equation of state
An equation of state is a recipe that ties together the few numbers describing a substance — usually its pressure, volume, temperature, and amount — into a single relationship. It captures the everyday fact that these quantities are not free to roam independently: heat a sealed bottle and its pressure climbs; squeeze a balloon and you change its volume. Fix some of them and the rest is settled.
Formally, an equation of state is a mathematical relation among the state variables of a system, valid at equilibrium. The most famous is the ideal gas law, pV = nRT, which links pressure, volume, amount, and temperature for a gas whose particles are tiny and barely interact. More elaborate equations of state, such as the van der Waals equation, add corrections to describe real gases and even liquids.
It matters because an equation of state lets you predict one property from the others without remeasuring everything. Tell it the temperature and amount of gas in a fixed tank and it returns the pressure. The honest caveat is that every equation of state is a model with a domain of validity; the ideal gas law fails when a gas is cold and dense, and you must reach for a better one.
A car tyre's pressure gauge reads higher after a long drive: the rubbing has warmed the trapped air, and the equation of state guarantees that, with the volume fixed, a rise in temperature means a rise in pressure.
Pin down some state variables and the equation of state fixes the rest.
No single equation of state is exact for all conditions. Each is a model: the ideal gas law is excellent for dilute, warm gases but breaks down near condensation, where richer equations like van der Waals do better.