Equations of change, for a multicomponent system

Table 8.4-1 The Differential Form of the Equations of Change for a Multicomponent System...

First a derivative is given of the equations of change for a pure fluid. Then the equations of change for a multicomponent fluid mixture are given (without proof), and a discussion is given of the range of applicability of these equations. Next the basic equations for a multicomponent mixture are specialized for binary mixtures, which are then discussed in considerably more detail. Finally diffusion processes in multicomponent systems, turbulent systems, multiphase systems, and systems with convection are discussed briefly. 

Here p is the chemical potential just as the pressure is a mechanical potential and the temperature Jis a thennal potential. A difference in chemical potential Ap is a driving force that results in the transfer of molecules tlnough a penneable wall, just as a pressure difference Ap results in a change in position of a movable wall and a temperaPire difference AT produces a transfer of energy in the fonn of heat across a diathennic wall. Similarly equilibrium between two systems separated by a penneable wall must require equality of tire chemical potential on the two sides. For a multicomponent system, the obvious extension of equation (A2.1.22) can be written... 

B Equation of Change for Temperature for a Multicomponent System Subtraction then gives ioi a. N ... 

Thus, the interrelationships provided by Eqs. 8.2-8 through 8.2-15 are really restrictions on the mixture equation of state. As such, these equations are important in minimizing the amount of experimental data necessary in evaluating the thermodynamic, properties of mixtures, in simplifying the description of multicomponent systems, and in testing the consistency of certain types of experimental data (see Chapter 10). Later in this chapter we show how the equations of change for mixtures and the Gibbs-Duhem equations provide a basis for the experimental determination of partial molar properties. 

Section 4.1 via Section 4.1.2 formally illustrates vapor-Uquid equilibria vis-a-vis distillation in a closed vessel along with bubble-point and dew-point calculations for multicomponent systems. How vapor-liquid equilibrium is influenced by chemical reactions in the liquid phase is treated in Section 5.2.1.2, where two subsections, 5.2.1.2.1 and 5.2.1.2.2, deal with reactions influencing vapor-Uquid equilibria in isotopic systems. We next encounter open systems in Chapter 6. The equations of change for any two-phase system (e.g. a vapor-Uquid system) are provided in Section 6.2.1.1 based on the pseudo-continuum approach for the dependences of species concentrations... 

If we compare Eqs. 5.1.14 with the conservation equation (Eq. 5.1.2) for a binary system and the pseudo-Fick s law Eq. 5.1.15, with Eq. 3.1.1 then we can see that from the mathematical point of view these pseudomole fractions and pseudofluxes behave as though they were the corresponding variables of a real binary mixture with diffusion coefficient D-. The fact that the are real, positive, and invariant under changes of reference velocity strengthens the analogy. If the initial and boundary conditions can also be transformed to pseudocompositions and fluxes by the same similarity transformation, the uncoupled equations represent a set of independent binary-type problems, n - 1 in number. Solutions to binary diffusion problems are common in the literature (see, e.g.. Bird et al., 1960 Slattery, 1981 Crank, 1975). Thus, the solution to the corresponding multicomponent problem can be written down immediately in terms of the pseudomole fractions and fluxes. Specifically, if... 

Our task here is to derive an expression that describes how the composition of a multicomponent mixture changes with time in a Loschmidt diffusion apparatus of the kind described in Section 5.5. The composition profile for a binary system is given by Eqs. 5.5.5 and 5.5.6) the solution to the binarylike multicomponent problem is given by the same expressions on replacing the binary diffusivity in those equations by the effective diffusivity. The average composition in the bottom tube after time Z, for example, is given by... 

In the following developments, we rely on the results of Section 6.2.1.1 and identify the equations of change of concentration of a species i in a countercurrent two-region/two-phase system we focus on two-phase systems. Next we consider the equations for operating lines in such devices. The multicomponent separation capability of sucb systems is treated next in tbe context of a two-pbase system. 

The meaning and also the limitation of the term possible variations must be considered. For the purposes of discussion, we center our attention on Equation (5.2) and consider a heterogenous, multicomponent system. The independent variables that are used to define the state are the entropy, volume, and mole numbers (i.e., amount of substance or number of moles) of the components. The statements of the condition of equilibrium require these to be constant because of the isolation of the system. Possible variations are then the change of the entropy of two or more of the phases subject to the condition that the entropy of the whole system remains constant, the change of the volume of two or more phases subject to the condition that the volume of the whole system remains constant, or the transfer of matter from one phase to another subject to the condition that the mass of the whole system remains constant. Such variations are virtual or hypothetical,... 

We choose the total system to be the condenser and the entire dielectric medium. The condenser is immersed in the medium which, for purposes of this discussion, is taken to be a single-phase, multicomponent system. The pressure on the system is the pressure exerted by the surroundings on a surface of the dielectric. In setting up the thermodynamic equations we omit the properties of the metal plates, because these remain constant except for a change of temperature. The differential change of energy of the system is expressed as a function of the entropy, volume, and mole numbers, but with the addition of the new work term. Thus,... 

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