Chapter 4

FUGACITY

Fundamental equations for closed system consisting of n moles:

nVdPnSdTnUd (1)

dPnVnSdTnHd (2)

nVdPdTnSnAd (3)

dPnVdTnSnGd (4)

HOMOGENEOUS OPEN SYSTEM

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

For a closed homogeneous system, we consider U to be a function only of S and V:

U = U(S, V) (5)

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

nU = nU(S,V, n1, n2, ....., nm) (6)

where m is the number of components.

The total differential of eq. (6) is

ii

n,V,Sin,Sn,V

dnnnU

dVV

nUdS

SnU

nUdijii

(7)

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,V,Sii n

nU

(8)

We may rewrite eq. (7) as

i

iidnnVdPnSdTnUd (9)

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

i

iidxdVPdSTdU (10)

Eqs. (9) and (10) are the fundamental equations for an open system corresponding to eq. (1) for a closed system.

Using similar derivations, we can get the following relations:

i

iidndPnVnSdTnHd (11)

(12)

(13)

i

iidnnVdPdTnSnAd

i

idndPnVdTnSnGd

It follows that:

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

nGnnA

nnH

nnU

(14)

(15)

(9)

(16)

(17)

(18)

(21)

(19)

(20)

Equations (19 – 21) can be written as

nSdnSdnSdnSd 321 (22)

nVdnVdnVdnVd 321 (23)

i3

i2

i1

i dndndndn (24)

Substituting eqs. (22 – 24) into eq. (18) gives:

nSdnSdnSdTnUd 321

nVdnVdnVdP 321

i

i3

i2

i1

i dndndn

i

2i

2i

2222 dnnVdPnSdT

i

ii dnnVdPnSdT

i

3i

3i

3333 dnnVdPnSdT

• 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

0nSnU

3 13 TT

0nSnU

1TT

TTT 21

0nVnU

2 12 PP

0nVnU

3 13 PP

0nV

nU

1PP

PPP 21

0

nnU

21

1

12

1

0

nnU

1

0

nnU

31

1

13

1

111

12

11

1

0

nnU

22

1

22

2

0

nnU

2

0

nnU

32

1

23

2

122

22

21

2

0

nnU

2m

1

m2

m

0

nnU

m

0

nnU

3m

1

m3

m

1mm

m2

m1

m

(25)

(26)

(27)

(28)

(29)

Thus, at equilibrium

TTT 21

PPP 21

12

11

1

22

21

2

m2

m1

m

(30)

jn,P,Tii n

nG

jn,P,Tii n

nMM

Eq. (14):

Eq. (30):

ii G (31)

Important relations for partial molar properties are:

i

iiMxM (32)

and Gibbs-Duhem equation:

0MxdTTM

dPPM

iii

x,Px,T

(33)

(34)

(35)

(36)

GIBB’S THEOREM

(37)

(38)

(39)

(32):

Equation (20) of Chapter 3:

dPTV

TdT

CdSP

P

(3.20)

For ideal gas:

dPT

VT

dTCdS

P

igiig

Pigi i

(40)P

dPR

TdT

CdS igP

igi i

For a constant T process

TdT

CdS igP

igi i (constant T)

P

p

P,TS

p,TS

igi

i

igi

iigi

TdT

RdS (constant T)

iii

iigi

igi ylnR

PyP

lnRpP

lnRp,TSP,TS

iigii

igi ylnRP,TSp,TS

According to eq. (37):

iigi

igi p,TSP,TS

whence

iigi

igi ylnRP,TSP,TS

iigi

igi ylnRSS

By the summability relation, eq. (32):

i

iigii

i

igii

igi ylnRSySyS

Or:

i

iii

igii

igi ylnyRSyS

(41)

This equation is rearranged as

i

iii

igii

igi ylnyRSyS

i i

ii

igii

igi y

1lnyRSyS

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.

(42)

Gibbs energy for an ideal gas mixture: igigig TSHG

Partial Gibbs energy :

igi

igi

igi STHG

In combination with eqs. (38) and (41) this becomes

iigii

igi

igi

igi ylnRTGylnRTTSHG

iigi

igi

igi ylnRTGG

or:

(43)

An alternative expression for the chemical potential can be derived from eq. (2.4):

dPVdTSdG igi

igi

igi

At constant temperature:

(2.4)

PdP

RTdPVdG igi

igi (constant T)

Integration gives:

PlnRTTG iigi (44)

Combining eqs. (43) and (44) results in:

PylnRTT iiigi (45)

Fugacity for Pure SpeciesThe origin of the fugacity concept resides in eq. (44), valid only for pure species i in the ideal-gas state.

For a real fluid, we write an analogous equation:

iii flnRTTG (46)

where fi is fugacity of pure species i.

Subtraction of eq. (44) from Eq. (46), both written for the same T and P, gives:

Pf

lnRTGG iigii (47)

Combining eqs. (3.41) with (47) gives:

Pf

lnRTG iRi

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

Pfi

i

(48)

(49)

Equation (48) can be written as

iRi lnRTG (50)

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

Pf igi (51)

Equation (3.50):

P

0i

Ri

PdP

1ZRTG

(constant T) (3.50)

Combining eqs. (50) and (3.50) results in:

P

0ii P

dP1Zln (constant T) (51)

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) (52)

FUGACITY COEFFICIENT DERIVED FROM VOLUME-EXPLICIT EQUATION OF STATE

Use equation (3.63):

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

(53)

VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES

Eq. (46) for species i as a saturated vapor:

Vii

Vi flnRTTG

(54)

For saturated liquid:

Lii

Li flnRTTG (55)

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

(56)

(57)

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:

(58)

sati

Li

Vi (59)

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. (53), evaluated at P = Pi

sat and Vi = Vi

sat. The fugacity is calculated using eq. (49).

ii

ii

i

i

i

iiiii bV

bVln

RTba

VbV

Zln1Zln

(53)

Pf ii (49)

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. (3.4) is integrated to give:

P

Pi

satii

sati

dPVGG (60)

iii flnRTTG

According to eq. (46):

satii

sati flnRTTG

( – )

sati

isatii f

flnRTGG (61)

Eq. (60) = Eq. (61):

P

Pisat

i

i

sati

dPVff

ln

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

ln

RT

PPVexpfactorPoynting

ff sat

iisat

i

i (62)

Remembering that:

sati

sati

sati Pf

The fugacity of a pure liquid is:

RT

PPVexpPf

satiisat

isatii (63)

Phase RuleFirst law of thermodynamics:

WQdU

thermodynamic properties (U, T, P, V) reflect the internal state or the thermodynamic state of the system.

Heat and work quantities, are not properties; they account for the energy changes that occur in the surroundings and appear only when changes occur in a system.

They depend on the nature of the process causing the change, and are associated with areas rather than points on a graph.

P1

P2

P

V V2V1

(P1, V1)

(P2, V2)

W

• The intensive state of a PVT system containing N chemical species and phases in equilibrium is characterized by the intensive variables, temperature T, pressure P, and N – 1 mole fractions for each phase.

• These are the phase-rule variables, and their number is 2 + (N – 1)().

• The masses of the phases are not phase-rule variables, because they have no influence on the intensive state of the system.

• An independent phase-equilibrium equation may be written connecting intensive variables for each of the N species for each pair of phases present.

• Thus, the number of independent phase-equilibrium equations is ( – 1)(N).

• The difference between the number of phase-rule variables and the number of independent equations connecting them is the number of variables that may be independently fixed.

Called the degrees of freedom of the system F:

F = 2 + (N – 1)() – ( – 1)(N)

F = 2 – + N (60)

• The intensive state of a system at equilibrium is established when its temperature, pressure, and the compositions of all phases are fixed. • These are therefore phase-rule variables, but they are

not all independent. • The phase rule gives the number of variables from this

set which must be arbitrarily specified to fix all remaining phase-rule variables number of degrees of freedom (F).

PURE HOMOGENEOUS FLUID

T, P, V

N = 1

= 1

F = 2 – + N

= 2 – 1 + 1 = 2

• The state of a pure homogeneous fluid is fixed whenever two intensive thermodynamic properties are set at definite values

N = 1 = 2F = 2 – + N = 2 – 2 + 1 = 1

VAPOR/LIQUID EQULIBRIUM FOR PURE SPECIES

TV, PV, VV

TL, PL, VL

The state of the system is fixed when only a single property is specified. For example, a mixture of steam and liquid water in equilibrium at 101.325 kPa can exist only at 373.15 K (100°C). It is impossible to change the temperature without also changing the pressure if vaporand liquid are to continue to exist in equilibrium

Intensive variables: TV, PV, VV, V, TL, PL, VL, L = 8

Independent equations:

TV = TL = T

PV = PL = P

V = L

VV = f(T, P) ……. biggest root of the eos

V = f(T, P, VV)

VL = f(T, P) ………. smallest root of the eos

L = f(T, P, VL)

Here we have 7 equations with 8 unkowns. It means that we must define one intensive variable (T or P)

Algorithm:

1. Input: T

2. Assume P

3. Calculate ZV and ZL (cubic equation)

4. Calculate VV and ZL

5. Calculate V (eq. 53 with V = VV)

6. Calculate L (eq. 53 with V = VL)

7. Calculate Ratio = V/ L

8. If Ratio 1, assume new value of P

9. Go to step no. 3