SOLUTION THERMODYNAMICS

(Theory)(Part 1)

Chapter 3

1

PdVTdSdU (1.3)

VdPTdSdH (1.8)

PdVSdTdA (1.11)

SdTVdPdG (1.14)

Fundamental equations for homogeneous closed system consisting of 1 mole:

Fundamental equations for homogeneous closed system consisting of n moles:

nVdPnSdTnUd (3.1)

dPnVnSdTnHd (3.2)

nVdPdTnSnAd (3.3)

dPnVdTnSnGd (3.4)

For a single-phase fluid in a closed system wherein no chemical reactions occur, the composition is necessarily constant, and therefore:

nV

PnG

n,T

nS

TnG

n,P

(3.5)

(3.6)

HOMOGENEOUS OPEN SYSTEM

An open system can exchange matter as well as energy with its surroundings.

For a closed homogeneous system, we consider G to be a function only of T and P:

G = g(T, P) (3.7)

In an open system, there are additional independent variables, i.e., the mole numbers of the various components present.

nG = g(T, P, n1, n2, ....., nm) (3.8)

where m is the number of components.

The total differential of eq. (3.8) is

ii

n,T,Pin,Pn,T

dnnnG

dTT

nGdP

PnG

nGdij

(3.9)

Where subscript ni refers to all mole numbers and subscript nj to all mole numbers other than the ith. Chemical potential is defined as:

ijn,T,Pii n

nG

(3.10)

2

n,n,T,P21

n,n,T,P1

dnnnG

dnnnG

nGd3132

i

in,T,Pi

dnnnG

nGdij

3

n,n,T,P3

dnnnG

21

For a three-component system:

We may rewrite eq. (3.9) as

i

iidnnVdPnSdTnUd (3.11)

For a system comprising of 1 mole, n = 1 and ni = xi

i

iidxdVPdSTdU (3.12)

Eqs. (3.11) and (3.12) are the fundamental equations for an open system corresponding to eq. (3.1) for a closed system.

Using similar derivations, we can get the following relations:

i

iidndPnVnSdTnHd (3.13)

(3.14)

(3.13)

i

iidnnVdPdTnSnAd

i

idndPnVdTnSnGd

It follows that:

jjjj n,P,Tin,V,Tin,P,Sin,V,Sii n

nGnnA

nnH

nnU

(3.16)

For a closed system undergoing a reversible process, the criterion for equilibrium is defined in:

0nVdPnSdTnUd

Within this closed system, each phase is an open system which is free to transfer mass to each other. Eq. (3.11) may be written for each phase:

i

1i

1i

11111 dnnVdPnSdTnUd (3.18)

i

2i

2i

22222 dnnVdPnSdTnUd (3.19)

(3.17)

Total change of internal energy is the sum of internal energy of each phase in the system:

21 nUdnUdnUd

i

1i

1i

1111 dnnVdPnSdT

i

2i

2i

2222 dnnVdPnSdT

i

ii dnnVdPnSdTnUd

(3.20)

The individual variation d(nS)(1), d(nS)(2), etc. are subject to the constraints of constant total entropy, constant total volume, and constant total moles of each species.

These may be written as:

0nSdnSdnSd 21

0nVdnVdnVd 21

0dndndn 2i

1ii

(3.21)

(3.22)

(3.23)

Equations (3.21 – 3.23) can be written as

21 nSdnSd (3.24)

21 nVdnVd (3.25)

2i

1i dndn (3.26)

21

11 dndn

22

12 dndn

23

13 dndn

2211 nSdTnSdTnUd

2211 nVdPnVdP

Eq. (3.20) for a two-phase 3-component system gives:

13

13

12

12

11

11 dndndn

33

33

22

22

21

21 dndndn

(3.27)

i

ii dnnVdPnSdTnUd

Substituting eqs. (3.24 – 3.26) into eq. (3.27)

2221 nSdTnSdTnUd

2221 nVdPnVdP

23

13

22

12

21

11 dndndn

23

33

22

22

21

21 dndndn

212212 nVdPPnSdTTnUd

23

13

23

22

12

22

21

11

21 dndndn

(3.28)

• All variations d(nS)(2), d(nV)(2), dn1(2), dn2

(2), etc., are truly independent.

• Therefore, at equilibrium in the closed system where d(nU) = 0, it follows that

0nSnU

2

12 TT

0nVnU

2 12 PP

0

nnU

21

1

12

1

0

nnU

22

1

22

2

0

nnU

23

1

32

3

(3.29)

(3.30)

(3.31)

(3.32)

(3.33)

Thus, at equilibrium

TTT 21

PPP 21

12

11

1

22

21

2

m2

m1

m

(3.34)

Equations Relating Molar and Partial Molar Properties

The definition of a partial molar property, Eq. (3.34), provides the means for calculation of partial properties from solution-property data.

Solution properties can be calculated from knowledge of the partial properties.

The derivation of this equation starts with the observation that the thermodynamic properties of a homogeneous phase are functions of T, P, and the numbers of moles of the individual species which comprise the phase.

Thus for thermodynamic property M:

,n,,n,n,P,TMnM i21

The total differential of nM is

ii

n,T,Pin,Pn,T

dnn

nMdT

TnM

dPP

nMnMd

ij

(3.35)

(3.36)

Because the first two partial derivatives on the right are evaluated at constant n and because the partial derivative of the last term is given by Eq. (3.34), this equation has the simpler form:

i

iix,Px,T

dnMdTTM

ndPPM

nnMd

(3.37)

Since ni = xi n

it follows that:

When dni is replaced by this expression, and d(nM) is replaced by the identity:

Equation (3.37) becomes:

iii dxndnxdn

dnMdMnnMd

i

iiix,Px,T

dxndnxMdTTM

ndPPM

ndnMdMn

The terms containing n are collected and separated from those containing dn to yield:

0dnMxM

ndxMdTTM

dPPM

dM

iii

iii

x,Px,T

In application, one is free to choose a system of any size (n), and to choose any variation in its size (dn).

Thus n and dn are independent and arbitrary.

The only way that the left side of this equation can be zero is for each term in brackets to be zero.

(3.38)

0dxMdTTM

dPPM

dMi

iix,Px,T

i

iiMxM

Therefore:

(3.39)

(3.40)

Eq. (3.34) is an important relations (summability relation) for partial molar properties are.

Since Eq. (3.40) is a general expression for M, differentiation yields a general expression for dM:

i

iii

ii dxMMdxdM (3.41)

Combining eqs. (3.39) and (3.41) yields Gibbs-Duhem equation :

0MdxdTTM

dPPM

iii

x,Px,T

(3.42)

For the important special case of changes at constant T and P, it simplifies to:

0Mdxi

ii (3.43)

Partial Properties in Binary Solutions

2211 MxMxM

(B)

(A)

Eq. (3.40) for a binary solution:

whence

22221111 dxMMdxdxMMdxdM

When M is known as a function of xl at constant T and P, the appropriate form of the Gibbs-Duhem equation is Eq. (3.43), expressed here as:

0MdxMdx 2211 (C)

Since x1+ x2 = 1, it follows that dx1 = – dx2. Eliminating dx2 in favor of dx1 in Eq. (B) and combining the result with Eq. (C) gives:

12221111 dxMMdxdxMMdxdM

12112211 dxMdxMMdxMdxdM

1211 dxMdxMdM

211

MMdxdM

(D)

or:

Elimination of Eq. (A) yields:

(3.44)

2211 MxMxM

11211 dx

dMMxMxM

12121 dx

dMxMxxM

121 dx

dMxMM

121 dx

dMxMM

2M

211

MMdxdM

Elimination of Eq. (A) yields:

(3.45)

2211 MxMxM 221

21 MxdxdM

MxM

11221 dx

dMxMxxM

112 dx

dMxMM

112 dx

dMxMM

1M

Thus for binary systems, the partial properties are readily calculated directly from an expression for the solution property as a function of composition at constant T and P.

Example 3.1

Describe a graphical interpretation of eqs. (3.44) and(3.45).

Solution

Values of dM/dx1 are given by the slope of tangent lines. One such tangent line is shown. at x1 = 1 intercept = I1

at x1 = 0 intercept = I2

As is evident from the figure that two equivalent expressions can be written for the slope:

1

2

1 xIM

dxdM

211

IIdxdM

and

The first eq. is solved for I2; it combines with the second to give I1:

112 dx

dMxMI and

111 dx

dMx1MI

Comparison of these expressions with eqs. (3.44) and (3.55) shows that:

The tangent intercepts give directly the values of two partial properties.

at x1 = 0 and

at x1 = 1 and

11 MI and 22 MI

22 MM 11 MM

11 MM 22 MM

3.2

SOLUTION 3.2

Molar volume of the solution is

2211 VxVxV

13 molcm025.24765.177.0632.383.0

Since the required volume is Vt = 2000 cm3, the total number of moles required is:

mol246.83025.24

2000VV

nt

Of this, 30% is methanol and 70% is water,

n1 = 24.794 mol n2 = 58.272 mol

The volume of each species is:

311

t1 cm1017VnV

322

t2 cm1053VnV

(3.46)

(3.47)

iy

jn,P,Tii n

nMM

(3.48)

GIBB’S THEOREM

(3.49)

(3.50)

(3.51)

(3.40):

Equation (1.30) of Chapter 1:

dPTV

TdT

CdSP

P

(1.30)

For ideal gas:

dPT

VT

dTCdS

P

igiig

Pigi i

(3.52)P

dPR

TdT

CdS igP

igi i

PRT

V ig

For a constant T process

(constant T)

P

p

P,TS

p,TS

igi

i

igi

iigi

PdP

RdS (constant T)

iii

iigi

igi ylnR

PyP

lnRpP

lnRp,TSP,TS

iigii

igi ylnRP,TSp,TS

According to eq. (3.49):

iigi

igi p,TSP,TS

PdP

RdSigi

whence

iigi

igi ylnRP,TSP,TS

iigi

igi ylnRSS

By the summability relation, eq. (3.49):

i

iigii

i

igii

ig ylnRSySyS

Or:

i

iii

igii

ig ylnyRSyS

(3.53)

This equation is rearranged as

i

iii

igii

ig ylnyRSyS

i i

ii

igii

ig

y1

lnyRSyS

the left side is the entropy change of mixing for ideal gases.

Since 1/yi >1, this quantity is always positive, in agree-ment with the second law.

The mixing process is inherently irreversible, and for ideal gases mixing at constant T and P is not accompanied by heat transfer.

(3.54)

Gibbs energy for an ideal gas mixture: igigig TSHG

Partial Gibbs energy :

igi

igi

igi STHG

In combination with eqs. (3.50) and (3.54) this becomes

iigii

igi

igi

igi ylnRTGylnRTTSHG

iigi

igi

igi ylnRTGG

or:

(3.55)

An alternative expression for the chemical potential can be derived from eq. (1.14):

dPVdTSdG igi

igi

igi

At constant temperature:

(1.14)

PdP

RTdPVdG igi

igi (constant T)

Integration gives:

PlnRTTG iigi (3.56)

Combining eqs. (3.55) and (3.56) results in:

PylnRTT iiigi (3.57)

Fugacity for Pure SpeciesThe origin of the fugacity concept resides in eq. (3.56), valid only for pure species i in the ideal-gas state.

For a real fluid, we write an analogous equation:

iii flnRTTG (3.58)

where fi is fugacity of pure species i.

Subtraction of eq. (3.56) from Eq. (3.58), both written for the same T and P, gives:

Pf

lnRTGG iigii (3.59)

Combining eqs. (3.53) with (3.59) gives:

Pf

lnRTG iRi

The dimensionless ratio fi/P is another new property, the fugacity coefficient, given the symbol i:

Pfi

i

(3.60)

(3.61)

Equation (3.50) can be written as

iRi lnRTG (3.62)

The definition of fugacity is completed by setting the ideal-gas-state fugacity of pure species i equal to its pressure:

Pf igi (3.63)

1igi

Equation (1.50):

P

0i

Ri

PdP

1ZRTG

(constant T) (1.50)

Combining eqs. (3.62) and (1.50) results in:

P

0ii P

dP1Zln (constant T) (3.63)

Fugacity coefficients (and therefore fugacities) for pure gases are evaluated by this equation from PVT data or from a volume-explicit equation of state.

An example of volume-explicit equation of state is the 2-term virial equation:

RTPB

1Z ii

RTPB

1Z ii

P

0

ii dP

RTB

ln (constant T)

Because the second virial coefficient Bi is a function of temperature only for a pure species,

RTPB

ln ii (constant T) (3.64)

FUGACITY COEFFICIENT DERIVED FROM PRESSURE-EXPLICIT EQUATION OF STATE

Use equation (1.62):

ii

ii

i

i

i

iiii

Ri

bVbV

lnRTb

aV

bVZln1Z

RTG

Combining eqs. (3.63) and (50) gives:

ii

ii

i

i

i

iiiii bV

bVln

RTba

VbV

Zln1Zln

(3.65)

Tugas II:Soal no. 3.38(a)dari buku Smith dkk (menghitung koefisien fugasitas fase uap dan cair)

VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES

Eq. (3.58) for species i as a saturated vapor:

Vii

Vi flnRTTG

(3.66)

For saturated liquid:

Lii

Li flnRTTG (3.67)

By difference:

Li

ViL

iVi f

flnRTGG

Phase transition from vapor to liquid phase occurs at constant T dan P (Pi

sat). According to eq. (4):

d(nG) = 0

Since the number of moles n is constant, dG = 0, therefore :

0GG Li

Vi

Therefore:

sati

Li

Vi fff

(3.68)

(3.69)

For a pure species, coexisting liquid and vapor phases are in equilibrium when they have the same temperature,

pressure, and fugacity

An alternative formulation is based on the corresponding fugacity coefficients

sati

satisat

i Pf

whence:

(3.70)

sati

Li

Vi (3.71)

FUGACITY OF PURE LIQUID

The fugacity of pure species i as a compressed liquid is calculated in two steps:

1. The fugacity coefficient of saturated vapor is determined from Eq. (3.65), evaluated at P = Pi

sat and Vi = Vi

sat. The fugacity is calculated using eq. (3.61).

ii

ii

i

i

i

iiiii bV

bVln

RTba

VbV

Zln1Zln

(3.65)

Pf ii (3.61)

2. the calculation of the fugacity change resulting from the pressure increase, Pi

sat to P, that changes the state from saturated liquid to compressed liquid.

An isothermal change of pressure, eq. (1.49) is integrated to give:

P

Pi

satii

sati

dPVGG (3.72)

iii flnRTTG

According to eq. (46):

satii

sati flnRTTG

( – )

sati

isatii f

flnRTGG (3.73)

dPVdTSdG

Eq. (3.72) = Eq. (3.73):

P

Pisat

i

i

sati

dPVff

lnRT

Since Vi, the liquid-phase molar volume, is a very weak function of P at T << Tc, an excellent approximation is often obtained when Vi is assumed constant at the value for saturated liquid, Vi

L:

satiisat

i

i PPVff

lnRT

RT

PPVexpfactorPoynting

ff sat

iisat

i

i (3.74)

Remembering that:

sati

sati

sati Pf

The fugacity of a pure liquid is:

RT

PPVexpPf

satiisat

isatii (3.75)

RTPPV

expff sat

iisat

i

i

RTPPV

expffsat

iisatii