Now we introduce a new concept: fugacity

• When we try to model “real” systems, the expression for the chemical potential that we used for ideal systems is no longer valid

• We introduce the concept of fugacity that for a pure component is the analogous (but is not equal) to the pressure

We showed that:

)ln()( PyRTTG iiigi

igi

PRTTG iigi ln)(

iii fRTTG ln)(

Pure component i, ideal gas

Component i in a mixtureof ideal gases

Let’s define:

For a real fluid, we define Fugacity of pure species i

Residual Gibbs free energy

iRi

iigii

Ri

RTG

P

fRTGGG

ln

ln

Valid for species iin any phase and any condition

Since we know how to calculate residual properties… (section 6.2)

P

dPZ

RT

G

RTG

P

ii

Ri

iRi

)1(ln

ln

0

Zi from an EOS, Virial, van der Waals, etc

Eqn. 6.49)

examples• From Virial EOS

• From van der Waals EOS

RT

PBiii ln

i

iiiii ZTR

Pa

RT

PbZZ

22ln1ln

General form, see eqn. 11.37 for cubic EOS. First solve for Zi in the vapor or in the liquid phase

))((1

ZZ

ZqZ

For the vapor phase:

r

r

T

P

RT

bP

r

r

T

T

bRT

Taq

)()(

For the liquid phase:

q

ZZZZ

1))((

3.52

3.56

See Table 3.1 for parameters

For cubic EOS

Page 98

Fugacities of a 2-phase system

lii

li

vii

vi

fRTTG

fRTTG

ln)(

ln)(

One component, two phases: saturated liquid and saturated vapor at Pi

sat and Tisat

What are the equilibrium conditions for a pure component?

Fugacity of a pure liquid at P and T

satisat

ili

li

sati

vi

sati

li

sati

sati

vil

i PPf

Pf

Pf

Pf

P

PfPf

)(

)(

)(

)()()(

Fugacity of a pure liquid at P and T

dPVRT

PPfP

P

li

sati

sati

li sat

i

1exp)(

example• For water at 300oC and for P up to 10,000 kPa (100 bar)

calculate values of fi and i from data in the steam tables and plot them vs. P

)(1

)(1

ln **

*

* iiii

ii

i

i SST

HH

RGG

RTf

f

Get Hi* and Si

* from the steam tables at 300oC and the lowest P, 1 kPa

At low P, steam is an ideal gas => fi* =P*

Then get values of Hi and Si at 300oC and at other pressures P and calculate fi (P)

Problem• For SO2 at 600 K and 300 bar, determine good

estimates of the fugacity and of GR/RT.

SO2 is a gas, what equations can we use to calculate f = /P

Find Tc, Pc, and acentric factor, Table B1, p. 680

Calculate reduced properties: Tr, Pr

Tr=1.393 and Pr=3.805

What equations can we use to determine i (gas phase)

Generalized correlations: fugacity coefficient

)(

lnlnln

)1(ln

)1(ln

)1(ln

ln

10

10

0

1

0

0

0

0

i

r

rP

r

rP

i

r

rP

ii

rc

P

ii

Ri

iRi

P

dPZ

P

dPZ

P

dPZ

PPPP

dPZ

RT

G

RTG

rr

r

Tables E13 to E16Lee-Kessler

High P, high T, gas: use Lee-Kessler correlation

• From tables E15 and E16 find 0 and 1

• 0 = 0.672; 1 = 1.354

• = 0 1

• f = P = 0.724 x 300 bar = 217.14 bar

• GR/RT = ln

Problem

• Estimate the fugacity of cyclopentane at 110oC and 275 bar. At 110 oC the vapor pressure of cyclopentane is 5.267 bar.

• At those conditions, cyclopentane is a high P liquid

dPVRT

PPfP

P

li

sati

sati

li sat

i

1exp)(

Find Tc, Pc, Zc,, Vc and acentric factor, , Table B1, p. 680

Calculate reduced properties: Tr, Prsat

Tr = 0.7486 and Prsat = 0.117

At P = Psat we can use the virial EOS to calculate isat

2.41

6.10

10

172.0139.0;

422.0083.0

)(exp

rr

r

rsati

TB

TB

BBT

P

Eqn. 11.68

Eqns. 3.65 and 3.66

isat = 0.9

P-correction term:

Get the volume of the saturated liquid phase, Rackett equation

7/2)1( rTcc

sat ZVV Vsat = 107.55 cm3/mol

dPVRT

PPfP

P

li

sati

sati

li sat

i

1exp)(

f = 11.78 bar

Eqn. 3.72, p. 109