Systems of More Than One Component

Problem The true vapor pressure of water is 23.76 mm. at 25 C. Calculate the vapor pressure when water vaporizes into a space already containing an insoluble gas at 1 atm. pressure, assuming ideal behavior. 

In using equation (27.36), it will be seen that Ap and p must be in the same units, the exact nature of which is immaterial further, the units of H are determined by those of Vi and AP, Thus, if Viis expressed in liters and AP in atm., R will be 0.082 liter-atm. deg. mole The specific volume of water may be taken, with sufficient accuracy, as 1.0, so that Vi is about 18 ml. or 0.018 liter mole  

When the water vaporizes into a vacuum, the vapor pressure p is equal to the total pressure, i.e., 23.76 mm. at 25 C or 298 K the external pressure is increased by approximately 1 atm., assuming the vapor pressure not to alter greatly, so that AP is 1.0. Hence, by equation (27.36), 

The effect of external pressure on the vapor pressure of a liquid is seen to be relatively small nevertheless, the subject has some significance in connection with the theory of osmotic pressure.  

Conditions of Equilibrium.—If a system of several phases consists of more than one component, then the equilibrium condition. of equal molar free energies in each phase requires some modification. Because each phase may contain two or more components in different proportions, it is necessary to introduce partial molar free energies, in place of the molar free energies. Consider a closed system of P phases, indicated by the letters a, 6,. .., P, containing a total of C components, designated by 1,2,. .., C, in equilibrium at constant temperature and pressure which are the same for all the phases. The chemical potentials, or partial molar free energies ( 26c), of the various components in the P phases may be represented by Mua), M2(a ,. ./ cca) Mi(6), M2(W, , A c(ft) Mi(P), M2(P), A c(P). Suppose various small 

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Fig. 9.9. Cyclic voltammetry in the investigation of systems of more than one component, showing the importance of the inversion potential in the identification of the peaks on the inverse scan.

A solution is t3q>lcally a system of more than one component. In actual cases, there are at least two substances that can adsorb. For a binary fluid mixture, including dilute solutions, adsorption of one type of molecule (say A) involves replacement of the other (B). Thus, adsorption from solution is essentially an exchange process. If one molecule of A replaces r molecules of B at the Interface, the adsorption equilibrium can be written as... 

It is, however, only in the case of systems of more than one component that any difficulty will be found for only in this case will a choice of components be possible. Take, for example, the dissociation of calcium carbonate into calcium oxide and carbon dioxide. At each temperature, as we have seen, there is a definite state of equilibrium. When equilibrium has been established, there are three different substances present—calcium carbonate, calcium oxide, and carbon dioxide and these are the constituents of the system between which equilibrium exists. Although these constituents take part in the equilibrium, they are not all to be regarded as components, for they are not mutually independent. On the contrary, the different phases are related to one another, and if two of these are taken, the composition of the third is defined by the equation... 

This chapter applies equilibrium theory to a variety of chemical systems of more than one component. Two different approaches will be used as appropriate one based on the relation Ilf = 11 for transfer equilibrium, the other based on v//r/ = 0 or AT = Wiof for reaction equilibrium. 

From the nature of things this scheme embraces only the systems of more than one component. System 1 does not occur as a colloid system. In 2 c 1, p. 9, wc mentioned systems 5 and 6, in 2 c 2 system 9 but one can hardly classify the gels under system 8 because in a gel which is constructed of a three-dimensional network of solid particles with liquid enclosed inside it, the distinction between dispersion medium and dispersed phase falls to the ground because the two are interconnected. 

When the system contains more than one component it is important to be able to explore the distribution of the different components both locally and at long range. One way in which this can be achieved is to evaluate the distribution function for the different species. For example in a binary mixture of components A and B there are four radial distribution functions, g (r), g (r), g (r) and g (r) which are independent under certain conditions. More importantly they would, with the usual definition, be concentration dependent even in the absence of correlations between the particles. It is convenient to remove this concentration dependence by normalising the distribution function via the concentrations of the components [26]. Thus the radial distribution function of g (r) which gives the probability of finding a molecule of type B given one of type A at the origin is obtained from... 

It is quite evident that in a multicomponent system wherein more than one component exhibits weight variations and that too at different temperature regions, the composition of the original compound may be estimated as depicted in Figure 11.2. 

The intersection points between coexistence lines and of coexistence lines with the boundaries of the diagram are calculated in the same way as already described for simple diagrams. However, in systems of more than one type 1 component quadrupl intersections can occur at which for example HSOq, SOq2, Cu2 and Cu2S coexist. In such a case four values of x will be found to be equal at the intersection point showing that it is the... 

Let us look now at vapor-liquid systems with more than one component. A liquid stream at high temperature and pressure is flashed into a drum, i.e., its pressure is reduced as it flows through a restriction (valve) at the inlet of the drum. This sudden expansion is irreversible and occurs at constant enthalpy. If it were a reversible expansion, entropy (not enthalpy) would be conserved. If the drum pressure is lower than the bubblepoint pressure of the feed at the feed temperature, some of the liquid feed will vaporize. 

Thermodynamic modelling of solution phases lies at the core of the CALPHAD method. Only rarely do calculations involve purely stoichiometric compounds. The calculation of a complex system which may have literally 100 different stoichiometric substances usually has a phase such as the gas which is a mixture of many components, and in a complex metallic system with 10 or 11 alloying elements it is not unusual for all of the phases to involve solubility of the various elements. Solution phases will be defined here as any phase in which there is solubility of more than one component and within this chapter are broken down to four types (1) random substitutional, (2) sublattice, (3) ionic and (4) aqueous. Others types of solution phase, such as exist in polymers or complex organic systems, can also be modelled, but these four represent the major types which are currently available in CALPHAD software programmes. 

For a pure substance, the critical temperature may be defined as the temperature above which the gas cannot be liquefied, regardless of the pressure applied. Similarily, the critical pressure of a pure substance is defined as the pressure above which liquid and gas cannot coexist, regardless of the temperature. These definitions of critical properties are invalid for systems with more than one component. 

In the next two chapters, we use thermodynamic relationships summarized in Chapter 1 la to delve further into the world of phase equilibria, using examples to describe some interesting effects. As we do so, we must keep in mind that our discussion still describes only relatively simple systems, with a much broader world available to those who study such subjects as critical phenomena, ceramics, metal alloys, purification processes, and geologic systems. In this chapter, we will limit our discussion to phase equilibria of pure substances. In Chapter 14, we will expand the discussion to describe systems containing more than one component. 

Polymeric materials consisting of more than one component are produced in even larger quantities and their practical importance increases. These materials are usually stronger and/or tougher than one-component systems. In the field of metallurgy this fact has been known for centuries already metal alloys are often as old as metals themselves. For polymers the same empirical fact proved to be true. 

For systems of more than one phase, the summation must include all phases and all components. For example, the total free energy of a system of tt phases is... 

For systems of more than one chemical component, the concentrations of each are additional variables. A typical two-component phase diagram displays the variation of vapor pressure or melting point with composition at a fixed pressure, for example. 

All industrial liquid systems are made up of more than one component, which makes the studies of mixed liquid systems important. The analyses of surface tension of liquid mixtures (for example, two or three or more components) has been the subject of studies in many reports." According to Guggenheim s model of liquid surfaces, the free energy of the molecule is... 

The foregoing considerations become modified for certain systems, especially those of more than one component, such as alloys, and here a special kind of change, known as an order-disorder transition, can occur. The phenomena shown by the alloy of copper and zinc known as jS-brass will introduce us to a matter of some general importance. At low temperatures the alloy consists of a regular lattice of copper atoms, one at each corner of a series of cubes, and of a similar lattice of zinc atoms, so disposed that each cube of the copper lattice has a zinc atom at its centre and each cube of the zinc lattice a copper atom. The two interpenetrating simple cubic lattices thus give what is called a body-centred cubic lattice. 

We have so far described a statistical mechanics of molecular liquids, implying that a system includes only one chemical species. However, in ordinary chemistry, a system contains more than one component, and major and minor components in the mixture are conventionally called solvent and solute , respectively. The vanishing limit of solute concentration, or infinite dilution, is of particular interest because it purely reflects the nature of solute-solvent interactions. The word solvation is most commonly used for describing properties concerning solute-solvent interactions at the infinite dilution limit. Here, we provide a brief outline of the way to obtain solvation properties, solvation structure and thermodynamics, from the RISM theory described in the previous sections [3]. It is straightforward to generalize the RISM equation to a mixture of different molecular species. The equation for a mixture can be written in a matrix notation as... 

Thus a plot of In bi versus In bt yields a straight line of slope Xi/Xj. Similarly, in a system of more than three components, all roots X< ( = 1,2,... f,. .. — 2) can be determined relative to X, i. A matrix A proportional to A can be built and a matrix K proportional to K calculated. Finally, all rate constants relative to one of them taken as unity can be determined. 

Percolation in a solid made up of more than one component (a composite system), the volume fraction of the different components can be varied. Percolation refers to the transitions which occur when the volume fraction of a component is such that there are connected paths of that... 

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