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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 cm^{3}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}

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

( P + a / Vmwhere a and b are either determined empirically for each individual compound or estimated from the relations.^{2})( Vm - b ) = R T P = pressure Vm = molar volume R = ideal gas constant T = temperature

a = 27 RThe first parameter, a, is dependent upon the attractive forces between molecules while the second parameter, b, is dependent upon repulsive forces.^{2}Tc^{2}-------- 64 Pc b = R Tc ---- 8 Pc Tc = critical temperature Pc = critical pressure

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 ) Twhere a and b are not identical to the a and b in the van der Waals equation.^{1/2}p = pressure a = empirical constant Vm = molar volume R = ideal gas constant b = empirical constant T = temperature

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 / Vrwhere 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.^{2}) ( Vr - 1/3 ) = 8/3 * Tr Pr = reduced pressure Tr = reduced temperature Vr = reduced volume

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

p = R T - a ----- ---- V - b T VA somewhat more accurate modified Berthelot equation is^{2}p = pressure a = empirical constant V = volume R = ideal gas constant b = empirical constant T = temperature

p = R T [ 1 + 9 p Tc ( 1 - 6 TcThe Dieterici equation is another two parameter equation which has been seldom used in recent years.^{2}) ] --- -------- ----- V 128 Pc T T^{2}p = pressure Pc = critial pressure V = volume R = ideal gas constant T = temperature Tc = critical temperature

p = R T eThe Clausius equation is a simple three parameter equation of state.^{-a / ( Vm R T )}------------------ Vm - b p = pressure a = empirical constant Vm = molar volume R = ideal gas constant b = empirical constant T = temperature

[ P + a ] ( Vm - b ) = R T ------------- T ( Vm + c )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.^{2}a = Vc - R Tc ---- 4 Pc b = 3 R Tc - Vc ------ 8 Pc c = 27 R^{2}Tc^{3}-------- 64 Pc P = pressure T = temperature R = real gas constant Vm = molar volume Tc = critical temperature Vc = critical volume Tc = critical temperature

p Vm = R T ( 1 + B(T) / Vm + C(T) / Vmwhere B is not identical to B' and etc.^{2}+ D(T) / Vm^{3}+ ... ) or p Vm = R T ( 1 + B'(T) / p + C'(T) / p^{2}+ D'(T) / p^{3}+ ... ) 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

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 = temperatureThe 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 ) TA some what more complex equation is the Beattie-Bridgeman equation^{2}Vm^{3}a = 6 Pc Tc Vc^{2}b = Vc / 4 c = 4 Pc Tc^{2}Vc^{3}p = pressure Vm = molar volume R = ideal gas constant T = temperature Pc = critical pressure Tc = critical temperature Vc = critical volume

P = R T d + ( B R T - A - R c / TBenedict, Webb and Rubin suggest the real gas equation of state^{2}) d^{2}+ ( - B b R T + A a - R B c / T^{2}) d^{3}+ R B b c d^{4}/ T^{2}P = pressure R = ideal gas constant T = temperature d = molal density a, b, c, A, B = empirical parameters

P = R T d + d^{2}{ R T [ B + b d ] - [ A + a d - a alpha d^{4}] - 1 [ C - c d ( 1 + gama d^{2}) exp ( - gama d^{2}) ] } -- T^{2}P = pressure R = ideal gas constant T = temperature d = molal density a, b, c, A, B, C, alpha, gama = empirical parameters

**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 Twhere the empirical constants are all positive and specific to each substance.^{2}- C4 p - C5 p T Vm = molar volume T = temperature p = pressure C1, C2, C3, C4, C5 = empirical constants

For constant pressure processes, this equation is often shortened to

Vm = Vmo ( 1 + A T + B Twhere A and B are positive.^{2}) Vm = molar volume Vmo = molar volume at 0 degrees C T = temperature A, B = empirical constants

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

**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 Twhere the empirical constants are all positive and specific to each substance.^{2}- C4 p - C5 p T Vm = molar volume T = temperature p = pressure C1, C2, C3, C4, C5 = empirical constants

For constant pressure processes, this equation is often shortened to

Vm = Vmo ( 1 + A T + B Twhere A and B are positive.^{2}) Vm = molar volume Vmo = molar volume at 0 degrees C T = temperature A, B = empirical constants

**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.

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).