For pure substances (one component systems), equilibrium between liquid and vapor phases takes place if specific (molar) Gibbs energy or fugacity values of the coexisting phases are equal:

The *equilibrium state* corresponds to the saturated vapor pressure curve p = p_{s}(T), which is limited by a triple point (T_{tr}P_{tr}) and a critical point (T_{c},p_{c}) of the substance. (In the presence of surface tension forces at the phase interface, pressures values p^{(1)} and p^{(g)} in coexisting phases are not equal.) The Clausius-Clapeyron equation follows from Eq. (1)

This equation gives the relationship between the saturation vapor pressure and specific (molar) thermodynamic properties of the substance: volumes ν', ν'', entropies s', s", enthalpies h', h" for the liquid and gaseous phases which are in equilibrium; r is a heat of vaporization.

Integrating Eq. (2) with the simplest assumptions that a vapor phase is a perfect gas, r = *const*, ν'' = RT/p_{s}
ν' (R is the universal gas constant), leads to the two parameter equation

(p_{0} = 101325 Pa, T_{b} is the normal boiling point temperature), which is a base for derivation of the majority of empirical correlations to describe the vapor pressure versus temperature. The best of them provide a discrepancy of 1-2% in describing experimental data of p_{s}(T) (the equations by Frost-Kalkwarf-Thodos, Lee-Kesler, etc.). To estimate thermodynamic properties of the phases which are in equilibrium, it is possible to use generalized equations such as Lee-Kesler, Gunn and Yamada for density, as well as the equations by Carruth and Kobayashi for heat of evaporation. Actually all of them have been based on the three parameter corresponding states law, therefore to calculate any of the properties mentioned above it is necessary that critical parameters T_{c}, p_{c} as well as Pitzer Ґ (acentric) factor need to be known. It should be noted, however, that such correlations are essentially less precise than experimental data to be described. So it is reasonable to use the correlations only for predicting the properties *a priori* as a first approximation.

An alternative way in describing thermodynamic properties is based on using an equation of state which may present the properties of both liquid and gaseous states. In general, it may be presented in the form

where ρ = 1/ν is the molar density, z the compressibility coefficient, and a,b, ... are individual parameters of the substance, which are fitted to the experimental data. If the equation of the form, Eq. (4), is available, then it makes possible to evaluate fugacity

and then solving the equilibrium Eq. (1)

to calculate the properties of the phases which are in equilibrium, p_{s} (T), ν' = 1/ρ', ν" = 1/ρ". The system of Eq. (5) is equivalent to the Maxwell Rule

Van der Waals was the first who have proposed the equation of state to discribe both liquid and vapor phases. Since then there have been proposed a lot of similar equations (by Redlich-Kwong, Soave, Wilson, et. al.), the parameters of which may be calculated if the critical point parameters of the substance are known. Also multiparameter equations of state have been proposed (for example, Benedict-Webb-Rubin), which are more precise. However, the parameters of the equations may not be defined without experimental data.

For multicomponent systems the conditions of vapor-liquid equilibrium are expressed by equality of the partial Gibbs Energy
of the components or appropriate fugacities

where
and
are the molar fractions of the components in liquid and vapor phases, hence

To describe the behavior of the closed v-component system when total mole fractions of the components are given by
, …,
it is necessary to incorporate the material balances equations

where z' and z" are a mole fraction of the liquid and the vapor phases.

The system of Equations (6)-(8) is a base for analysis of vapor-liquid equilibrium in mixtures (solutions). The equations under consideration allow the calculation of a bubble point line as well as a dew point line. In contrast to the case of a pure substance, for the multicomponent two phase system these lines do not coincide as soon as the system is a polyvariant one (a number of degrees of freedom f = ν > 1), and the lines bound a field of two-phase equilibrium. At the bubble point, the liquid phase fraction z' = 1 and the phase composition of the liquid phase and the total composition of the system are equal:
= χ_{i}. Under that condition Eqs. (6) to (8) define a bubble point temperature T_{b}(p, χ) if the pressure is known or a bubble point pressure p_{b}(T, χ) if temperature is known; in both cases the vapor phase composition χ
, …, χ
is a subject of calculations also. Conversely, at the dew point which relates to a starting point of condensation, it is known that z" = 1, χ
= χ_{i}. Then the liquid phase composition χ
, …, χ
may be defined from Eqs. (6) to (8) together with either temperature T_{d}(p, χ) or pressure p_{d}(T, χ) which correspond to a dew point. By way of example, Figure 1 shows a typical shape of the curve which bounds the two-phase vapor-liquid region of a binary system for a given composition χ ≡ χ_{2}, χ_{1} = 1 − χ. The system comprises substances with saturation pressure dependence p_{s1}(T), p_{s2}(T); a smooth line presents bubble points and dash ones relates to dew points. Having defined the two-phase region, it is possible, on the basis of Eqs. (6) to (8), to calculate the compositions of
, χ
and fractions z', z" of the phases under equilibrium for any T, p and the given total composition of the system χ_{i}. In addition, the Duhem theorem has to be satisfied; that is, the equations shall be independent.

If a liquid phase is an ideal solution and a gaseous phase is a mixture of ideal gases, then the calculations may be executed in a very simple way. In this case the fugacities of the components, which appear in Eq. (6), may be calculated according to the Raoult's Law
= p_{si}(T)
,
. As a result, Eq. (6) may be rearranged into the form, which is more convenient:

where K_{i}(T, p) is a distribution coefficient of a component i

Having eliminated the phases compositions from the equation of material balances, Eq. (8) by means of Eq. (7) and (9), we obtain an equation to calculate the fraction of each phase, for example, in the form of

The phase composition may then be estimated as

Parameters of the bubble point curve P_{b}(T, χ) or T_{b}(p, χ) and the dew point curve one p_{d}(T, χ) or T_{d}(p, χ) are the solutions of the following equations, accordingly,

where
= p_{si}(T)/p_{b} or p_{si}(T_{b})/p;
= p_{si}(T)/p_{d} or p_{si}(T_{d})/p. Compositions of new phases being formed in the process are obtained from the equations

in an explicit form. The curves p_{b}(T, χ) and p_{d}(T, χ) may be obtained from Eq. (13) if Eq. (10) is taken into account:

A p-χ diagram which illustrates the correlations given above for the binary ideal solution (χ = χ_{2}) is presented in Figure 2. For this case it is possible to solve Eqs. (11) and (12) in the explicit form:

and the bubble point line (Eq. (15)) is linear. In the T-χ-diagram (Figure 3) both the bubble point line and the dew point line, Eq. (16), are nonlinear.

To calculate vapor-liquid equilibria for multicomponent nonideal systems, the phase equilibrium, Eqs. (6), are usually presented in the same form as Eq. (9). However, in contrast to the ideal case (Eq. (10)), for this case distribution coefficients depend not only on temperature and pressure but also on phase composition. If, for example, the excess thermodynamic functions were used to describe a liquid state and an equation of state was used for a gas phase description, then,

where
and γ_{i} are the fugacity of the pure i-component and its activity coefficient in the liquid phase; φ_{i} =
is the fugacity coefficient of the component in a gas state. As a result, the set of Equations (9), (11), and 12 as well as Equations (13), (14) become essentially nonlinear, and appropriate iterative methods should be used to find a solution. It is not difficult to estimate
, if the pure components properties are available for the liquid phase:

As for the methods of calculations of the activity coefficients γ_{i}, there exists a vast literature on the subject. More often than not the methods are based on the molecular theory and follow the goal of calculating γ_{i} in multicomponent mixtures in terms of the pure component properties. In general, the problem is impossible to solve at the current level of knowledge, that is why, as a rule, the suggested formulas include the empirical so-called "binary parameters" which characterize the molecular interactions for the binary systems which comprise the mixture components. Experimental data related to thermodynamic properties or vapor-liquid equilibria in the binary systems need to be known to adjust the values of the binary parameters. Often used are the correlations developed by van Laar, Hildebrand and Scatchard, as well as those which have been obtained on the basis of two-liquid theory—NRTL, quazi-chemical theory UNIQUAC, The UNIFAC Method allows the calculation of activity coefficient using the fundamental idea that γ_{i} may be defined due to the combinatorial contribution of the molecular functional groups interacting in the solution. The method may be applied for a system which has not been studied experimentally.

Fugacity coefficients of the components for the gas phase φ_{i}, presented in Eq. (17), may be calculated if the equation of state for the mixture is available. It is a common assumption that if all the pure components may be described by an equation of the form of Eq. (4) (Redlich-Kwong, Benedict-Webb-Rubin, etc.) then the equation is valid to describe the mixture comprising the same components. The parameters of the equation a, b,... are the combinations of the individual parameters for the components a_{i}, b_{i}, ... . The rules for such combinations are empirical ones, for example

In the most prevalent cases, mixing rules involve an adjustable binary parameter I_{ij}, as it is possible to see in Eq. (19).

Nonideal solutions are different from ideal ones not only quantitatively but qualitatively. For example, the bubble point line for nonideal solutions in the p-χ diagram is not linear (as it was for ideal case, see Figure 2) however the shape of the T-χ diagram is still the same as for an ideal solution (Figure 3). Also, the deviations from ideality may induce new phenomena. Even the simplest model of a nonideal solution such as the Margules equation predicts an *azeotrophy*. An essence of this phenomena is that in particular T and p both a liquid and a vapor phase composition become equal to the total mixture composition:
=
= χ_{i}; 1 ≤ i ≤ ν. As a result, the material balance equations become linear and may be satisfied with any value of the mole fraction of the phases which are in equilibrium. Otherwise there is an indifferent state and the boiling or condensation processes take place at T = *const* and p = *const*, similar to pure substances or monovarint systems, however f > 1. At the azeotrope point dew points and boiling points lines have a common extremum: for instance Figure 4 presents the lines in the T-χ-diagram for a binary solution forming an azeotrope. The azeotrope point is influenced by pressure and, therefore, temperature (Figure 4).

Often azeotroping is a forerunner of another phenomenon which occurs in nonideal solutions with the strong positive deviation from ideality (γ_{i} > 1): this is a limited miscibility of the components in the liquid state. In this case, a liquid mixture splits into two separate liquid phases which have different compositions. In general, the phenomena takes place for the certain compositions which form an immisibility gap. Figure 5 presents an immissibility gap for a binary system with an upper critical solution temperature (broken line). The bubble point (smooth) and dew points (dash lines) curves are also shown. If the total composition of the system is within the limits of the immiscibility gap, then vapor phase is formed by evaporation of the both liquid phases at the constant temperature T_{E}(p) providing a vapor phase composition χ_{E}(p). The system exists as a mono-variant until a composition is reached when there is one of the liquid phases: then, the bubble point continues to rise in temperature with increasing c. If the total composition of the system equals χ_{E}, the system represents a geteroazeotrope mixture which evaporates at constant temperature T_{E} from the beginning to the end of the immiscibility range.

The method of calculation of vapor-liquid equilibria in strongly nonideal, multicomponent systems which is based on excess thermodynamics functions and a separate description of the phases compositions describes the complex phenomena mentioned above with a high accuracy. However, it has some shortcomings which are impossible to avoid. It is evident from Figure 1 that, for any combination of T and p, in the two-phase region the second component, if it is pure, should exist in the gas state. At the same time, to calculate phase equilibria with the method under consideration, the liquid state properties of the component must be known. This kind of information may be obtained only by extrapolation—a method which is impossible to justify. The situation becomes even more complicated if the temperature of the system is higher than the critical temperature of one of the components as is shown at Figure 1. For this case, the value of p_{S2}(T) appearing in Eq. (18) has no physical meaning. Also, there is no way to describe the situation in which the system changes into a supercritical state and the line bounding the two-phase region in Figure 6 does not touch the right axis χ = 1. Of course, the method under consideration is invalid to predict critical points of the solution, i.e., those in which compositions of the phases both vapor and liquid become equal, that is the point C in Figure 6. The dash line in the figure is a critical curve which joins the critical points of the mixtures of all the possible total compositions. Problems of this kind do not appear if calculations of equilibria are based on the application of the equations of state. If the parameters of Eqs. (4) and (19) have been adjusted to describe both the liquid and gas states of the given multicomponent system as well as the properties of pure components, then instead of Eq. (17) it is possible to obtain an equation to calculate the distribution coefficients:

Mole densities of the phases which are in equilibrium r', r" may be found from the equations

It is necessary, however, to point out that even for the modern equations of state, discrepancy of the calculated and experimental data is higher than for the excess functions method described above.

#### REFERENCES

Reid, R. C, Prausnitz, J. M. and Poling, B. E. (1987) *The Properties of Gases and Liquids*, McGraw-Hill, New York.

Prigogine, I. and Defay, R. (1954) *Chemical Thermodynamics*, Longmans, London.

Walas, S. M. (1985) *Phase Equilibrium in Chemical Engineering*, Butterworth Publisher.