Chapter 18

Chapter 18

Heat Engines, Entropy, & the 2nd Law of Thermodynamics

Heat Engines

A heat engine is a device that converts internal energy to other useful forms, such as kinetic energy .

A heat engine carries some working substance through which cyclic processes during which1. Energy is transferred from a source at a high

temperature2. Work is done by the engine3. Energy is expelled by the engine to a source

at a lower temperature

Heat Engines

A process that utilizes heat energy input (Qh) to enable a working substance perform work output.

Because the working substance goes through a cycle, Eint =0. From the 1st law,

Eint = Q + W = 0 Qnet = - W = Weng

Qh

Weng

Qc

Heat Engine

Hot re

serv

oir a

t T

h

Cold

rese

rvoir a

t T

c

Weng = |Qh|-|Qc| = Qnet

Qh

Weng

Qc

Heat Engine

Hot re

serv

oir a

t T

h

Cold

rese

rvoir a

t T

c

Heat Engines

If the working substance is a gas, the net work done by the engine fora cyclic process is the area enclosed by the curve representing the processon a PV diagram.

The thermal efficiency Efficiency = = Weng/|Qh|

= (|Qh| - |Qc| ) /|Qh|

=1 - |Qc|/|Qh|

Area=Weng

o

P

V

Heat engines

Hot reservoir

Condenser

Low T, low P gas

High T, high P gas

The 2nd Law of Thermodynamics

The Kelvin-Planck statement of the 2nd law of thermodynamics:

It is impossible to construct a heatengine that, operating in a cycle, produces on effect other than the absorption of energy from a reservoir and the performance of an equal amount of work.

impossible to achieve =100%

The Carnot (“ideal”) engine

A reversible process is one for which the system can be return to its initial conditions along the same path and for which every point along the path is an equilibrium state.

A process dose not satisfy these requirements is irreversible.

The Carnot (“ideal”) engine

A heat engine operating in an ideal, reversible cycle — called a Carnot cycle — between two energy reservoirs is the most efficient engine possible.– An “ideal” reversible heat

engine (no heat engine can be more efficient than a Carnot engine). Sadi Carnot

(1796-1832)

A 4 stage engine that operates between 2 temperature reservoirs (Th and Tc) consisting of

2 isothermal phases2 adiabatic phases

A->BIsothermalexpansion

B->CAdiabaticexpansion

C->DIsothermal

compression

D->AAdiabatic

compression

A->B: isothermal expansion at Th. The gas absorbs Qh from the reservoir and does work WAB in raising the piston.

A 4 stage engine that operates between 2 temperature reservoirs (Th and Tc) consisting of

2 isothermal phases2 adiabatic phases

A->BIsothermalexpansion

B->CAdiabaticexpansion

C->DIsothermal

compression

D->AAdiabatic

compression

B->C: adiabatic expansion. No energy enters or leaves the system by heat. T falls from Th to Tc and the gas does work WBC in raising the piston.

A 4 stage engine that operates between 2 temperature reservoirs (Th and Tc) consisting of

2 isothermal phases2 adiabatic phases

A->BIsothermalexpansion

B->CAdiabaticexpansion

C->DIsothermal

compression

D->AAdiabatic

compression

C->D: isothermal compression at Tc. The gas expels Qc to the reservoir and the work done on the gas is WCD.

A 4 stage engine that operates between 2 temperature reservoirs (Th and Tc) consisting of

2 isothermal phases2 adiabatic phases

A->BIsothermalexpansion

B->CAdiabaticexpansion

C->DIsothermal

compression

D->AAdiabatic

compression

D->A: adiabatic compression. No energy enters or leaves the system by heat. T increases from Tc to Th and the work done on the gas is WDA.

The Carnot (“ideal”) engine

For the Carnot (“ideal”) engine:

Efficiency = Carnot = (|Qh|– |Qc|)/|Qh|

since |Qh| / |Qc| = Th/Tc the efficiency can be written as

Carnot = [(Th-Tc)/Th ].100%

= 1-Tc/Th

Heat Pumps & Refrigerators

Transfer some energy into a device!

How to move energy from the cold reservoir to the hot reservoir?

Qh

W

Qc

Heat pump

Hot re

serv

oir a

t T

h

Cold

rese

rvoir a

t T

c

Heat Pumps & RefrigeratorsThe coefficient of performance, COP

energy transferred to hot reservoir

work done onpump

h

COP heat pump

Q

W

c cCarnot

h c h c

Q TCOP refrigerator

Q Q T T

cQCOP refrigeratorW

hCarnot

h c

TCOP heat pump

T T

Heat Pumps

Refrigerators

Evaporator

CondenserLow T, low P liquid

High T, high P liquid

Expansion valve

Heat engine

Hot reservoir

Hot reservoir

hQ

cQW

Vo

p

2T

W1T

A

B

CD

1p

2p

4p

3p

1V 4V 2V 3V

21 TT

Heat pump

Hot reservoir

Cold reservoir

hQ

cQW

Vo

p

2T

W1T

A

B

CD

21 TT

Heat Pumps

The 2nd Law of Thermodynamics

2nd Law: thermodynamic limit of heat engine efficiency1. Heat only flows spontaneously from high T to

cold T2. A heat engine can never be more efficient

that a “Carnot” engine operating between the same hot & cold temperature range

3. The total entropy of the universe never decreases

Entropy

Entropy is a measure of the disorder (or randomness) of a system.

For a reversible the change in entropy is measured as the ratio of heat gained to temperature

dS = dQr/T– When heat energy is gained by a system, entropy is

gained by the system (and lost by the surrounding environment)

– When heat is lost by a system, entropy is lost by the system (and gained by the surrounding environment)

Entropy is a state function (like energy). Changes in entropy occur independent of path

taken by the system.

Entropy

Multiplicity = Entropy = k ln (k is Boltzmann's constant ) High-probability macrostates are disordered

macrostates. Low-probability macrostates are ordered macrostates.

Entropy

All physical processes tend toward more probable states for the system andits surroundings. The more probable state is always one of higher disorder.

Entropy & The 2nd Law

For the Carnot engine |Qh| / Th= |Qc| /Tc

Qh / Th= -Qc /Tc or Qh / Th + Qc /Tc =0

S=0 For a system taken through an arbitrary

reversible cycle,

(reversible path)f f

r

i i

dQS dS

T

0f i

i

r r r

f

dQ dQ dQS

T T T

the line integral is path independent

Entropy Changes in a Free Expansion

This process is neither reversible nor quasi-static.

The wall is insulating, Q=0.

The work done by gas is W=0.

From the 1st Law, Eint = Q + W = 0

Eint,i = Eint,f Ti = Tf

Vacuum

Vf

Vi

Entropy Changes in a Free Expansion

Find an equivalent reversible path that share the same initial and final states.

An isothermal, reversible expansion, in which the gas pushes slowly against a piston:

Vacuum

Vf

Vi

1

f f

i i

rr

dQ

TQS

Td

1 1 1ln

f

i

f V fri V

i

VnRT dVS dQ PdV dV nR nR

T T T V V V

dQr = -dW = PdV

Vf>Vi, S>0

Entropy Changes in Irreversible Processes

The total entropy of an isolated system that undergoes a change cannot decrease.

S≥0The net entropy change by the universe

due to a thermodynamic process:Suniverse = Sgained - Slost

= Qcold/Tcold - Qhot/Thot

The total entropy of the universe (Suniverse) will never decrease, it will either – Remain unchanged (for a reversible process)– Increase (for an irreversible process)

The 2nd Law of Thermodynamics

2nd Law: thermodynamic limit of heat engine efficiency1. Heat only flows spontaneously from high T to

cold T2. A heat engine can never be more efficient

that a “Carnot” engine operating between the same hot & cold temperature range

3. The total entropy of the universe never decreases