System divariant

If the micelles are regarded as a phase, then adding an excess solubilizate phase means there are three phases (the third is the intermicellar bulk phase). The total number of components is three (solvent, surfactant, and solubilizate), so the presence of three phases makes the system divariant. That would mean that surfactant concentration would be constant at constant temperature and pressure— but, in fact, the maximum additive concentration (MAC) changes with total surfactant concentration. Even if it were postulated that the increase in the MAC with surfactant concentration above the CMC is due to an increase in the total micellar phase, the concentration of solubilizate in the micellar phase should still remain constant, because the concentration is an intensive property of the system and is therefore homogeneous throughout the micellar phase. 

It should be noted that we have here considered the system at constant pressure. If we are not considering the system at isobaric conditions, the invariant equilibrium becomes univariant, and a univariant equilibrium becomes divariant, etc. A... 

For three-component (C = 3) or ternary systems the Gibbs phase rule reads Ph + F = C + 2 = 5. In the simplest case the components of the system are three elements, but a ternary system may for example also have three oxides or fluorides as components. As a rule of thumb the number of independent components in a system can be determined by the number of elements in the system. If the oxidation state of all elements are equal in all phases, the number of components is reduced by 1. The Gibbs phase rule implies that five phases will coexist in invariant phase equilibria, four in univariant and three in divariant phase equilibria. With only a single phase present F = 4, and the equilibrium state of a ternary system can only be represented graphically by reducing the number of intensive variables. 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C ... 

At 0 three phases exist in equilibrium, hence F = 0 or 0 is an invariant point. On elevation of the temperature the water phase disappears and the system becomes divariant. Along the line OA the surface film at first solid is observed to melt. No abrupt melting point can however be noted and no break is observed in the line OA. 

Lower a small crystal of Glauber salt into the solution. Explain what happens. Draw curves of the solubility of anhydrous sodium sulphate and its crystallohydrates. Acquaint yourself with the phase diagram of the sodium sulphate-water system. In what parts of the diagram is the system invariant, monovariant, or divariant ... 

Using published data, draw a phase diagram of the sodium sulphate-water system. Proceeding from the phase rule, determine which parts of the diagram will characterize invariant, monovariant, and divariant systems. What is meant by the transition point ... 

Consider a mixture of water and alcohol in the presence of the vapor. This system of two phases and two components is divariant (see Phase Rule). Now choose some fixed pressure and study the composition of the system at equilibrium as a function of temperature. The experimental results are shown schematically in Fig. I. 

Multivariant systems may also become indifferent under special conditions. In all considerations the systems are to be thought of as closed systems with known mole numbers of each component. We consider here only divariant systems of two components. The system is thus a two-phase system. The two Gibbs-Duhem equations applicable to such a system are... 

Other divariant systems composed of two phases and containing two components that become univariant and hence indifferent under special... 

Equation (5.78) is applicable to the systems that have been described in the immediate neighborhood of the indifferent state. It shows the divariant character of these systems. If the pressure is held constant, this equation becomes... 

The derivatives (dP/dT)S3t and (dxt/dT)sat may be determined experimentally or by solution of the set of Gibbs-Duhem equations applicable to each phase, provided we have sufficient knowledge of the system. If the system is multivariant, a sufficient number of intensive variables—the pressure or mole fractions of the components in one or more phases—must be held constant to make the system univariant. Thus, for a divariant system either the pressure or one mole fraction of one of the phases must be held constant. When the pressure is constant, Equation (9.9) becomes... 

Figure 8.1. Univariant curves for the dissociation of calcite, magnesite, and dolomite in system CaO-MgO-CC>2. Compatibility diagrams are shown for each divariant area. F=CC>2, C=calcite, D=dolomite, M=magnesite, L=lime, P=periclase. (After Harker and Tuttle, 1955.)...

Divariant relations between Cu2 S (s.s.) and MoS2 split the system into sulfur-rich and sulfur-poor portions. The X-phase in the latter region coexists stably with all other phases. At 700 °C (Fig. 18), the X-phase coexists with Cu2S however, below 685 5 °C tie lines have switched and metallic copper coexists with MoS2. 

C Since the X-phase is unstable, molybdenite (MoS2) coexists with all the phases of the system, and divariant relations exist in the diagram with MoS2 and Cu, MoS2 and Cu2 xS (s.s.), and with MoS2 and CuS. 

Thus, the considered system is divariant, and the solubility depends on two variables, temperature and pressure. 

The state of a system with two phases, and hence two degrees of freedom (divariant system), is not determined until two of its variables have been arbitrarily specified. The most important case is the coexistence of water vapour with a solution of calcium chloride. Here temperature and concentration determine the vapour pressure. Hence the vapour pressure of the solution at constant temperature is a function of its concentration. From this again it follows that the vapour pressure of a solution is always different from the vapour pressure of the. pure solvent at the same temperature. 

The differential heat of solution is of theoretical importance in the calculation of the variation of solubility with pressure. If a saturated solution containing solid solute is subjected to a pressure greater than the vapour pressure of the solution, the gaseous phase disappears and the system becomes divariant (two phases and two components). The concentration of the saturated solution (i.e. the solubility) is then a function of the pressure as well as of the temperature. When a condensed system of this kind is subjected to a further change in pressure the solid solute and the solution will not remain in equilibrium unless the temperature is changed simultaneously. As in the analogous case of the variation with pressure of the melting point of a pure substance (p. 221), the Clausius equation assumes the form — L/ ... 

Consider first the two phase system, solution + solid. Here we have c —2, (f> = 2, r =0 so that the phase rule gives w = 2, i.e. the system is divariant. We can therefore fix two of the intensive variables arbitrarily, for example the mole fraction in the solution and the pressure p. The temperature at which equilibrium is established is then a function of these two variables. For many purposes we may consider systems at constant pressure (for example atmospheric pressure). The equilibrium diagram can then be represented in two dimensions as in fig. 13.4 in which CE gives the equilibrium temperature T between solution and solid A as a function of the concentration of the solution Similarly DE corresponds to equilibrium between the solution P. W. Bridgman, J. Chem. Phys., 5, 964 (1937). 

SO that w — 2, and the system is divariant. We can therefore vary two intensive variables arbitrarily and study the variation of the other variables in terms of them. In this example we shall be interested in two cases, first that in which we fix the temperature and study the variation of pressure and vapour composition in terms of the mole fraction in the liquid, which we take as the other independent variable and secondly where the total pressure is fixed and the equilibrium temperature and vapour composition are given as functions of the liquid concentration. 

The line Kk corresponds to the three phase system solution + solid A + sohd B this is a divariant system but when p is fixed the representative points for the solution lie on a line. The three lines k K, k K and k K meet at K which is the ternary eutectic point at which the four phases, liquid, solid A, solid B and solid C are in equilibrium. This system is monovariant, but at a given pressure there is only one point representing this state, namely K. 

As an example of this behaviour we may take the divariant system composed of a mixture of carbon disulphide and benzene and their vapours. The total masses of the two components in the closed system are given initially. We know from the phase rule that if T and p are fixed the physico-chemical state of the system is determined, that is to say the mole fractions or weight fractions of the components in both liquid and vapour phase are determined. As we have seen these weight fractions are, in general, different. If we know these weight fractions and the initial masses of the components and m% we can calculate the masses of the two phases from the equations... 

The above property for the divariant system water + alcohol can be generalized for polyvariant systems. For any system which, for a certain set of values of the intensive variables T, p, w. .. w, leads to a set of c equations... 

Case 1 w — 2. All closed divariant systems of the second category can in general attain an indifferent state if one exists. 

In petrology, a buffered system usually refers to a system which is divariant, that is, which has the same number of components as phases, and which therefore has all its properties fixed at a given temperature and pressure. For example, a rock consisting of (pure) gypsum and anhydrite will have a fixed and invariable water activity at a given T and P. The relationship is... 

Degrees of freedom can also be described as the number of intensive variables that can be changed (within limits) without changing the number of phases in a system. This point of view is perhaps more useful to someone looking at a phase diagram thus divariant, univariant, and invariant systems correspond to areas, lines, and points in a P-T projection. We prefer however to emphasize the fact that coexisting phases reduee the number of independent variables, and that some systems have all their properties determined. This fact is very useful, as we will elaborate on below, and its explanation in terms of the Phase Rule is a very beautiful example of the interface between mathematics and physical reality. 

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