Thermodynamics multicomponent systems

The preceding material of this section has focused on the most important phenomenological equation that thermodynamics gives us for multicomponent systems—the Gibbs equation. Many other, formal thermodynamic relationships have been developed, of course. Many of these are summarized in Ref. 107. The topic is treated further in Section XVII-13, but is worthwhile to give here a few additional relationships especially applicable to solutions. 

Thermodynamic treatments may, of course, be extended to multicomponent systems. See Ref 117 for an example. 

Examples of ideal binary systems ate ben2ene—toluene and ethylben2ene—styrene the molecules ate similar and within the same chemical families. Thermodynamics texts should be consulted before making the assumption that a chosen binary or multicomponent system is ideal. When pressures ate low and temperatures ate at ambient or above, but the solutions ate not ideal, ie, there ate dissimilat molecules, corrections to equations 4 and 5 may be made ... 

From the definition of a partial molar quantity and some thermodynamic substitutions involving exact differentials, it is possible to derive the simple, yet powerful, Duhem data testing relation (2,3,18). Stated in words, the Duhem equation is a mole-fraction-weighted summation of the partial derivatives of a set of partial molar quantities, with respect to the composition of one of the components (2,3). For example, in an / -component system, there are n partial molar quantities, Af, representing any extensive molar property. At a specified temperature and pressure, only n — 1) of these properties are independent. Many experiments, however, measure quantities for every chemical in a multicomponent system. It is this redundance in reported data that makes thermodynamic consistency tests possible. 

The N equations represented by Eq. (4-282) in conjunction with Eq. (4-284) may be used to solve for N unspecified phase-equilibrium variables. For a multicomponent system the calculation is formidable, but well suited to computer solution. The types of problems encountered for nonelectrolyte systems at low to moderate pressures (well below the critical pressure) are discussed by Smith, Van Ness, and Abbott (Introduction to Chemical Engineering Thermodynamics, 5th ed., McGraw-Hill, New York, 1996). 

During this period of intensive development of unit operations, other classical tools of chemical engineering analysis were introduced or were extensively developed. These included studies of the material and energy balance of processes and fundamental thermodynamic studies of multicomponent systems. 

McMillan, W. G. Mayer, J. E. (1945). The statistical thermodynamics of multicomponent systems. Journal of Chemical Physics, 13, 276-305. 

Fortunately, few of these variables are truly independent. Geochemists have developed a variety of numerical schemes to solve for equilibrium in multicomponent systems, each of which features a reduction in the number of independent variables carried through the calculation. The schemes are alike in that each solves sets of mass action and mass balance equations. They vary, however, in their choices of thermodynamic components and independent variables, and how effectively the number of independent variables has been reduced. 

Thermodynamic data (enthalpy of reaction, specific heat, thermal conductivity) for simple systems can frequently be found in date bases. Such data can also be determined by physical property estimation procedures and experimental methods. The latter is the only choice for complex multicomponent systems. 

Bernard, G. Hocine, R. Lupis, C.H.P. Thermodynamic conditions for spinodal decomposition in a multicomponent system. Transactions of the metallurgical society of AIME. 1967, 239, 1600-1604. 

In developing the thermodynamic framework for ECES, we attempted to synthesize computer software that would correctly predict the vapor-liquid-solid equilibria over a wide range of conditions for multicomponent systems. To do this we needed a good basis which would make evident to the user the chemical and ionic equilibria present in aqueous systems. We chose as our cornerstone the law of mass action which simply stated says "The product of the activities of the reaction products, each raised to the power indicated by its numerical coefficient, divided by the product of the activities of the reactants, each raised to a corresponding power, is a constant at a given temperature. ... 

The mathematical basis of classic thermodynamics was developed by J. Willard Gibbs in his essay [1], On the Equilibrium of Heterogeneous Substances, which builds on the earlier work of Kelvin, Clausius, and Helmholtz, among others. In particular, he derived the phase mle, which describes the conditions of equilibrium for multiphase, multicomponent systems, which are so important to the geologist and to the materials scientist. In this chapter, we will present a derivation of the phase rule and apply the result to several examples. 

Bertrand G. L., Acree W. E. Jr., and Burchfield T. (1983). Thermodynamical excess properties of multicomponent systems Representation and estimation from binary mixing data. J. Solution. Chem., 12 327-340. 

Wood B. I (1987). Thermodynamics of multicomponent systems containing several solid solutions. In Reviews in Mineralogy, vol. 17, P. H. Ribbe (series ed.), Mineralogical Society of America. 

A thermodynamic parameter (dV/dnB)T,F,n g which describes how the volume of component S in a multicomponent system depends on the change in its amount expressed in mol. Hpiland recently summarized the partial molar volumes of numerous biochemical compounds in aqueous solution. See Dalton s Law of Partial Pressures Concentrations Molecular Crowding... 

The basic problem in determining phase equilibria in multicomponent systems is the existence of a large number of variables, necessitating extensive experimental work. If ten measurements are considered satisfactory for acceptable characterization of the solubility in a two-component system in a particular temperature range, then the attainment of the same reliability with a three-component system requires as many as one hundred measurements. Therefore, a reliable correlation method permitting a decrease in the number of measurements would be extremely useful. Two different methods - the first of them based on geometrical considerations, and the second on thermodynamic condition of phase equilibria - are presented and their use is demonstrated on worked examples. 

Two different methods have been presented in this contribution for correlation and/or prediction of phase equilibria in ternary or mul> ticomponent systems. The first method, the clinogonial projection, has one disadvantage it is not based on concrete concepts of the system but assumes, to a certain extent, additivity of the properties of individiial components and attempts to express deviations from additivity of the properties of individual components and attempts to express deviations from additivity by using geometrical constructions. Hence this method, although simple and quick, needs not necessarily yield correct results in all the cases. For this reason, the other method based on the thermodynamic description of phase equilibria, reliably describes the behaviour of the system. Of cource, the theory of concentrated ionic solutions does not permit a priori calculation of the behaviour of the system from the thermodynamic properties of pure components however, if a satisfactory equation is obtained from the theory and is modified to express concrete systems by using few adjustable parameters, the results thus obtained are still substantially more reliable than results correlated merely on the basis of geometric similarity. Both of the methods shown here can be easily adapted for the description of multicomponent systems. 

Equation (3.20) implies that the system will be thermodynamically stable if the addition of an infinitely small amount of any component leads to a decrease in chemical potentials of all the other constituent components. The fulfilment of the second inequality in equation (3.20) is a sufficient condition for the stability of the multicomponent system with respect to mutual diffusion. 

There is, however, another statement of the necessary and sufficient condition of thermodynamic stability of the multicomponent system in relation to mutual diffusion and phase separation that is less stringent than equation (3.20) because it may be fulfilled not for every component of the multicomponent system. For example, in the case of the ternary system biopolymeri + biopolymer2 + solvent, it appears enough to fulfil only two of the inequalities (Prigogine and Defay, 1954)... 

Goranson, R. W., Thermodynamic Relations in Multicomponent Systems, Washington,... 

The objective of this review is to characterize the excimer formation and energy migration processes in aryl vinyl polymers sufficiently well that the excimer probe may be used quantitatively to study polymer structure. One such area of application in which some measure of success has already been achieved is in the analysis of the thermodynamics of multicomponent systems and the kinetics of phase separation. In the future, it is likely that the technique will also prove fruitful in the study of structural order in liquid crystalline polymers. 

The determination of the thermodynamic functions of individual components in a multicomponent system is based either on the total molar thermodynamic quantity (1) as in Equation 1... 

The diffusional flux of each species in a multicomponent system may be derived from nonequilibrium thermodynamics (27), and may be represented by the expression ... 

---