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# 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.

### 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

$\left(P + \frac{a}{V^2}\right)\left(V-b\right) = RT$

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

$a = 3P_c V_c^2$
$b = \frac{V_c}{3}$
$R = \frac{8P_c V_c}{3T_c}$

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

$\frac{PV}{RT} = 1 + \frac{B}{V} + \frac{C}{V^2} + \frac{D}{V^3} + \dots$

$B = -V_c$
$C = \frac{V_c^2}{3}$
$R = \frac{P_c V_c}{T_c}$

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

$P = \frac{RT}{V-b} - \frac{a}{\sqrt{T}V\left(V+b\right)}$

$a = \frac{0.42748R^2T_c^{2.5}}{P_c}$
$b = \frac{0.08664RT_c}{P_c}$
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

$P = \frac{RT}{V-b} - \frac{a\alpha}{V\left(V+b\right)}$

R = Ideal Gas constant (8.31451 J/mol·K)
$a = \frac{0.42747R^2T_c^2}{P_c}$
$b = \frac{0.08664RT_c}{P_c}$
$\alpha = \left(1 + \left(0.48508 + 1.55171\omega - 0.15613\omega^2\right) \left(1-T_r^{0.5}\right)\right)^2$

$T_r = \frac{T}{T_c}$

Where ω is the acentric factor for the species.

for hydrogen:

$\alpha = 1.202 \exp\left(-0.30288T_r\right)$

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

$P=\frac{RT}{V-b} - \frac{a\alpha}{V^2+2bV-b^2}$

R = Ideal Gas constant (8.31451 J/mol·K)
$a = \frac{0.45724R^2T_c^2}{P_c}$
$b = \frac{0.07780RT_c}{P_c}$
$\alpha = \left(1 + \left(0.37464 + 1.54226\omega - 0.26992\omega^2\right) \left(1-T_r^{0.5}\right)\right)^2$

$T_r = \frac{T}{T_c}$

Where ω is the acentric factor for the species.

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

1. The parameters should be expressible in terms of the critical properties and the acentric factor.
2. The model should provide reasonable accuracy near the critical point, particularly for calculations of the Compressibility factor[?] and liquid density.
3. The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature pressure and composition.
4. 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

$P=\rho RT + \left(B_0 RT-A_0 - \frac{C_0}{T^2} + \frac{D_0}{T^3} - \frac{E_0}{T^4}\right) \rho^2 + \left(bRT-a-\frac{d}{T}\right) \rho^3 + \alpha\left(a+\frac{d}{T}\right) \rho^6 + \frac{c\rho^3}{T^2}\left(1 + \gamma\rho^2\right)\exp\left(-\gamma\rho^2\right)$

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).