Equation of state

An equation of state is a fundamental thermodynamic equation used in physics and chemistry. It relates state variables to describe the physical condition of matter. These variables include pressure, volume, temperature, and internal energy. Scientists use these equations to understand pure substances and mixtures. They apply them to liquids, gases, and solids. These models even help describe the matter inside stars.

However, no single equation can accurately predict every property under all conditions. The search for a universal equation has lasted for three centuries.

To understand how these equations work, we must look at their variables. The general form often links pressure, volume, and temperature. Many modern equations are formulated using Helmholtz free energy. This is a way to describe the energy available in a system. The number of independent variables depends on the number of substances and phases present. This relationship is connected to the Gibbs phase rule. In many cases, scientists use empirical parameters. These are values adjusted to match real measurement data.

One famous example is the ideal gas law. It correlates the density of gases and liquids to temperature and pressure. This law works well for weakly polar gases at low pressures. It also works at moderate temperatures. However, the law becomes inaccurate at high pressures or low temperatures. It cannot predict when a gas will condense into a liquid. For elementary particles, scientists use a quantum ideal gas law. This version accounts for quantum effects like mass and spin. It uses Fermi-Dirac or Bose-Einstein statistics to describe how particles behave.

The history of these equations began over 300 years ago. In 1662, Robert Boyle used a J-shaped glass tube for his experiments. He trapped air with mercury and measured its volume. He discovered that gas volume varies inversely with pressure. This is known as Boyle's law. In 1787, Jacques Charles found that gases expand over temperature intervals. Joseph Louis Gay-Lussac later showed a linear relationship between volume and temperature in 1802. In 1834, Émile Clapeyron combined these findings into the first ideal gas law.

In 1873, J. D. van der Waals revolutionized the field. He assumed that constituent molecules occupy a finite volume. This led to the first equation of state based on molecular size. His work started the era of cubic equations of state. These can be rewritten as cubic functions of volume. Famous examples include the Redlich–Kwong and the Soave modifications. Today, process engineers use the Peng Robinson equation of state. These tools are vital in the petroleum and pharmaceutical industries.

There are several other complex types of equations. Virial equations of state, or the Kamerlingh Onnes equation, are derived from statistical mechanics. They use coefficients to describe interactions between pairs or triplets of molecules. Physically based equations of state are also used today. They model molecular size, shape, and attraction. They are often more accurate for liquids and solids than cubic equations. Some use perturbation theory to model dispersive interactions. Others, like the Statistical Associating Fluid Theory (SAFT), describe hydrogen bonding and chain formation.

These mathematical models connect many different scientific fields. In cosmology, scientists use a perfect fluid equation of state. This helps them model the universe and radiation fields. In astrophysics, these equations describe the interior of neutron stars. They also help describe dense matter like quark–gluon plasmas. Whether studying a tiny pharmaceutical molecule or a massive star, these equations provide a way to predict behavior. They turn complex physical observations into organized, mathematical patterns.