1

TOPIC 4

NON-IDEAL FLUID BEHAVIOR

2

• 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.

3

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.

4

Supercritical

P-T phase diagram for a pure material.

C - critical point

5

P-V phase diagram for a pure material. C - critical point.

V

nRTP

At high T we expect the isotherms to conform to the ideal gas law, i.e., P is inversely proportional to V.

6

P-V phase diagram for pure H2O.

7

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.

8

Schematic isotherms in the two-phase field for a pure fluid.

A

XY

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).

B

P Q

9

MOST GENERAL EQUATION OF STATE

dPP

VdT

T

VdV

TP

dPdTV

dV

Two special cases:a) Incompressible fluid

= = 0dV/V = 0 (no equation of state exists)

V = constantb) and are temperature- and pressure-independent

10

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.

11

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!

12

I. THE COMPRESSIBILITY FACTOR

13

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.

14

The constants for the two versions of the virial equation are related by the equations:

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.

RT

BB ' 2

2

'RT

BCC

3

323'

RT

BBCDD

15

SOME APPROXIMATIONS

Low pressure (0 - 15 bars at T < Tc)

Becomes valid over greater pressure ranges as temperature increases. Easily solved for volume.

Moderate pressure (0 - 50 bars)

Only B and C are generally well known. At higher pressures, other EOS’s are required.

RT

BP

RT

PVZ 1

21

V

C

V

B

RT

PVZ

16

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:

RT

BPZ 1

17

The compressibility factor as a function of pressure for various gases. Z measures the deviation from the ideal gas law.

18

II. EQUATIONS OF STATE

19

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.

20

VAN DER WAALS EQUATION (1873)

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:

RTbVV

aP

2

2V

a

bV

RTP

21

P/Pc

V/Vc

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 one-phase region, but shows unrealistic oscillations in the two-phase region. The theory fails because it only considers two-body interactions.

22

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.,

Solving these equations we get:

0

TV

P0

2

2

TV

P

bVc 3

bR

aTc 27

8

227b

aPc 375.0

8

3

c

ccc RT

VPZ

23

CRITICAL CONSTANTS OF GASES - IGas 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

24

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 gases.

25

We can rearrange the previous equations to get the van der Waal parameters in terms of the critical parameters:

where

Note that, the actual measured value of Vc is not used to calculate a and b!

23 ccVPa 3

cVb

c

cc P

RTV

8

3

26

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 CH4 2.253 4.278

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

a - dm6 atm mol-2; b - 10-2 dm3 mol-1

27

SOME OTHER EOSs

Bertholet (1899)

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)

Keyes (1917)

, , A and l are correction factors.

RTbVVT

aP

2

VRT

a

ebV

RTP

2)( lV

A

eV

RTP

V

28

BEATTIE-BRIDGEMAN (1927)

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.

22

)1(

V

ABV

V

RTP

VaAA 10 VbBB 10 3TV

c

29

Rearrangement of the Beattie-Bridgeman equation gives:

Where

This shows the B-B equation to be simply a truncated form of the virial equation.

432 VVVV

RTP

200 T

RcARTB

20

00 T

cRBaAbRTB 2

0

T

bcRB

30

BEATTIE (1930)

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.

RT

ABV 1

aAA 10 bBB 10 3T

c

P

RT perfect gas volume

31

SOME MORE EOS’s

Jaffé (1947) - a modification of the Dieterici EOS

Wohl (1949)

VRTc ebV

TTR

bV

RTP

12

1

2

224

eP

TR

c

c2eP

RTb

c

c

3V

c

bVVbV

RTP

26 ccVP 4cVb 34 ccVPc

32

McLeod (1949)

where

and a, A, B, and c are constants.

Benedict-Webb-Rubin (1940) - specifically devised for hydrocarbons. Useful for both liquids and gases. Define

RTbV )'(

2V

aP 2' cBAb

Vd 1

2

2

236

322000

1 deT

dcdda

dabRTdTCARTBRTdP

33

Martin-Hou (1955)

Introduces the reduced temperature: Tr = T/Tc

Like many others, this EOS is also a version of the virial EOS.

55

44

3

239333

2

238222

)()(

)(

)(

)(

)(

)(

bV

TB

bV

A

bV

eCTBA

bV

eCTBA

bV

RTP

rr TT

34

35

REDLICH-KWONG (1949)This has been one of the most useful to geology

where

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.

)()( 21

bVVT

a

bV

RTP

c

c

P

TRa

5.2242748.0

c

c

P

RTb

08664.0 3333.0

3

1cZ

36

MODIFICATIONS OF THE R-K EOS

de Santis et al. (1974)

but b is a constant and a(T) = a0 + a1T.

Peng and Robinson (1976)

where

and

is a parameter for the fluid called the acentric factor.

)()( 21

bVVT

a

bV

RTP

)2(

)(

)( 22 bVbV

Ta

bV

RTP

221

11)( rTmT

226992.054226.137464.0 m

37

Carnahan and Starling (1969) - “hard-sphere” model

Kerrick and Jacobs (1981) - Hard-Sphere Modified Redlich-Kwong (HSMRK)

a(P,T) = an empirically-derived polynomial.

3

32

)1(

1

y

yyy

V

RTP

V

by

4

)(

),(

)1(

12

13

32

bVVT

TPa

y

yyy

V

RTP

2)()()(),(V

TeV

TdTcTPa

3321)( TzTzzTz where z = c, d, or e

38

LEE AND KESLER (1975)

21

20

21

12314

5

11211

2

313

11211

314

213

11211

20

02304

5

10201

2

303

10201

304

203

10201

1

1

r

r

V

rrrr

r

r

rr

r

rrr

V

rrrr

r

r

rr

r

rrr

eVVT

c

V

Tdd

V

TcTcc

V

TbTbTbb

eVVT

c

V

Tdd

V

TcTcc

V

TbTbTbbZ

39

DUAN, MOLLER AND WEARE (1992)

2

225421 rV

rrrrrr

eVV

F

V

E

V

D

V

C

V

BZ

33

221

rr TaTaaB 36

254

rr TaTaaC

39

287

rr TaTaaD 312

21110

rr TaTaaE

3 rTF c

cr RT

VPV

cr P

PP

cr T

TT

This is just a modified form the the Lee and Kesler EOS.

40

CALCULATING FUGACITY COEFFICIENTS BY INTEGRATING AN

EOSUsing the van der Waals equation:

2V

a

bV

RTP

P

PPRT

V

0

1ln

RT

VP

RTV

a

bV

b

bV

Vln

2lnln

41

Using the original Redlich-Kwong equation:

)()( 21

bVVT

a

bV

RTP

P

PPRT

V

0

1ln

RT

VP

bV

b

V

bV

bRT

a

V

bV

bRT

a

bV

b

bV

V

lnln

ln2

lnln

23

23

42

Using the HSMRK EOS of Kerrick and Jacobs (1981)

VbRT

e

V

bV

bRT

e

bRT

e

V

bV

bRT

d

bVRT

d

bV

V

bRT

c

bVVRT

e

bVVRT

d

bVRT

c

RT

VP

y

yyy

23

22

2

3

32

23

23

23

23

23

23

23

23

23

ln

2ln

ln

)()(

)(ln

)1(

398ln

43

Activities in CO2-H2O mixtures predicted by a MRK EOS after Kerrick & Jacobs (1981).

44

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).

45

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).

46

Effect of 12 wt. % NaCl on the CO2-H2O solvus at 1 kbar. After Bowers & Helgeson (1983).

47

III. CORRESPONDING STATES

48

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.

49

The reduced compressibility factor vs. the reduced pressure

501.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

Reduced pressure

Reduced pressure

Z Z

Generalized compressibilitychart. Medium- and high-pressure range.

RT

PVZ

51

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

Corresponding states: Tr = (773 K)/(405.5 K) = 1.91; Pr = (2000 atm)/(111 atm) = 18.02.

11

molL0310.0atm)(2000

)K773)(molLatm080256.0(

P

RTV

)K773)(molLatm080256.0(

Vatm)(200062.1

1Z

1molL050.0 V

52

53

54

55

56

Measured compressibility factors for H2O vs. those obtained from corresponding state theory

57

Measured compressibility factors for CO2 vs. those obtained from corresponding state theory

58Generalized density correlation for liquids. r = /c

59

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:

but the L-V curve terminates at the critical point where Tr = Pr. So a = b and

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.

r

satr T

baP log

r

satr TaP 11log

60

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 Pr

sat = -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:

can be determined from Tc, Pc and a single vapor pressure measurement at Tr = 0.7.

000.1log 7.0 rTsat

rP

61

Slope -3.2(n-octane)

Approximate temperature-dependence of reduced vapor pressure

62

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 0.12 n-heptane 0.350

N2 0.037 H2S 0.100 propane 0.153

F2 0.048 SO2 0.251 m-xylene 0.331

63

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:

The quantity in brackets is the reduced 2nd virial coefficient.

r

r

c

c

T

P

RT

BP

RT

BPZ

11

10 BBRT

BP

c

c

64

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!

6.10 422.0

083.0rT

B 2.41 172.0

139.0rT

B

65

Generalized correlation for Z0. Based on data for Ar, Kr and Xe from Pitzer’s correlation.

66

Generalized correlation for Z1 based on Pitzer’s correlation.

67

EXAMPLE -1 What is the volume of SO2 at P = 500 atm and T =

500°C?

According to the ideal gas law:

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

11

molL124.0atm)(500

)K773)(molLatm080256.0(

P

RTV

)K773)(molLatm080256.0(

Vatm)(500055.1

1Z -1mol L0.131V

68

Line defining the region where generalized second virial coefficients may be used. The line is based on Vr 2.

saturation

69

EXAMPLE - 2What is the volume of SO2 at P = 150 atm and T =

500°C?

According to the ideal gas law:

Using the acentric factor: =0.273; Tr = 773/430.8 = 1.79; Pr = 150/77.8 = 1.93.

11

molL414.0atm)(150

)K773)(molLatm080256.0(

P

RTV

083.079.1

422.0083.0

6.10 B

124.079.1

172.0139.0

2.41 B

70

Vc = 0.122 L mol-1

Vr = 0.392/0.122 = 3.25

049.0)124.0(273.0083.010

BB

RT

BP

c

c

947.079.1

93.1)049.0(11

r

r

c

c

T

P

RT

BPZ

)K773)(molLatm080256.0(

Vatm)(150947.0

1Z

-1mol L0.392V

71

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.

72

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.

rP

rr

dPP

Z0

1ln

73

74

75

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

76

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 = 51 cm3 mol-1; assume to be constant.

f2 = 8.36 atm

)(lnln 1212 PPRT

Vff

77

THREE-PARAMETER CORRELATIONS FOR FUGACITY ETC.

Fugacity:

Enthalpy:

Entropy:

Density:

Tables for these correlations can be found in Pitzer

(1995) Thermodynamics. McGraw-Hill.

)1()0(

logloglog

P

f

P

f

P

f

)1(0)0(00

ccc RT

HH

RT

HH

RT

HH

PR

SS

R

SS

R

SSln

)1(0)0(00

)91.089.1()1()1(85.01 31

rrc

TT

78

IV. GASEOUS MIXTURES

79

IDEAL GAS MIXTURES

• Mixture as a whole obeys:• 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.

RTVP

80

DALTON’S LAW VS. AMAGAT’S LAW

• Dalton’s Law: Pi = XiPT

• Amagat’s Law: Vi = XiVT

These two laws are mutually exclusive at a given pressure and temperature.

iT PP At constant VT and T

iT VV At constant PT and T

81

THERMODYNAMICS OF IDEAL MIXING - REVISITED

We have previously shown that:

using Dalton’s Law we can derive:

and for entropy we have:

i

iimixideal XXRTG ln

i T

iimixideal P

PXRS ln

i T

iimixideal P

PXRTG ln

82

NON-IDEAL MIXTURES OF NON-IDEAL GASES

For a perfect gas mixture:

For an ideal mixture of real gases:

For a real mixture of real gases:

iToii

oii XRTPRTPRT lnlnln

io

ioii

oii XRTfRTfRT lnlnln

Tiio

iii PXfXf

ioii fRT ln

Tiiio

iiii PXfXf

Lewis Fugacity Rule

Correction for non-ideal mixing

Correction for non-ideal gas

83

DALTON’S LAW AND GENERALIZED CHARTS

Calculate reduced pressure according to:

ic

iir P

PP

,,

RTZnVP AATA RTZnVP BBTB RTZnVP CCTC

RTZn

RTZXZXZXn

RTZnZnZnVPPP

mixT

CCBBAAT

CCBBAATCBA

84

AMAGAT’S LAW AND GENERALIZED CHARTS

Calculate reduced pressure according to:

ic

Tir P

PP

,,

RTZnVP AAAT RTZnVP BBBT RTZnVP CCCT

RTZn

RTZXZXZXn

RTZnZnZnVVVP

mixT

CCBBAAT

CCBBAACBAT

85

PSEUDOCRITICAL CONSTANTS

A

B

XA1 XA

2

L-V c urve fo r B

L-V c urve fo r A

P

T

86

KAY’S METHOD

Assumes a linear critical curve between the critical points for A and B.

i

icic PXP ,'

i

icic TXT ,'

When answers are near the critical point for the mixture, we cannot be certain that we are not dealing with a liquid-vapor mixture.

87

JAFFÉ’S METHOD

For binary mixtures only.

BcBBAcAA

mixc

c PXPXP

T,,

'

'

BABABBAA

mixc

c XXXXP

T 3

4122 3

13

1

'

'

ic

ici P

T

,

,

88

MIXING CONSTANTS IN EQUATIONS OF STATE

Van der Waals and simple Redlich-Kwong EOS:

n

jjjmix bXb

1

n

j

n

kjkkjmix aXXa

1 1

21

kjjk aaa Use if no mixture dataare available.

89

Beattie-Bridgeman EOS:

2

1,0,0

n

jjjmix AXA

n

jjjmix aXa

1

n

jjjmix bXb

1

n

jjjmix BXB

1,0,0

n

jjjmix cXc

1

90

Benedict-Webb-Rubin EOS:2

1,0,0

n

jjjmix AXA

3

1

3

n

jjjmix aXa

3

1

3

n

jjjmix bXb

n

jjjmix BXB

1,0,0

3

1

3

n

jjjmix cXc

2

1,0,0

n

jjjmix CXC

3

1

3

n

jjjmix X

2

1

n

jjjmix X

91

Virial Equation of State:

Z = 1 + B/V + C/V2 + D/V3 + …

n

i

n

jijjimix BXXB

1 1

n

i

n

j

n

kijkkjimix CXXXC

1 1 1

92

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.

93

C. All other compounds

Tc = 1.027 Tb + 159

Critical Pressure:

where Tc is in K and Vc is in cm3 g-1.

Critical Volume:

where is a parameter called the Sugten Parachor.

9

8.20

c

cc V

TP

2.1)0.11377.0( PVc

P

94

SUGTEN PARACHOR VALUES FOR ATOMS AND STRUCTURAL UNITSC 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

iicompound PnP