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Equations of State

# Equations of State

David Young
Cytoclonal Pharmaceutics Inc.

An equation of state is a formula describing the interconnection between various macroscopically measurable properties of a system. This document only adresses the behavior of physical states of matter, not the conversion from one state to another.

For physical states of matter, this equation usually relates the thermodynamic variables of pressure, temperature, volume and number of atoms to one another.

In materials science the important properties are often what are termed "mechanical properties" rather than physical properties. Examples of mechanical properties would be hardness and ductility. Mechanical properties will not be addressed here.

Gas - There are several types of gases with slightly different behaviors. These are ideal gasses, real gasses, super critical fluids, plasmas and critical opalescent conditions. The ideal gas law is often used as the first order description of any gas although this practice is questionable in the case of critical opalescent conditions.

Ideal Gas - Although no gas is truly ideal, many gasses follow the ideal gas law very closely at sufficiently low pressures. The ideal gas law was originally determined empirically and is simply

p V = n R T

p = absolute pressure (not gage pressure)
V = volume
n = amount of substance (usually in moles)
R = ideal gas constant
T = absolute temperature (not F or C)

where some values for R are

8.3145 J mol-1 K-1

0.0831451 L bar K-1 mol-1

82.058 cm3 atm mol-1 K-1

0.0820578 L atom mol-1 K-1

1.98722 cal mol-1 K-1

62.364 L Torr K-1 mol-1
Real Gas - Real gas laws try to predict the true behavior of a gas better than the ideal gas law by putting in terms to describe attractions and repulsions between molecules. These laws have been determined empirically or based on a conceptual model of molecular interactions or from statistical mechanics.

A well known real gas law is the van der Waals equation

( P + a / Vm2 )( Vm - b ) = R T

P = pressure
Vm = molar volume
R = ideal gas constant
T = temperature
where a and b are either determined empirically for each individual compound or estimated from the relations.
a = 27 R2 Tc2
--------
64  Pc

b = R Tc
----
8 Pc

Tc = critical temperature
Pc = critical pressure
The first parameter, a, is dependent upon the attractive forces between molecules while the second parameter, b, is dependent upon repulsive forces.

Another two parameter real gas equation is the Redlich-Kwong equation. It is almost always more accurate than the van der Waals equation and often more accurate than some equations with more than two parameters. The Redlich-Kwong equation is

( p +         a         ) ( Vm - b ) = R T
------------------
Vm ( Vm + b ) T1/2

p = pressure
a = empirical constant
Vm = molar volume
R = ideal gas constant
b = empirical constant
T = temperature
where a and b are not identical to the a and b in the van der Waals equation.

Equations of state in terms of reduced variables give reasonable results without any empirically determined constants for a specific substance. However, these are not generally as accurate as equations using empirical constants. One such equation is

( Pr + 3 / Vr2 ) ( Vr - 1/3 ) = 8/3 * Tr

Pr = reduced pressure
Tr = reduced temperature
Vr = reduced volume
where reduced pressure and temperature are the unitless quantities obtained by dividing the value by the critical value. In the case of reduced volume, molar volume is divided by critical molar volume.

A two parameter equation which is no longer used much is the Berthelot equation

p =  R T  -  a
-----   ----
V - b   T V2

p = pressure
a = empirical constant
V = volume
R = ideal gas constant
b = empirical constant
T = temperature
A somewhat more accurate modified Berthelot equation is

p = R T [ 1 +  9 p Tc  ( 1 - 6 Tc2 ) ]
---       --------       -----
V        128 Pc T         T2

p = pressure
Pc = critial pressure
V = volume
R = ideal gas constant
T = temperature
Tc = critical temperature
The Dieterici equation is another two parameter equation which has been seldom used in recent years.

p = R T e-a / ( Vm R T )
------------------
Vm - b

p = pressure
a = empirical constant
Vm = molar volume
R = ideal gas constant
b = empirical constant
T = temperature
The Clausius equation is a simple three parameter equation of state.

[ P +       a       ] ( Vm - b ) = R T
-------------
T ( Vm + c )2

a = Vc - R Tc
----
4 Pc

b = 3 R Tc - Vc
------
8 Pc

c = 27 R2 Tc3
--------
64 Pc

P = pressure
T = temperature
R = real gas constant
Vm = molar volume
Tc = critical temperature
Vc = critical volume
Tc = critical temperature
The virial equation is popular because the constants are readily obtained using a perturbative treatment such as from statistical mechanics. The virial coeficients are also readily fitted to experimental data because it is a linear curve fit.

p Vm = R T ( 1 + B(T) / Vm + C(T) / Vm2 + D(T) / Vm3 + ... )

or
p Vm = R T ( 1 + B'(T) / p + C'(T) / p2 + D'(T) / p3 + ... )

p = pressure
Vm = molar volume
R = ideal gas constant
T = temperature
B, C, D, .. = constants for a given temperature
B', C', D', .. = constants for a given temperature
where B is not identical to B' and etc.

The equation of state created by Peng and Robinson has been found to be useful for both liquids and real gasses.

p =  R T   -          a ( T )
------   ----------------------------
Vm - b   Vm ( Vm + b ) + b ( Vm - b )

p = pressure
a = empirical constant
Vm = molar volume
R = ideal gas constant
b = empirical constant
T = temperature
The Wohl equation is formulated in terms of critial values making it a bit more convenient for situations where no real gas constants are available

[ p +         a        -   c   ] ( Vm - b ) = R T
---------------   ------
T Vm ( Vm - b )   T2 Vm3

a = 6 Pc Tc Vc2

b = Vc / 4

c = 4 Pc Tc2 Vc3

p = pressure
Vm = molar volume
R = ideal gas constant
T = temperature
Pc = critical pressure
Tc = critical temperature
Vc = critical volume
A some what more complex equation is the Beattie-Bridgeman equation

P = R T d + ( B R T - A - R c / T2 ) d2 + ( - B b R T + A a - R B c / T2 ) d3

+ R B b c d4 / T2

P = pressure
R = ideal gas constant
T = temperature
d = molal density
a, b, c, A, B = empirical parameters
Benedict, Webb and Rubin suggest the real gas equation of state

P = R T d + d2 { R T [ B + b d ] - [ A + a d - a alpha d4 ]

- 1  [ C - c d ( 1 + gama d2 ) exp ( - gama d2 ) ] }
--
T2

P = pressure
R = ideal gas constant
T = temperature
d = molal density
a, b, c, A, B, C, alpha, gama = empirical parameters

Supercritical Fluids - Supercritical fluids are well described by real and ideal gas laws.

Critical Opalescence - Critical behavior is generally described using real gas equations which have constants defined in a way which ensures that the slope of reduced pressure vs. reduced volume is zero at the critical point. These give reasonable estimates of the relationships between pressure, volume and temperature but do not describe the opalescence or unique chemical properties very near the critical point.

Plasma - The physical behavior of plasmas is most often described by the ideal gas law equation which is quite reasonable except at very high pressures.

Liquid - Liquids are much less compressible than gasses. Even when a liquid is described with an equation similar to a gas equation, the constants in the equation will result in much less dramatic changes in volume with a change in temperature. Like wise at constant volume, a temperature change will give a much larger pressure change than seen in a gas.

A common equation of state for both liquids and solids is

Vm = C1 + C2 T + C3 T2 - C4 p - C5 p T

Vm = molar volume
T = temperature
p = pressure
C1, C2, C3, C4, C5 = empirical constants
where the empirical constants are all positive and specific to each substance.

For constant pressure processes, this equation is often shortened to

Vm = Vmo ( 1 + A T + B T2 )

Vm = molar volume
Vmo = molar volume at 0 degrees C
T = temperature
A, B = empirical constants
where A and B are positive.

The equation of state created by Peng and Robinson has been found to be useful for both liquids and real gasses.

p =  R T   -          a ( T )
------   ----------------------------
Vm - b   Vm ( Vm + b ) + b ( Vm - b )

p = pressure
a = empirical constant
Vm = molar volume
R = ideal gas constant
b = empirical constant
T = temperature

Superfluid - Superfluids are physically liquids although they have interesting properties, which are quantum mechanical in origin. Since this is still an active area of research and not completely understood, a reference to an introductory article is given, but no equations will be presented here.

Suspension - Suspensions behave physically most like liquids.

Colloid - A colloid being a type of suspension is also physically most like a liquid.

Liquid Crystal - Depending upon the temperature, liquid crystals may be crystalline, glassy, flexible thermoplastics or ordered liquids. At sufficiently high temperatures, a true liquid phase will exist. Most of the physical properties of these are the same as non liquid crystal compounds. One exception is that as liquid crystal compounds are added to a solvent the viscosity increases as expected until the concetration becomes high enough to form a liquid crystal phase, when the viscosity drops.

Visceoelastic - Since visceoelastics behave like solids on short time scales and like liquids over a long period of time, equations for liquids and solids could be used. Most of the usefulness of visceoelastics is based on their mechanical properties rather than their physical properties.

Solid - The volume of a solid will generally change very little with a change in temperature. However, most solids are very incompressible so a constant volume heating will give a very large pressure change for even a small change in temperature. Crystals, glasses and elastomers are all types of solids.

A common equation of state for both liquids and solids is

Vm = C1 + C2 T + C3 T2 - C4 p - C5 p T

Vm = molar volume
T = temperature
p = pressure
C1, C2, C3, C4, C5 = empirical constants
where the empirical constants are all positive and specific to each substance.

For constant pressure processes, this equation is often shortened to

Vm = Vmo ( 1 + A T + B T2 )

Vm = molar volume
Vmo = molar volume at 0 degrees C
T = temperature
A, B = empirical constants
where A and B are positive.

Crystal - Crystals are solids which are often very hard. The equations above are used for describing the physical properties of crystals.

Glass - Glasses are generally very brittle. The equations above are useful for describing the physical behavior until the stress becomes too great and the material shatters.

Elastomer - An elastomer is an amorphous solid which can be deformed with out breaking. The change in volume is generally negligible with deformation. However, the cross sectional area may change considerably. For changes in temperature and pressure, elastomers can be considered to be solids although much softer than other solids.

Superplastic - The unique ability of superplastics to stretch is a mechanical property. Physically, superplastics are treated as solids.

Bose-Einstein Condensate - At the time of this writing, the first reports of having made a Bose-Einstein condensate have just been released. No measurements of physical properties have yet been made. Considering various aspects of the theory predicting the existence of this state lead to the conclusions that it might be a solid or a very supercooled gas or one very large single atom.

Refractory - Refractory materials behave physically as solids.

## Further Information

For an introductory chemistry text see
L. Pauling "General Chemistry" Dover (1970)

A physical chemistry text for non-chemists is
P. W. Atkins "The Elements of Physical Chemistry" Oxford University Press (1993)

A physical chemistry text for undergraduate chemistry majors is
I. N. Levine "Physical Chemistry" McGraw-Hill (1995)

A review of real gas equations is
K. K. Shah, G. Thodos Industrial and Engineering Chemistry, vol 57, no 3, p. 30 (1965)

An introductory article about superfluids is
O. V. Lounasmaa, G. Pickett Scientific American, p. 104, June (1990)

A mathematical treatment can be found in
D. L. Goodstein "States of Matter" Dover (1985)

Properties of high molecular weight solids (most commonly polymers) are discussed in
H. R. Allcock, F. W. Lampe "Contemporary Polymer Chemistry" Prentice-Hall (1990)

Solid state properties are covered in
A. R. West "Solid State Chemistry and its Applications" John Wiley & Sons (1992)

A review article is
M. Ross, D. A. Young, Ann. Rev. Phys. Chem. 44, 61 (1993).

 Modified: Thu Jan 4 02:29:25 2001 GMT Page accessed 9238 times since Thu Jan 4 04:27:04 2001 GMT