Equation of State: The Perfect Gas

The assumed properties of a perfect gas are closely matched by those of actual gases in many circumstances, although no actual gas is perfect. The molecules of a perfect gas would behave like tiny, perfectly elastic spheres in random motion and would influence one another only when they collided. Their total volume would be negligible in comparison with the space in which they moved. From these hypotheses the kinetic theory of gases indicates that, for equilibrium conditions, the absolute pressure p, density ‘ρ’, the volume V occupied by mass m and the absolute temperature T are related by the expression

p = ρRT

                                                                              or           pV = mRT                      (1)

in which ‘R’ is a constant called the gas constant, the value of which is constant for the gas concerned and ‘V’ is the volume occupied by the mass m of the gas. The absolute pressure is the pressure measured above absolute zero (or complete vacuum) and is given by

p_abs = p_gage + p_atm 

The absolute temperature is expressed in ‘kelvin’ i.e., K, when the temperature is measured in °C and it is given by

T°(abs) = T K = 273.15 + t°C

No actual gas is perfect. However, most gases (if at temperatures and pressures well away both from the liquid phase and from dissociation) obey this relation closely and hence their pressure, density and (absolute) temperature may to a good approximation, be related by Eq.1.

Similarly, air at normal temperature and pressure behaves closely in accordance with the equation of state. It may be noted that the gas constant R is defined by Eq. 1 as p/ρT and therefore, its dimensional expression is (FL/Mθ). Thus in SI units the gas constant R is expressed in Newton-metre per kilogram per kelvin i.e., (N.m/kg. K). Further, since 1 joule = 1 newton × 1 metre, the unit for R also becomes joule per kilogram per kelvin i.e., (J/kg. K). Again, since 1 N = 1 kg × 1 m/s², the unit for R becomes (m²/s² K).

In metric gravitational and absolute systems of units, the gas constant R is expressed in kilogram (f)-metre per metric slug per degree C absolute i.e., [kg(f)-m/msl deg. C abs.] and dyne-centimetre per gram (m) per degree C absolute i.e., [dyne-cm/gm(m) deg. C abs.] respectively. For air the value of R is 287 N-m/kg K, or 287 J/kg K, or 287 m²/s² K.

In metric gravitational system of units, the value of R for air is 287 kg(f)-m/msl deg. C abs. Further, since 1 msl = 9.81 kg (m), the value of R for air becomes (287/9.81) or 29.27 kg(f)-m/kg(m) deg. C abs.

Since specific volume may be defined as reciprocal of mass density, the equation of state may also be expressed in terms of specific volume of the gas as

pv = RT

in which v is specific volume.

The equation of state may also be expressed as

p = wRT

in which w is the specific weight of the gas. The unit for the gas constant R then becomes (m/K) or (m/deg. C abs). It may be shown that for air the value of R is 29.27 m/K. For a given temperature and pressure, Eq. 1 indicates that ρR = constant. By Avogadro’s hypothesis, all pure gases at the same temperature and pressure have the same number of molecules per unit volume. The density is proportional to the mass of an individual molecule and so the product of R and the ‘molecular weight’ M is constant for all perfect gases. This product MR is known as the universal gas constant. For real gases it is not strictly constant but for monatomic and diatomic gases its variation is slight. If M is the ratio of the mass of the molecule to the mass of a hydrogen atom, MR = 8310 J/kg K.

Any equation that relates p, ρ and T is known as an equation of state and equation of state is therefore termed the equation of state of a perfect gas. Most gases, if at temperatures and pressures well away both from the liquid phase and from dissociation, obey this relation closely and so their pressure, density and (absolute) temperature may, to a good approximation, be related by Eqn. 1. For example, air at normal temperatures and pressures behaves closely in accordance with the equation. But gases near to liquefaction – which are then usually termed vapours – depart markedly from the behaviour of a perfect gas. Equation 1 therefore does not apply to substances such as non-superheated steam and the vapours used in refrigerating plants. For such substances, corresponding values of pressure, temperature and density must be obtained from tables or charts.

It is usually assumed that the equation of state is valid not only when the fluid is in mechanical equilibrium and neither giving nor receiving heat, but also when it is not in mechanical or thermal equilibrium. This assumption seems justified because deductions based on it have been found to agree with experimental results.

Calorically Perfect Gas

A gas for which the specific heat capacity at constant volume, cv, is a constant is said to be calorically perfect. The term perfect gas, used without qualification, generally refers to a gas that is both thermally and calorically perfect.

Changes of State

A change of density may be achieved both by a change of pressure and by change of temperature. If the process is one in which the temperature is held constant, it is known as isothermal. On the other hand, the pressure may be held constant while the temperature is changed. In either of these two cases there must be a transfer of heat to or from the gas so as to maintain the prescribed conditions. If the density change occurs with no heat transfer to or from the gas, the process is said to be adiabatic.

If, in addition, no heat is generated within the gas (e.g. by friction) then the process is described as isentropic, and the absolute pressure and density of a perfect gas are related by the additional expression p/e^γ  = constant, where γ = cp/cv and cp and cv represent the specific heat capacities at constant pressure and constant volume respectively. For air and other diatomic gases in the usual ranges of temperature and pressure γ = 1.4.