Equation of state in the context of "Ideal gas"

In cosmology, the equation of state of a perfect fluid is characterized by a dimensionless number ${\displaystyle w}$, equal to the ratio of its pressure ${\displaystyle p}$ to its energy density ${\displaystyle \rho }$: $${\displaystyle w\equiv {\frac {p}{\rho }}.}$$It is closely related to the thermodynamic equation of state and ideal gas law.

The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:

$${\displaystyle pV=nRT}$$where ${\displaystyle p}$, ${\displaystyle V}$ and ${\displaystyle T}$ are the pressure, volume and temperature respectively; ${\displaystyle n}$ is the amount of substance; and ${\displaystyle R}$ is the ideal gas constant.It can also be derived from the microscopic kinetic theory, as was achieved (independently) by August Krönig in 1856 and Rudolf Clausius in 1857.

The National Ignition Facility (NIF) is a laser-based inertial confinement fusion (ICF) research facility, located at Lawrence Livermore National Laboratory in Livermore, California, United States. NIF's mission is to achieve fusion ignition with high energy gain. It achieved the first instance of scientific breakeven controlled fusion in an experiment on December 5, 2022, with an energy gain factor of 1.5. It supports nuclear weapon maintenance and design by studying the behavior of matter under the conditions found within nuclear explosions.

NIF is the largest and most powerful ICF device built to date. The basic ICF concept is to squeeze a small amount of fuel to reach the pressure and temperature necessary for fusion. NIF hosts the world's most energetic laser, which indirectly heats the outer layer of a small sphere. The energy is so intense that it causes the sphere to implode, squeezing the fuel inside. The implosion reaches a peak speed of 350 km/s (0.35 mm/ns), raising the fuel density from about that of water to about 100 times that of lead. The delivery of energy and the adiabatic process during implosion raises the temperature of the fuel to hundreds of millions of degrees. At these temperatures, fusion processes occur in the tiny interval before the fuel explodes outward.

The Navier–Stokes equations (/nævˈjeɪ ˈstoʊks/ nav-YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Sir George Gabriel Stokes, Bt. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of conservation of mass. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are an elliptic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable).

In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz energy during a process is equal to the maximum amount of work that the system can perform in a thermodynamic process in which temperature is held constant. At constant temperature, the Helmholtz free energy is minimized at equilibrium.

In contrast, the Gibbs free energy or free enthalpy is most commonly used as a measure of thermodynamic potential (especially in chemistry) when it is convenient for applications that occur at constant pressure. For example, in explosives research Helmholtz free energy is often used, since explosive reactions by their nature induce pressure changes. It is also frequently used to define fundamental equations of state of pure substances.

Johannes Diderik van der Waals (Dutch: [joːˈɦɑnəz ˈdidərɪk fɑn dər ˈʋaːls] ; 23 November 1837 – 8 March 1923) was a Dutch theoretical physicist who received the Nobel Prize in Physics in 1910 "for his work on the equation of state for gases and liquids". Van der Waals started his career as a schoolteacher. He became the first physics professor of the University of Amsterdam when its status was upgraded to Municipal University in 1877.

His name is primarily associated with the van der Waals equation, an equation of state that describes the behavior of gases and their condensation to the liquid phase. His name is also associated with van der Waals forces (forces between stable molecules), with van der Waals molecules (small molecular clusters bound by van der Waals forces), and with the van der Waals radius (size of molecules). James Clerk Maxwell once said that, "there can be no doubt that the name of Van der Waals will soon be among the foremost in molecular science."