Lecture 10

Advanced Thermodynamics

Note 10Solution Thermodynamics: Theory

Lecturer: 郭修伯

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

nGdPP

nGnGdnPnT

)()()()()(,,

i

inTPinPnT

dnnnGdT

TnGdP

PnGnGd

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

xPTGS

,

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

dnMdTTMndP

PMnnMd

,,

)(

i

iinPnT

dnMdTTMndP

PMnnMd

,,

)(

i

iiinPnT

ndxdnxMdTTMndP

PMnMdnndM )(

,,

0,,

dnMxMndxMdTTMdP

PMdM

iii

iii

nPnT

0,,

i

iinPnT

dxMdTTMdP

PMdM and 0

iiiMxM

Calculation of mixture properties from partial properties

0 i

iiMnnM

0,,

iii

nPnT

MdxdTTMdP

PM

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

MMdxdM

1

21 dxdMxMM

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

molVVn

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

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 xx311 20180600 xxH

121 dx

dHxHH

31

211 4060420 xxH

121 xx312 40600 xH

01 x

molJH 4201

11 x

molJH 6402

Relations among partial properties

• Maxwell relation:

i

iidnGdTnSdPnVnGd )()()(

nTnP PS

TV

,,

jnTPinP

i

nnS

TG

,,,

)(

jnTPinT

i

nnV

PG

,,,

)(

ixP

i STG

,i

xT

i VPG

,

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

PyPR

pPRpTSPTS lnlnln),(),(

iig

iigi yRPTSPTS ln),(),(

For those independent of pressure, e.g., ),(),(),( PTHpTHPTH ig

iiigi

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 i

igi ln)(

From integration of PRTdPP

RTdPVdG igi

igi 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)(

PfRTGG iig

ii ln iRi RTG ln P

fii

Residual Gibbs energy Fugacity coefficient

).()1(ln0

TconstP

dPZP

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 li

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 )( processisothermaldPVGGP

P isatii sat

i

P

P isati

isat

i

dPVRTf

f 1ln

Since Vi is a weak function of P

RTPPV

ff sat

il

isat

i

i )(ln

sati

sati

sati Pf RT

PPVPfsat

il

isati

satii

)(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

)(1ln ** ii

i

i GGRTf

f

iii TSHG

)(1ln *

*

* iiii

i

i SST

HHRf

f

The low pressure (say 1 kPa) at 300°C:

gJH i 8.3076*

gKJSi 3450.10* kPaPf i 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

PPVPfsat

il

isati

satii

)(exp

Fig 11.3

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:

PyfRTi

iigii

ˆln

igii

Ri MMM The partial residual property:

igii

Ri GGG

iR

i RTG ln

iii fRTT ˆln)(

PyRTT iiigi ln)(

inTPi

i GnnG

j

,,

)(

Pyf

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

dTRTnGnGd

RTRTnGd 2)(1

i

iidndTnSdPnVnGd )()()(

TSHG

ii

i dnRTGdT

RTnHdP

RTnV

RTnGd 2 ),,( inTPf

RTnG

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

dnRTGdT

RTnHdP

RTnV

RTnGd 2

iii

RRR

dndTRTnHdP

RTnV

RTnGd ln2or

ii

Ri

RRR

dnRTGdT

RTnHdP

RTnV

RTnGd 2

iii

RRR

dndTRTnHdP

RTnV

RTnGd ln2

Fix T and composition:xT

RR

PRTG

RTV

,

)/(

Fix P and composition:xP

RR

TRTGT

RTH

,

)/(

Fix T and P:jnTPi

R

i nRTnG

,,

)/(ˆln

Develop a general equation for calculation of values form compressibility-factor data.

iln

jnTPi

R

i nRTnG

,,

)/(ˆln

PR

PdPnnZ

RTnG

0)(

P

nTPii P

dPn

nnZ

j

0,,

)(ˆln

ii

ZnnZ

)(1

inn

P

ii PdPZ

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:

RTBPZ 1

i j

ijji ByyB

2222211212211111 ByyByyByyByyB

RTnBPnnZ

22 ,1,,11

)(1)(

nTnTPnnB

RTP

nnZZ

22 ,10

,11

)()(1ˆlnnT

P

nTnnB

RTPdP

nnB

RT

22 ,10

,11

)()(1ˆlnnT

P

nTnnB

RTPdP

nnB

RT

2222211212211111 ByyByyByyByyB

22111212 2 BBB nny ii /

1222111ln yB

RTP

Similarly: 1221222ln yB

RTP

For multicomponent gas mixture, the general form:

)2(

21ˆln ijik

i jjikkk yyB

RTP

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

RTP

1835.0)6.20()4.0(0.105)200)(14.83(

30ˆln 212

21222 yB

RTP

9511.01

8324.02

Generalized correlations for the fugacity coefficient

rP

rr

rii Tconst

PdPZ

0).()1(ln

10 ZZZ

r rP

r

P

r

r

r

r TconstP

dPZP

dPZ0 0

10 ).()1(ln

or10 lnlnln

rP

r

r

PdPZ

0

00 )1(ln rP

r

r

PdPZ

0

11ln

with

Table E1:E4 or Table E13:E16

)(1 10 BBTPZ

r

r

)(ln 10 BBTP

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(

21ˆln ijik

i jjikkk yyB

RTP

)( 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

RTP

0172.0ˆln 1221222 yB

RTP

987.01

983.02

The ideal solution

• Serves as a standard to be compared:

iiid

i xRTGG ln

cf. iigi

igi yRTGG ln

iP

i

xP

idiid

i xRTG

TGS ln

,

iiidi xRSS ln

T

i

xT

idiid

i PG

PGV

,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

dnRTGdT

RTnHdP

RTnV

RTnGd 2

The fundamental excess-property relation

Table 11.1

(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

aTG

xP

E

,2

2

bTaT

G

xP

E

ln,

integration

cbTTTTaG E )ln(integration

From Table 11.1:xP

EE

TGS

,

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

fRTGˆ

ln

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

dnRTGdT

RTnHdP

RTnV

RTnGd 2

iii

EEE

dndTRTnHdP

RTnV

RTnGd ln2

xT

EE

PRTG

RTV

,

)/(

xP

EE

TRTGT

RTH

,

)/(

jnTPi

E

i nRTnG

,,

)/(ln

Experimental accessible values:activity coefficients from VLE data,VE and HE values come from mixing experiments.

i

ii

E

xRTG 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.

Fig 11.4