Equation of state

In physics and chemistry, an equation of state is a special kind of thermodynamic equation. It helps us understand how things like pressure, volume, and temperature work together. These equations can tell us about liquids, gases, and solids, as well as what happens inside stars.

Most of these equations use something called Helmholtz free energy. Even though scientists have made many equations to describe how materials behave, none can perfectly predict everything. People have been looking for a perfect equation for over 300 years, but it is still hard to find. Equations of state are important for understanding how materials act in different forms.

Overview

Right now, there isn’t one single equation that can perfectly describe all materials under every condition. One well-known example is the ideal gas law, which connects how dense gases and liquids are with their temperatures and pressures. This works fairly well for gases that aren’t very sticky and at normal pressures and temperatures.

An equation of state is a way to describe how matter behaves under different conditions. It usually connects pressure, volume, and temperature, shown as a math formula. These equations can also help us understand solids, like how they change shape, and even the conditions inside stars, such as neutron stars. People use these equations in many jobs, like making things in factories, working with oil, or creating medicines.

Scientists can use any units they like, but many prefer the SI units, where temperature is measured in Kelvin.

Historical background

Equations of state started about three hundred years ago with the ideal gas law. One of the earliest was Boyle's law. In 1662, Robert Boyle used a special glass tube with mercury to trap air. He found that when pressure goes up, the space the air takes up goes down. This is called Boyle’s law.

Later, in 1787, Jacques Charles found that gases like oxygen and nitrogen expand by about the same amount when heated. This became known as Charles's law. More discoveries helped scientists learn how gases act under different conditions.

Ideal gas law

The ideal gas law is a simple way to describe how gases behave. It shows the relationship between pressure, volume, and temperature. For a gas, this law is written as p V = n R T, where p is pressure, V is volume, n is the number of moles of gas, R is a constant, and T is temperature.

There is also a quantum version of this law for very small particles, like atoms or molecules. This version helps us understand how these tiny particles behave under certain conditions.

Cubic equations of state

Main article: Cubic equations of state

Cubic equations of state are called that because they are written using a cubic function of volume. They began with the van der Waals equation of state, and now there are many versions. These equations are still used in engineering, such as the Peng Robinson equation and the Soave Redlich Kwong equation.

Virial equations of state

Virial equation of state

Main article: Virial expansion

The virial equation of state is a way to describe how gases behave under different conditions. It is useful because it connects to how matter moves and changes. This equation is also called the Kamerlingh Onnes equation.

When there is a lot of space for gas molecules, all gases act similarly to ideal gases. By studying how groups of molecules interact, we can make our description more accurate. Each part of the equation shows how different groups of molecules affect the overall behavior.

The BWR equation of state

Main article: Benedict–Webb–Rubin equation

The BWR equation of state is another way to describe how gases behave. It uses different values that can be found in reference materials. This equation has been used to study certain types of fluids. There are also several changes and extensions to the original BWR equation.

The Benedict–Webb–Rubin–Starling equation of state is a changed version of the BWR equation.

The Lee–Kesler equation of state is based on a principle that helps us understand how different gases behave similarly under certain conditions. It is a change to the BWR equation.

Physically based equations of state

There are many equations that help us understand how matter behaves under different conditions. Most of these equations use something called Helmholtz free energy, which depends on factors like temperature and density. These equations help scientists understand how molecules interact and affect the properties of liquids, gases, and solids.

One common method to build these equations is called perturbation theory, which helps model how molecules attract each other. An important idea in this area is the statistical associating fluid theory (SAFT), which helps describe how molecules stick together, like in water with hydrogen bonds.

Multiparameter equations of state

Multiparameter equations of state are special formulas that help us understand pure fluids very accurately. These formulas are based on real experiments and often use something called Helmholtz free energy. They can describe both liquids and gases.

These equations use the idea of "reduced" temperature and density, which are usually based on the critical values of the fluid. They don’t need complex calculations and can find properties using simple thermodynamic rules. Some of these equations use up to 50 specific numbers for each fluid but can still describe the fluid’s behavior very well. They are available for about 50 common industrial fluids, including water, and there are also models for mixtures.

List of further equations of state

Stiffened equation of state

When we study water under very high pressures, such as in some special situations, a specific equation is often used. This equation helps us understand how water behaves in these extreme conditions.

Landau–Stanyukovich–Zeldovich–Kompaneyets equation of state

The Landau–Stanyukovich–Zeldovich–Kompaneets equation is a model used to describe the relationship between pressure and energy in explosive materials and high-pressure gases. It is used in studying explosions and shock waves.

Morse oscillator equation of state

An equation has been created for the Morse oscillator. It describes how pressure relates to certain properties of the system.

Ultrarelativistic equation of state

For certain very energetic fluids, a special equation shows how pressure relates to mass density and the speed of sound.

Ideal Bose equation of state

The equation for an ideal Bose gas shows how pressure, volume, and temperature are related, especially near a special temperature where a new state of matter begins to form.

Jones–Wilkins–Lee equation of state for explosives (JWL equation)

The Jones–Wilkins–Lee equation is used to describe the behavior of materials after an explosion. It helps scientists understand how pressure changes with density and energy in these situations.

Others

• Tait equation for water and other liquids
• Murnaghan equation of state
• Birch–Murnaghan equation of state
• Stacey–Brennan–Irvine equation of state
• Modified Rydberg equation of state
• Adapted polynomial equation of state
• Johnson–Holmquist equation of state
• Mie–Grüneisen equation of state
• Anton-Schmidt equation of state
• State-transition equation