NONIDEAL FLUID BEHAVIOR

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TOPIC 4

• NON-IDEAL FLUID BEHAVIOR

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• 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.

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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.

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Supercritical
C - critical point
P-T phase diagram for a pure material.
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P-V phase diagram for a pure material. C -
critical point.
At high T we expect the isotherms to conform to
the ideal gas law, i.e., P is inversely
proportional to V.
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P-V phase diagram for pure H2O.
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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.

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Schematic isotherms in the two-phase field for a
pure fluid.
For fluid of density A, the proportion of vapor
is Y/(XY) and the proportion of liquid is
X/(XY). For fluid of density B, the proportion
of vapor is P/(PQ) and the proportion of liquid
is Q/(PQ).
A
B
P
Q
X
Y
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MOST GENERAL EQUATION OF STATE
Two special cases a) Incompressible fluid ? ?
0 dV/V 0 (no equation of state exists) V
constant b) ? and ? are temperature- and
pressure-independent
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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.

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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!
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I. THE COMPRESSIBILITY FACTOR
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THE COMPRESSIBILITY FACTOR

• Z ? PV/RT 1 BP CP2 DP3
• 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.

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• 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.

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SOME APPROXIMATIONS

• Low pressure (0 - 15 bars at T lt 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 EOSs are required.

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

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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 pressures 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

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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.
P/Pc
V/Vc
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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

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CRITICAL CONSTANTS OF GASES - I
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CRITICAL CONSTANTS OF GASES - II
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.
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• 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!

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VAN DER WAALS CONSTANTS FOR GASES
a - dm6 atm mol-2 b - 10-2 dm3 mol-1
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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.

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BEATTIE-BRIDGEMAN (1927)

• a, b, c, A0, B0 are constants
• We usually dont 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.

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• Rearrangement of the Beattie-Bridgeman equation
  gives
• Where
• This shows the B-B equation to be simply a
  truncated form of the virial equation.

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BEATTIE (1930)
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.

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SOME MORE EOSs

• Jaffé (1947) - a modification of the Dieterici
  EOS
• Wohl (1949)

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• 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

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• Martin-Hou (1955)
• Introduces the reduced temperature Tr T/Tc
• Like many others, this EOS is also a version of
  the virial EOS.

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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.

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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.

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• Carnahan and Starling (1969) - hard-sphere
  model
• Kerrick and Jacobs (1981) - Hard-Sphere Modified
  Redlich-Kwong (HSMRK)
• a(P,T) an empirically-derived polynomial.

where z c, d, or e
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LEE AND KESLER (1975)
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DUAN, MOLLER AND WEARE (1992)
This is just a modified form the the Lee and
Kesler EOS.
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CALCULATING FUGACITY COEFFICIENTS BY INTEGRATING
AN EOS

• Using the van der Waals equation

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• Using the original Redlich-Kwong equation

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Using the HSMRK EOS of Kerrick and Jacobs (1981)
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Activities in CO2-H2O mixtures predicted by a MRK
EOS after Kerrick Jacobs (1981).
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Predicted H2O and CO2 activities in H2O-CO2-CH4
mixtures at 400C 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).
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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).
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Effect of 12 wt. NaCl on the CO2-H2O solvus at
1 kbar. After Bowers Helgeson (1983).
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III. CORRESPONDING STATES
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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.

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The reduced compressibility factor vs. the
reduced pressure
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Reduced pressure
Generalized compressibility chart. Medium- and
high-pressure range.
3.0
2.8
2.6
2.4
Z
Z
2.2
2.0
1.8
1.6
1.4
1.2
1.0
Reduced pressure
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• EXAMPLE Calculate the specific volume of NH3 at
  500C 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.

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Measured compressibility factors for H2O vs.
those obtained from corresponding state theory
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Measured compressibility factors for CO2 vs.
those obtained from corresponding state theory
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Generalized density correlation for liquids. ?r
?/?c
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PITZERS 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.

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

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Slope ? -3.2 (n-octane)
Approximate temperature-dependence of reduced
vapor pressure
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ACENTRIC FACTORS FOR GASES
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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.

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• 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!

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Generalized correlation for Z0. Based on data for
Ar, Kr and Xe from Pitzers correlation.
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Generalized correlation for Z1 based on Pitzers
correlation.
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EXAMPLE -1

• What is the volume of SO2 at P 500 atm and T
  500C?
• 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

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

• What is the volume of SO2 at P 150 atm and T
  500C?
• According to the ideal gas law
• Using the acentric factor ? 0.273 Tr
  773/430.8 1.79 Pr 150/77.8 1.93.

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• Vc 0.122 L mol-1
• Vr 0.392/0.122 3.25

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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.

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• 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 Ts and
  Ps
• 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.

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USE OF TWO-PARAMETER GENERALIZED FUGACITY CHARTS

• EXAMPLE 1 Calculate the fugacity of CO2 at 600C
  (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

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• EXAMPLE 2 What is the fugacity of liquid Cl2 at
  25C and 100 atm? The vapor pressure of Cl2 at
  25C 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

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THREE-PARAMETER CORRELATIONS FOR FUGACITY ETC.

• Fugacity
• Enthalpy
• Entropy
• Density
• Tables for these correlations can be found in
  Pitzer (1995) Thermodynamics. McGraw-Hill.

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IV. GASEOUS MIXTURES
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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.

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DALTONS LAW VS. AMAGATS LAW

• Daltons Law Pi XiPT
• Amagats Law Vi XiVT
• These two laws are mutually exclusive at a given
  pressure and temperature.

At constant VT and T
At constant PT and T
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THERMODYNAMICS OF IDEAL MIXING - REVISITED

• We have previously shown that
• using Daltons Law we can derive
• and for entropy we have

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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

Lewis Fugacity Rule
Correction for non-ideal mixing
Correction for non-ideal gas
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DALTONS LAW AND GENERALIZED CHARTS

• Calculate reduced pressure according to

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AMAGATS LAW AND GENERALIZED CHARTS

• Calculate reduced pressure according to

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PSEUDOCRITICAL CONSTANTS
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KAYS METHOD

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

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

• For binary mixtures only.

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MIXING CONSTANTS IN EQUATIONS OF STATE

• Van der Waals and simple Redlich-Kwong EOS

Use if no mixture data are available.
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• Beattie-Bridgeman EOS

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• Benedict-Webb-Rubin EOS

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Virial Equation of StateZ 1 B/V C/V2
D/V3
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PREDICTION OF CRITICAL CONSTANTS

• Critical Temperature
• I. All compounds with Tboil (1 atm) lt 235 K and
  all elements Tc 1.70Tb - 2.00.
• II. All compounds with Tboil (1 atm) gt 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.

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• 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.

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SUGTEN PARACHOR VALUES FOR ATOMS AND STRUCTURAL
UNITS