Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Chapter 15 Introduction to Phase and Chemical Chapter 15 Introduction to Phase and Chemical EquilibriumEquilibrium

OUTLINEOUTLINE•• criteria for phase and chemical equilibrium• determination of thermophysical properties• What is the metastable equilibrium• simultaneous reactions• ionization

EQUILIBRIUMEQUILIBRIUMNo possibility that the system can do any work when it is isolatNo possibility that the system can do any work when it is isolated from its ed from its surroundings surroundings

enginesusystem1 susystem1

wRequirements Requirements A.A. Two subsystems have the same temperature (thermally equilibrium)Two subsystems have the same temperature (thermally equilibrium)B.B. No unbalanced mechanical forces between the two systems (mechaniNo unbalanced mechanical forces between the two systems (mechanical)cal)

Establishment of general equilibrium criteriaEstablishment of general equilibrium criteriaA.A. Here assumes the simple compressible substances undergo chemicalHere assumes the simple compressible substances undergo chemical reactionsreactionsB.B. Gibbs function is very importantGibbs function is very importantC.C. The general criteria : minimum of Gibbs functionThe general criteria : minimum of Gibbs function

EXAMPLE EXAMPLE –– Natural gas well problemNatural gas well problem

GOAL: GOAL: 평형에평형에관한관한정량적정량적관계유도관계유도

A.A. Uniform well temperatureUniform well temperatureB.B. SteadySteady--state flowsstate flows

Mass flow = 0Mass flow = 0Uniform temperatureUniform temperature

0

2 2

0

i e

i e

i e

rev

T T T constm m

V V

W

= = =

=

=

=

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

EQUILIBRIUM BETWEEN EQUILIBRIUM BETWEEN TWO PHASES AT A PURE SUBSTANCETWO PHASES AT A PURE SUBSTANCE

•• pure substance at equilibriumpure substance at equilibrium•• Consider a control mass consisting of two phases (same P and T)Consider a control mass consisting of two phases (same P and T) –– thermal and thermal and mechanical equilibriummechanical equilibrium•• Then, how about the chemical equilibrium ?Then, how about the chemical equilibrium ?

Phase 1 : Phase 1 : nn1 1 and Phase : and Phase : nn22Positive change of moles in phase 1 = Negative change of moles iPositive change of moles in phase 1 = Negative change of moles in phase 2n phase 2

( )( )

2 2

2 2

2 2

2 2

0

0

0

i erev i i i o i e e e o e

i ei i i e e e

i e i e

T T

T

V VW m h gZ T s m h gZ T s

V Vm g gZ m g gZ

m g g g Z gZ

dg gZ

dg

⎛ ⎞ ⎛ ⎞= + + − − + + −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞

= + + − + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= − + −

+ =

∴ =No work and uniform temperature

1 2dn dn= −

1 2 1 2 2 1

1 21 2

, , , , , , , ,T n n P n n P T n P T n

G G G GdG dP dT dn dnP T n n∂ ∂ ∂ ∂⎞ ⎞ ⎞ ⎞= + + +⎟ ⎟ ⎟ ⎟∂ ∂ ∂ ∂⎠ ⎠ ⎠ ⎠

1 2( , , , )G f T P n n=

1 1 2 2 1 1 2

1 2

( ) 0dG g dn g dn dn g gg g

= + = − =

=

, 0T PdG =

g g

f f

f g

Tds dh vdP

Tds dh

g g

= −

=

=

∫ ∫( ) ( )f f f g g g

g f g f

fg fg

fg fg

dg v dP s dT dg v dP s dT

dP v v dT s s

s hdPdT v Tv

= − = = −

− = −

= = ClapeyronClapeyron equationequation

예예) ) 포화증기와포화증기와포화액체의포화액체의주어진주어진온도에서의온도에서의값은값은동일동일

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

WHAT IS METASTABLE EQUILIBRIUM?WHAT IS METASTABLE EQUILIBRIUM?

,,,

A B C D

A A

B B

C C

D D

A B C D

dn ddn ddn ddn d

ν ν ν ν

ν εν εν εν ε

+ +

= −

= −= +

= +

CHEMICAL EQUILIBRIUM (VERY IMPORTANT)CHEMICAL EQUILIBRIUM (VERY IMPORTANT)

•• if point a is reached in the divergent section of the nozzle, nif point a is reached in the divergent section of the nozzle, no condensation o condensation occurs until point b is reachedoccurs until point b is reached•• At point b, abrupt condensation occurs : condensation shockAt point b, abrupt condensation occurs : condensation shock•• MetastableMetastable state : astate : a와와 bb사이는사이는증기지만증기지만실제로실제로온도는온도는포화온도보다포화온도보다낮다낮다•• Dotted lines : possible Dotted lines : possible metastablemetastable states for solidstates for solid--liquidliquid--vapor equilibrium vapor equilibrium •• MetastableMetastable state : state : 상변화에서상변화에서항상항상존재가능성이존재가능성이있음있음..

homogeneous chemical reactionhomogeneous chemical reaction

, 0T PdG =

AssumptionsAssumptions-- Two phases are in equilibrium at given T and P (constant)Two phases are in equilibrium at given T and P (constant)-- A small quantity of one component be transferred from one A small quantity of one component be transferred from one phase to the otherphase to the other-- reaction process toward the right ( infinitesimal amount)reaction process toward the right ( infinitesimal amount)

decreasesdecreases

increasesincreasesdε : change of the degree of reaction: change of the degree of reaction

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

( )

,

,

T P C C D D A A B B

T P

C C D D A A B B

C D A BC D A B

dG G dn G dn G dn G dndG

G d G d G d G d

G G G G d

ν ε ν ε ν ε ν ε

ν ν ν ν ε

= + − −

= + − −

= + − −

0 00 0

0 00 0

0 0 0 0 0

ln ln

ln ln

C DC C D D

TPA B

A A B B

C C D D A A B B

y P y Pg RT g RTP P

dG dy P y Pg RT g RTP P

G g g g g

ν νε

ν ν

ν ν ν ν

⎧ ⎫⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞+ + +⎪ ⎪⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠⎝ ⎠⎪ ⎣ ⎦ ⎪⎣ ⎦= ⎨ ⎬⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎪ ⎪− + − +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎪ ⎪⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎩ ⎭

∆ = + − −

0, 0ln

C D A BC D

A A

C DT P

A A

y y PdG G RT dPy y

ν ν ν νν ν

ν ν ε+ − −⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= ∆ + ⎢ ⎥⎨ ⎬⎜ ⎟

⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

Evaluation of the change in the Gibbs function (process to the rEvaluation of the change in the Gibbs function (process to the right in the amount )ight in the amount )dε

, , , , , ,B B BA A AT P n T P n T P n

G H TS

G H STn n n

= −

⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂= −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

For ideal gases only in this lectureFor ideal gases only in this lecture

0

0

0 00

0

00

0 0 00 0

ln

ln ln

A A

A A A ATP ATP

AATPA ATP Y P

A AATP ATPA ATP

G H TS h h

y PS s s RP

y P y PG h Ts RT g RTP P

=

= − = =

⎛ ⎞= = − ⎜ ⎟⎝ ⎠⎛ ⎞ ⎛ ⎞= − + = +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

For compound A, For compound A,

Change in Gibbs function of each component in the mixtureChange in Gibbs function of each component in the mixture

The partial The partial molalmolalGibbs functionGibbs function

At temperature and the standardAt temperature and the standard--state pressurestate pressureOnly function of temperatureOnly function of temperatureAssume that the reactants are initially separated and the producAssume that the reactants are initially separated and the products are ts are finally separated at T and reference pressurefinally separated at T and reference pressure

Change in Gibbs Change in Gibbs funcionfuncion

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Chemical equilibrium state : Chemical equilibrium state :

0, 0ln

C D A BC D

A A

C DT P

A A

y y PdG G RT dPy y

ν ν ν νν ν

ν ν ε+ − −⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= ∆ + ⎢ ⎥⎨ ⎬⎜ ⎟

⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

, 0T PdG =

0

0

0

ln lnC D A BC D

A A

C D A BC D

A A

C D

A A

C D

A A

y y P G KP RTy y

y y PKPy y

ν ν ν νν ν

ν ν

ν ν ν νν ν

ν ν

+ − −

+ − −

⎡ ⎤ ∆⎛ ⎞ = − =⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

⎛ ⎞= ⎜ ⎟⎝ ⎠

Chemical equilibrium constant (Chemical equilibrium constant (화학평형상수화학평형상수))Negative and large Negative and large lnln K leads to a very small value of KK leads to a very small value of KIt means the large and positive change in Gibbs functionIt means the large and positive change in Gibbs function

Small K : reaction is shifted to the left (the Small K : reaction is shifted to the left (the values of RHS are smaller than those of LHS)values of RHS are smaller than those of LHS)K = 1 : zero change in Gibbs functionK = 1 : zero change in Gibbs function

NOTE :NOTE :A. Effects of T on reactionsA. Effects of T on reactions

For higher T, but same shift in Gibbs function, For higher T, but same shift in Gibbs function, lnln K is smaller, K is closer to unityK is smaller, K is closer to unity(reaction is centered)(reaction is centered)For lower T, the reaction is shifted toward the side with the smFor lower T, the reaction is shifted toward the side with the smallest Gibbs functionallest Gibbs function

B. Influence of P : only if B. Influence of P : only if

In the case that moles of products > those of reactants In the case that moles of products > those of reactants P/P0 >1 : reduces the RHS concentrations (because of a fixed K aP/P0 >1 : reduces the RHS concentrations (because of a fixed K at a given T)t a given T)

ArgumentsArgumentsA higher pressure pushes the reaction toward the side with fewerA higher pressure pushes the reaction toward the side with fewer molesmolesA lower pressure pushes the reaction toward the side with more mA lower pressure pushes the reaction toward the side with more molesoles

0

lnG KRT∆

− =

0C D A Bν ν ν ν+ − − ≠

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Table A.11

0 0 0 0 0, ,

0 0 0

ln

III I II

III I II

III III RHS III LHS I II

III I IIIII I II

a bIII I II

LHS aLHS bLHSRHS aRHS bRHS

G G G a G b G

G G GK a b aK bKRT RT RT

K K K

= +

= +

∆ = ∆ − ∆ = ∆ + ∆

∆ ∆ ∆= − = − − = +

=

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

SIMULTANEOUS REACTIONS SIMULTANEOUS REACTIONS

1

1

2

2

01 10

02 20

ln

ln

C D A BC D

A B

M N A LM N

A L

C DTP

A B

M N

A L

y y PdG G RT dPy y

y y PG RT dPy y

ν ν ν νν ν

ν ν

ν ν ν νν ν

ν ν

ε

ε

+ − −

+ − −

⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= ∆ + ⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞+ ∆ + ⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

1

2

1 21 2 1

2 1

1 2 2

, ,

, ,, , ,

A B C D

A L M N

A A A B B

L L C C

D D M M N N

A B C D

A L M N

dn d d dn d

dn d dn ddn d dn d dn d

ν ν ν ν

ν ν ν ν

ν ε ν ε ν ε

ν ε ν εν ε ν ε ν ε

+ +

+ +

= − − = −

= − = +

= + = + = +

•• So far, it was assumed that there was only a single reactionSo far, it was assumed that there was only a single reaction•• Here deals with simultaneous reactions, (more than one chemicaHere deals with simultaneous reactions, (more than one chemical reaction)l reaction)•• More general situationMore general situation

( ) ( )1 2

,

, 1 1

T P i i

T P C C D D A A B B M M N N A A B B

dG G dn

dG G G G G d G G G G dν ν ν ν ε ν ν ν ν ε

=

= + − − + + − −

∑

•• Consider a mixture of substances A,B,C,D,L,M, and NConsider a mixture of substances A,B,C,D,L,M, and N

Change in Gibbs function of each component in the mixtureChange in Gibbs function of each component in the mixture

00ln i

iiy PG g RTP

⎛ ⎞= + ⎜ ⎟⎝ ⎠

Therefore, the partial Therefore, the partial molalmolal Gibbs functions in terms of Gibbs functions in terms of

The change in Gibbs function can be rewritten as The change in Gibbs function can be rewritten as

1

2

0 0 0 0 01

0 0 0 0 02

C D A BC D A B

M N A LM N A L

G g g g g

G g g g g

ν ν ν ν

ν ν ν ν

∆ = + − −

∆ = + − −

The standardThe standard--state change in Gibbs function for each reactionstate change in Gibbs function for each reaction

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

1

1

2

2

0 01 1

1 2

1 0

2 0

ln , ln

C D A BC D

A B

M N A LM N

A L

C D

A B

M N

A L

G GK KRT RT

y y PKPy y

y y PKPy y

ν ν ν νν ν

ν ν

ν ν ν νν ν

ν ν

+ − −

+ − −

∆ ∆= − = −

⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

1

1

2

2

01 10

02 20

ln

ln

C D A BC D

A B

M N A LM N

A L

C DTP

A B

M N

A L

y y PdG G RT dPy y

y y PG RT dPy y

ν ν ν νν ν

ν ν

ν ν ν νν ν

ν ν

ε

ε

+ − −

+ − −

⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞= ∆ + ⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫⎡ ⎤⎪ ⎪⎛ ⎞+ ∆ + ⎢ ⎥⎨ ⎬⎜ ⎟⎝ ⎠⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

The change in Gibbs function of the system at constant The change in Gibbs function of the system at constant T, PT, P for infinitesimal degrees of for infinitesimal degrees of reaction of both reactions 1 and 2reaction of both reactions 1 and 2

Chemical Equilibrium ConditionChemical Equilibrium Condition , 0T PdG =

Reactions 1 and 2 are independent : bracketed terms must be zeroReactions 1 and 2 are independent : bracketed terms must be zero

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University

IONIZATIONIONIZATION

2 2N N

N N e

N N e

+ −

+ ++ −

+

+

Consider the equilibrium of systems that are made of ionized gasConsider the equilibrium of systems that are made of ionized gases, or plasmaes, or plasmaMolecular dissociation at high temperatureMolecular dissociation at high temperature

1 1 1

0

0

0 0 0 0

ln

A e

A

AA e

A A ey y PK

y P

GKRT

G g g g

+ −

+ −

+ −

+ −

+

⎛ ⎞= ⎜ ⎟⎝ ⎠

∆= −

∆ = + −

At about 3000 to 10 000 KAt about 3000 to 10 000 K

At higher temperature, atoms lose an electronAt higher temperature, atoms lose an electron

Free electronFree electron

Lecture note for general thermodynamics, 2003

School of Mechanical Engineering, ChungAng University