models for the excess gibbs energy - tongji...

Models for the Excess Gibbs Energy

Models for the Excess Gibbs Energy

• GE/RT is a function of T, P, and composition,

• Therefore,

• But for liquids at low to moderate pressures it is a very weak function of P. Therefore the pressure dependence of activity coefficients is usually neglected.

i

E

i RTG ln

i

ii

E

xRT

Gln

Models for the Excess Gibbs Energy

• Thus,

• For binary systems, the function most often represented by a power series in x1

N

E

xxxgRT

G,,, 21

2

11

21

cxbxaRTxx

GE

Models for the Excess Gibbs Energy

• Since x2 = 1 - x1, therefore,

• This is know as the Redlich/Kister expansion which is the most commonly used polynomial in regular and subregular solution models

2

2121

21

xxCxxBARTxx

GE

Regular Solution

• Recall

• Let

• Regular solution is an non ideal solution with small deviation from ideal solution

• Regular solution is defined as a solution possesses an enthalpy of mixing but no entropy of mixing.

i

RS

i

RS

i GGG

ii

id

i xRTGG ln

Regular Solution

• That is

• Therefore,

RS

i

id

i

RS

i

id

i

RS

i HHHGG

idRS

RSRSRS

STH

STHG

Enthalpy of Regular Solution

• The word regular implies that the molecules mix in a completely random manner, which means that there is no segregation or preference.

• Consider two neighbors in a solution. The probability that one of the neighbors is A or B is simply

B

BA

BB

A

BA

AA

xNN

NP

xNN

NP

Enthalpy of Regular Solution

• Therefore the probability that a given ``bond'' is A—B type is

• If each atom has z nearest neighbors, the number of bonds, total, is

BA

BA

BABAABAB xx

NN

NNPPP 2

22

2

,2

zBNN

zB

total

BA

total

Enthalpy of Regular Solution

• The bond density of A-B type is:

• If the energy per bond is w(AB) then the enthalpy density (due to the A-B bonds) is:

• Similarly

BA

total

ABAB xzxBPB

BAAB

RS

AB xxzwH

BBBBzRS

BB

AAAAAAAzRS

AA

xxwH

xPxxwH

2

2

2 ,

Enthalpy of Regular Solution

• Therefore,

• Since

• Therefore,

puresol

mixing HHH

ABBA xxxx 1 and 1

RS

BA

BBAAABBA

RS

AB

wxxz

wwwxxz

H

2

22

Regular Solution

• According to the definition of Regular solution, entropy of mixing equals to ideal solution

• Therefore,

BA

RS

BA

id

AB

RS

AB

RS

AB

xxRTwxxz

STHG

lnln2

BA

RS

BA

RSExxwxx

zG

2

,

Regular Solution

2

,

A

BAABA

B

BAABA

B

RS

AB

A

RS

AB

RSE

B

x

xxxxx

x

xxxxx

x

HxHH

Gibbs-Duhem

Regular Solution

• Since

• Therefore,

i

E

i

RSE

i RTGG ln,

2,

2,

ln

ln

AB

RSE

B

BA

RSE

A

xRTRT

G

xRTRT

G

Otto Redlich AND A. T. Kister Expansion

• Recall

• For binary system,

i

ii

E

xRT

Gln

2211 lnln xxRT

GE

2

1

1

ln

dx

RT

Gd

E

Otto Redlich AND A. T. Kister Expansion

• Therefore,

• Let

1

12

1

11

ln

1ln

dx

RT

Gd

xRT

G

dx

RT

Gd

xRT

G

E

E

E

E

Used in HE,RS

RT

GQ

E

Otto Redlich AND A. T. Kister Expansion

• Since Q = 0 for x = 0 and x1 = 1, each term must contain the factor x1(1 – x1).

• Introduce the expansion,

• The coefficients may be determined by

• This is know as the Redlich/Kister expansion

2

1111 12121 xDxCBxxQRT

GE

1

2

1

1

11

ln

1

ln

)1( xxxx

Q

Otto Redlich AND A. T. Kister Expansion

• Or

• Case 1, ideal solution

• Case 2, B ≠ 0, C,D,…=0, a form of regular

• Case 3, B ≠ 0, C ≠ 0, D,E,…=0, subregular

• ···

0ln2

1

11621ln 111

12

1 xxCxBdx

dQ

Criteria for Equilibrium

Huang, Min

Chemistry Department

Tongji University

Criteria for Phase Equilibrium

• Assume at the beginning that we know

nothing about the conditions under which

two phases can be in equilibrium.

• The only things we know from the Second

Law are the criteria for equilibrium under

certain conditions. That is,

• That is: the entropy seeks a maximum, Gibbs free energy seeks a minimum

0 ,0 ,, PTVE dGdS

Criteria for Phase Equilibrium

• Consider a multiphase multicomponent system.

– Each phase comprises a different subsystem.

– Repartitionings of extensive variables can be accomplished by shuffling portions of the extensive variables between the different phases,

• Repartitioning of the energy would correspond to changing the E(a) ’s but keeping the total fixed.

a

a

1

EE

Criteria for Phase Equilibrium

• We have

• where is the number of moles of species i in phase a.

a

a

a

a

a

a

1

1

1

ii nn

VV

SS

a

in

Criteria for Phase Equilibrium

• Form the definition of d E as the first-order variational displacement of E,

v different phases, r different species

• From the Second Law, at the equilibrium, E must be it’s minimum, therefore, any deviation must

v r

i

ii nVpSTE1a

aaaaaa dddd

0,, inVSEd

Criteria for Phase Equilibrium

• Consider repartition processes, while keeping the total S, V, ni fixed.

• Which requires

aaa

inVS ,,

rin

V

S

i ,,2,1for 0

0

0

1

1

1

a

a

a

a

a

a

d

d

d

Criteria for Phase Equilibrium

• Consider a two-phase system

Criteria for Phase Equilibrium

• The consistency of S, V, ni corresponds to

21

2

2

1

2

2

1

1

1

21

21

rr nn

nn

nn

VV

SS

dd

dd

dd

dd

dd

Criteria for Phase Equilibrium

• Therefore, the first order displacement of E at constant S, V, ni is,

• Since these variables are independent and their variations are uncoupled, the only solution to the

1

1

21

121121

,,

0

i

r

i

ii

inVS

n

VppSTTE

d

ddd

0,, inVSEd

Criteria for Phase Equilibrium

which will guarantees is

• For multiphase,

0,, inVSEd

ri

pp

TT

ii ,,2,1 ,21

21

21

ri

ppp

TTT

v

iii

v

v

,,2,1 ,,

,

,

21

21

21

Further Discuss

• Since (1) = (2) guarantees mass equilibrium, it is interesting to study what action a gradient in produces.

Further Discuss

• Consider the system is prepared initially with (1) > (2) , Mass flow will bring it to equilibrium with . If no work is done on the total system and there is no heat flow into and out the system, S > 0.

• Assuming displacements from equilibrium are small

• S > 0, when (1) > (2), implies n(1) < 0, that is matter flows from high to low .

21

finalfinal

. since

21

121

22

11

nn

nTT

nT

nT

S

Further Discuss

• The gradients in (or more precisely, gradients in /T) produce mass flow.

• is a generalized force.

• In the variant statement of the Second Law, we see a similar form,

• is a generalized force that causes heat to flow from high T to low T.

T/

T/1

The Phase Rule

• Suppose v phases are coexisting in equilibrium. The conditions for equilibrium

• where 1 ≤ a < ≤ and 1 ≤ I ≤ r

• There are r(-1) independent equations which couple together 2 + (r-1) different intensive variables (T, p, and the mole fractions for each phase).

aaa 1111 ,,,,,,,, riri xxpTxxpT

The Phase Rule

• Hence, the thermodynamic degrees of freedom (the number of independent intensive thermodynamic variables) is

• This is the Gibbs phase rule.

vr

vrrvf

2

)1()1(2

Topics in Phase Equilibria

Binary Solid-liquid Equilibria of n-

decanol and n-dodecanol

Marcos Serra Gómez-Nicolau

Tutor: Dr. Huang Min

Coordinator: Dr. Hu Zhong-Hua

Department of Chemistry

Tongji University

January 2010

Index

• Introduction

• Thermodynamics

• CALPHAD

• Thermo-Calc

• Possible causes of discrepancy

• Conclusions

• Appendix: Direct Contact Melt Crystallization

Introduction

Capric alcohol, 1-decanol, decan-1-ol, decyl alochol

Lauric alcohol, lauryl alcohol, 1-dodecanol, dodecan-1-ol

Natural: from fats, oils and waxes of plant or animal origin

Synthetic: from petrochemicals

CH3(CH2)nCH2OH

n = 4-20

Introduction

Fatty alcohols from natural sources

decanol dodecanol

osmanthus absolute

(9.20%) violet leaf (3.60%)

coriander leaf oil (0.89-

2.09%)

cochlospermum planchonii

oil (0.70%)

ambrette seed oil (0.60%) cochlospermum tinctorium

oil (0.70%)

frankincense oil from

Somalia (0.40%) ambrette seed oil (0.30%)

citronella oil from Zimbabwe

(0.33%)

coriander leaf oil (0.09-

0.18%)

kachur oil (0.10%)

• Plant or animal origin

• Wax esters from the sperm oil of whale

• Plant or animal origin

Ambrette seed

Coriander leaf

Violet leaf

• Surfactants (ca. 70-75%)

• Oil additives

• Cosmetics and pharmaceutical preparations

• Polymer processing

• Preservatives for food

Introduction

Uses of fatty alcohols

Emulsions and microemulsions

(cosmetics creams and lotions)

Fragrance Materials = fragrance

compound + water + alcohol

(creams, deodorants, lotions…)

Decanol Dodecanol

Odour type Fatty Waxy

Odour strength Medium Medium

Odour description at 100% Fatty waxy floral orange sweet

clean watery

Earthy soapy waxy fatty honey

coconut

Taste description Aldehydic waxy green fatty tart

perfumistic

Soapy waxy aldehydic earthy

fatty

Thermodynamics Real binary liquid mixture Gibbs energy

For the calculation of phase equilibria, it is necessary to minimize the total Gibbs

Energy:

),,,(),,(),( 0 xTTGxTPGxTGG CmPT

minimum1

F

i

ii GnG

Thermodynamics

C

i

ii PTGxxPTG1

),(),,(

i

C

i

i

C

i

ii

IMxxRTPTGxxPTG ln),(),,(

11

Ideal mixture:

Real mixture: ex

i

C

i

i

C

i

ii GxxRTPTGxxPTG

ln),(),,(11

Real binary liquid mixture Gibbs energy

Thermodynamics

ex

i

C

i

i

C

i

ii GxxRTPTGxxPTG

ln),(),,(11

2

212121

0

2121 xxCxxBAxxxxaxxGn

i

i

i

ex

Redlich-Kister polynomial expansion:

Temperature-dependent parameters:

TVVA 21

Variables to be optimized

with Thermo-Calc

Real binary liquid mixture Gibbs energy

T

T

i

T

T

iTiTiiii

trtr

trtrdT

T

CpTdTCpSTHSTHPTG

*

*

,,),(

Thermodynamics

Real binary liquid mixture Gibbs energy

ex

i

C

i

i

C

i

ii GxxRTPTGxxPTG

ln),(),,(11

3

4

2

321

*TCTCTCCCpi

tr

Ti

TiT

HS tr

tr

,

,

4433221,

443322

3423211

,

1

4321262

32lnln

trtrtrtrTi

trtrtrtr

Ti

i

TC

TC

TC

TCHTC

TC

TC

TTC

TC

TCTCCTr

HTTCG

tr

tr

Thermodynamics

360

370

380

390

400

410

420

430

440

450

460

470

480

490

500

510

520

530

280 310 340 370 400 430 460 490 520 550 580

T (K)

Cp

(J

·K-1

·mo

l-1)

440

450

460

470

480

490

500

510

520

530

540

550

560

570

580

590

600

610

300 320 340 360 380 400 420 440 460 480

T (K)

Cp

(J

·K-1

·mo

l-1)

360

370

380

390

400

410

420

430

440

450

460

470

480

490

500

510

520

530

280 310 340 370 400 430 460 490 520 550 580

T (K)

Cp

(J

·K-1

·mo

l-1)

Růžička (2004)

Zábranský (1990)

Cees (2003)

Equation

Dependence of heat capacities with temperature

20093225.0

7.32270589055.904223.00156.215.3163 T

TTnTnCp

3

4

2

321

*TCTCTCCCpi

Thermodynamics

y = -1,5251E-06x3 - 3,6229E-03x2 + 4,2556E+00x - 4,7247E+02

R2 = 9,9752E-01

440

450

460

470

480

490

500

510

520

530

540

550

560

570

580

590

600

610

300 320 340 360 380 400 420 440 460 480

T (K)C

p (

J·K

-1·m

ol-1

)

y = 1,2252E-05x3 - 1,8980E-02x

2 + 9,7947E+00x - 1,1869E+03

R2 = 9,9316E-01

360

370

380

390

400

410

420

430

440

450

460

470

480

490

500

510

520

530

280 310 340 370 400 430 460 490 520 550 580

T (K)

Cp

(J·K

-1·m

ol-1

)

Dependence of heat capacities with temperature

y = -1,5251E-06x3 - 3,6229E-03x2 + 4,2556E+00x -

4,7247E+02

R2 = 9,9752E-01

440

450

460

470

480

490

500

510

520

530

540

550

560

570

580

590

600

610

300 320 340 360 380 400 420 440 460 480

T (K)

Cp

(J·K

-1·m

ol-1

) Růžička (2004)

Zábranský (1990)

Cees (2003) [8]

Equation (2)

Polynomial (Růžička (2004))

3522.3*10·2252.110·8980.17947.910·1869.1 TTTCpDE

36232*10·5251.110·6229.32556.410·7247.4 TTTCp DO

Thermodynamics

Gibbs energy functions

4433221,

443322

3423211

,

1

4321262

32lnln

trtrtrtrTi

trtrtrtr

Ti

i

TC

TC

TC

TCHTC

TC

TC

TTC

TC

TCTCCTr

HTTCG

tr

tr

3522.3*10·2252.110·8980.17947.910·1869.1 TTTCpDE

36232*10·5251.110·6229.32556.410·7247.4 TTTCp DO

5463323310·0606.110·0210.110·1633.38974.410·9210.5ln10·1869.1

TTTTTTG DE

4473423210·7320.210·2709.110·0382.61278.210·2066.2ln10·7247.4

TTTTTTG DO

KDETm 1.280)(

1·66.37)(

molkJDEH fus

KDOTm 2.300)(

1·17.40)(

molkJDOH fus

CALPHAD

ex

i

C

i

i

C

i

ii GxxRTPTGxxPTG

ln),(),,(11

assessment

ex

binGBinary:

(A, B)

Ternary:

(A, B, C)

Quaternary:

(A, B, C, D)

Extrapolation and

assessment:

ex

binG

ex

terG

Extrapolation and

assessment:

ex

ter

ex

bin GG

ex

quaG

ex

binG

ex

terG

ex

ter

ex

bin GG

ex

quaG

Thermo-Calc

What is Thermo-Calc software?

Thermo-Calc is a general and flexible software for all kinds of thermodynamic and phase

diagram calculations, which is based upon a powerful Gibbs Energy Minimizer. It is

especially designed for systems with strongly non-ideal phases.

Thermo-Calc also provides a unique tool (the PARROT module) for critical assessment

based upon varied experimental data such as EOS, phase equilibria, phase diagrams and so

forth.

Thermo-Calc

Optimization Module (PARROT)

Gibbs Energy System Module

(GES)

Assessment

Thermo-Calc

Variable Value RSD

V11 3.57729450·104 1.38374758·101

V12 -1.32979526·102 2.88416718

V15 -1.14258537·103 7.16573950·101

V16 5.70769803·101 1.40175298·10-2

Reduced sum of squares: 1.0359·101

21xAxGex

TVVAphaseSolid

TVVAphaseLiquid

1615

1211

:

:

Thermo-Calc

Possible causes of discrepancy

Possible causes of discrepancy

1. Fusion enthalpies

1

1

·74.37)(

·79.28)(

molkJDOH

molkJDEH

fus

fusvs

1·66.37)(

molkJDEH fus

1·17.40)(

molkJDOH fus

4463323310·7192.910·0210.110·1633.38974.410·8893.5ln10·1869.1

TTTTTTG DE

4473423210·4890.210·2709.110·0382.61278.210·1985.2ln10·7247.4

TTTTTTG DO

Reduced sum of squares: 4.8785

Variable Value RSD

V11 8.03627948·103 2.83742795

V12 -3.31208687·101 2.48241825

V15 -3.23535636·103 2.67013956·104

V16 2.65182804·102 1.69713840·104

Possible causes of discrepancy

1. Fusion enthalpies

Group coefficients

Acyclic hydrocarbon groups Group value Gi Ci Cj

Primary sp3 carbon atom CH3[C] 4.38 1.0 1.0

Secondary sp3 carbon atom CH2[C2] 2.25 1.0 1.0

Tertiary sp3 carbon atom CH[C3] -3.87 1.0 0.69

Quaternary sp3 carbon atom C[C4] -9.25 1.0 0.67

Group coefficient (CK)

Functional group (k) Group

value (Gk) C1 C2 C3 C4

Hydroxyl group

Alcohol 0.27 1.0 12.6 18.9 26.4

Reduced sum of squares: 3.8364·10-3

Variable Value RSD

V11 2.78843867·104 6.63622105

V12 -1.01597519·102 6.62042326

V15 2.73701059·101 7.77898557·106

V16 5.56925163·101 1.33566753·104

11··9.24)(

molKcalDES fus

11·4.29)(

molKcalDOS fus

tr

Ti

TiT

HS tr

tr

,

,

1

1

·558.36)(

·181.29)(

molkJDOH

molkJDEH

fus

fus

k

kKk

j

jjj

i

iiifus GCnGCnGCnS

Possible causes of discrepancy

2. Heat capacities

100

150

200

250

300

350

400

450

500

550

200 250 300 350 400 450 500

T (K)

Cp

(J·K

-1·m

ol-1

)

old formula

Růžička (2004)

360

370

380

390

400

410

420

430

440

450

460

470

290 300 310 320 330 340 350 360 370 380

T (K)

Cp

(J·K

-1·m

ol-1

)

y = -5,7498E-05x3 + 5,6853E-02x2 - 1,7466E+01x +

2,0515E+03

R2 = 1,0000E+00360

370

380

390

400

410

420

430

440

450

460

470

290 300 310 320 330 340 350 360 370 380

T (K)

Cp

(J·K

-1·m

ol-1

)Růžička (2004)

cubic regression

y = 1,1836E+00x + 2,1886E+01

R2 = 9,9872E-01

360

370

380

390

400

410

420

430

440

450

460

470

290 300 310 320 330 340 350 360 370 380

T (K)

Cp

(J·K

-1·m

ol-1

)

Růžička (2004)

linear regression

Possible causes of discrepancy

2. Heat capacities

100

140

180

220

260

300

340

380

420

460

500

540

580

620

200 250 300 350 400 450 500

T (K)

Cp

(J·K

-1·m

ol-1

)

old formula

Růžička (2004)

new linear formula

new cubic formula

TCpDE 1836.110·1886.21*

TCp DO 3169.110·5191.41*

4212110·4901.110·9180.510·4229.3ln10·1886.2

TTTTG DE

4212110·1420.310·5845.610·5875.5ln10·5191.4

TTTTG DO

Reduced sum of squares: 8.6490·10-3

Variable Value RSD

V11 3.45701098·104 5.35821836

V12 -1.28415582·102 5.24342738

V15 3.39538287·10-4 1.74739330·1011

V16 5.35786127·101 3.90996426·103

Possible causes of discrepancy

2. Heat capacities

y = -4,6445E-03x2 + 4,2927E+00x - 4,9432E+02

R2 = 9,9660E-01

350

370

390

410

430

450

470

490

510

290 320 350 380 410 440 470 500

T (K)

Cp

(J·K

-1·m

ol-

1)

Růžička (2004)

quadratic formula

223*10·9432.42927.410·6445.4

TTCpDE

223*10·5974.59433.410·4073.5

TTCpDO

15.3273910·7408.714635.285.2361ln32.494342

TTTTTG DE

86.3320210·0122.94717.225.2644ln74.559342

TTTTTG DO

Reduced sum of squares: 4.33899·10-3

Variable Value RSD

V11 2.80666395·104 2.11481408·10-2

V12 -1.02775126·102 2.08823286·10-2

V15 1.82918841·10-2 5.76074009

V16 6.12723859·101 1.37118907·10-1

Possible causes of discrepancy

3. Experimental data

265

270

275

280

285

290

295

300

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Mass fraction of dodecanolT

(K

)

265

270

275

280

285

290

295

300

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Mass fraction of dodecanolT

(K

)

Possible causes of discrepancy

3. Experimental data points

Reduced sum of squares: 2.3402·102

Variable Value RSD

V11 4.30371467·104 3.20019842·10-1

V12 -1.60697579·102 3.12444640·10-1

V15 -1.24197073·10-

2

1.04699276·102

V16 4.69106820·101 4.73816471·10-1

Possible causes of discrepancy

3. Experimental data points

Reduced sum of squares: 6.3462·102

Variable Value RSD

V11 2.12541399·104 9.73327408·10-1

V12 -8.25529414·101 9.51768718·10-1

V15 1.44869057·105 4.25482680·101

V16 -4.58345239·102 4.51677061·101

( )ij ij ij i j

A B x x

1

1 1

[ ( )]c c

E

m i j ij ij i j

i j i

G x x A B x x

1 2 1,2 1,2 1 2[ ( )]

E

mG x x A B x x

Subregular Solution Model 罗婕

4. Subregular Model

Level rule

Continue Suspension Crystallization

Continue Suspension Crystallization

• OC-MC Binary Solid Liquid Equilibrium using regular solution model

王文静

Continue Suspension Crystallization

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