Trivariant systems

Multivariant systems. Trivariant systems, page 118.— xoz. Theory of double salts, 118.—10a. Surface of solubility of a double salt at a given pressure, 119.—103, Case in which the solution may furnish two distinct salts, 120.—Z04, Conditions in which the two precipitates are simultaneously in equilibrium with the solution,... 

By the variance, or number of degrees of freedom of the system, we mean the number of independent variables which must be arbitrarily fixed before the state of equilibrium is completely determined. According to the number of these, we have avariant, univariant, bivariant, trivariant,. . . systems. Thus, a completely heterogeneous system is univariant, because its equilibrium is completely specified by fixing a single variable— the temperature. But a salt solution requires two variables— temperature and composition—to be fixed before the equilibrium is determined, since the vapour-pressure depends on both. 

System having degrees of freedom three, two, one or zero are known as trivariant, bivariant, univariant (or monovariant) and non-variant systems, respectively. 

Fig. 24. Schematic phase relations of the Fe-Cu-Mo-S system at 700 °C. Note Between the two ternary compounds (X-phase and Y-phase) occurs a complete quaternary solid solution series (indicated simply as a solid bar). All trivariant regions are specified in the text thus the bivariant phase assemblages and the univariant volumes can be deduced...

When the number of independent components exceeds by unity the number of phases into which the sj stem is divided (c= +l), the variance is equal to 3 the system is Trivariant. 

In order to know completely the composition of the phases into which a trivariant system in equilibrium is divided, it does not suflSce to know the temperature and the pressure it is necessary to add a third quantity. ... 

Let C, C, C".. . be the precipitates that may be observed. Under the given pressure ic the states of equilibrium between the liquid mixture of the three independent components and the single solid precipitate C, states in which the system is trivariant, are represented by the various points on a limited surface S that is called the domain of the precipUate C. 

In general a solution formed of four independent components may precipitate various solids of definite composition, as C, C, . .. By a method similar to that used for trivariant systems, we may reach the following conclusions ... 

If the representative point is on the boundary line of the two surfaces /S, /S of the two substances C, C, the three coordinates of this point represent the three concentrations of a solution which may, under atmospheric pressure and at 15 C., remain in equilibrium in contact with a solid precipitate composed of C and C in such a state of equilibrium one system of four independent components is divided into three phases, so that it is no longer quadrivariant but trivariant. 

We have in fact here a system formed of three independent components water 0 and the two salts 1 and 2 this system is divided into two phases, the liquid solution for which we shall continue to indicate by S, S2, the two concentrations, and the mixed crystals C this system is therefore trivariant when the temperature and pressure only are given, the composition of each of the two phases capable of remaining in equilibrium in contact with each other is not completely determined it becomes entirely determined if to the temperature and pressure there is added another ven quantity, for example, one of the concentrations s of the solution. 

Formation in solution of a racemic compound.—The precipitation within a solution of one of the substances we have just studied leads to the study of the equilibrium of a system no longer bivariant, but trivariant this study is, from the experimental point of view, much less advanced than the preceding it has given rise nevertheless to several interesting researches among this number is the analysis of the conditions of formation of the double racemate of sodium and ammonium, anal3rsis for which we are indebted to Van t Hoff and van Deventer. ... 

The classification, borrowed from the phase rule, of systems into numovariantf hivariard, trivariant, etc., is therefore of extreme utility it arranges in admirable order a great number of questions in the discussion of chemical equilibria it is, however, neither the... 

All points on the diagram lying above the curve CED represent conditions under which the solution can exist alone at the given pressure p the one phase system is a trivariant system and all the variables T, p and can be varied arbitrarily. 

Let us consider a solution of three substances A, B and C which do not react together, and suppose the solution is in equilibrium with one of the solid components. Then c = 3, r = 0, = 2 whence — 3 and the system is trivariant. We can thus consider the pressure and composition of the solution (p, Xj, x ) and see how the equilibrium temperature changes with these variables. Let us take the pressure as constant,... 

Example Consider a two-phase system of three non-reacting components. We have r =0, r" — 3, so that r = 3 it will therefore be trivariant w = 3. 

Note what we have gained because the Hamiltonian operator determines the wave function for the system from the variational principle, it follows that any property, Q, of the ground state of an electronic system may be written as a function of the number of electrons, N, and a functional of a real-valued trivariate function, v(r), which we call the external potential. We denote this functional Q[v(r) IV]. Unfortunately, no expression for Q[v(r) IV] with computational utility comparable to Eq. (1) is known. However, the fact that properties of a system can be expressed as a function of N and a functional of a single trivariate function, v(f), does suggest that there might be a computational useful theory in terms of a trivariate function. 

A great deal of the power and usefulness of the phase rule in geochemistry comes from its demonstration of which systems are divariant, and which therefore have all their properties fixed at a given T and P. Changing the concentration of any component of a trivariant or multivariant solution will change all the properties of the solution, even if T and P do not change. However, consider a divariant system at fixed values of T and P. 

A system with / = 1 is monovariant, with / = 2 it is bivariant, with / = 3 it is trivariant, etc. In the phase space, the boundary between two distinct phases is described by a line for a monovariant system, a plane for bivariant system, and a cube for a trivariant system. 

The systems with degrees of freedom of zero, one, two and three are referred to as the non-variant, monovariant, divariant and trivariant systems, respectively. For example, if we have a system with one species and treat only the pressure, / = 3—/), therefore the condition for coexistence of gas, liquid and solid phases is / = 0, which is referred to as the triple point. ... 

A brief comment is in order on the maximum additive concentration. The above discussion pertains when the solubilizate concentration is less than its solubility. When solubility is exceeded, an excess solubilizate phase appears in the system, which then has two phases and is trivariant. In this case, the system is fixed by specifying the total surfactant concentration at constant temperature and pressure. This prediction from the phase rule is supported by the observed changes of the phenothiazine MAC with the concentration of sodium dodecyl sulfate (Fig. 9.1). ... 

We deduce that

independent components, with at least a solid phase. There will be three variables to fix, for example, temperature, pressure, and the composition of one phase. The composition of each phase thus will be a function of these three variables. 

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