byeong-joo lee multi-component homogeneous system: solution thermodynamics byeong-joo lee postech -...

Byeong-Joo Lee http://cmse.postech.ac.kr

Multi-component Multi-component Homogeneous Homogeneous

System: System: Solution Solution ThermodynamicsThermodynamicsByeong-Joo LeeByeong-Joo Lee

POSTECH - MSEPOSTECH - [email protected]@postech.ac.kr


Thermodynamic Properties of Gases Thermodynamic Properties of Gases - mixture of ideal gases- mixture of ideal gases

Mixture of Ideal Gases

Definition of Mole fraction: xi

Definition of partial pressure: pi

Partial molar quantities:

Pxp ii

1 mole of ideal gas @ constant T:

1

212 ln),(),(

P

PRTTPGTPG

PRTddPP

RTVdPdG ln

PRTTGTPG o ln)(),( PRTGG o ln

,,,,

'

kj nnPTii n

QQ

iiG

i

compT

i VP

G

,iiQnQ '


Thermodynamic Properties of Gases Thermodynamic Properties of Gases - mixture of ideal gases- mixture of ideal gases

Heat of Mixing of Ideal Gases

io

i HH

0' io

ii

iii

mix HnHnH

PRTxRTGG iio

i lnln

T

TG

T

TG io

i

)/()/(

Entropy of Mixing of Ideal Gases

Gibbs Free Energy of Mixing of Ideal Gases

mixmixmix STHG '''

iii

io

ii

iii

mix xRTnGnGnG ln'

iii

mix xRnS ln'


fRTddG ln

1P

f as 0P

For Equation of state P

RTV

fRTdVdP ln dPRTP

fd

ln

RT

P

P

f

P

f

PPP

0

lnlnid

RTP

P

P

RT

PV

RT

Pe

P

f 1/

Thermodynamic Properties of Gases Thermodynamic Properties of Gases - Treatment of nonideal gases- Treatment of nonideal gases

Introduction of fugacity, f

fRTGG o ln

※ actual pressure of the gas is the geometric mean of the fugacity and the ideal P ※ The percentage error involved in assuming the fugacity to be equal to the

pressure is the same as the percentage departure from the ideal gas law


dPPRT

VdP

RTP

fd

1

ln

RT

PVZ dP

P

Z

P

fd

1ln

dPP

Z

P

f PP

PPP

1ln

0

PdRTP

fdRTfRTddG lnlnln

JRTP

fRTG 112971137376150lnln

150

Thermodynamic Properties of Gases Thermodynamic Properties of Gases - Treatment of nonideal gases- Treatment of nonideal gases

Alternatively,

Example) Difference between the Gibbs energy at P=150 atm and P=1 atm for 1 mole of nitrogen at 0 oC


Solution Thermodynamics Solution Thermodynamics - Mixture of Condensed Phases- Mixture of Condensed Phases

Vaper A: oPA

Condensed Phase A

Vaper B: oPB

Condensed Phase B

+ →

Vaper A+ B: PA + PB

Condensed Phase A + B

condensedA

ovaporA

o GG condensedB

ovaporB

o GG condensedA

vaporA GG

condensedB

vaporB GG

io

ii

iii

mix GnGnG 'i

oi

ii p

pRTn ln for gas


oiie kpr )(

Solution Thermodynamics Solution Thermodynamics - ideal vs. non-ideal solution- ideal vs. non-ideal solution

oiii pxp iiie kpxr )(

oiiii pxkp o

iiie

iei px

r

rp

)(

)('iiie kpxr )('

Ideal Solution

Nonideal Solution


Solution Thermodynamics Solution Thermodynamics - Thermodynamic Activity- Thermodynamic Activity

oi

ii f

fa

Thermodynamic Activity of a Component in Solution

oi

ii p

pa

1

212 ln),(),(

P

PRTTPGTPG

→ ix for ideal solution

Draw a composition-activity curve for an ideal and non-ideal solution

Henrian vs. Raoultian


),,,,,('' 21 cnnnPTQQ

i

nnPTi

c

innTnnP

dnn

QdP

P

QdT

T

QdQ

ijkjkj

,,1,,,,,,

''''

▷ Partial Molar Quantity

ij nnPTii n

QQ

,,

'

kk

c

kPT dnQdQ

1

,'

kk

c

k

nQQ

1

'

01

kk

c

k

Qdn

Solution Thermodynamics Solution Thermodynamics - Partial Molar Property- Partial Molar Property

01

kk

c

k

Qdx

Gibbs-Duhem EquationGibbs-Duhem Equation ▷ Molar Properties of

Mixture

kk

c

k

dXQdQ

1

k

c

kk QXQ

1


''' QQQ omix

kko

c

k

o nQQ

1

'

Qo

kk

c

kkk

ok

c

kmix nQnQQQ

11

)('

Solution Thermodynamics Solution Thermodynamics - Partial Molar Quantity of Mixing- Partial Molar Quantity of Mixing

definition of solution and mechanical mixing

is a pure state value per molewhere

왜 partial molar quantity 를 사용해야 하는가 ?


Solution Thermodynamics Solution Thermodynamics - Partial Molar Quantities- Partial Molar Quantities

)lnln(' BBAAphaseref

Bo

Bphaseref

Ao

A ananRTGnGnG

)lnln( BBAABo

BAo

Am axaxRTGxGxG

)lnln(' BBAABo

BAo

A ananRTGnGnG

)lnln()lnln( BBAABBAABo

BAo

Am xxRTxxxxRTGxGxG

ABBABBAABo

BAo

Am LxxxxxxRTGxGxG )lnln(


222 )1(

dX

dQXQQ exampl

e) 21XaXH mix

• Partial molar & Molar Gibbs energy

Gibbs energy of mixing vs. Gibbs energy of formation

• Graphical Determination of Partial Molar Properties: Tangential Intercepts

• Evaluation of a PMP of one component from measured values of a PMP of the other

02211 QdXQdX 21

21 Qd

X

XQd

22

2

1

2

021

2

01

2

2

2

2

dXdX

Qd

X

XQd

X

XQ

X

X

X

X

212 aXH

Solution Thermodynamics Solution Thermodynamics - Partial Molar Quantities- Partial Molar Quantities

Evaluation of Partial Molar Properties in 1-2 Binary System• Partial Molar Properties from Total Properties

)lnln( BBAABo

BAo

Am axaxRTGxGxG ii

oi

Mi aRTGGG ln

example)


jjjj nPTknVTknPSknVSkk n

G

n

F

n

H

n

U

,,,',,,',','

''''

k

nPTkk G

n

G

j

,,

'

jj nPTk

k

nPSkk n

HH

n

H

,,,,'

''

kkG knP

kk

TS

,

knT

kk

PV

,

knP

kkk

TTH

,

kk nT

k

nP

kkk

PP

TTU

,,

knT

kkk

PPF

,

22

2

1

2

01

2

2

dXdX

d

X

XX

X

dPVdTSdGd kkkk

Solution Thermodynamics Solution Thermodynamics - Chemical Potential as a Partial Molar Quantity- Chemical Potential as a Partial Molar Quantity

※ Relationships among Partial Molar Quantities: Chapter 5 에서 언급한 Thermodynamic Relationship 들이 모두 적용됨


kkk xRTaRT lnln

kkk xax )1lim(

ko

kkk xrax )0lim(

Solution Thermodynamics Solution Thermodynamics - Non-Ideal Solution- Non-Ideal Solution

▷ Activity Coefficient

▷ Behavior of Dilute Solutions

2

)ln()/(

T

H

T

R

T

TG Mii

Mi

Mi

i HT

R

)/1(

)ln(


1.Gibbs energy of formation 과 Gibbs energy of mixing 의 차이는 무엇인가 ?

2. Solution 에서 한 성분이 Henrian 또는 Raoultian 거동을 한다는 것을 무엇을 의미하는가 ? Molar Gibbs energy 가 다음과 같이 표현되는 A-B 2 원 Solution phase 에서 각 성분은 dilute 영역에서는 Henrian 거동을 , rich 영역에서는 Raoultian 거동을 보인다는 것을 증명하시오 .

LxxxxxxRTGxGxG BABBAABo

BAo

Am }lnln{

ExampleExample


i

ii QdX 0

0lnln BBAA adXadX

BA

BA ad

X

Xad loglog

BA

BXXata

XataXXA adX

Xa

AAB

ABAA

log)(loglog

1log

Solution Thermodynamics Solution Thermodynamics - Use of Gibbs-Duhem Relation - I- Use of Gibbs-Duhem Relation - I


0lnln BBAA dXdX

BA

BA d

X

Xd lnln

BA

BXXat

XatXXA dX

XAAB

ABAA

ln)(ln

ln

1ln

Solution Thermodynamics Solution Thermodynamics - Use of Gibbs-Duhem Relation - II- Use of Gibbs-Duhem Relation - II


2)1(

ln

i

ii X

AB

XX

XBABA dXXXAA

A

1ln

BA

BA d

X

Xd lnln

BABABB dXXdXX 2

Solution Thermodynamics Solution Thermodynamics - Introduction of - Introduction of αα-function-function


Fe-Ni Fe-CuFe-Ni Fe-Cu

Solution Thermodynamics Solution Thermodynamics - Composition Dependence of - Composition Dependence of αα-function-function


• Margules, 1895.Margules, 1895.

333

1222

11ln BBBA XXX

333

1222

11ln AAAB XXX

• Hildebrand, 1929. Hildebrand, 1929. (using van Laar Equation)(using van Laar Equation)

2'ln AB XRT 2'ln BA XRT

Solution Thermodynamics Solution Thermodynamics - Regular Solution Model- Regular Solution Model

ABBABBAABo

BAo

Am xxxxxxRTGxGxG )lnln(


ABABBBBBAAAABA EWEWEWE

221

AAA NzxW 2

21

BBB NzxW

BAAB xNzxW

])2[(2 BBAAABBABBBAAABA EEExxExEx

NzE

Solution Thermodynamics Solution Thermodynamics - Quasi-Chemical Model, Guggenheim, 1935.- Quasi-Chemical Model, Guggenheim, 1935.


Sn-In Sn-BiSn-In Sn-Bi


ABBAxsm xxG


Sn-Zn Fe-NiSn-Zn Fe-Ni

Solution Thermodynamics Solution Thermodynamics - Sub-Regular Solution Model- Sub-Regular Solution Model

])([ 10ABABABBA

xsm xxxxG


• Composition and temperature dependence of Composition and temperature dependence of ΩΩ

• Extension into ternary and multi-component systemExtension into ternary and multi-component system

• Sublattice ModelSublattice Model

• Inherent InconsistencyInherent Inconsistency

ABBABBAABo

BAo

Am xxxxxxRTGxGxG )lnln(


byeong-joo lee multi-component homogeneous system: solution thermodynamics byeong-joo lee postech -...

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