models for the excess gibbs energy - tongji...
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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
:
:
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
mX-pX oX-pX
oX-mX pX-mX-oX Ternary
郭艳姿、郭思斯