What are Equations of State?

In thermodynamics, an equation of the state is an equation that relates the parameters like pressure, temperature and a particular volume of a material. For an incompressible material with such parameters (such as solids or liquids), the simplest equation of state states that the real volume is unchanged.

There are many state equations, some simple and some very complicated. The theoretical equation of state is the most basic and well-known equation for substances in the gaseous phase. Within a properly chosen field, this equation correctly predicts the P-V-T behavior of a gas.

Gas and vapor (especially in terms of parameters) are sometimes used interchangeably. When a substance's vapor phase is above the critical temperature, it is commonly referred to as a gas. Vapors are generally referred as a gas that is on the brink of condensing.

Ideal Gas Equation of State

During his experimentation with a vacuum chamber in 1662, Robert Boyle found that the pressure of gases is inversely proportional to their volume.

J. Charles and J. Gay-Lussac of France discovered in 1802, that at low temperatures, the volume of a gas is proportional to its temperature. Thus, taking all of it into consideration, the following equation was established.

or

Here the constant of proportionality, ‘R’, is called the gas constant.

The above equation is known as the theoretical equation of state or simply the gas (ideal) relation and any gas that obeys it is known as an ideal gas or a perfect gas. The parameter P is the absolute strain, $T$is the absolute temperature and $V$is the real volume in this equation. Another form of equation of a state is also available in the form of density.

Units of R are: $kJ/kg.K$ or $kPa.m^3 /kg.K$

The equation of state can also be expressed as:

Where, $\overline{V}$(also expressed in parameters like density, molar volume, critical pressure, phase behavior, interaction parameter) is the specific molar volume and is the volume parameter per mole (measured in m3/kmol). A bar over a property signifies unit-molecular values.

The properties of the thermodynamics ideal- gas in two distinct states are compared to one another as follows:

An ideal- gas is a hypothetical pure fluid that follows the relationship ($PV=RT$) and is also expressed in the form of density. Experiments have shown that the defined gas relation strongly approximates the P-V-T action of real gases at low density.

The density of gas reduces at low pressures and high temperatures (Pressure-volume-temperature), and the gas behaves like an ideal- gas under these conditions.

Gases such as, argon, oxygen, nitrogen, neon, helium, and even heavy gases such as carbon dioxide, may be viewed as ideal-gases with marginal error in the range of practical importance (often less than 1 percent). However, thick gases such as water vapor in steam power plants and refrigerant vapor in refrigerators should not be called ideal-gases.

Compressibility Factor—A Measure of Deviation from Ideal Gas Behavior

The perfect gas equation is very simple and, as a result, very straightforward to use. However, when gases deviate greatly from gas law activity near the saturation area and the critical stage, this deviation from ideal gas law behavior at a given temperature and pressure can be correctly accounted for by introducing a correction factor known as the compressibility factor, Z (at high pressure, free energy, molar volume, pure fluid) which is defined as:

$$\begin{array}{l}Z=\frac{PV}{RT}\\ or\\ PV=ZRT\end{array}$$

It can also be expressed as:

where,

$V_{ideal}=\frac{RT}{P}$ and $Z=1$for an ideal- gases. In the case of the actual gases coefficient, Z can be more or less than 1. The greater the distance between Z and unity, the more the gas deviates from ideal gas behavior.

At low pressures and high temperatures, gases closely obey the gas equation. But, precisely, what is low pressure or high temperature? Is 100°C considered a low temperature? It most emphatically is for most liquids, but not for air. At this temperature and ambient high pressure, air (or nitrogen) can be treated as an ideal-gas with an error of less than 1%.

This is because that nitrogen is well above its critical temperature (147°C) and outside of the saturation zone. Many compounds, though, will remain in the solid phase at these temperatures and pressures. As a result, a substance's pressure or temperature is high or low in comparison to its critical temperature or pressure.

Gases behave differently at varying temperatures and pressures, but they behave very closely when temperatures and pressures are normalized to their critical temperatures and pressures. The normalization is carried out as follows (always using absolute pressure and temperature):

and

The reduced pressure is referred to as P_R, and the reduced temperature is referred to as T_R. At the same reduced pressure and temperature, all gases have about the same Z factor. This is referred to as the theory of corresponding states.

When P and v, or T and v, are used instead of P and T, the generalized compressibility map will also be used to calculate the third property, although it would be time-consuming trial and error. As a result, another reduced property known as the pseudo-reduced specific volume V_R must be described as follows:

It should be noted that V_R is represented differently than P_R and T_R. Instead of V_cr, it is related to T_cr and P_cr. Lines of constant V_R are now applied to the compressibility tables, allowing one to quantify T or P without resorting to time-consuming iterations.

Types of Equations of State

Other equations of states are:

Van-der Waals equation of state

The Van-der Waals equation of state, proposed in 1873, has two constants determined by the behavior of a material at the critical point. It is given by:

Van-der Waals hoped to strengthen the gas (ideal) equation of state by adding two results not found in the gas model, which were, intermolecular attraction forces and the space filled by the molecules themselves.

The intermolecular forces are represented by the term a, and the volume filled by the gas molecules is represented by the term b. At air pressure and temperature, the space currently filled by molecules in a room is just about one-thousandth of the overall volume.

If the strain rises, the space filled by the molecules becomes a larger proportion of the total volume. Van-der Waals suggested correcting this by replacing v in the pure gas relation with the quantity v b, where b describes the number of gas molecules per unit mass.

The two constants in this equation are determined by observing that the critical isotherm on a P-v diagram has a horizontal inflection point at the critical point.

 As a result, at the critical point, the first and second derivatives of P with respect to v must be zero. In other words,

and

The constants a and b are calculated by conducting differentiations and removing V_cr:

and

Redlich-Kwong equation

Virial equation of state

A virial equation is a more generalized version of a state equation. It is written as follows:      

where A, B, C are all empirically defined temperature functions known as virial coefficients.

Volumetric data for fluids may be used for a variety of uses, ranging from fluid metering to tank scaling. Tables will, of course, be presented with data for V as a function of T and P. However, using equations to express the functional relation $f\left(P,V,T\right)=0$ is much more compact and convenient. The virial equations of state for gases are ideal for this reason.

Isotherms for gases and vapors, which lie to the right of the saturated-vapor isotherm, are relatively straightforward curves under which V decreases as P rises. In this case, the product PV for a given T differs much more slowly than any of its members, and thus can be expressed analytically as a function of P. This means that PV for an isotherm can be expressed as a power series in P:

If we define, b = aB′ , c = aC′, etc.,

then,

where,$B\prime ,C\prime$ and so on are constants for a given temperature and material.

Context and Applications

This topic is useful for undergraduate and postgraduate courses, especially for Bachelors and Masters in Chemistry and various courses of Chemical Engineering.