The p-v-T behavior of ideal gases can be determined through the kinetic theory of gases assuming that i) the gas consists of a large number of equi-sized molecules which move randomly in all directions, ii) all collisions between the molecules and the molecules and container walls are elastic, iii) the volume of the molecules is negligible, iv) attractive forces between the molecules are negligible and v) the kinetic energy of the molecules is proportional to the absolute temperature, A direct result of this model is the Ideal Gas Law
which includes the gas laws postulated by Charles, Boyle and Avogadro. In Eq. (1), is the molar volume (m^{3}/mol), V is the volume (m^{3}), N is the number of moles and is the Universal Gas Constant, J/(mol K).
In a mixture of ideal gases, the momentum transfer between each molecular species and the container walls will be independent of the presence of other molecules. The total pressure can then be determined by Dalton's Law
as the sum of all partial pressures. The Ideal Gas Law can be applied to each component of a gas mixture as if it alone were present.
At low temperature and high pressure, the assumption of negligible molecular volume and intermolecular attraction becomes increasingly approximate. The deviation is expressed by the compressibility factor
Typical effects of temperature and pressure on the compressibility factor are shown in Figure 1.
Various prediction methods for the p-v-T behavior of nonideal gases have been recommended in the literature. The most common method is the Van-der-Waals Equation
The value of a corrects the Ideal Gas Law for the intermolecular attraction, the value of b for the finite molecular volume. The Van-der-Waals constants vary between gases and are listed below. A generalized, but less accurate formulation of the Van-der-Waals Equation may be obtained from the Law of Corresponding States
with the reduced properties p_{r}, v_{r} and T_{r} being the actual pressure, volume and temperature divided by the appropriate critical value. Critical properties are also given below.
Other equations of state have been proposed to improve the prediction of the Van-der-Waals Equation. The most common are: The Redlich-Kwong Equation which is recommended for hydrocarbons
the Peng-Robinson Equation
the Berthelot Equation
the Dieterici Equation
the Virial Equation
The constants in Eqs. (6)-(10) can be found in Tables. They can also be estimated from the critical properties and the knowledge that the critical point is the point of inflection of the isotherms in the pressure-volume diagram.
Two fluids are in corresponding states if they have the same reduced properties p_{r}, v_{r} and T_{r} Fluids in corresponding states deviate from ideal behavior to the same extent, i.e., have the same Compressibility Factor z.
According to the law of Corresponding States, p-v-T data for all gases can be obtained from a plot of compressibility factor as a function of reduced properties. The most common diagram is shown in Figure 2.
For reduced equations of state and compressibility plots, pseudocritical constants of gas mixtures are used. The critical temperature T_{c} of a gas mixture can be obtained with good accuracy from Kay's Law
with y_{1} being the mole fraction of component i in the gas mixture. The critical pressure p_{c} is more accurately determined using the Prausnitz and Gunn method
For more information refer to Reid, Prausnitz and Sherwood.
REFERENCES
Reid, R., Prausnitz, J. and Sherwood, T. (1977) The Properties of Gases and Liquids, McGraw-Hill Book Company.
References
- Reid, R., Prausnitz, J. and Sherwood, T. (1977) The Properties of Gases and Liquids, McGraw-Hill Book Company.
Heat & Mass Transfer, and Fluids Engineering