Multicomponent thermodynamics

Equation (12.53) gives the desired evaluation of the general thermodynamic derivative V in a system of/degrees of freedom, expressed in terms of known geometrical quantities. As in the two-dimensional case, other expressions for V would be possible in other special choices of basis. Equation (12.53) is suitable for machine computation in multicomponent thermodynamic systems of arbitrary complexity. 

Clegg SL, Brimblecombe P (1995) Application of a multicomponent thermodynamic model to activities and thermal properties of 0 40 mol kg 1 aqueous sulfuric acid from < 200 to 328K. J Chem Eng Data 40 43-64 Clegg SL, Rard JA, Pitzer KS (1994) Thermodynamic properties of 0-6 mol kg-1 aqueous sulfuric acid from 273.15 to 328.15K. Chem Soc Faraday Trans 90 1875-1894... 

When the presence of other components cannot be neglected, the system can still be treated approximately by considering the membrane in the presence of the additional component to be a new effective medium however, this is at best a rough approximation since it neglects the bulk-flow contribution and thermodynamic influence of additional components on the flux of component A. Multicomponent thermodynamic issues are beyond the scope of this discussion, but have been treated (38). 

The starting point of the multicomponent thermodynamic theory of capillarity is the classical work of Gibbs [2]. In the subsequent development, great attention has been paid to the phase interfaces, both from thermodynamic [4] and molecular [16] points of view. With regard to thermodynamic behavior of the bulk phases, few important problems have been studied, such as the phase rule [17] or generalization of the Kelvin equation to a multicomponent case [4,18]. Much more is known about capillary equihbrium in a single-component case [19-21]. Recently, the interest in multicomponent capillary equihbria has appeared in connection with the petroleum appUcations [22-25],... 

The prediction of volumes of denatured proteins in the presence of denaturants, or of protein volumes in salt or other multicomponent solutions, requires application of the laws of multicomponent thermodynamics, i.e., the introduction of proper interaction parameters or of other reasonable assumptions. 

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. 

However, in the study of thermodynamics and transport phenomena, the behavior of ideal gases and gas mixtures has historically provided a norm against which their more unruly brethren could be measured, and a signpost to the systematic treatment of departures from ideality. In view of the complexity of transport phenomena in multicomponent mixtures a thorough understanding of the behavior of ideal mixtures is certainly a prerequisite for any progress in understanding non-ideal systems. 

Ideal Adsorbed Solution Theory. Perhaps the most successful approach to the prediction of multicomponent equiUbria from single-component isotherm data is ideal adsorbed solution theory (14). In essence, the theory is based on the assumption that the adsorbed phase is thermodynamically ideal in the sense that the equiUbrium pressure for each component is simply the product of its mole fraction in the adsorbed phase and the equihbrium pressure for the pure component at the same spreadingpressure. The theoretical basis for this assumption and the details of the calculations required to predict the mixture isotherm are given in standard texts on adsorption (7) as well as in the original paper (14). Whereas the theory has been shown to work well for several systems, notably for mixtures of hydrocarbons on carbon adsorbents, there are a number of systems which do not obey this model. Azeotrope formation and selectivity reversal, which are observed quite commonly in real systems, ate not consistent with an ideal adsorbed... 

Many simple systems that could be expected to form ideal Hquid mixtures are reasonably predicted by extending pure-species adsorption equiUbrium data to a multicomponent equation. The potential theory has been extended to binary mixtures of several hydrocarbons on activated carbon by assuming an ideal mixture (99) and to hydrocarbons on activated carbon and carbon molecular sieves, and to O2 and N2 on 5A and lOX zeoHtes (100). Mixture isotherms predicted by lAST agree with experimental data for methane + ethane and for ethylene + CO2 on activated carbon, and for CO + O2 and for propane + propylene on siUca gel (36). A statistical thermodynamic model has been successfully appHed to equiUbrium isotherms of several nonpolar species on 5A zeoHte, to predict multicomponent sorption equiUbria from the Henry constants for the pure components (26). A set of equations that incorporate surface heterogeneity into the lAST model provides a means for predicting multicomponent equiUbria, but the agreement is only good up to 50% surface saturation (9). 

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). 

The thermodynamic consistency test for binary systems described above can be extended to ternary (and higher) systems with techniques similar to those described by Herington (H3). The necessary calculations become quite tedious, and unless extensive multicomponent data are available, they are usually not worthwhile. 

In this section, a general formulation will be given for the effect of bubble residence-time and bubble-size distributions on simultaneous and thermodynamically coupled heat- and mass-transfer in a multicomponent gas-liquid dispersion consisting of a large number of spherical bubbles. Here one can... 

Unfortunately, relatively little work has been done on the solution thermodynamics of concentrated polymer solutions with "gathering". The definitive work on the subject Is the article of Yamamoto and White (17). The corresponding-states theory of Flory (11) does not account for gathering. We therefore restrict our consideration here to multicomponent solutions where the solvents and polymer are nonpolar. For such solutions, gathering Is unlikely to occur. 

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. 

Vynckier, E., and Froment, G. F., Modeling of the kinetics of complex processes based upon elementary steps , in Kinetic and Thermodynamic Lumping of Multicomponent Mixtures (G. Astaiita and S. I. Sandler, Eds.) Elsevier, Amsterdam (1991) 131-161. 

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

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