chapter 10 spontaneity, entropy, and free energy

Chapter 10

Spontaneity, Entropy, and Free Energy

Concept for second law of thermodynamic

熱機原理壓縮機原理

亂度

Isothermal expansion device

One-Step Expansion (No Work)

Mass M1 is removed from the pan, the gas will expand, moving the piston to the right end of the cylinder.

P1→1/4P1, V1→4V1,

No work is done. W0=0Free expansion

One-Step Expansion

M1 is replaced by M1/4.

11111

1

1

1

4

3)4)(

4(

4

VPVVP

W

VPW

PP

ex

ex

Two-Step Expansion

P1→1/2P1, V1→2V1

112

1111

1

1111

1

224

4

22

22

1

VPW

VP)VV(

PW

tep Work for s

VP)VV(

PW

tep Work for s

''

'

1/2P1→1/4P1, 2V1→4V1

PV diagram two-step expansion

The PV diagram six-step expansion

Infinite-Step Expansion

)V

VnRTln(Wq

V1.4P1.4nRTnRTln4W

)V

4VnRTln()

V

VnRTln()lnVnRT(lnVW

V

dVnRTW

dVPW

.increments smallmally infinitesiby changed is P

ΔVPW

1

2revrev

11rev

1

1

1

212rev

V

Vrev

V

V exrev

ex

i

n

1iin

2

1

2

1

(dV: V→0 )△

)(2

1

2

1

W

W

V

VPdVdWPdVdW

)(V

nRTP

1

2ln12

1 x

xdx

x

x

x

Reversible expansion

Reversible Process

Reversible process: the system is always infinitesimally close to equilibrium, and an infinitesimal change in conditions can reverse the process to restore both system and surroundings to their initial states.

Heat Engines

A heat engine converts some of the random molecular energy of heat flow into macroscopic mechanical energy.

qH: the working substance from a hot body

-w: the performance of work by the working substance on the surroundings

-qC: the emission of heat by the working substance to a cold body

The Second Law of Thermodynamics Kelvin-Planck statement for heat engine

It is impossible to extract an amount of heat qH from a hot reservoir and use it all to do work W. Some amount of heat qC must be exhausted to a cold reservoir.

This is sometimes called the "first form" of the second law, and is referred to as the Kelvin-Planck statement of the second law.

Heat Efficiency

100

1

0

0ΔU

0q and 0 wso

0q- and 0w- ,0q engineheat a

c

cw

e , qq

q

q

q

qqe

qq-wwqqwq

tion, e of operaFor a cycl

q

w

q

-w

utenergy inp

workiency eheat effic

For

Hc

cc

cc

HH

H

HH

HH

The Second Law of ThermodynamicsClausius statement for refrigerator

It is not possible for heat to flow from a colder body to a warmer body without any work having been done to accomplish this flow. Energy will not flow spontaneously from a low temperature object to a higher temperature object.

The statements about refrigerators apply to air conditioners and heat pumps which embody the same principles.

Carnot’s Principle

No heat engine can be more efficient than a reversible heat engine when both engines work between the same pair of temperature H and C.

Isothermal Process: the temperature of the system and the surroundings remain constant at all times. (q=-w)Adiabatic: a process in which no energy as heat flows into or out of the system. (∆U=w)

Carnot cycle

four stage reversible sequence consisting of

1. isothermal expansion at high temperature T2

2. adiabatic expansion

3. isothermal compression at low temperature T1

4. adiabatic compression

V

dVnR

T

dq

T

dT(T)C

V

dVnR

T

dq

T

dT(T)C

dVV

nRTdq(T)dTC

PdVdqdwdqdU

(T)dTC and dUV

nRTP

V

V

V

V

1

4

3

2

4

3

2

1

)()()()(T

T V

T

T

T

T VV

T

T VV T

dTTC

T

dTTC

T

dTTC

T

dTTC

T

dT(T)C

0)(

)()(

)()()(

1

4

3

2

T

dTTC

T

dTTC

T

dTTC

T

dTTC

T

dTTC

T

dTTC

V

T

T V

T

T V

T

T V

T

T

T

T VV

H

C

H

C

C

H

0 0

H

CH

H

CHrev

C

C

H

HT

T

T

T

T

T

T

T

T

T

T

T

T

TT

q

qqe

T

q

T

q

T

dq

T

dq

T

dq

T

dq

T

dq

T

dq

T

dq

T

dq

0

0

4

3

2

1

1

4

3

2

4

3

2

1

0 0

Adiabatic Process

)( )()()(

R)C-C( )()()()(

)ln()ln()ln(11

0

1

2

1)1(

2

1

1

2

vp)(

2

1

1

2

2

1

1

2

2

1

1

2

1

22

1

2

1

v

pC

C

CCCRC

v

V

V

T

Tv

vvv

C

C

V

V

V

V

T

T

V

V

T

T

V

V

T

T

V

VR

V

VR

T

TC

VRdT

TC

dVV

RdT

T

CdV

V

nRTdTnCPdVdTnC

)qw (ΔU

v

p

vpvv

12 1

11 2

11 1 2 2 2 2 2 1

1 1 2 211 2 1 1 1 2

1 1 2 2

1 1 2 2

T V

T V

for ideal gas

PV PV T PV VPV PV

T T T PV V

For isothermal process PV PV

For adiabatic process PV PV

Isothermal process

1

2

1

2

ln

ln12

1

2

1

V

VnRTq

V

VnRTdV

VnRTdVPW

V

V

V

V

∆U=0, q=-w

Adiabatic process

1

)( 11

12

2

1

2

1

VVKW

dVV

KdVPW

KPVV

V

V

V

No heat transfer (q=0) , ∆U=w

Adiabatic Process

Process in which no heat transfer takes place

WTTnRU )(2

312

Application of Carnot Cycle

P (atm) V (L)

3 10

1.5 20

1 25.5

2 12.75

Calculate Q, U, W First law:

△U = QH – QL + W

W = QL - QH

Spontaneous Process and Entropy

Spontaneous Process: A process occurs without outside intervention.

Entropy: In qualitative terms, entropy can be viewed as a measure of randomness or disorder of the atoms or molecules in a substance.

Definition of Entropy

S=kBlnΩ

kB: Boltzmann’s constant

Ω : the number of microstates

corresponding to a given state

For one particle

S1=kBlnΩ1

S2=kBlnΩ2

∆S=S2-S1= kBlnΩ2-kBlnΩ1=kBln(Ω2/Ω1)

∆S= kBln(2Ω1/Ω1)=kBln2

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Ω

4顆粒子都有 2種選擇，微觀態數： 24=16

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1 2

1 2

2 2

1 1

2

1

2

1

1 ln( )

ln( )

V V

V V

V

V

VFor mole of gas S R

V

VFor n mole of gas S nR

V

Definition of entropy in term of probability

Entropy for Isothermal Process

1

2

1

2

ln

ln12

1

2

1

V

VnRTq

V

VnRTdV

VnRTdVPW

V

V

V

V

T

qΔS

)V

V(nRT) and q

V

V(nRΔS

wqΔUwq

rev

rev

1

2

1

2 lnln

0

Entropy and Physical ChangesTemperature Dependence

2

1

2

121

2

1

2

121

2

1

2

121

1

2

1

2

ln

constant VFor

ln

constant PFor

T

T vv

T

T

p

TTvv

p

T

Tp

T

T

p

TTpp

T

T

revT

TTTrev

rev

T

TnC

T

dTnC

T

dqSdTnCdq

T

TnC

T

dTnC

T

dqSdTnCdq

T

dqdSS

T

dqdS

T

qS

Entropy and Physical ChangesChange of State

Change of state from solid to liquid

qrev=ΔHfusion

T=melting point in K

Change of state from liquid to gas

qrev=ΔHvaporization

T=boiling point in K

T

qS rev

The Second Law of thermodynamicsThe Third Statement

In any spontaneous process, there is always an increase in the entropy of the universe.

dq/T is the differential of a state function S that has the property ∆Suniv ≥ 0 for any process

Entropy and Second Law of Thermodynamics

ΔSuniv= ΔSsys+ΔSsurr

T

HSsurr

Gibbs Free EnergyAt constant T and P

0 spontaneous

0 equilibrium

0 non-spontaneous

surr surr sys univ

G HG H T S S

T TH G

S S S ST T

G

G

G

△Suniv>0, so G<0△

Free Energy and Chemical Reactions

0 0 0

0 0 0

reaction products reactantsS S S

G H T S

Third Law of Thermodynamics The entropy of a perfect crystal at 0 K is zero.

It is impossible to reach a temperature of absolute zero

It is impossible to have a (Carnot) efficiency equal to 100% (this would imply Tc = 0).

0lim 0T

S

(a) T=0 K, S=0

(b) T>0 K, S>0

The Dependence of Free Energy on Pressure

3 3 2 2 2 2

3 2 2 3 2 2

3 2 2

0

2 2 3

0 0 0

0 0 0

0 2 3

0

ln( )

3 2

2[ ln( )] [ ln( )] 3[ ln( )]

(2 3 ) [2 ln( ) ln( ) 3ln( )]

[ln( ) ln( ) ln( )]

NH NH N N H H

NH N H NH N H

reaction NH N H

reaction

G G RT P

N H NH

G G RT P G RT P G RT P

G G G RT P RT P P

G G RT P P P

G G

3

2 2

3

2 2

2

3

20

3

ln[ ]( )( )

ln( ) (Q= )( )( )

NH

N H

NHreaction

N H

PRT

P P

PG G RT Q

P P

Free Energy and Equilibrium

0

0

0 ln( )

ln( )

products reactants reaction

reaction

At equilibrium

G G G G RT Q

G RT K

The Temperature Dependence of K

0 0 0

0 0 0 0

0 0

22

0 0

11

02

1 2 1

ln( )

1ln( ) ( )

ln( )

ln( )

1 1ln( ) [ ]

G RT K H T S

H S H SK

RT R R T R

H SK

RT R

H SK

RT R

K H

K R T T

Free Energy and Work

max

P and Tconstant at process aFor

wG

dwdGPdVdwPdVdG

dwPdVdwdwdw

PdVdwVdPPdVdwdG

PVwTSPVwTSGTSPVUH-TSG

wTSUdwTdSdUTdSdwdUdq

TdSdqT

dq

T

dqdS

otherother

otherotherVP

irrev

revirrevirrevrev

rev