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Equations of state for real gases and liquids
(Fluids)
Introduction
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• search for a mathematical relationship between
  the variables Pressure, Temperature and Volume
  started about 300 years ago
• development of the ideal gas law until mid of
  19th century (R. Boyle, J.L. Gay-Lussac, B.P.E.
  Clapeyron)
• starting point for further development van der
  Waals Equation of state (vdW EOS)
• first EOS that could describe the liquid and the
  vapor phase qualitatively
• from mid of the 20th century until today a couple
  of hundred equations suggested

• practical interest
• a connection between variables P,T,v important
  since they can be easily measured

Introduction
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PVT- Verhalten (Reinstoff)
Zusammenhang zwischen Druck P, Temperatur T und
Volumen V Aggregatzustände in dreidimensionalem
Diagramm stabile und damit mögliche Zustände als
Flächen.
fest (s), flüssig (L), gasförmig (V) L und V
ähnlich in der molekularen Ordnung -gt
fluid homogene Gebiete, Mehrphasengebiete,
Linien kennzeichnen Phasenübergänge
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P-V Projektion (Reinstoff)
Isothermen Isotherme Zustandsänderung A-B-C im
2-Phasegebiet Mengen an Dampf und Flüssigkeit
nach dem Hebelgesetz Siede- und Tauline treffen
sich im kritischen Punkt Sattelpunkt der Isotherme
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P-T Projektion (Reinstoff)
Siede- und Tauline falle zu einer Kurve zusammen
-gt Dampfdruckkurve Schmelz- und Erstarrungslinie
-gtSchmelzkurve Phasenübergang zwischen Feststoff
und Dampf -gtSublimationskurve alle Kurven
treffen sich im Tripelpunkt Dampfdruckkurve endet
im kritischen Punkt
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important application of EOS fugacity
coefficient
fugacity coefficient
pure component
mixture
The fugacity coefficient is calculated by
integration from the ideal gas to the real
pressure

pressure dependence of the fugacity coefficient
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The Equation of State (EoS) relates the state
variables The form for a thermal EoS is
or
because of practical reasons, only v is used
instead of V

• There are several classes of EoS (based on
  their theory)
• virial equations of state
• cubic equations of state
• EoS from perturbation theory

Introduction
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a theoretical equation virial (lat.) force
The virial EoS is a Taylor expansion of the
compressibility factor in terms of density at the
ideal gas density
2nd virial coefficient
3rd virial coefficient

1st virial coefficient
virial equation of state
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theoretical Leiden Form
empirical Berlin Form
assuming that each of member of the Taylor
expansion adds the same value to the series, both
forms can be set equal

with the compressibility factor
Virial EoS truncated after 2nd virial coefficient
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Leiden Form
2nd Virial coefficient
3rd Virial coefficient
4th Virial coefficient
proven fact from experiments Z is function T and
r
B,C,D functions of T and since Z(T, r) not an
equal function for all gases
B,C,D for mixtures must also be a function of
the composition
rigorous theoretical foundation from statistical
mechanics provides analytic relations between
virial coefficients and interactions between
molecules
B interaction between pairs of molecules
C,D interaction between clusters of three (four)
molecules
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Temperature dependence of the virial coefficients

• significant advantage of virial equation very
  large database virial coefficients
• second virial coefficient by far most well
  studied
• values for C are known with some accuracy for a
  number of fluids
• few values for D reported

B and C can be calculated from statistical
mechanics when a pair potential is known -gt
valid only for very simple fluids!! e.g. krypton
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Temperature dependence of the virial coefficients
B positive for high temperatures, varies slowly
at lower T large negative values C positive at
high T, slowly varying, rapidly varying at lower
T with negative values

• B(T) at TB B(T)0
• attractive and repulsive forces are equally
• TB is about 3x Tcr
• e.g. Nitrogen Tcr330K TB770K

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the only theoretical mixing rule we have
written for a binary mixture
mixture
pure
pure
(1) 2nd virial coefficient of the pure substances
1 or 2 ? tabulated in the Dymond/Smith
book1 ? available from PvT-measurements ?
from empirical estimation methods ? calculable
from Tsonopoulos or Hayden/OConnell theoretical
approaches 2 (2) cross virial coefficient ?
tabulated in the Dymond/Smith book ? estimable
from the data on Bii using combining rules

1 Dymond, J.H. Smith, E.B. The virial
coefficients of gases Clarendon Press Oxford
1969. 2 Poling, B.E. Prausnitz, J.M.
OConnell, J.P. The Properties of Gases and
Liquids McGraw Hill, New York u.a. 2001.
Virial EoS and mixtures
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ideal gas law
van-der-Waals-equation of state
forces
volume of the molecule
a,b describe the component.
cubic equations of state van-der-Waals
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the pressure-temperature dependence is similar to
an ideal gas
at the critical point we have 2 equations
van-der-Waals EoS difference to ideal gas and at
the critical point
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the result is
the component-specific parameters can be replaced
from the EoS -gt generalized EoS
determination of the parameters with critical
data!
van-der-Waals EoS difference to ideal gas and at
the critical point

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Mixtures the mixture is considered to be ONE
FLUID -gt vdW one fluid mixing rules
mixing (composition)
combining the pure parameters
binary
van-der-Waals EoS mixing rules

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reduced variables
since vr is a function of Tr and Pr
zf(Tr,Pr)
-gt generalized EoS principle of corresponding
states (two parameter)
z and other variables have the same values for
all components when written in reduced form no
other component specific parameters than Tcr and
Pcr
Experiments show only valid for simple,
spherical molecules (Ar, Kr, Xe)
-gt third Parameter
van-der-Waals EoS corresponding states
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The acentric factor is the empirical attempt to
find a 3rd parameter for the principle of
corresponding states by Pitzer.
parameter written in form of vapor pressure so
that acentric factor 0 for simple Fluids (Ar,
Kr, Xe)
The reduced vapour pressure curves of the noble
gases show the logarithmic value of -1 at 70 of
the critical temperature, Tr0,7
This experimental value is taken to decribe the
non-spherical character of all other components

The acentric factor is used in several
correlation, e.g. in equations of state typical
values 0 0.4
acentric factor
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plot of reduced vapour pressure curves
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