Chapter 3 The second law A spontaneous direction of change: the direction of change that does not...

Chapter 3 The second lawChapter 3 The second law A spontaneous direction of change: the direction of change that does not require work to be done to bring it about.

Clausius statement: No cyclic process is possible in which the sole result is the transfer energy from a cooler to a hotter body.

All real process is irreversible process.

3.1 The second law

Second kind of perpetual motion machine

A machine that converts heat into with 100 percent efficiency.

It is impossible to built a second kind of perpetual motion machine.

Kelven statement : No cyclic process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.

3.1.1 Carnot principle

1. Efficiency of heat engine

def work performed

heat absorbed h

w

Q

1 2

1 1

def W Q Q

Q Q

2. Carnon cycleProcess AB ： (i)

A

B

V

VnRTQ ln11

CD ： (ii)2 2lnD

C

VQ nRT

V

BC ： (iii)1 1

1 2B CTV T V p

V

C

C

DA ： (iv)

1 11 2A DTV T V

(iii),（ iv)

CB

A D

VV

V V

therefore (v)1 2 1 2( - )ln B

A

VQ Q nR T T

V

1 2

1 1

def W Q Q

Q Q

1

21

1

21

1 T

TT

Q

QQ

Q

W

0

2

2

1

1 T

Q

T

Q

3. Carnot principle

No heat engine can be more efficient than a reversible heat engine when both engines work between the same pair of temperature T1 and T2.

1 2r

1

T T

T

Reversible engine (=)

Irreversible engine(<)

Reversible engine (=)

Irreversible engine(<)

1

21

T

TT

02

2

1

1 T

Q

T

Q

3.1.2 Entropy

1. Definition of entropy

0≤+2

2TQ

1

1TQ

Reversible engine (=)

Irreversible engine(<)

≤0δ

exT

Q

Reversible engine (=)

Irreversible engine(<)

≤0δ

exT

Q

Reversible engine (=)

Irreversible engine(<)

Clausius inequality

Reversible process

Irreversible process

0ex

ir T

Q

0r T

Q

Entropy S

(1) S is a state function

(2) S is an extensive property

(3) unit is J·K － 1

T

QS rδ def

d * Reversible process

2

1

2r

2 1 1

δdS

S

QS S S S

T * Reversible process

Reversible process0r T

Q

2. Clausius inequality

Bir ir

Aex ex

δ δor d

Q QS S

T T

B Air r

A Bex

δ δ0

Q Q

T T (Irreversible

cycle)B B

ir r

A Aex

δ δQ QS

T T therefore

B

Aex

δ

T

QS Irreversible

Reversible

ex

δd

T

QS Irreversible

Reversible

3 . The principle of the increase of entropy

For an adiabatic process ， δQ ＝０，

B

Aex ex

irreversible irreversibleδ δor d

reversible reversible

Q QS S

T T

The entropy of a closed system must increase in an irreversible adiabatic process.

0ad S

0d ad S IrreversibleReversible

IrreversibleReversible

4 . The entropy criterion of equilibrium

For an isolated system ， δQ ＝０，

B

Aex ex

irreversible irreversibleδ δor d

reversible reversible

Q QS S

T T

reversible

leirreversib0dor

reversible

leirreversib0 isoiso SS

Since all real process is irreversible, when processes are occurring in an isolated system, its entropy is increasing.

Thermodynamic equilibrium in an isolated system is reached when the system’s entropy is maximized.

5. Calculation of entropy change of surrounding

Note: δQsu ＝ - δQsy

sysu

ex

δQS

T

sysu

ex

(-δ )d

QS

T

Tex=constantsy

suex

QS

T

3.1.3 Calculation of entropy change of system

T

Q

T

QS rrδ

S=0

Reversible adiabatic process.

2

1

r12

δd2

1 T

QSSSS S

S Reversible process

Reversible isothermal process.

T

VpU

T

QS

ddδd r

V

VnR

T

TnCS V dd

d m, (perfect gas)

)lnln(

)lnln(

)lnln(

1

2m,

1

2m,

2

1

1

2

m,

1

2

1

2

m,

V

VC

p

pCn

p

pR

T

TCn

V

VR

T

TCnS

pV

p

V

IsochoricIsobaricIsothermalAdiabatic reversible irreversibleliquid and solid

1. p, V, T change

(1). p, V, T change

(2) mixing of different perfect gas at constant temperature and constant pressure

mixing at constant T, p

2

212

1

211mix lnln

V

VVRn

V

VVRnS

at constant T, p 221

21

21

1 , yVV

Vy

VV

V

mixS = － (n1 Rlny1 ＋ n2Rlny2)

y1 ＜ 1 ， y2 ＜ 1 ， so mixS ＞０

n1

T, p, V1

n2

T, p,V2

n1 +n2

T, p, V1+V2

2. The phase transition

(1) The phase transition at transition temperature

At constant T, p , the two phase in the system are in equilibrium,

the process is reversible, and W′ ＝０， so Qp ＝ H ，

T

HnS

)transitionphase ofenthalpy(m

fusHm ＞０， vapHm ＞０，

Sm(s) ＜ Sm(l) ＜ Sm(g)

(2) irreversible phase transition

Irreversible B(,T1,p1) B(,T2,p2)

S=?

B(,Teq,peq) B(, Teq,peq) reversible

S2

S1 S3

S ＝ S １＋ S ２＋ S ３

T/TnCST

T p d)l,OH( 2

m,1

eq

1

T/TnCST

HnS

T

T p d)g,OH( , 2

m,3mvap

2

2

eq

S ＝ S １＋ S ２＋ S

３

For example

Irreversible

S =?H2O （ l, 90 ,100kPa)℃ H2O （ g, 90 , ℃

100kPa)S1 S3

Reversible

S2H2O （ l, 100 , 100kPa)℃ H2O （ g, 100 , ℃

100kPa)

3.2 The third law

3.2.1 The third law

Nernst heat theorem: The entropy change accompanying any physical or chemical transformation approaches zero as the temperature approaches zero: S0, T0.

The entropy of all perfect crystalline substance is zero at 0 K.

-1

0KJ0)(lim

TS

T

S *(perfect crystalline ， 0 K) ＝０Ｊ · Ｋ -

１

3.2.2 Conventional molar entropy and standard molar entropy

Second law T

T

QSTS

0K

rδ)K0(*)(*

Third law S *( ０Ｋ )= ０

T

T

QTS

0K

mr,*m

δ)B,(

Sm* （ B ， T)— conventional molar entropy of substanc

e at temperature T 。

S m(B,,T) —standard molar entropy

(p ＝ 100kPa)

3.2.3 The entropy change of chemical reaction

For reaction aA ＋ b B →yY ＋ zZ

rS m(T) ＝ Σν Ｂ S m （ B, ,T )

rS m (298.15K) ＝ Σν Ｂ S m （ B, ,298.15K )

rS m (T) = yS m (Y, ,T ) ＋ z S m (Z, ,T )

－ a S m (A, ,T ) － b S m (B, ,T )

1S

2S 4

S3

S

)( 1mr TS

aA + bB yY + zZ

)( 2mr TSaA + bB yY + zZ

r S m (T1) = S 1 ＋ S 2 ＋ r S m (T2) ＋ S 3 ＋ S 4

T p

T

TCSTS

298.15K

m,Bmrmr

d)B().15K298()(

3.3 Helmholtz and Gibbs energies

3.3.1 Helmholtz energy A

reversible

leirreversibδd

exT

QS

reversible

leirreversib

exT

QS

Tex （ S2 － S1 ）≥ Q

at constant T , T2S2 － T1S1 ＝ Δ(TS)

Q ＝ ΔU － W

Δ(TS ）≥ ΔU － W

－ Δ(U － TS ）≥－W

The maximum work done by a closed system in an isothermal

process is obtained when the process is carried out reversible.

TSUA def

Helmholtz energy A is an extensive state function

reversible

irreversible A T≥ W

reversible

irreversible A T ≤ W

reversible

irreversible dA T ≤ W

In a closed system doing volume work only at constant T and

V, the Helmholtz energy A decrease in a spontaneous change.

At constant T and V W ＝ 0

δW′ ＝ 0

reversible

irreversible dA T, V ≤ 0 reversible

irreversible A T, V ≤ 0

In a closed system capable doing only volume work, the constant temperature and volume equilibrium condition is the minimization of the Helmholtz energy A. dA T, V = 0

Helmholtz energy criterion

3.3.2 Gibbs energy G

at constant T Δ(TS)≥ΔU － W

Δ(H － TS ）≤ W′

at constant p p1 ＝ p2 ＝ pex

W ＝－ pex （ V2 － V1 ）＋ W ′

＝－ p2V2 ＋ p1V1 ＋ W′

＝－ Δ(pV) ＋ W′

Δ(TS)≥ΔU ＋ Δ(pV) － W′

－［ ΔU ＋ Δ(pV)-Δ(TS) ］≥－ W′

－ Δ(U+pV-TS)≥ － W′

G H － TS ＝ U ＋ pV － TS ＝ A ＋ pV def

Gibbs energy G is an extensive state function

reversible

irreversible G T ,p≤ W

reversible

irreversible dG T, p ≤ W

The maximum possible non-volume work done by a closed

system in a constant temperature and pressure process is

equal to the G

reversible

irreversible G T ,p≤ W

reversible

irreversible dG T, p ≤ W

W′ ＝ 0, δW′ ＝ 0

reversible

irreversible G T ,p≤ 0

reversible

irreversible dG T, p ≤ 0

In a closed system doing volume work only at constant T

and p, the Gibbs energy G decrease in a spontaneous change.

In a closed system capable doing only volume work, the constant temperature and pressure equilibrium condition is the minimization of the Gibbs energy G . dG T, p = 0

Gibbs energy criterion

3.3.3 Calculation of A and G

(1).change p ,V at constant temperature

at constant T, reversible, dAT ＝ δWr ＝－ pdV ＋ δW r '

If δWr ' ＝ 0 ， then dAT ＝－ pdV

VpAV

VT d2

1 closed system, change p ,V at constant T,

W′ ＝ 0 reversible process

reversible

irreversible dA T≤ W

1

2

1

2 lnlnp

pnRT

V

VnRTAT For perfect gas

ΔA＝ ΔU－ TΔS ΔG＝ ΔH－ TΔS

pVGp

pT d2

1

G ＝ A ＋ pV, dG ＝ dA ＋ pdV ＋ Vdp

at constant T, reversible, δW′r ＝ 0, dA ＝－ pdV

1

2

1

2 lnlnV

VnRT

p

pnRTAG TT

dGT ＝ Vdp

closed system, change p ,V at constant T, W′ ＝ 0 reversible process

1

2

1

2 lnlnV

VnRT

p

pnRTGT For an perfect gas

(2) phase transition

(i) reversible phase transition

(ii) irreversible phase transition

reversible phase transition at constant T and p

ΔU = ΔH-Δ(pV) = ΔH- pΔV = ΔH- nRT

ΔA = ΔU -Δ(TS) = ΔH - nRT - TΔS = - nRT

G =H - (TS)

A = U - (TS) ΔG =ΔH-Δ(TS) = ΔH- TΔS

ΔA = ΔU -Δ(TS) = ΔU - TΔS

ΔH =TΔS

reversible vaporization or sublimation at constant T and p, and vapor is an perfect gas

ΔG= ΔH- TΔS = 0

3.3.4 Funtdantal relations of thermodynamic functions

U, H, S, A, G, p, V and T

H = U + pV

H

U pV

pVATS

TS G

A = U - TS

G = H - TS = U + pV -TS = A +pV

1. Master equation of thermodynamic

If δWr ′ ＝ 0 ， then δWr ＝－ pdV ，

dU=δQr+δWr ，For a reversible change

δQr ＝ TdS

dU = TdS pdV

dH = TdS + Vdp

dA = SdT pdV

dG = SdT + Vdp

H =U+pV

A =U-TS

G =H-TS

closed system of constant composition ， reversible process ， volume work only.

Master equation of thermodynamic

dU = TdS － pdV

dH = TdS + Vdp

dA = － SdT － pdV

dG = － SdT + Vdp

TS

U

V

pV

U

S

TS

H

p

V

p

H

S

ST

A

V

p

V

A

T

ST

G

p

Vp

G

T

dyy

Zdx

x

ZdZ

xy

dZ = M dx + N dy

xy y

ZN

x

ZM

,

dZ = M dx + N dy

yxx

N

y

M

dU = TdS － pdVSV V

T

S

p

dH = TdS + VdppS

S

V

p

T

VT T

p

V

S

pTT

V

p

S

dA = － SdT － pdV

dG = － SdT + Vdp

2. Maxwell’s relation

3. Gibbs - helmholtz equation

2T

TT

GTG

Tpp

2

)/(

T

H

T

TG

p

2

)/(

T

U

T

TA

V

2

T

GTS

2T

H

pT

TG

)/(

3.4 Chemical potential 3.4.1 Chemical potential

Total differential

B

)BC,C(,,B

A

)AC,C(,,A)B(,)B(,

d

dddd

nn

G

nn

Gp

p

GT

T

GG

npT

npTnTnp

G ＝ f (T ， p ， nA ， nB……)

Consider a system that consists of a single homogeneous phase whose composition can be varied

ST

G

np

)B(,

Vp

G

nT

)B(,

B

B

)BC,C(,,B

d d d- d nn

GpVTSG

npT

)BCC,(,,BB

def

nPTn

G

B

BBdddd npVTSG

B

BBdddd nVpTSA

B

BBdd-dd nVpSTU

B

BBdddd npVSTH

)BCC,(,,BB

npS

n

H

)BCC,(,,BB

nVT

n

A)BCC,(,,B

B

nVS

n

U

)BCC,(,,BB

nPT

n

G

Chemical potential is an intensive state function.

B of pure substance

G* (T ， p ， nB) = nB G *m,B (T ， p ， )

),(*B,mB

,B

pTGn

G

pT

α

αB

B

αB

α

αα

α

αα dddd nVpSTU

Heterogeneous system

α B

αB

αBdnpdVTdSdU

α B

αB

αBdnVdpTdSdH

α B

αB

αBdnpdVSdTdA

α B

αB

αBdnVdpSdTdG

(1) Condition of phase equilibrium

mequilibriu

sspontaneou0β

BαB

0ddd BαB

βB nnn

T ， pB() B( )

dnB

0d αB

α B

αB n

β

B

α

B Condition of phase equilibrium

β

B

α

B spontaneous

3.4.2 Equilibrium criterion of substance

(2) Condition of chemical reaction equilibrium

Chemical reaction B0 B

B

BBB

BB mequilibriusspontaneou 0dd n

ΣBμB ＜ 0 ， dξ ＞ 0 spontaneous

Homogeneous system dξ

reversible

leireverisibn 0d α

Bα B

αB

For example aA ＋ bB ＝ yY ＋ zZ

aμA+bμB ＝ yμY+zμZ

ΣBμB ＝ 0 ， equilibrium

BB

defA

A—potential of chemical reaction

A ＜ 0 left spontaneous 。

A ＝ 0 at equilibrium ；

A ＞ 0 right spontaneous ；

A= (aμA+bμB) -( yμY+zμZ )

aA ＋ bB ＝ yY ＋ zZ