Equation of state

In physics, equations of state attempt to describe the relationship between temperature, pressure, and volume for a given substance or mixture of substances. The ideal gas law, shown below, is one of the simplest equations of state. Although reasonably accurate for gases at low pressures and high temperatures, it becomes increasingly inaccurate at higher pressures and lower temperatures.

Despite its shortcomings, the ideal gas law is used extensively in many fields of science and engineering. Due to its simple form, straightforward solutions to a number of problems involving the equation of state can be obtained if the system of interrest can be assumed to behave as an ideal gas. The solutions become much more complicated and difficult to use for the cases where more accurate (and complicated) equations of state must be used.

Using statistical mechanics, the ideal gas law can be derived by assuming that a gas is composed of a large number of small molecules, with no attractive or repulsive forces. In reality gas molecules do interact with attractive and repulsive forces. In fact it is these forces that result in the formation of liquids.

A major weakness of the ideal gas law is its failure to predict the formation of liquid. Most other equations of state do predict the formation of a liquid phase. Usually these equations are cubic in volume and when solved will have either one or three real roots. When there is one real root, there is no liquid phase and the solution corresponds to the volume of the gas phase. When three real roots exist, one solution corresponds to the gas phase and one to the liquid phase. The intermediate root is an artefact and has no real meaning.

Table of contents

1 Examples of Equations of State
2 Ideal Gas Law
3 Van der Waals equation
4 The Virial Equation
5 Redlich-Kwong Equation of State
6 The Soave Equation
7 The Peng-Robinson Equation of State
8 The BWRS Equation of State
9 Stiffened equation of state
10 History
11 Boyle's law (1662)
12 Charles' law (1787)
13 Dalton's law of partial pressures (1801)
14 The Ideal gas law (1834)
15 van der Waals Equation of State (1873)
16 Amagat's law (1880)

Examples of Equations of State

In the following equations the variables are defined as follows, any consistent set of units can be used although SI units are preferred:

P = Pressure

V = Molar volume, the volume of 1 mole of gas or liquid

T = Temperature (K)

Ideal Gas Law

R = Ideal Gas Constant (8.31451 J/mol·K)

The ideal gas law may also be expressed as follows

where is the density, the adiabatic index, and e the internal energy. This form is purely in terms of intensive quantities and is useful when simulating the Euler equations because it expresses the relationship between internal energy and other forms of energy (such as kinetic), thus allowing simulations to obey the First Law.

Van der Waals equation

Where a, b and R are constants that depend on the specific material. They can be calculated from the critical properties 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 Virial Equation

Although usually not the most convenient equation of state, the Virial Equation is important because it can be derived directly from statistical mechanics. If appropriate assumptions are made about the mathematical form of intermolecular forces, theoretical expressions can be developed for each of the coefficients. In this case B corresponds to interactions between pairs of molecules, C to triplets, and so on.

Redlich-Kwong Equation of State

R = Ideal Gas constant (8.31451 J/mol·K)

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. Although, 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 is less than about one-half of the ratio of the temperature to the critical temperature.

The Soave Equation

R = Ideal Gas constant (8.31451 J/mol·K)

Where ω is the acentric factor for the species.

for hydrogen:

In 1972 Soave replaced the a/√(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.

The Peng-Robinson Equation of State

R = Ideal Gas constant (8.31451 J/mol·K)

Where ω is the acentric factor for the species.

The Peng-Robinson Equation was developed in 1976 in order to satisfy the following goals:

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 BWRS Equation of State

ρ = the molar density

Values of the various parameters for 15 substances can be found in:

K.E. Starling, Fluid Properties for Light Petroleum Systems. Gulf Publishing Company (1973).

Stiffened equation of state

When considering water under very high pressures (typical applications are underwater nuclear explosions, sonic shock lithotripsy, and sonoluminescence) the stiffened equation of state is often used:

where is the internal energy per unit mass, is an empirically determined constant typically taken to be about 6.1, and is another constant, representing the molecular attraction between water molecules. The magnitude of the correction is about 20000 atmospheres.

The equation is stated in this form because the speed of sound in water is given by .

Thus water behaves as though it is an ideal gas that is already under about 20000 atmospheres pressure, and explains why water is commonly assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres, the water behaves as an ideal gas would do when changing from 20001 to 20002 atmospheres.

This equation mispredicts the specific heat capacity of water but few alternatives are available for severely nonisentropic processes such as strong shocks.

History

Boyle's law (1662)

Boyle's Law was perhaps the first expression of an equation of state. In 1662 Robert Boyle, an Irishman, 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:

PV = constant

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.

Charles' law (1787)

In 1787 the French physist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 degree interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:

V₁/T₁ = V₂/T₂

Dalton's law of partial pressures (1801)

The Ideal gas law (1834)

In 1834 �mile Clapeyron combined Boyle's Law and Charles' law into the first statement of the ideal gas law. Initially the law was formulated as PV=R(T+267) (with temperature expressed in degrees celsius). However, later work revealed that the number should actually be 273.2, giving:

PV=R(T+273.2)