NON-IDEAL FLUID BEHAVIOR
1
• Homogeneous fluids are normally divided into
two classes, liquids and gases (vapors).
• Gas: A phase that can be condensed by a reduction
of temperature at constant pressure.
• Liquid: A phase that can be vaporized by a
reduction of pressure at constant temperature.
• The distinction cannot always be made
unambiguously, and the two phases become
indistinguishable at the critical point.
2
THE CRITICAL POINT
• Critical point: The maximum pressure and
temperature where a pure material can exist in
vapor-liquid equilibrium. Beyond Tc and Pc, the
designation of gas vs. liquid is arbitrary.
• At the critical point, the meniscus between phases
slowly fades and dissappears.
• If one moves around the critical point, it is
possible to get from the liquid to the vapor field
without crossing a phase boundary.
3
Supercritical
C - critical point
P-T phase diagram for a pure material.
4
P-V phase diagram for a pure
material. C - critical point.
P
nRT
V
At high T we expect the
isotherms to conform to
the ideal gas law, i.e., P
is inversely proportional
to V.
5
P-V phase diagram for pure
H2O.
6
THE P-V DIAGRAM
• We can use the lever rule on a P-V diagram to
determine the proportion of vapor vs. liquid at any
given pressure.
• The bending of the isotherms in the vapor field
from the ideal hyperbolic shape as the critical
point is approached indicates non-ideality.
• The P-V diagram illustrates the difficulty in
developing an equation of state for all regions for
a pure substance. However, this can be done for
the vapor phase.
7
Schematic isotherms in the two-phase field for a pure fluid.
A
B
P
Y
Q
For fluid of density
A, the proportion
of vapor is
Y/(X+Y) and the
proportion of liquid
is X/(X+Y).
For fluid of density
B, the proportion of
vapor is P/(P+Q)
and the proportion
of liquid is
Q/(P+Q).
X
8
MOST GENERAL EQUATION OF
STATE
 V 
 V 
dV  
dT



 dP
 T  P
 P T
dV
 dT  dP
V
Two special cases:
a) Incompressible fluid
==0
dV/V = 0 (no equation of state exists)
V = constant
b)  and  are temperature- and pressure-independent
9
VIRIAL EQUATION OF STATE
The most generally applicable EOS
PV = a + bP + cP2 + …
a, b, and c are constants for a given temperature and
substance.
In principle, an infinite series is required, but in
practice, a finite number of terms suffice.
At low P, PV  a + bP. The number of terms
necessary to accurately describe the PVT
properties of gases increases with increasing
pressure.
10
The limit of PV as P  0 is independent of the gas.
T = 273.16 K = triple point of water
lim (PV)T, P  0 = (PV)T* = 22.414 (cm3 atm g-mol-1) = a
So a is the same for all gases. It is in fact, RT!
11
I. THE COMPRESSIBILITY FACTOR
12
THE COMPRESSIBILITY FACTOR
Z  PV/RT = 1 + B’P + C’P2 + D’P3 + …
or
Z = 1 + B/V + C/V2 + D/V3 + …
The virial equation of state is the only one which has
a firm basis in theory. It follows from statistical
mechanics. It can be used to represent both liquids
and gases.
The term B/V arises due to pairwise interactions of
molecules.
The term C/V2 arises due to interactions among three
molecules, etc.
13
The constants for the two versions of the virial
equation are related by the equations:
B
B' 
RT
C  B2
C' 
RT 2
D  3BC  2 B 3
D' 
RT 3
Disadvantages of the virial equation of state:
1) Cumbersome, many variables
2) Not much predictive value
3) Difficult to use for mixtures
4) Only really useful when convergence is rapid, i.e.,
at low to moderate pressures.
14
SOME APPROXIMATIONS
Low pressure (0 - 15 bars at T < Tc)
PV
BP
Z
 1
RT
RT
Becomes valid over greater pressure ranges as
temperature increases. Easily solved for volume.
Moderate pressure (0 - 50 bars)
PV
B C
Z
 1  2
RT
V V
Only B and C are generally well known. At higher
pressures, other EOS’s are required.
15
Compressibility factor
diagram for methane.
Note two things:
1) All isotherms
originate at Z = 1where
P  0.
2) The isotherms are
nearly straight lines at
low pressure, in
accordance with the
truncated virial
equation:
BP
Z  1
RT
16
The compressibility factor
as a function of pressure
for various gases. Z
measures the deviation
from the ideal gas law.
17
II. EQUATIONS OF STATE
18
THE OBJECTIVE IN THE SEARCH
FOR AN EOS
The objective is to find a single equation of
state:
1) whose form is appropriate for all gases
2) that has relatively few parameters
3) that can be readily extrapolated
4) that can be adapted for mixtures
This objective has only been partially
fulfilled.
19
VAN DER WAALS EQUATION (1873)
a 

 P  2 V  b   RT
V 

The a term accounts for forces of attraction between
molecules (long-range forces).
The b term accounts for the non-zero volume of
molecules (short-range repulsion).
At low pressure’s real gases are easier to compress
than ideal gases; at higher pressures they are more
difficult to compress (see Z plot).
An alternate form is:
RT
a
P
V  b

V2
20
The van der Waals
isotherms (labelled with
values of T/Tc.
The van der Waals
equation predicts the
shape of the isotherms
fairly well in the onephase region, but shows
unrealistic oscillations in
the two-phase region. The
theory fails because it
only considers two-body
interactions.
P/Pc
21
V/Vc
THE VAN DER WAALS
PARAMETERS
We can determine how to calculate the a and b parameters
by setting the 1st and 2nd derivatives of the van der
Waals equation to zero at the critical point (an inflection
point), i.e.,
 2 P 
 P 
 2   0

 0
 V T
 V T
Solving these equations we get: Vc  3b
8 a
Tc 
27 bR
a
Pc 
27b 2
PcVc 3
Zc 
  0.375
RTc 8
22
CRITICAL CONSTANTS OF GASES - I
Gas Pc (atm) Vc (cm3mol) Tc (K)
Zc
He
2.26
57.8
5.21
0.306
Ne
26.9
41.7
44.4
0.308
Ar
48.0
73.3
150.7
0.285
Xe
58.0
119
289.8
0.290
H2
12.8
65.0
33.2
0.305
O2
50.1
78.0
154.8
0.308
N2
33.5
90.1
126.3
0.291
F2
55
----
144
---
Cl2
76.1
124
417.2
0.276
Br2
102
135
584
0.287
23
CRITICAL CONSTANTS OF GASES - II
Gas Pc (atm) Vc (cm3mol) Tc (K)
Zc
CO2
72.7
94.0
304.2
0.275
H2O
218
55.3
647.4
0.227
NH3
111
72.5
405.5
0.242
CH4
45.8
99
191.1
0.289
C2H4
50.5
124
283.1
0.270
C2H6
48.2
148
305.4
0.285
C6H6
48.6
260
562.7
0.274
The van der Waals equation does better than the ideal gas law
but is not great. No two-parameter EOS can predicts all these
24
gases.
We can rearrange the previous equations to get the
van der Waal parameters in terms of the critical
parameters:
a  3PcVc2
where
b
Vc
3
3RTc
Vc 
8Pc
Note that, the actual measured value of Vc is not
used to calculate a and b!
25
VAN DER WAALS CONSTANTS FOR GASES
Gas
a
b
Gas
a
b
He
0.03412 2.370
Cl2
6.493 5.622
Ne
0.2107
1.709
CO2
3.592 4.267
Ar
1.345
3.219 H2O 5.464 3.049
Kr
2.318
3.978 NH3 4.170 3.707
Xe
4.194
5.105
H2
0.2444
2.661 C2H4 4.471 5.714
O2
1.360
3.183 C2H6 5.489 6.380
N2
1.390
3.913 C6H6 18.00 1.154
CH4
2.253 4.278
a - dm6 atm mol-2; b - 10-2 dm3 mol-1
26
SOME OTHER EOSs
a 

V  b  RT
Bertholet (1899)  P 
2 
TV 

The higher the temperature, the less likely particles
will come close enough to attract one another
significantly. a and b are different from VdW.
Dieterici (1899)
RT
P
e
V  b
 a 


 RTV 
RT
A
Keyes (1917) P 

2
V
(V  e ) V  l 
, , A and l are correction factors.
27
BEATTIE-BRIDGEMAN (1927)
RT (1   )
A
V  B   2
P
2
V
V
A  A0 1  a V 
B  B0 1  b V 
c

VT 3
a, b, c, A0, B0 are constants
We usually don’t know V, but we know P, so an
iterative approach is required: calculate A, B and 
with an assumed V value and compute P. If Pcalc 
Pexp, then adjust V accordingly and recalculate P.
28
Rearrangement of the Beattie-Bridgeman equation
gives:
RT



P
 2 3 4
V
V
V
V
Rc
  RTB0  A0  2
Where
T
RB0bc
RB0c

   RTB0b  aA0  2
2
T
T
This shows the B-B equation to be simply a
truncated form of the virial equation.
29
BEATTIE (1930)
A
V    B 1    
RT
A  A0 1  a  
B  B0 1  b  
RT

P
c

T 3
perfect gas volume
a, b, and c are the same as for the B-B EOS
One can use the Beattie equation to obtain a first
guess for the Beattie-Bridgeman equation, which
is more accurate because it allows for the variation
of A, B and  with volume.
30
SOME MORE EOS’s
Jaffé (1947) - a modification of the Dieterici EOS
  



RT
RTTc  
RT
V
P

1  e  

V  b V  b 

RTc
4 R 2Tc2

b
2
Pc e 2
Pc e
1
Wohl (1949)
2
RT

c
P

 3
V  b V V  b V
  6 PcVc 2
b  Vc 4
c  4 PcVc3
31
McLeod (1949)
where
(V  b' )  RT
a
 P 2
V
b '  A  B  c  2
and a, A, B, and c are constants.
Benedict-Webb-Rubin (1940) - specifically devised
for hydrocarbons. Useful for both liquids and
gases. Define
d 1
V


P  RTd  B0 RT  A0  C0 T 2 d 2  bRT  a d 3
3
2



cd
1


d
6
 d 2 
 ad  
e

2
T


32
Martin-Hou (1955)
Introduces the reduced temperature: Tr = T/Tc
RT
A2  B2T  C2 ( eTr
P

(V  b)
(V  b) 2
A4
B5T


4
(V  b)
(V  b)5
238
)
A3  B3T  C3 ( eTr

(V  b)3
239
Like many others, this EOS is also a version of the
virial EOS.
33
)
34
REDLICH-KWONG (1949)
This has been one of the most useful to geology
RT
a
P
 12
(V  b) T V (V  b)
where
0.42748R 2Tc2.5
0.08664 RTc
1
a
b
Z c   0.3333
Pc
Pc
3
The R-K EOS is quite accurate for many purposes,
particularly if the a and b parameters are adjusted
to fit experimental data. However, there have been
a number of attempts at improvement.
35
MODIFICATIONS OF THE R-K EOS
RT
a
de Santis et al. (1974) P 
 12
(V  b) T V (V  b)
but b is a constant and a(T) = a0 + a1T.
Peng and Robinson (1976)
RT
a (T )
P
 2
(V  b) (V  2bV  b2 )
2
1
where
2
 (T )  1  m 1  T


r

m  0.37464  1.54226  0.26992
and
 is a parameter for the fluid called the acentric factor.
2
36
Carnahan and Starling (1969) - “hard-sphere” model
RT
P
V
 1  y  y2  y3 


3
(1  y )


b
y
4V
Kerrick and Jacobs (1981) - Hard-Sphere Modified
Redlich-Kwong (HSMRK)
RT  1  y  y 2  y 3 
a ( P, T )

  1
P
3
2
V 
(1  y )
T
V (V  b)

a(P,T) = an empirically-derived polynomial.
a ( P, T )  c(T )  d (T )
V
 e(T )
V2
z (T )  z1  z2T  z3T 3 where z = c, d, or e
37
LEE AND KESLER (1975)
 b01  b02Tr1  b03Tr2  b04Tr3 c01  c02Tr1  c03Tr3
Z  1 

2
V
V
r
r

1
02 r
5
d 01  d T

Vr
c04 
0 
 3 2   0  2  e
Tr Vr 
Vr 
 0

 V2
 r




 b11  b12Tr1  b13Tr2  b14Tr3 c11  c12Tr1  c13Tr3
  1 

2
V
V
r
r

1
12 r
5
d11  d T

Vr
c14 
1 
 3 2  1  2  e
Tr Vr 
Vr 
 1

 V2
 r








38
DUAN, MOLLER AND WEARE (1992)
B C
D
E
F 
 
Z  1   2  4  5  2    2 e
Vr Vr Vr Vr Vr 
Vr 
 

 V2
 r




B  a1  a2Tr2  a3Tr3
C  a4  a5Tr2  a6Tr3
D  a7  a8Tr2  a9Tr3
E  a10  a11Tr2  a12Tr3
F  T
3
r
VPc
Vr 
RTc
P
Pr 
Pc
T
Tr 
Tc
This is just a modified form the the Lee and Kesler EOS.
39
CALCULATING FUGACITY
COEFFICIENTS BY INTEGRATING AN
EOS
Using the van der Waals equation:
RT
a
P
 2
V  b V
1
V
ln    
  P
RT P 
0
P
b
2a
PV
 V 
ln   ln 

 ln

RT
 V  b  V  b V RT
40
Using the original Redlich-Kwong equation:
RT
a
P
 12
(V  b) T V (V  b)
1
V
ln    
  P
RT P 
0
P
b
2a
 V 
V  b 
ln   ln 

ln 


3
2
 V  b  V  b RT b  V 
a  V  b 
b 
PV
 ln 
  ln



3
RT
RT 2 b   V  V  b 
41
Using the HSMRK EOS of Kerrick and Jacobs (1981)
8 y  9 y2  3y3
PV
c
ln  
 ln

3
3
(1  y )
RT RT 2 (V  b)
d
e


3
3
2
RT V (V  b) RT 2V 2 (V  b)
V 
d
 c

ln

3
3
2
2
V

b
RT
b
RT
Vb


V b
e
 d

ln


3
3
2
2
2
2
V
RT
b
RT
2
b


V b
e
 e

ln


3
3
3
2
2 2
V
RT
b
RT
bV


42
Activities in CO2-H2O mixtures predicted by a MRK EOS after
Kerrick & Jacobs (1981).
43
Predicted H2O and CO2 activities in H2O-CO2-CH4 mixtures at 400°C
and 25 kbar. Calculated for XCH4 = 0.0, 0.05 and 0.20. Dotted curves
imply a miscibility gap of H2O-rich liquid and CO2-rich vapor.
After Kerrick
& Jacobs
(1981).
44
CO2-H2O solvus at 1 kbar. The solid line is a fit of a MRK EOS to
experimental data (solid dots). After Bowers & Helgeson (1983).
45
Effect of 12 wt. % NaCl on
the CO2-H2O solvus at 1
kbar. After Bowers &
Helgeson (1983).
46
III. CORRESPONDING
STATES
47
PRINCIPLE OF CORRESPONDING
STATES
Reduced variables of a gas are defined as:
Pr = P/Pc Tr = T/Tc
Vr = V/Vc
Principle of corresponding states - real gases in the
same state of reduced volume and temperature
exert approximately the same pressure. Another
way to say this is, real gases in the same reduced
state of temperature and pressure have the same
reduced compressibility factor.
This fact can be used to calculate PVT properties of
gases for which no EOS is available.
48
The reduced compressibility factor vs. the reduced pressure
49
Reduced pressure
Generalized compressibility
chart. Medium- and highpressure range.
PV
Z
RT
3.0
2.8
2.6
2.4
Z
Z
2.2
2.0
1.8
1.6
1.4
1.2
1.0
50
Reduced pressure
EXAMPLE: Calculate the specific volume of NH3 at
500°C and 2 kbar using the reduced Z chart and
compare to the ideal gas law prediction.
Ideal gas law
RT (0.080256 atm L mol 1 )( 773 K )
V 

 0.0310 L mol 1
P
(2000 atm)
Corresponding states: Tr = (773 K)/(405.5 K) = 1.91;
Pr = (2000 atm)/(111 atm) = 18.02.
(2000 atm) V
Z  1.62 
(0.080256 atm L mol 1 )( 773 K)
V  0.050 L mol 1
51
52
53
54
55
Measured compressibility factors for H2O vs. those obtained from
corresponding state theory
56
Measured compressibility factors for CO2 vs. those obtained from
corresponding state theory
57
Generalized density correlation for liquids. r = /c
58
PITZER’S ACENTRIC FACTOR
The acentric factor of a material is defined with
reference to its vapor pressure.
The vapor pressure of a subtance may be expressed as:
log Prsat  a  b
Tr
but the L-V curve terminates at the critical point where
Tr = Pr. So a = b and
log Prsat  a1  1 
Tr 

If the principle of corresponding states were exact, all
materials would have the same reduced-vapor
pressure curve, and the slope a would be the same for
all materials. However, the value of a varies.
59
The linear relation is only approximate; a is not defined
with enough precision to be used as a third parameter
in generalized correlations.
Pitzer noted that Ar, Kr and Xe all lie on the same
reduced-vapor pressure curve and this passes through
log Prsat = -1 at Tr = 0.7. We can then characterize the
location of curves for other gases in terms of their
position relative to that for Ar, Kr and Xe.
The acentric factor is:    log Prsat Tr 0.7  1.000
 can be determined from Tc, Pc and a single vapor
pressure measurement at Tr = 0.7.
60
Slope  -3.2
(n-octane)
Approximate temperature-dependence of reduced vapor pressure
61
ACENTRIC FACTORS FOR GASES
Gas

Gas

Gas

Ne
0
Cl2
0.073
methane
0.011
Ar
-0.004
Br2
0.132
ethylene
0.087
Kr
-0.002 CO2 0.223
ethane
0.100
Xe
0.002
CO
0.049
benzene
0.212
H2
-0.22
NH3 0.250
toluene
0.257
O2
0.021
HCl
N2
0.037
H2S 0.100
F2
0.048
SO2 0.251 m-xylene 0.331
0.12
n-heptane 0.350
propane
0.153
62
PRINCIPLE OF CORRESPONDING
STATES - REVISITED
Restatement of principle of corresponding states:
All fluids having the same value of  have the same
value of Z when compared at the same Tr and Pr.
The simplest correlation is for the second virial
coefficients:
 BPc  Pr
BP
Z  1

 1  
RT
 RTc  Tr
The quantity in brackets is the reduced 2nd virial
coefficient.
 BPc 

  B 0  B1
 RTc 
63
0.422
B  0.083  1.6
Tr
0
0.172
B  0.139  4.2
Tr
1
The range where this correlation can be used safely is
shown on the chart on the next slide.
For the range where the generalized 2nd virial
coefficient cannot be used, the generalized Z charts
may be used:
Z = Z0 + Z1
These correlations provide reliable results for non-polar
or only slightly polar gases. The accuracy is ~3%. For
highly polar gases, the accuracy is ~5-10%. For gases
that associate, even larger errors are possible.
The generalized correlations are not intended to be
substitutes for reliable experimental data!
64
Generalized correlation
for Z0. Based on data
for Ar, Kr and Xe from
Pitzer’s correlation.
65
Generalized correlation
for Z1 based on Pitzer’s
correlation.
66
EXAMPLE -1
What is the volume of SO2 at P = 500 atm and T =
500°C?
According to the ideal gas law:
RT (0.080256 atm L mol 1 )( 773 K )
V 

 0.124 L mol 1
P
(500 atm)
Using the acentric factor:  =0.273; Tr = 773/430.8 =
1.79; Pr = 500/77.8 = 6.43.
From the charts: Z0 = 0.97; Z1 = 0.31.
Z = 0.97 + 0.273(0.31) = 1.055
(500 atm) V
Z  1.055 
(0.080256 atm L mol 1 )( 773 K)
V  0.131 L mol -1
67
saturation
Line defining the region where generalized second virial
coefficients may be used. The line is based on Vr  2.
68
EXAMPLE - 2
What is the volume of SO2 at P = 150 atm and T =
500°C?
According to the ideal gas law:
RT (0.080256 atm L mol 1 )( 773 K )
V 

 0.414 L mol 1
P
(150 atm)
Using the acentric factor:  =0.273; Tr = 773/430.8 =
1.79; Pr = 150/77.8 = 1.93.
0.422
B  0.083 
 0.083
1.6
1.79
0
0.172
B  0.139 
 0.124
4.2
1.79
1
69
 BPc 

  B0  B1  0.083  0.273(0.124)  0.049
 RTc 
 BPc  Pr
1.93
  1  ( 0.049)
Z  1  
 0.947
1.79
 RTc  Tr
(150 atm) V
Z  0.947 
(0.080256 atm L mol 1 )( 773 K)
V  0.392 L mol -1
Vc = 0.122 L mol-1
Vr = 0.392/0.122 = 3.25
70
CORRESPONDENCE PRINCIPLE
FOR FUGACITY
• Correspondence principles and generalized
charts exist for fugacity and other
thermodynamic properties.
• For fugacity, both two- and three-parameter
generalized charts have been developed.
• Again, these are to be used only in the
absence of reliable experimental data.
71
ln   
Pr
0
 Z 1

 dPr
 Pr 
I. We can use this equation together with the generalized
Z charts.
1) Look up Pc and Tc of gas
2) Calculate Pr and Tr values for desired T’s and P’s
3) Make a Table of Z from the generalized charts at various
values of Tr and Pr. Of course, we must have Pr values from
0 to the pressure of interest at each temperature.
4) Graph (Z-1)/Pr vs. Pr for each Tr.
5) Determine the area under the the graph from Pr = 0 to Pr =
Pr to get ln .
II. Used generalized fugacity charts.
72
73
74
USE OF TWO-PARAMETER
GENERALIZED FUGACITY CHARTS
EXAMPLE 1: Calculate the fugacity of CO2 at 600°C
(873 K) and 1200 atm.
Tc = 304.2 K; Pc = 72.8 atm
Tr = 2.87; Pr = 16.48
From the chart   1.12
so
f = (1.12)(1200) = 1344 bars
75
EXAMPLE 2: What is the fugacity of liquid Cl2 at 25°C
and 100 atm? The vapor pressure of Cl2 at 25°C is 7.63
atm.
For the vapor coexisting with liquid:
Tc = 417 K; Pc = 76 atm
Tr = 0.71; Pr = 0.10
from the chart   0.9
f = (0.9)(7.63) = 6.87 atm
Now we must correct this to 100 atm.
V
ln f 2  ln f1 
( P2  P1 )
RT
V = 51 cm3 mol-1; assume to be constant.
f2 = 8.36 atm
76
THREE-PARAMETER CORRELATIONS
FOR FUGACITY ETC.
Fugacity:
log
f 
f 
  log 
P 
P
H H H H
 
RTc
 RTc
0
Enthalpy:
S S S S

 
R
 R 
0
Entropy:
Density:
0
0
( 0)
( 0)



f 

   log 
P

( 0)
(1)
H H
  
 RTc
0



(1)
(1)
S S
  ln P
  
 R 
0

 1  0.85(1  Tr )  (1  Tr ) (1.89  0.91 )
c
1
3
Tables for these correlations can be found in Pitzer
(1995) Thermodynamics. McGraw-Hill.
77
IV. GASEOUS MIXTURES
78
IDEAL GAS MIXTURES
• Mixture as a whole obeys: PV  RT
• Two such mixtures are in equilibrium with each
other through a semi-permeable membrane when
the partial of each component is the same on each
side of the membrane.
• There is no heat of mixing.
The gas mixture must therefore consist of freely
moving particles with negligible volumes and
having negligible forces of interaction.
79
DALTON’S LAW VS. AMAGAT’S
LAW
• Dalton’s Law:
Pi = XiPT
PT   Pi
• Amagat’s Law:
At constant VT and T
Vi = XiVT
VT  Vi
At constant PT and T
These two laws are mutually exclusive at a given
pressure and temperature.
80
THERMODYNAMICS OF IDEAL
MIXING - REVISITED
We have previously shown that:
Gideal mix  RT  X i ln X i
i
using Dalton’s Law we can derive:
Gideal mix
Pi
 RT  X i ln
PT
i
and for entropy we have:
Sideal mix
Pi
  R  X i ln
PT
i
81
NON-IDEAL MIXTURES OF NONIDEAL GASES
For a perfect gas mixture:
i  io  RT ln Pi  io  RT ln PT  RT ln X i
For an ideal mixture of real gases:
i  io  RT ln f i  io  RT ln f i o  RT ln X i
f i  X i f i o  X ii PT Lewis Fugacity Rule
For a real mixture of real gases:
i  io  RT ln f i
f i  X i  i f i o  X i  ii PT
Correction for
non-ideal mixing
Correction for
non-ideal gas
82
DALTON’S LAW AND
GENERALIZED CHARTS
Calculate reduced pressure according to:
Pi
Pr ,i 
Pc ,i
PAVT  nAZ ART
PBVT  nB Z B RT
PCVT  nC ZC RT
PA  PB  PC VT  n AZ A  nB Z B  nC ZC RT
 nT  X A Z A  X B Z B  X C Z C RT
 nT Z mix RT
83
AMAGAT’S LAW AND
GENERALIZED CHARTS
Calculate reduced pressure according to:
PT
Pr ,i 
Pc ,i
PTVA  nAZ ART
PTVB  nB Z B RT
PTVC  nC ZC RT
PT VA  VB  VC   n A Z A  nB Z B  nC Z C RT
 nT  X A Z A  X B Z B  X C Z C RT
 nT Z mix RT
84
PSEUDOCRITICAL CONSTANTS
P
B
1
A
X
A
XA2
L-V curve for A
L-V curve for B
T
85
KAY’S METHOD
Assumes a linear critical curve between the critical
points for A and B.
Pc '   X i Pc ,i
i
Tc '   X iTc ,i
i
When answers are near the critical point for the
mixture, we cannot be certain that we are not
dealing with a liquid-vapor mixture.
86
JAFFÉ’S METHOD
For binary mixtures only.
 Tc ' 

  X A A Pc , A  X B B Pc ,B
 P'
 c mix


 Tc ' 
1
1 3
2
2
3
   X A A  X B B  1 4  A   B3 X A X B
 Pc ' mix
Tc ,i
i 
Pc ,i
87
MIXING CONSTANTS IN
EQUATIONS OF STATE
Van der Waals and simple Redlich-Kwong EOS:
n
bmix   X j b j
j 1
n
n
amix   X j X k a jk
j 1 k 1
a jk  a j ak 
1
2
Use if no mixture data
are available.
88
Beattie-Bridgeman EOS:


  X j A0, j 
 j 1

n
A0,mix
n
amix   X j a j
j 1
n
B0,mix   X j B0, j
j 1
2
n
bmix   X j b j
j 1
n
cmix   X j c j
j 1
89
Benedict-Webb-Rubin EOS:


  X j A0, j 
 j 1

n
A0,mix
2


C0,mix   X j C0, j 
 j 1



bmix   X j 3 b j 
 j 1

 mix
2
3


  X j 3  j 
 j 1

n
B0,mix   X j B0, j
j 1
n
n
n


amix   X j 3 a j 
 j 1

n


cmix   X j 3 c j 
 j 1

n
3
3
3


 mix   X j  j 
 j 1

n
2
90
Virial Equation of State:
Z = 1 + B/V + C/V2 + D/V3 + …
n
n
Bmix   X i X j Bij
i 1 j 1
n
n
n
Cmix   X i X j X k Cijk
i 1 j 1 k 1
91
PREDICTION OF CRITICAL
CONSTANTS
Critical Temperature:
I. All compounds with Tboil (1 atm) < 235 K and all
elements: Tc = 1.70Tb - 2.00.
II. All compounds with Tboil (1 atm) > 235 K.
A. Containing halogens or sulfur
Tc = 1.41Tb + 66 - 11F
F = No. of fluorine atoms
B. Aromatics and napthenes
Tc = 1.41Tb + 66 - r(0.388Tb - 93)
r = ratio of non-cyclic carbon atoms to total carbon atoms.
92
C. All other compounds
Tc = 1.027 Tb + 159
Critical Pressure:
20.8Tc
Pc 
Vc  9
where Tc is in K and Vc is in cm3 g-1.
Critical Volume:
Vc  (0.377 P  11.0)1.2
where P is a parameter called the Sugten
Parachor.
93
SUGTEN PARACHOR VALUES FOR
ATOMS AND STRUCTURAL UNITS
C
4.8
S
48.2 triple bond 46.6
H
17.1
F
25.7
16.7
N
12.5
Cl
54.3
11.6
P
37.7
Br
68.0
8.5
O
20.0
I
91.0
6.1
O (esters) 60.0 double bond 23.2
Pcompound   ni Pi
94