equivalent The form you should use depends on the

application

Perfect Heat Pump Takes energy from

the cold reservoir Expels an equal

amount of energy to the hot reservoir

No work is done This is an

impossible heat pump

The Second Law of Thermodynamics, Summary Establishes which processes do and which do

not occur Some processes can occur in either direction

according to the First Law They are observed to occur only in one

direction Real processes proceed in a preferred

direction This directionality is governed by the Second

Law

Entropy Entropy, S, is a state variable related to

the Second Law of Thermodynamics The importance of entropy grew with the

development of statistical mechanics A main result is isolated systems tend

toward disorder and entropy is a natural measure of this disorder

Microstates vs. Macrostates A microstate is a particular

configuration of the individual constituents of the system

A macrostate is a description of the conditions from a macroscopic point of view It makes use of macroscopic variables

such as pressure, density, and temperature for gases

Microstates vs. Macrostates, cont For a given macrostate, a number of

microstates are possible The number of microstates associated

with a given macrostate is not the same for all macrostates

When all possible macrostates are examined, it is found that the most probable macrostate is that with the largest number of possible microstates

Microstates vs. Macrostates, Probabilities High-probability macrostates are disordered

macrostates Low-probability macrostates are ordered

macrostates All physical processes tend toward more

probable macrostates for the system and its surroundings The more probable macrostate is always one of

higher disorder

Entropy Entropy is a measure of disorder of a state Entropy can be defined using macroscopic

concepts of heat and temperature

Entropy can also be defined in terms of the number of microstates, W, in a macrostate whose entropy is S

S = kB ln W

reversibledQdS

T

Entropy and the Second Law The entropy of the Universe increases

in all real processes This is another statement of the Second

Law of Thermodynamics The change in entropy in an arbitrary

reversible process isf f

r

i i

dQS dS

T

S For A Reversible Cycle S = 0 for any reversible cycle In general,

This integral symbol indicates the integral is over a closed path

0T

dQr

Entropy Changes in Irreversible Processes To calculate the change in entropy in a

real system, remember that entropy depends only on the state of the system

Do not use Q, the actual energy transfer in the process Distinguish this from Qr, the amount of

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

Qr is the correct value to use for S

In general, the total entropy and therefore the total disorder always increase in an irreversible process

The total entropy of an isolated system undergoes a change that cannot decrease This is another statement of the Second

Law of Thermodynamics

Entropy Changes in Irreversible Processes, cont

Entropy Changes in Irreversible Processes, final If the process is irreversible, then the

total entropy of an isolated system always increases In a reversible process, the total entropy of

an isolated system remains constant The change in entropy of the Universe

must be greater than zero for an irreversible process and equal to zero for a reversible process

Heat Death of the Universe Ultimately, the entropy of the Universe should

reach a maximum value At this value, the Universe will be in a state of

uniform temperature and density All physical, chemical, and biological

processes will cease The state of perfect disorder implies that no

energy is available for doing work This state is called the heat death of the Universe

S in Free Expansion Consider an adiabatic free expansion The process is neither reversible or

quasi-static W = 0 and Q = 0 Need to find Qr, choose an isothermal

process1f f

rri i

dQS dQ

T T

S in Free Expansion, cont For an isothermal process, this becomes

Since Vf > Vi, S is positive This indicates that both the entropy and the

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

ln f

i

VS nR

V

Atmosphere as a Heat Engine The Earth’s atmosphere can be

modeled as a heat engine The energy from the Sun undergoes

various processes Reflected Absorbed by the air or the surface

Energy Balance in the Atmosphere

Work Done on the Atmosphere One process is missing from the

previous energy balance The various processes result in a small

amount of work done on the atmosphere This work appears as the kinetic energy of

the prevailing winds in the atmosphere

Schematic, Atmosphere as a Heat Engine The amount of

incoming solar energy converted to kinetic energy of the winds is about 0.5%

It is temporarily kinetic energy, eventually radiated into space As radiation

Efficiency of the Atmospheric Heat Engine The warm reservoir is the surface The cold reservoir is empty space The efficiency can be calculated:

0.5%0.8%

64%eng

h

We

Q