Advanced Thermodynamics Note 10 Solution Thermodynamics: Theory Lecturer: 郭郭郭
Jan 06, 2016
Compositions
• Real system usually contains a mixture of fluid.• Develop the theoretical foundation for applications of
thermodynamics to gas mixtures and liquid solutions• Introducing
– chemical potential
– partial properties
– fugacity
– excess properties
– ideal solution
Fundamental property relation
• The basic relation connecting the Gibbs energy to the temperature and pressure in any closed system:
– applied to a single-phase fluid in a closed system wherein no chemical reactions occur.
• Consider a single-phase, open system:
dTnSdPnVdTT
nGdP
P
nGnGd
nPnT
)()()()(
)(,,
i
i
nTPinPnT
dnn
nGdT
T
nGdP
P
nGnGd
j,,,,
)()()()(
jnTPii n
nG
,,
)(
Define the chemical potential:
i
iidndTnSdPnVnGd )()()(
The fundamental property relation for single-phase fluid systems of constant or variable composition:
When n = 1, i
iidxSdTVdPdG ,...),...,,,,( 21 ixxxTPGG
xPT
GS
,
xTP
GV
,
The Gibbs energy is expressed as a function of its canonical variables.
Solution properties, MPartial properties, Pure-species properties, Mi
iM
Chemical potential and phase equilibria
• Consider a closed system consisting of two phases in equilibrium:
i
ii dndTnSdPnVnGd )()()( i
ii dndTnSdPnVnGd )()()(
)()( nMnMnM
i
iii
ii dndndTnSdPnVnGd )()()(
ii
ii dndn
Mass balance:
Multiple phases at the same T and P are in equilibrium when chemical potential of each species is the same in all phases.
Partial properties
• Define the partial molar property of species i:
– the chemical potential and the particle molar Gibbs energy are identical:
– for thermodynamic property M:
jnTPii n
nMM
,,
)(
ii G,...),...,,,,( 21 innnTPMnM
i
iinPnT
dnMdTT
MndP
P
MnnMd
,,
)(
i
iinPnT
dnMdTT
MndP
P
MnnMd
,,
)(
i
iiinPnT
ndxdnxMdTT
MndP
P
MnMdnndM )(
,,
0,,
dnMxMndxMdTT
MdP
P
MdM
iii
iii
nPnT
0,,
i
iinPnT
dxMdTT
MdP
P
MdM and 0
iiiMxM
Calculation of mixture properties from partial properties
0 i
iiMnnM
0,,
iii
nPnT
MdxdTT
MdP
P
M
i
iii
ii dxMMdxdM
The Gibbs/Duhem equation
Partial properties in binary solution
• For binary system2211 MxMxM
22221111 dxMMdxdxMMdxdM
Const. P and T, using Gibbs/Duhem equation
2211 dxMdxMdM
121 xx
211
MMdx
dM
121 dx
dMxMM
112 dx
dMxMM
The need arises in a laboratory for 2000 cm3 of an antifreeze solution consisting of 30 mol-% methanol in water. What volumes of pure methanol and of pure water at 25°C must be mixed to form the 2000 cm3 of antifreeze at 25°C? The partial and pure molar volumes are given.
2211 VxVxV molcmV /025.24)765.17)(7.0()632.38)(3.0( 3
molV
Vn
t
246.83025.24
2000
moln 974.24)246.83)(3.0(1
moln 272.58)246.83)(7.0(2
3111 1017)727.40)(974.24( cmVnV t 3
222 1053)068.18)(272.58( cmVnV t
Fig 11.2
The enthalpy of a binary liquid system of species 1 and 2 at fixed T and P is:
Determine expressions for and as functions of x1, numerical values for the pure-species enthalpies H1 and H2, and numerical values for the partial enthalpies at infinite dilution and
1H 2H
1H
2H
)2040(600400 212121 xxxxxxH
)2040(600400 212121 xxxxxxH
121 xx
311 20180600 xxH
121 dx
dHxHH
31
211 4060420 xxH
121 xx312 40600 xH
01 x
mol
JH 4201
11 x
mol
JH 6402
Relations among partial properties
• Maxwell relation:
i
iidnGdTnSdPnVnGd )()()(
nTnP P
S
T
V
,,
jnTPinP
i
n
nS
T
G
,,,
)(
jnTPinT
i
n
nV
P
G
,,,
)(
i
xP
i ST
G
,
i
xT
i VP
G
,
PVUH
dTSdPVGd iii
iii VPUH
Ideal-gas mixture
• Gibbs’s theorem– A partial molar property (other than volume) of a
constituent species in an ideal-gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature but at a pressure equal to its partial pressure in the mixture.
igi
igi VM ),(),( i
igi
igi pTMPTM
For those depend on pressure, e.g., TconstPRddS igi .ln
iii
iigi
igi yR
Py
PR
p
PRpTSPTS lnlnln),(),(
iig
iigi yRPTSPTS ln),(),(
For those independent of pressure, e.g.,
),(),(),( PTHpTHPTH igii
igi
igi
i
igii
ig HyH i
igii
ig UyU
),(),( iigi
igi pTMPTM
i
iii
igii
ig yyRSyS ln
igi
igi
igi STHG i
igi
igi
igi yRTTSHG ln
),(),( PTHPTH igi
igi
iig
iigi yRPTSPTS ln),(),(
iigi
igi
igi yRTGG ln
PyRTT iiigi ln)( or
PyyRTTyGi
iii
iiig ln)(
PRTTG iigi ln)(
From integration of PRTdPP
RTdPVdG ig
iigi ln
Fugacity and fugacity coefficient
• Chemical potential:
– provides fundamental criterion for phase equilibria
– however, the Gibbs energy, hence μi, is defined in relation to the internal energy and entropy - (absolute values are unknown).
• Fugacity:– a quantity that takes the place of μi
jnTPii n
nG
,,
)(
iii fRTTG ln)(
With units of pressure
iii fRTTG ln)(
PRTTG iigi ln)(
P
fRTGG iig
ii ln iRi RTG ln P
fii
Residual Gibbs energy Fugacity coefficient
).()1(ln0
TconstP
dPZ
P
ii
VLE for pure species
• Saturated vapor:
• Saturated liquid:
vii
vi fRTTG ln)(
lii
li fRTTG ln)(
li
vil
ivi f
fRTGG ln
VLE
0ln l
i
vil
ivi f
fRTGG
sati
li
vi fff sat
ili
vi
For a pure species coexisting liquid and vapor phases are in equilibrium when they have the same temperature, pressure, fugacity and fugacity coefficient.
Fugacity of a pure liquid
• The fugacity of pure species i as a compressed liquid:
sati
isatii f
fRTGG ln )( processisothermaldPVGG
P
P isatii sat
i
P
P isati
isat
i
dPVRTf
f 1ln
Since Vi is a weak function of P
RT
PPV
f
f sati
li
sati
i )(ln
sati
sati
sati Pf RT
PPVPf
sati
lisat
isatii
)(exp
For H2O at a temperature of 300°C and for pressures up to 10,000 kPa (100 bar) calculate values of fi and φi from data in the steam tables and plot them vs. P.
For a state at P: iii fRTTG ln)(
For a low pressure reference state: ** ln)( iii fRTTG
)(1
ln ** ii
i
i GGRTf
f
iii TSHG
)(
1ln *
*
* iiii
i
i SST
HH
Rf
f
The low pressure (say 1 kPa) at 300°C:
gJH i 8.3076*
gKJSi 3450.10* kPaPfi 1**
For different values of P up to the saturated pressure at 300°C, one obtains the values of fi ,and hence φi . Note, values of fi and φi at 8592.7 kPa are obtained
Fig 11.3
Values of fi andφi at higher pressure:RT
PPVPf
sati
lisat
isatii
)(exp
Fugacity and fugacity coefficient: species in solution
• For species i in a mixture of real gases or in a solution of liquids:
• Multiple phases at the same T and P are in equilibrium when the fugacity of each constituent species is the same in all phases:
iii fRTT ˆln)(
Fugacity of species i in solution (replacing the particle pressure)
iii fff ˆ...ˆˆ
igR MMM The residual property:
Py
fRT
i
iigii
ˆln
igii
Ri MMM The partial residual property:
igii
Ri GGG
iR
i RTG ln
iii fRTT ˆln)(
PyRTT iiigi ln)(
i
nTPii G
n
nG
j
,,
)(
Py
f
i
ii
ˆˆ
The fugacity coefficient of species i in solution
For ideal gas, 0R
iG
1ˆ
ˆ Py
f
i
ii Pyf ii ˆ
Fundamental residual-property relation
dTRT
nGnGd
RTRT
nGd
2)(
1
i
iidndTnSdPnVnGd )()()(
TSHG
ii
i dnRT
GdT
RT
nHdP
RT
nV
RT
nGd
2 ),,( inTPfRT
nG
G/RT as a function of its canonical variables allows evaluation of all other thermodynamic properties, and implicitly contains complete property information.
The residual properties:
ii
Ri
RRR
dnRT
GdT
RT
nHdP
RT
nV
RT
nGd
2
iii
RRR
dndTRT
nHdP
RT
nV
RT
nGd ln
2or
ii
Ri
RRR
dnRT
GdT
RT
nHdP
RT
nV
RT
nGd
2
iii
RRR
dndTRT
nHdP
RT
nV
RT
nGd ln
2
Fix T and composition:
xT
RR
P
RTG
RT
V
,
)/(
Fix P and composition:
xP
RR
T
RTGT
RT
H
,
)/(
Fix T and P:
jnTPi
R
i n
RTnG
,,
)/(ˆln
Develop a general equation for calculation of values form compressibility-factor data.
iln
jnTPi
R
i n
RTnG
,,
)/(ˆln
PR
P
dPnnZ
RT
nG0
)(
P
nTPii P
dP
n
nnZ
j
0,,
)(ˆln
ii
Zn
nZ
)(
1
in
n
P
ii P
dPZ
0)1(ˆln
Integration at constant temperature and composition
Fugacity coefficient from the virial E.O.S
• The virial equation:
– the mixture second virial coefficient B:
– for a binary mixture:
• n mol of gas mixture:
RT
BPZ 1
i j
ijji ByyB
2222211212211111 ByyByyByyByyB
RT
nBPnnZ
22 ,1,,11
)(1
)(
nTnTPn
nB
RT
P
n
nZZ
22 ,10
,11
)()(1ˆlnnT
P
nTn
nB
RT
PdP
n
nB
RT
22 ,10
,11
)()(1ˆlnnT
P
nTn
nB
RT
PdP
n
nB
RT
2222211212211111 ByyByyByyByyB
22111212 2 BBB nny ii /
1222111ln yB
RT
P Similarly: 12
21222ln yB
RT
P
For multicomponent gas mixture, the general form:
)2(
2
1ˆln ijiki j
jikkk yyBRT
P
wherekkiiikik BBB 2
Determine the fugacity coefficients for nitrogen and methane in N2(1)/CH4(2) mixture at 200K and 30 bar if the mixture contains 40 mol-% N2.
molcmBBB
3
22111212 6.200.1052.35)8.59(22
0501.0)6.20()6.0(2.35)200)(14.83(
30ˆln 212
22111 yB
RT
P
1835.0)6.20()4.0(0.105)200)(14.83(
30ˆln 212
21222 yB
RT
P
9511.01
8324.02
Generalized correlations for the fugacity coefficient
rP
rr
rii Tconst
P
dPZ
0).()1(ln
10 ZZZ
r rP
r
P
r
r
r
r TconstP
dPZ
P
dPZ
0 0
10 ).()1(ln
or10 lnlnln
rP
r
r
P
dPZ
0
00 )1(ln rP
r
r
P
dPZ
0
11ln
with
Table E1:E4 or Table E13:E16
)(1 10 BBT
PZ
r
r
)(ln 10 BBT
P
r
r
),,( OMEGAPRTRPHIBFor pure gas
For pure gas
Estimate a value for the fugacity of 1-butene vapor at 200°C and 70 bar.
127.1rT
731.1rP
191.0
627.00 and 096.11
10 lnlnln
Table E15 and E16
638.0 barPf 7.44)70)(638.0(
For gas mixture:
)2(
2
1ˆln ijiki j
jikkk yyBRT
P
)( 10 BBP
RTB ij
cij
cijij
2ji
ij
)1()( ijcjcicij kTTT
Prausnitz et al. 1986cij
cijcijcij V
RTZP
2cjci
cij
ZZZ
33/13/1
2
cjci
cij
VVV
Empirical interaction parameter
Estimate and for an equimolar mixture of methyl ethyl ketone (1) / toluene (2) at 50°C and 25 kPa. Set all kij = 0.
1 2
2ji
ij
)1()( ijcjcicij kTTT
cij
cijcijcij V
RTZP
2cjci
cij
ZZZ
33/13/1
2
cjci
cij
VVV
)( 10 BBP
RTB ij
cij
cijij 22111212 2 BBB
0128.0ˆln 1222111 yB
RT
P
0172.0ˆln 1221222 yB
RT
P
987.01
983.02
The ideal solution
• Serves as a standard to be compared:
iiid
i xRTGG ln
cf. iigi
igi yRTGG ln
i
P
i
xP
idiid
i xRT
G
T
GS ln
,
iiid
i xRSS ln
T
i
xT
idiid
i P
G
P
GV
,i
idi VV
iiiiid
iid
iidi xRTTSxRTGSTGH lnln i
idi HH
i
iii
iiid xxRTGxG ln
ii
iii
iid xxRSxS ln
ii
iid VxV
ii
iid HxH
i
idii
id MxM
iiid
i xRTGG ln
iii fRTT ˆln)(
The Lewis/Randall Rule
• For a special case of species i in an ideal solution:id
iiid
iidi fRTTG ˆln)(
iii fRTTG ln)(
iiid
i fxf ˆThe Lewis/Randall rule
iidi ˆ
The fugacity coefficient of species i in an ideal solution is equal to the fugacity coefficient of pure species i in the same physical state as the solution and at the same T and P.
Excess properties
• The mathematical formalism of excess properties is analogous to that of the residual properties:
– where M represents the molar (or unit-mass) value of any extensive thermodynamic property (e.g., V, U, H, S, G, etc.)
– Similarly, we have:
idE MMM
ii
Ei
EEE
dnRT
GdT
RT
nHdP
RT
nV
RT
nGd
2
The fundamental excess-property relation
(1) If CEP is a constant, independent of T, find expression for GE, SE, and HE as
functions of T. (2) From the equations developed in part (1), fine values for GE , SE, and HE for an equilmolar solution of benzene(1) / n-hexane(2) at 323.15K, given the following excess-property values for equilmolar solution at 298.15K:CE
P =-2.86 J/mol-K, HE = 897.9 J/mol, and GE = 384.5 J/mol
From Table 11.1:xP
EEP T
GTC
,
2
2
.constaC EP T
a
T
G
xP
E
,
2
2
bTaT
G
xP
E
ln
,
integration
cbTTTTaGE )ln(
integration
From Table 11.1:xP
EE
T
GS
,
bTaS E ln
caTTSGH EEE
integration
86.2aC EP
ca )15.298(9.897
cba )15.198()15.298()15.298ln()15.298((5.384
We have values of a, b, c and hence the excess-properties at 323.15K
The excess Gibbs energy and the activity coefficient
• The excess Gibbs energy is of particular interest:idE GGG
iii fRTTG ˆln)(
iiiid
i fxRTTG ln)(
ii
iEi fx
fRTG
ˆln
ii
ii fx
f
iE
i RTG ln
The activity coefficient of species i in solution.A factor introduced into Raoult’s law to account for liquid-phase non-idealities.For ideal solution,
c.f. iR
i RTG ln
1,0 iE
iG
ii
Ei
EEE
dnRT
GdT
RT
nHdP
RT
nV
RT
nGd
2
iii
EEE
dndTRT
nHdP
RT
nV
RT
nGd ln
2
xT
EE
P
RTG
RT
V
,
)/(
xP
EE
T
RTGT
RT
H
,
)/(
jnTPi
E
i n
RTnG
,,
)/(ln
Experimental accessible values:activity coefficients from VLE data,VE and HE values come from mixing experiments.
i
ii
E
xRT
G ln
),.(0ln PTconstdxi
ii Important application in phase-equilibrium thermodynamics.
The nature of excess properties
• GE: through reduction of VLE data
• HE: from mixing experiment
• SE = (HE - GE) / T
• Fig 11.4
– excess properties become zero as either species ~ 1.
– GE is approximately parabolic in shape; HE and TSE exhibit individual composition dependence.
– The extreme value of ME often occurs near the equilmolar composition.