An equation of state is a mathematical relation between properties that characterize a material’s physical state. In thermodynamics, it commonly connects pressure, volume, temperature, and the amount or composition of matter. Such relations describe gases, liquids, solids, and mixtures under specified conditions. They may be derived from microscopic theory, fitted to measurements, or constructed by combining both approaches. The familiar ideal-gas law is one example; more elaborate equations account for interactions and departures from ideal behavior. (www1.grc.nasa.gov)
State variables and thermodynamic description
For a homogeneous fluid of fixed composition in thermodynamic equilibrium, a common representation is
[ p=p(T,\rho), ]
where (p) is pressure, (T) is absolute temperature, and (\rho) is mass or molar density, according to the convention used. Alternatively, the relation may be written implicitly as (f(p,V,T,n)=0), with (V) denoting volume and (n) the amount of substance. Mixture equations also require composition variables. These quantities describe the state rather than the path by which the material reached it. (www1.grc.nasa.gov)
A pressure–volume–temperature relation does not necessarily specify every thermodynamic property. Predicting internal energy or heat capacity generally requires additional information about temperature dependence. A more complete description expresses a thermodynamic potential in its natural variables, allowing other properties to be obtained through differentiation. This distinction separates a mechanical equation of state from a complete thermodynamic property model. (ocw.mit.edu)
Ideal-gas equation
For an ideal gas,
[ pV=nRT, ]
where (R) is the molar gas constant and (T) is measured in kelvins. The equivalent microscopic expression is
[ pV=Nk_{\mathrm B}T, ]
with (N) the number of particles and (k_{\mathrm B}) the Boltzmann constant. The model neglects intermolecular interactions and the volume occupied by individual particles. It approximates many gases when sufficiently dilute, but becomes inadequate when interactions or molecular size substantially affect their behavior. (goldbook.iupac.org)
Departures from the ideal-gas law are often expressed through the compressibility factor,
[ Z=\frac{pV_m}{RT}, ]
where (V_m=V/n) is molar volume. An ideal gas has (Z=1). For real fluids, (Z) varies with temperature, density, and composition; it is not a universal constant associated with a substance. (tsapps.nist.gov)
Real-fluid models
The van der Waals equation introduces corrections for attraction and excluded volume:
[ p=\frac{RT}{V_m-b}-\frac{a}{V_m^2}. ]
The parameter (a) represents attractive interactions, while (b) accounts approximately for the space unavailable because particles have finite size. Despite its simplicity, the model exhibits liquid–vapor coexistence and a critical point. It provides a qualitative explanation of real-fluid behavior rather than a generally precise property correlation. (sites.esm.psu.edu)
A different approach is the virial expansion, written in molar-density form as
[ Z=1+B(T)\rho_m+C(T)\rho_m^2+\cdots, ]
where (\rho_m=n/V). The virial coefficients encode effects of particle interactions. At sufficiently low density, only a few terms may be needed; truncation becomes less reliable as density increases. The expansion connects measurable departures from ideality with microscopic interaction models. (sites.esm.psu.edu)
Engineering models also include cubic equations and multiparameter formulations. Their complexity reflects different objectives: compact calculations over a useful operating range, or high accuracy across extensive regions of a fluid’s phase diagram. No single simple equation describes all substances equally well. (tsapps.nist.gov)
Free-energy formulations and microscopic foundations
A fundamental equation may express Helmholtz free energy as (A(T,V,N)). For a simple system,
[ p=-\left(\frac{\partial A}{\partial V}\right){T,N}, \qquad S=-\left(\frac{\partial A}{\partial T}\right){V,N}, ]
where (S) is entropy. Internal energy follows from (U=A+TS). A consistent free-energy formulation therefore generates pressure, thermal properties, and other equilibrium quantities from one mathematical description. (tsapps.nist.gov)
In statistical mechanics, the canonical partition function (\mathcal Z) provides the connection to microscopic states:
[ A=-k_{\mathrm B}T\ln\mathcal Z. ]
Differentiating this free energy yields an equation of state. Its form depends on particle interactions and the accessible microscopic states. Thus an equation of state can embody molecular information even when expressed entirely in macroscopic variables. (rotskoff.github.io)
Phase equilibrium and stability
An equation of state can support calculations of phase transitions, but algebraic solutions alone do not establish equilibrium. Below its critical temperature, the van der Waals model produces multiple volume solutions and a nonphysical, mechanically unstable portion of an isotherm. Equilibrium liquid and vapor states are selected using thermodynamic conditions rather than by accepting every mathematical root. (ocw.mit.edu)
At coexistence, phases have equal temperature, pressure, and the relevant chemical potentials. The Maxwell equal-area construction identifies coexistence pressure in the van der Waals model. Near the critical point, large fluctuations and nonclassical critical behavior limit the accuracy of simple analytic models. (ocw.mit.edu)
Applications and validity
Equations of state underpin property calculations for refrigeration, natural-gas processing, and other industrial fluid systems. Reference models for water and refrigerants commonly use Helmholtz-energy formulations fitted to experimental measurements. NIST’s REFPROP database implements such models to calculate density, enthalpy, entropy, heat capacities, sound speed, and phase-equilibrium properties. (nist.gov)
Accuracy is specific to the substance, property, and temperature–pressure region. A model that reproduces density well may predict derivative properties less accurately. Reliable formulations therefore specify their validity ranges and uncertainties. Viscosity and thermal conductivity require separate transport-property models; an equilibrium equation of state does not by itself determine them. (tsapps.nist.gov)