# Equation of State

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## equation of state

[i′kwā·zhən əv ′stāt]## Equation of State

an equation that relates the pressure *p*, volume *V*, and temperature *T* of a physically homogeneous system in a state of thermodynamic equilibrium: *f(p, V, T)* = 0. This equation is called an equation of state, in contrast to a thermodynamic equation of state, which gives the internal energy *U* of a system as a function of any two of the three parameters *p, V*, and *T*. An equation of state makes it possible to express pressure in terms of volume and temperature *p = p(V, T*), and to determine the unit work δ*A* = *p*δ*V* performed during an infinitesimal expansion δ*V* of the system.

An equation of state is an essential supplement to the laws of thermodynamics, since it allows the laws to be applied to real substances. Such an equation cannot be derived by means of the laws of thermodynamics alone but is determined or calculated theoretically by the methods of statistical mechanics on the basis of concepts of the structure of matter. Only the existence of thermodynamic equations of state follows from the first law of thermodynamics, but the relationship between the equations of state and thermodynamic equations of state (∂*U*/∂*V*)_{T} = *T*(∂*p*/∂*T*)_{V} – *P* follows from the second law of thermodynamics. From the relationship, it follows that for an ideal gas the internal energy does not depend on volume (∂*U*/∂*V*)_{T} = 0. Thermodynamics shows that to calculate both equations of state and thermodynamic equations of state, it is sufficient to know any one of the thermodynamic potentials as a function of its parameters. For example, if the Helmholtz free energy *F* is known as a function of *T* and *V*, the equations of state are found by differentiation:

*p* = –(∂*F*/∂*V*)*T*, *U* = –*T*^{2}(∂/∂*T*)(*F/T*)_{V}

Examples of equations of state for gases include the following: the Clapeyron equation for an ideal gas *pv* = *RT*, where *R* is the gas constant and *v* is the volume of 1 mole of gas; van der Waals’ equation (*p* + *a/v*^{2}) × (*v – b*) = *RT*, where *a* and *b* are constants that depend on the nature of the gas and take into account both the effect of the intermolecular attractive forces and the finiteness of the volume; and the virial equation of state for a nonideal gas *pv/RT* = 1 + *B(T)/v* + *C*(*T*)/*v*^{2} + ..., where *B(T)*, *C(T)*,... are the second, third, and higher-order virial coefficients, the values of which depend on the forces of molecular interaction (*see*GASES). The virial equation is the most reliable and theoretically valid equation of state for gases and makes it possible to explain numerous experimental results in terms of simple models of molecular interaction. Various empirical equations of state based on experimental data about specific heat and compressibility have also been proposed. An equation of state for nonideal gases indicates the existence of a critical point (with parameters *p*_{cr}, *V*_{cr}, and *T*_{cr}) at which the gas phase and the liquid phase become identical. If an equation of state is presented as a reduced equation of state, that is, as an equation in the dimen-sionless variables *p/p*_{cr}, *V/V*_{cr}, and *T/T*_{cr}, then at temperatures that are not too low, the equation varies little for different substances, in accordance with the law of corresponding states.

For equilibrium radiation or a photon gas, the equation of state is given by Planck’s radiation law for the mean energy density.

For liquids, a general equation of state has not yet been theoretically obtained because of the complexity of taking into account all the characteristics of molecular interaction. Although van der Waals’ equation has been used for the qualitative evaluation of the behavior of liquids, it is essentially inapplicable below the critical point, where the liquid and gas phases can coexist. Equations of state that well describe the properties of a number of simple liquids can be obtained from approximate theories of the liquid state, such as free-volume theory or electron-hole theory (*see*LIQUID). Knowledge of the probability distribution of the relative location of the molecules, called the two-point correlation function, makes it possible in principle to calculate an equation of state for a liquid; however, this problem is very complicated and has not yet been entirely solved, even with the aid of computers.

For solids, an equation of state defines the dependence of the moduli of elasticity on temperature and pressure. Such an equation can be obtained on the basis of the theory of thermal motion in crystals, which considers phonons and their interaction, but no general equation of state has yet been found for solids.

For magnetic media, the unit work performed during magnetization is equal to δ*A* = **–HδM,** where **M** is the magnetic moment and **H** is the magnetic field strength. Consequently, the relation **M** = **M**(**H**, *T*) is a magnetic equation of state.

For electrically polarizable media, the unit work performed during polarization is equal to δ*A* = – **EδP**, where **P** is the polarization and **E** is the electric field strength. Consequently, an equation of state has the form **P** = **P**(**E**, *T*).

### REFERENCES

Hill, T.*Statisticheskaia mekhanika*. Moscow, 1960. (Translated from English.)

Vukalovich, M. P., and I. I. Novikov.

*Uravnenie sostoioniia real’nykh gazov*. Moscow-Leningrad, 1948.

Mason, E., and T. Spurting.

*Vinal’noe uravnenie sostoianiia*. Moscow, 1972. (Translated from English.)

Leibfried, G., and W. Ludwig.

*Teoriia angarmonicheskikh effektov v kristallakh*. Moscow, 1963. (Translated from English.)

D. N. ZUBAREV