Virial expansion

The virial expansion is a mathematical model used in thermodynamics. It describes the equation of state for a gas. An equation of state shows how pressure relates to other properties. In this model, pressure is expressed as a power series of density. Density is a measure of how much matter is packed into a space. This expansion is vital for understanding how gases behave in local equilibrium. It allows scientists to account for the complex ways molecules interact with one another.

To understand the mechanism, we look at the compressibility factor, denoted as Z. The virial expansion represents Z as a series of terms. The first term is a constant value of 1. This ensures the model acts like an ideal gas when density approaches zero. The subsequent terms are called virial coefficients. The second coefficient, B, represents bimolecular attraction. This is the force that pulls two molecules together. The third coefficient, C, represents repulsion. This describes the force among three molecules in close contact.

Scientists have studied these coefficients for over a century. The model was first proposed by Kamerlingh Onnes in 1902. He published his findings in Amsterdam. Since then, researchers have created extensive tables of these values. For example, Dymond and Wilhoit published major compilations in 1969 and 2003. These books list coefficients for many pure gases and mixtures. You can also find this data in the National Institute of Standards and Technology's Thermo Data Engine Database. This data helps scientists predict how different fluids will act under various conditions.

Many different equations of state have been developed to describe fluids. Johannes van der Waals proposed a famous equation in 1873. His model showed that the second virial coefficient decreases as temperature lowers. However, his higher coefficients did not account for temperature changes correctly. Other models, like those by Dieterici or Redlich-Kwong, also have limitations. Some models suffer from a mathematical singularity at zero volume. In contrast, the Beattie-Bridgeman equation from 1928 is more closely related to virial equations. It is more accurate for both gases and liquids.

Further improvements were made to these mathematical models over time. The Benedict-Webb-Rubin equation was introduced in 1940. It provides better representations of isotherms below the critical temperature. In 1972, Kenneth Starling offered even more improvements. His equations use exponential terms to correct the third virial coefficient. These terms help represent the liquid phase correctly. When these exponential terms are expanded, they contribute to the third and eighth virial coefficients. This complexity allows for a much more precise description of fluid behavior.

A simplified version is the cubic virial equation of state. This model is useful because it is simple yet effective. It avoids the singularity issues found in the van der Waals equation. In this model, the coefficients B and C can be solved in a closed form. When applying critical conditions, the model yields a specific value. This value is 0.333, which differs from the 0.375 found in the van der Waals model. This precision is important when studying the saturation region of a fluid.

The saturation region is where gas and liquid phases coexist. This happens between the critical point and the triple point. In this state, the fluid exists under a specific saturation pressure. The gas and liquid have different molar volumes and densities. A valid equation of state must correctly show the isotherm crossing the saturation line. The cubic virial equation can find the coefficients B and C using these properties. By using the volumes of saturated gas and liquid, scientists can compute these values. These results align well with more complex models like those from Starling.