Example Equations of State
liquid phase, to the left of
the critical temperature
isotherm and the saturated
liquid curve.
1 of 9
super-critical fluid
phase, to the right of
the critical temperature
isotherm and above
the critical pressure
critical temperature isotherm
Dotted curve represented
by points ALC is the sat'd
liquid curve.
Dotted curve represented
by points CVB is the sat'd
vapor curve.
gas phase, below the
critical pressure and to
the right of the
saturated vapor curve
Note that the solid region is
not shown in this diagram.
critical pressure
vapor phase, subregion of the gas
phase that is below
critical temperature
isotherm and above
saturated vapor curve
saturated liquid and saturated
vapor phases coexisting at
equilibrium.
These units are inverse
density. To get molar volume
(mL / mol), just multiple by the
molecular weight of carbon
dioxide (44.0095).
Data used in the above PVT diagram were extracted from the "Thermophysical Properties of Fluid
Systems" by E.W. Lemmon, M.O. McLinden and D.G. Friend in NIST Chemistry WebBook, NIST
Standard Reference Database Number 69, Editors: P.J. Linstrom and W.G. Mallard, National Institute
of Standards Technology, Gaithersburg MD, 20899, http://webbook.nist.gov, (retrieved Sept. 22, 2010).
Click here to view the pressure-temperature
(PT), pressure-enthalpy (PH), and pressureinternal-energy (PU) diagrams for pure CO2.
ALCVB.
F
The specific volume at Point F is defined by the reverse lever rule, which relates vapor
fraction and specific volumes of the sat'd liquid and vapor at Points L and V, as follows:
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© 2007, Michael E. Hanyak, Jr., All Rights Reserved
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Example Equations of State
2 of 9
For carbon dioxide,
the limit is greater
than 454 cm3/g.
To compare the "NIST data" PVT diagram with some example equation-of-state PVT diagrams,
click here to download, open an Excel file, and view worksheets like "ig-PVT" and "ig-nist".
v07.07.15
© 2007, Michael E. Hanyak, Jr., All Rights Reserved
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Example Equations of State
3 of 9
; that is,
when the fugacity coefficients of the sat'd liquid and sat'd vapor are equivalent.
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© 2007, Michael E. Hanyak, Jr., All Rights Reserved
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Example Equations of State
4 of 9
0.42748
0.08664
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Example Equations of State
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© 2007, Michael E. Hanyak, Jr., All Rights Reserved
5 of 9
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Example Equations of State
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A modified version of this equation of
state exists in the Aspen HYSYS
simulator. It is labeled MBWR.
The Aspen HYSYS simulator has several fluid packages that are cubic equations of state. Some are Soave-RedlichKwong (SRK) and Peng-Robinson (PR), as well as several modified version of these two equations of state. For
general simulation calculations in Aspen HYSYS, we will used Peng-Robinson Stryjek-Vera (PRSV), a modified
version of the PR equation of state, in this introductory course about chemical engineering.
The bottom line with respect to any equation of state is how well it predicts quantities like molar volume, specific
enthalpy, and distribution coefficients for the mixture of chemical compounds that you are working with. Most
process simulators like Aspen HYSYS have documentation on how to pick the appropriate fluid package to
predict thermodynamic quantities. It is important for you to check the validity of a fluid package that you select.
To access the Aspen HYSYS documentation about fluid packages at Bucknell University, select Start/Run...
from the Windows desktop, type \\eng-file1\engapps\Hysys-Docs in the "Open" text box and click the OK
button. Open the file AspenHYSYSSimulationBasis.pdf, select "Property Methods and Calculations", then
select "Selecting Property Methods", and finally read about the different property packages.
v07.07.15
© 2007, Michael E. Hanyak, Jr., All Rights Reserved
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Example Equations of State
7 of 9
A generalized correlation to find the molar
volume of a chemical mixture. When the
number (nc) of chemical components in the
mixture is one, the models below define the
molar volume for a pure chemical compound.
In the Apsen HYSYS simulator, the fluid package named Lee-Kesler-Plocker is an equation of state that represents the
generalized compressibility figures found in the Felder and Rousseau textbook [2005, pp. 207-212].
v07.07.15
© 2007, Michael E. Hanyak, Jr., All Rights Reserved
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Example Equations of State
8 of 9
Peng-Robinson Cubic Equation of State
Molar Volume of a Pure Substance
P=
RT
aα
− 2
ˆ
ˆ
V − b V + 2bVˆ − b 2
or
) 0
z 3 + ( B − 1) z 2 + ( A − 3B 2 − 2 B ) z + ( B 3 + B 2 − AB=
2
( RTc )
RTc
PVˆ
a 0.45724=
b 0.07780
z
=
=
Pc
Pc
RT
)(
(
)
1 + 0.37464 + 1.54226ω − 0.26992ω 2 1 − T 
α=
r

2
T
Tr =
Tc
Fugacity Coefficient of a Pure Substance
A
 z + 2.4142 B 
ln 

2.8284 B  z − 0.4142 B 
aα P
bP
=
A =
B
2
RT
( RT )
ln φ = ( z − 1) − ln( z − B ) −
φV = φ L
Vapor-Liquid Equilibrium for a Pure Substance:
Molar Volume of a Mixture of Substances
P=
RT
aα
− 2
ˆ
ˆ
V − b V + 2bVˆ − b 2
=
aα
nc
or
) 0
z 3 + ( B − 1) z 2 + ( A − 3B 2 − 2 B ) z + ( B 3 + B 2 − AB=
nc
nc
=
b ∑
x b
( aα )
( aα )
( aα ) ( aα )
∑∑ x x=
j
k
j
jk
k 1
=j 1 =
( RT )
b
0.45724
=
2
aj
=
j
jk
=j 1
c, j
j
Pc , j
)
1 + 0.37464 + 1.54226ω − 0.26992ω 2 1 − T 
αj =
j
j
r, j

kk
RTc , j
PVˆ
z
0.07780
=
Pc , j
RT
)(
(
jj
2
Tr , j =
T
Tc , j
Fugacity Coefficient of the i-th Substance in a Mixture of Substances
 Bi
2
A
ˆ Bi ( z − 1) − ln( z − B) +
ln φ=
 −
i
2.8284 B  B a α
B
=
A
nc
nc
x x A
A
∑∑=
j
k
jk
jk
  z + 2.4142 B 
x j ( a α )ij  ln 

j =1
  z − 0.4142 B 
nc
∑
=
A jj Akk
A jj
k 1
=j 1 =
=
B
( a α ) jj P
( RT )2
bi P
bP
=
Bi =
( a α ) jj a j α j
RT
RT
Vapor-Liquid Equilibrium for the i-th Substance in a Mixture:
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© 2008, Michael E. Hanyak, Jr., All Rights Reserved
yi φˆiV P = xi φˆiL P
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Example Equations of State
9 of 9
Both the Peng-Robinson (PR) and the Soave-Redlich-Kwong (SRK)
equations of state predict the vapor-liquid dome region and the liquid region
with molar volumes and vapor pressures having errors on the order of 2%.
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© 2010, Michael E. Hanyak, Jr., All Rights Reserved
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