Wikipedia: Equations of state

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Equations of State

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, the ideal gas law becomes increasingly inaccurate at higher pressures and lower temperatures©

Using Statistical Mechanics, the ideal gas law can be derived by assuming that a gas is composed of a large number of small hard sphere 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©

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

PV = RT

R = Ideal Gas Constant ¥8©31451 J/mol*K¤

van der Waals Equation of State

¥P + a/V²¤¥V-b¤ = RT

Where a, b and R are constants that depend on the specific material© They can be calculated from the critical properties as:

a = 3P_cV_c²
b = V_c/3
R = 8P_cV_c/3T_c

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

PV/RT = 1 + B/V + C/V² + D/V³ + ©©©

B = -V_c
C = V_c²/3
R = P_cV_c/T_c

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

P = RT/¥V-b¤ - a/√¥T¤V¥V+b¤

a = 0©42748R²T_c^2©5/P_c
b = 0©08664RT_c/P_c
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

P = RT/¥V-b¤ - aα/{V¥V+b¤}

R = Ideal Gas constant ¥8©31451 J/mol*K¤
a = 0©42747R²T_c²/P_c
b = 0©08664RT_c/P_c
α = {1 + ¥0©48508 + 1©55171ω - 0©15613ω²¤¥1-T_r^0©5¤}²

T_r = T/T_c

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

The BWR Equation of State

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