Thermodynamics of Real Gases

Useful though it may be, the ideal gas approximation ignores the finite size of the molecules and the intermolecular forces. Consequently, as the gas becomes denser, the ideal gas equation does not predict the relation between the volume, pressure and temperature with good accuracy one has to use other equations of state that provide a better description. If the molecular size and forces are included in the theory, one refers to it as a theory of a real gas.  

As a result of molecular forces, the total internal energy U, the relation between the molar heat capacities Cp and Cy, the equation for adiabatic processes and other themodynamic quantities will differ from those for the ideal gas. In this section we shall see how the thermodynamic quantities of a real gas can be obtained from an equation of state that takes molecular size and forces into account. 

The van der Waals equation, which takes into account the intermolecular forces and molecular size, and the critical constants pc, Vmc and Tc were introduced in Chapter 1  

To write this in a convenient form, first we note that, for a fixed N, as the volume Vq — oo, the density approaches zero, and 1/reai approaches the energy of an ideal gas, (/ideau given by (6.1.3). Hence equation. (6.2.6) can be written as 

If [S(p/ )/57 )], can be calculated using the equation of state, explicit expressions for Uka niay be derived. For example, using the van der Waals equation, we can obtain the energy of a real gas. From (6.2.1) it is easy to see that p/T = NR/ V - Nb)- a N/V) l/T). Substituting this expression into 

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Real gases follow the ideal-gas equation (A2.1.17) only in the limit of zero pressure, so it is important to be able to handle the thermodynamics of real gases at non-zero pressures. There are many semi-empirical equations with parameters that purport to represent the physical interactions between gas molecules, the simplest of which is the van der Waals equation (A2.1.50). However, a completely general form for expressing gas non-ideality is the series expansion first suggested by Kamerlingh Onnes (1901) and known as the virial equation of state ... 

With the above quantities, aU the thermodynamics of real gases can be described, once the real gas parameters, such as the van der Waals constants or the virial coefficients, are known. 

Hea.t Ca.pa.cities. The heat capacities of real gases are functions of temperature and pressure, and this functionaHty must be known to calculate other thermodynamic properties such as internal energy and enthalpy. The heat capacity in the ideal-gas state is different for each gas. Constant pressure heat capacities, (U, for the ideal-gas state are independent of pressure and depend only on temperature. An accurate temperature correlation is often an empirical equation of the form ... 

From this equation, the temperature dependence of is known, and vice versa (21). The ideal-gas state at a pressure of 101.3 kPa (1 atm) is often regarded as a standard state, for which the heat capacities are denoted by CP and Real gases rarely depart significantly from ideaHty at near-ambient pressures (3) therefore, and usually represent good estimates of the heat capacities of real gases at low to moderate, eg, up to several hundred kPa, pressures. Otherwise thermodynamic excess functions are used to correct for deviations from ideal behavior when such situations occur (3). 

The behavior of real gases is discussed in the previous section on Thermodynamics and Heat Transfer. ... 

Even though the van der Waals equation is not as accurate for describing the properties of real gases as empirical models such as the virial equation, it has been and still is a fundamental and important model in statistical mechanics and chemical thermodynamics. In this book, the van der Waals equation of state will be used further to discuss the stability of fluid phases in Chapter 5. 

In mixtures of real gases the ideal gas law does not hold. The chemical potential of A of a mixture of real gases is defined in terms of the fugacity of the gas, fA. The fugacity is, as discussed in Chapter 2, the thermodynamic term used to relate the chemical potential of the real gas to that of the (hypothetical) standard state of the gas at 1 bar where the gas is ideal ... 

Many equations have been suggested to express the behavior of real gases. In general, there are those equations that express the pressure as a function of the volume and temperature, and those that express the volume as a function of the pressure and temperature. These cannot usually be converted from one into the other without obtaining an infinite series. The most convenient thermodynamic function to use for those in which the volume and temperature are the independent variables is the Helmholtz energy. The... 

The thermodynamic functions of real gases The equation of state... 

The changes of the thermodynamic functions on mixing of real gases... 

The fugacity, considered as the thermodynamically effective pressure /, equals the measured pressure p exactly in the case of ideal gases only. In the case of real gases both values differ by the so called activity coefficient Yj the value of which depends on the given state of the gas ... 

Equation (144a) states that molar refraction is a constant, independent of structure and of the thermodynamic state of the medium. Numerous studies have shown this to be closely correct, at certain special conditions, in a first approximation, since in general Rm is a function of tepiperature and density, or pressure in the case of real gases. The non-constancy of has been explained theoretically by Yvon as well as by Mazur and Mandel.3 We shall deal here only with deviations from constancy in Rm due to Kirkwood-Yvon translational statistical fluctuations, omitting for simplidty the fact that molecular interactions perturb the intrinsic polarizabilities in accordance with the Jansen-Mazm model. For a molecule immersed in a statistically inhomogeneous medium, this in place of (145) leads to ... 

In thermodynamics, the pressure-volume temperature relationships of real gases are described by the equation of state. The compressibility factor from the reduced pressure and temperature can be rearranged from Redlich and Kwong [9] to two constant state equations in the form... 

In practice, the most important set of thermodynamic variables is of course T, P, pA, employed in (6.34). However, relation (6.33) is also useful and has enjoyed considerable attention in osmotic experiments where pB is kept constant. This set of variables provides relations which bear a remarkable analogy to the virial expansion of various quantities of real gases. We demonstrate this point by extracting the first-order expansion of the osmotic pressure n in the solute density pA. This can be obtained by the use of the thermodynamic relation... 

J. A. Beattie The Computation of the Thermodynamic Properties of Real Gases and Mixtures of Real Gases, Chem. Rev., 44 141 (1949). 

D. S. Viswanath and G.-J. Su Generalized Thermodynamic Properties of Real Gases, Part II. Generalized Benedict-Webb-Rubin Equation of State for Real Gases, AIChE J., 11(2) 205 (1965). 

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