Virial expansion

Gas can push on things.

We use math to see how it pushes. This helps us know how gas moves. A man named Onnes first thought of this.

It shows how gas acts when it is crowded. This happens when many tiny bits are close. We can learn how bits pull or push each other. This helps us study both gas and liquid. It is a smart way to learn. Can you wonder about the air?

Gas can push on things. This push is called pressure. Scientists use math to study how gas acts. One way to do this is a virial expansion.

A scientist named Kamerlingh Onnes first proposed this. It is a way to find pressure using density. Density is how crowded the gas is. The math uses special numbers called virial coefficients.

These numbers tell us about tiny particles. The second coefficient shows how two particles pull on each other. This is called attraction. The third coefficient shows how three particles push away. This is called repulsion.

Many scientists have studied these numbers for a long time. They have made big lists of these numbers for many fluids. This helps us understand both gases and liquids. Some math models are very simple. Others are more complex to be more accurate. A cubic virial equation is a simple model. It helps us find the right numbers for many different fluids. This makes it easier to study how they change.

Scientists use special math to study how gases behave. This math is called the virial expansion. It is a model for equations of state. These equations help us find the pressure of a gas. The math looks at how crowded the gas is. This crowding is called density. When a gas is in local equilibrium, its pressure changes with density. The expansion uses a power series to show this relationship.

This math works by using special numbers. These are called virial coefficients. The first coefficient is always the number 1. This makes sure the math works like an ideal gas when density is very low. The second coefficient, called B, shows how two particles pull on each other. This is known as bimolecular attraction. The third coefficient, called C, shows how three particles push away. This is called repulsion among molecules in close contact.

A scientist named Kamerlingh Onnes first proposed this idea. He shared his work in 1902 in Amsterdam. Since then, many people have studied these coefficients. For over a century, scientists have made long lists of these numbers. Two famous books by Dymond and Wilhoit were published in 1969 and 2003. You can also find these numbers in the Thermo Data Engine Database. This database is run by the National Institute of Standards and Technology.

Many different math models have tried to explain these gases. Johannes van der Waals proposed a famous equation in 1873. Later, scientists like Dieterici and Redlich-Kwong made other models. Some models, like the Beattie-Bridgeman equation from 1928, are very close to virial equations. Other models, like the Benedict-Webb-Rubin equation from 1940, are very accurate. Kenneth Starling made more improvements in 1972. Each new model helps us understand fluids better.

You can use a simple version called a cubic virial equation. This model is easy to use. It does not have the problems that the van der Waals equation has. It can help find the right numbers for many different fluids. This is helpful in the saturation region. This is the place where gas and liquid exist together. In this region, the gas and liquid are in equilibrium. The math helps us find the properties of both states at once.

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.