Ideal-Gas Mixtures

Let us consider an ideal gas mixture of N species, subject to R independent reaction with ratesThe internal energy function is 

The following two theorems show that the function S defined by Eq.(1.6.3) satisfies the requirements (hi) and (iv) of Postulate 1.5.1. 

Theorem 1.6.1. The maximum of. .., in any given invariant manifold f (wq) is attained in the interior of the manifold. 

We recall that r uo)y the image of r uo) in the space, is a closed bounded and connected set. The boundary dr uQ) consists of all points where at least one of the variables c, . Cjy, T vanishes. First, we verify that the maximum of S cannot be attained at a point Jr where T=0, because as S — co. Next we consider a point where 

The outward unit normal on dr uo) at the point is — vJ Va where Va = (v i,, v k). The directional derivative of S along that normal is 

There exist several reference states of solutions referred to as ideal state, for which we can say something on the behavior of the thermodynamic functions of the system. The most important ideal states are the ideal-gas mixtures, the symmetric ideal solutions and the dilute ideal solution. The first arises from either the total lack of interactions between the particles (the theoretical ideal gas), or because of a very low total number density (the practical ideal gas). The second arises when the two (or more) components are similar. We shall discuss various degrees of similarities in sections 5.2. The last arises when one component is very dilute in the system (the system can consist of one or more components). Clearly, these are quite different ideal states and caution must be exercised both in the usage of notation and in the interpretations of the various thermodynamic quantities. Failure to exercise caution is a major reason for confusion, something which has plagued the field of solution chemistry. 

As in the case of a one-component system, ideal-gas (IG) mixtures also enjoy having a simple and solvable molecular theory, in the sense that one can calculate all the thermodynamic properties of the system from molecular properties of single molecules. We also have a truly molecular theory of mixtures of slightly nonideal gases, in which case one needs in addition to molecular properties of single molecules, also interactions between two or more molecules. 

As in the case of a one-component system, we should also make a distinction between the theoretical ideal gas, and the practical ideal gas. The former is a system of noninteracting particles the latter applies to any real system at very low densities. Occasionally, the former serves as a model for the latter. For instance, to obtain the equation of state of an ideal gas 

The theoretical ideal-gas partition function for a system of c components of composition N=NU N2. Nc contained in a volume V at temperature T is 

The chemical potential of the species i can be obtained by direct differentiation with respect to AT , i.e., 

Every equation that provides a linear re at on among thermodynamic properties of a constant-composition so ut on has as its counterpart an equation connecting the corresponding partial properties of each species in the soiution. 

We demonstrate tliis by example. Consider the equation that defines enthalpy  

If n moles of an ideal-gas mixture occupy a total volume V at temperature T, tire pressure is  

If the Hi moles of species i in tliis mixture occupy tlie same total volume alone at the same temperature, tlie pressure is  

The partial molar volume of species i in an ideal-gas mixture is found from Eq. (1. T) applied to the volume, superscript ig denoting an ideal-gas value  

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The principle of corresponding states enables the enthalpy of a liquid mixture to be expressed starting from that of an ideal gas mixture and a reduced correction for enthalpy ... 

The partial pressure p- of a component in an ideal-gas mixture is thus... 

Given this experimental result, it is plausible to assume (and is easily shown by statistical mechanics) that the chemical potential of a substance with partial pressure p. in an ideal-gas mixture is equal to that in the one-component ideal gas at pressure p = p. 

Note that this has resulted in the separation of pressure and composition contributions to chemical potentials in the ideal-gas mixture. Moreover, the themiodynamic fiinctions for ideal-gas mixing at constant pressure can now be obtained ... 

Phase transitions in binary systems, nomially measured at constant pressure and composition, usually do not take place entirely at a single temperature, but rather extend over a finite but nonzero temperature range. Figure A2.5.3 shows a temperature-mole fraction T, x) phase diagram for one of the simplest of such examples, vaporization of an ideal liquid mixture to an ideal gas mixture, all at a fixed pressure, (e.g. 1 atm). Because there is an additional composition variable, the sample path shown in tlie figure is not only at constant pressure, but also at a constant total mole fraction, here chosen to be v = 1/2. 

So little systematic information is available about transport in liquids, or strongly non-ideal gaseous mixtures, that attention will be limited throughout to the behavior of ideal gas mixtures. It is not intende thereby, to minimize the importance of non-ideal behavior in practice. 

An ideal gas is a model gas comprised of imaginary molecules of zero volume that do not interact. Each chemical species in an ideal gas mixture therefore has its own private properties, uninfluenced by the presence of other species. The partial pressure of species i (i = 1,2,... , N) in an ideal gas mixture is defined by equation 142 ... 

For the Gibbs energy of an ideal gas mixture, — T the parallel relation for partial properties is equation 149 ... 

Thus the fugacity of species / in an ideal gas mixture is equal to its partial pressure. 

If the hquid phase is an ideal solution, the vapor phase an ideal gas mixture, and the hquid-phase properties independent of pressure, then 7, = 1,... 

A substance is in the ideal gas state when the volume of its molecules is a zero fraction of the total volume taken up by the substance and when the individual molecules are far enough apart from each other so that there is no interaction between them. Although this only occurs at infinite volume and zero pressure, in practice, ideal gas properties can be used for gases up to a pressure of two atmospheres with little loss of accuracy. Thermal properties of ideal gas mixtures may be obtained by mole-fraction averaging the pure component values. 

The partial molar property, other than the volume, of a constituent species in an ideal gas mixture is equal to the corresponding molar property of the species as a pure ideal gas at the mixture temperature hut at a pressure equal to its partial pressure in the mixture. 

Instantaneous heat transfer between solid eatalyst and the reaeting ideal gas mixture... 

Assuming an ideal gas mixture at amiospheric pressure, calculate llie mole fraction and ppm of a component if its partial pressure is 19 mniHg. 

The volume fractions and mole fractions become identical in ideal gas mixtures at fixed conditions of pressure and temperature. In an isolated, nonreactive system, the molar composition does not vary with temperature. 

Two relations are postulated to describe the P-V-T behavior of ideal gas mixtures ... 

Thus, in an ideal gas mixture, the mole fraction of each component is identical with its volume fraction (by Amagat s law) or the ratio of its partial pressure to the total pressure (by Dalton s law). For both laws to be applicable simultaneously, the mixture and its components must behave ideally. 

If the vapor phase behaves as an ideal gas mixture, then by Dalton s law of partial pressures,... 

Fig. 10. The mole fraction of carbon dioxide in saturated solutions in air at — 110°C (above the lower critical end point). The full line is the experimental curve of Webster and the dashed curves are 1, an ideal gas mixture 2, an ideal gas mixture with Poynting s correction and 3, the solubility calculated from Eq. 8 and the principle of corresponding states.

From the definition of an ideal gas mixture, we shall have for the free energy of the mixture of i gases in the volume Y the expression m j/ = = XtiiMfy/ri... 

It therefore follows that /ree energy of an ideal gas mixture is a function of the same form as that of a simple gas. Hence, in virtue of Massieu s theorem, an ideal gas mixture behaves thermally and mechanically exactlylike a simpler gas. 

Examples.—(1) Prove that the entropy of an ideal gas mixture is the sum of the entropies of the components at the same temperature, each occupying the whole volume of the mixture. 

The definition of an Ideal Gas Mixture given in 122, although it leads to results in entire accord with those established by... 

Special cases of such phases are ideal gas mixtures, and the limiting case of a pure substance in any state of aggregation. 

The transformation of numerical concentrations c, to volumetric molecular concentrations f, cannot be effected in the case of solutions so readily as with ideal gas mixtures ( 121), on account of the changes of density. 

In the case of a solution of moderate concentration we may perhaps assume the same expression (cf. van Laar, Thermodynamik und Cliemie Thermodyn. Potential Planck, loc. cit.)t whenever the solutions can legitimately be considered as brought, by suitable changes of temperature and pressure with unchanged composition, into ideal gas mixtures ( 185). 

As indicated in the next section, for an ideal gas mixture, at constant pressure (Ca + Cg), is constant (equation 10.9) and hence ... 

As previously noted, the equilibrium constant is independent of pressure as is AG. Equation (7.33) applies to ideal solutions of incompressible materials and has no pressure dependence. Equation (7.31) applies to ideal gas mixtures and has the explicit pressure dependence of the F/Fq term when there is a change in the number of moles upon reaction, v / 0. The temperature dependence of the thermodynamic equilibrium constant is given by... 

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