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).
Return to table of contents.
Boyle's law (1662)
Boyle's Law was perhaps the first expression of an equation of state. In 1662, the noted Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was carefully measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:
The above relationship has also been attributed to Edme Mariotte and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.
[edit]Charles's law or Law of Charles and Gay-Lussac (1787)
In 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:
[edit]Dalton's law of partial pressures (1801)
Dalton's Law of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.
Mathematically, this can be represented for n species as:
[edit]The ideal gas law (1834)
In 1834 Émile Clapeyron combined Boyle's Law and Charles' law into the first statement of the ideal gas law. Initially the law was formulated as pVm=R(TC+267) (with temperature expressed in degrees Celsius), where R is the gas constant. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0 °C = 273.15 K, giving:
[edit]Van der Waals equation of state
In 1873, J. D. van der Waals introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules.[2] His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich–Kwong equation of state and the Soave modification of Redlich-Kwong.
[edit]Major equations of state
For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form:
In the following equations the variables are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.
= pressure (absolute)
= volume
= number of moles of a substance
= = molar volume, the volume of 1 mole of gas or liquid
= absolute temperature
= ideal gas constant (8.314472 J/(mol·K))
= pressure at the critical point
= molar volume at the critical point
= absolute temperature at the critical point
[edit]Classical ideal gas law
The classical ideal gas law may be written:
The ideal gas law may also be expressed as follows
where ρ is the density, γ = Cp / Cv is the adiabatic index (ratio of specific heats), e = CvT is the internal energy per unit mass (the "specific internal energy"), Cv is the specific heat at constant volume, and Cp is the specific heat at constant pressure.
[edit]Cubic equations of state
[edit]Van der Waals equation of state
The Van der Waals equation of state may be written:
where Vm is molar volume, and a and b are substance-specific constants. They can be calculated from the critical properties pc,Tc and Vc (noting that Vc is a the molar volume at the critical point) as:
Also written as
Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation a is called the attraction parameter and b the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.
The van der Waals equation may be considered as the ideal gas law, “improved” due to two independent reasons:
Molecules are thought as particles with volume, not material points. Thus V cannot be too little, less than some constant. So we get (V − b) instead of V.
While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write (p + something) instead of p. To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface. While the force acting on each surface molecule is ~ρ, the force acting on the whole element is ~ρ2~.
[edit]Redlich–Kwong equation of state
Introduced in 1949 the Redlich–Kwong equation of state was a considerable improvement over other equations of the time. It is still of interest primarily due to its relatively simple form. While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating vapor-liquid equilibria. However, it can be used in conjunction with separate liquid-phase correlations for this purpose.
The Redlich–Kwong equation is adequate for calculation of gas phase properties when the ratio of the pressure to the critical pressure (reduced pressure) is less than about one-half of the ratio of the temperature to the critical temperature (reduced temperature):
[edit]Soave modification of Redlich-Kwong
Where ω is the acentric factor for the species.
for hydrogen:
In 1972 Soave replaced the 1/√(T) term of the Redlich-Kwong equation with a function α(T,ω) involving the temperature and the acentric factor. The α function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for these materials.
Note especially that this replacement changes the definition of a slightly, as the Tc is now to the second power.
[edit]Peng-Robinson equation of state
In polynomial form:
where, ω is the acentric factor of the species, R is the universal gas constant and Z=PV/(RT) is compressibility factor.
The Peng-Robinson equation was developed in 1976 in order to satisfy the following goals:[3]
The parameters should be expressible in terms of the critical properties and the acentric factor.
The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factor and liquid density.
The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature pressure and composition.
The equation should be applicable to all calculations of all fluid properties in natural gas processes.
For the most part the Peng-Robinson equation exhibits performance similar to the Soave equation, although it is generally superior in predicting the liquid densities of many materials, especially nonpolar ones. The departure functions of the Peng-Robinson equation are given on a separate article.