Systems, mono variant

The equilibrium interfaces of fluid systems possess one variant chemical potential less than isolated bulk phases with the same number of components. This is due to the additional condition of heterogeneous equilibrium and follows from Gibbs phase rule. As a result, the equilibrium interface of a binary system is invariant at any given P and T, whereas the interface between the phases a and /3 of a ternary system is (mono-) variant. However, we will see later that for multiphase crystals with coherent boundaries, the situation is more complicated. 

For example, it may happen that with the same chemical substances different mono variant systems may be constituted to each one of these mono variant systems will correspond a curve of transformation tensions, but these curves will not be superposable. 

It is no longer the same if there is introduced into the system enough mercury so that a part of this substance remains in the liquid state the system, formed of two independent components, oxygen and mercury, and divided into three phases, red oxide of mercury, mixture of oxygen and mercury vapor, and liquid mercury, is a mono variant system it admits of a curve of transformation tensions C at each temperature T the curve C has a corresponding transformation tension P whose value is inde- pendent of the masses of mercury and oxygen which the system contains. 

These phenomena are very sharply produced when the temperature does not exceed a certain limit, near the fusing-point of gold beyond this limit the two oxides exist in contact as a fused mass instead of forming two solid phases, th form but one liquid phase from mono variant the system becomes bivariant at a given temperature there can be no longer any definite dissociation tension. 

Consider a mono variant system let M (Fig. 39) be a point of the transformation curve of this system d, P are abscissa and ordinate of this point. 

If one of the four phases is supposed to be excluded, a mono-variant system is obtained according as the excluded phase is one or another of the four possible phases, there may be formed four distinct monovariant systems. 

X34. The zeolites are solid solutions, 157.—X35. The existence of a dissociation tension does not always prove the existence of a definite compound. Dissociation of palladium hydride, 158.—X36. Robin s law, 169.—137. Moutier s law, 163.—X38. False equilibria in mono-variant systems, 164.—X39. Another form of Moutier s law, 165.—... 

Consider a system consisting of a number of phases made up of several different components, and suppose that the number of variables and conditions of constraint is such that the system has one degree of freedom. If we assign in addition a value to any one of the variables which characterize the state of the system (such as temperature, pressure, or the concentration of one of the components in one of the phases) the system will come to a perfectly definite state of equilibrium. Such an equilibrium is called a monovariant equilibrium. Rooseboom, to whom many important investigations on the phase rule and its applications are due, used the term complete equilibrium for an equihbrium of this kind. Nemst also adopts this terminology, although he raises objections to it, since mono variant equilibria are in no way more complete than nonvariant or multivariant equilibria. It would be more appropriate to use the term complete equilibria for nonvariant equilibria in which the number of phases is a maximum. (See Nernst, LeJirhuch, 6th ed. p. 473.)... 

A given mono variant closed system thus has the remarkable property that its state is completely determined when, in addition to one intensive variable, we know the total volume in particular the mass of each phase is determined. 

These two theorems are general and include as particular cases the theorems established in chap. XVIII, 6 and in chap. XXIII. They do not however apply to mono variant or invariant systems. Thus the eutectic point, which is certainly an indifferent point, does not represent, mathematically, an extreme value of or p for it is the point of intersection of two curves each of which refers to a two-phase system e.g. solution + ice or solution-f salt) under constant pressure. Only at the eutectic do three phases (solution + salt + ice) coexist. A mono variant three-phase system does not have an isobaric curve. 

Hence the system is mono variant. The solid phase consists only of... 

Let us consider a mono variant system having (f> phases. If, from this system, we remove one or more of these phases we are left with a system of (f>s phases (0g< ) which constitutes what we shall call a sub-system of the mono variant system being considered. The mono variant system itself we shall call the parent system. 

This follows immediately since all the projections of mono variant lines must, because of the theorem proved in 17, be tangents at the same point to the same lines of indifference, corresponding to the common sub-system. 

If the sub-system also is monovariant, then it is indifferent in all its states and the indifferent line of the sub-system is also its equilibrium line. It then follows from the theorem of 17 that the projection of the mono variant line of the parent system on the T,p) plane is coincident with the projection of the monovariant line of the sub-system. 

The Morey-Schreinemakers theorem when applied to this case may be written The monovariant curves of all mono variant systems which have a monovariant sub-system in common, are coincident when projected on the T, p) plane. 

We have already seen that all the states of a mono variant system are indifferent states. On the other hand, if a poly variant system is in an indifferent state, then its properties are in many ways analogous to those of mono variant systems. 

In the same way, for an indifferent state of a closed poly variant system, the temperature is sufficient to determine p and the composition of the phases, but not the masses of the individual phases. Furthermore, as we have seen, the law governing the variations hp and hT along an indifferent line, are of just the same form as the law which relates Sp and ST along the equilibrium states of a monovariant system. However, a profound difference is apparent between monovariant systems, and indifferent states of a pol3rvariant system when we consider the possibility of a closed system moving along the line of indifference. A closed mono variant system can clearly traverse its indifferent line, for this is simply its equilibrium line on the other hand, for a polyvariant closed system the ability to move along the indifferent line is exceptional as we shall now proceed to show. 

The Uquidus surface in the Fe-Cr-ZrCr2-ZrFc2 region has been reported by [1963Sve]. It looks incomplete because only one mono variant line (from the Fe rich to the Cr rich binary eutectics) is reported. It has been completed here and adapted to the accepted binary and quasibinary systems in order to be shown in Fig. 3. Notice however that it has to be considered poorly reliable. [1978Hao] confirmed that no ternary eutectic is present in the Fe rich comer of the phase diagram. 

N(Li = L2-G) and R (Li = G-L2) are the critical endpoints in binary subsystems LN (Li = L2-G-S), NN (Li = L2-G) and NR (Li = L2 = G) are the nonvariant critical points in ternary systems. One-dotted-dashed line is the ternary mono variant critical curve Li = L2-G, two-dotted-dashed line is the ternary mono variant critical curve Li = G-L2 dashed line is the approximate compositions of liquid phase in equilibria L1-L2-G-S and L-G-S. 

It should be recognised that appreciable shifts in properties are sometimes made possible by special compounding variations. For instance, the heat resistance of natural rubber vulcanisates may be improved considerably by variation of the vulcanising recipe. The normal sulfur vulcanisation system is capable of many variants which will govern the chemical nature of sulfur crosslinks, i.e., whether it is essentially a mono, di or polysulfide linkage. The nature of sulfur crosslinks can have considerable influence on the heat and chemical resistance of vulcanisates. 

There are many variants of analytical flow systems, e.g., segmented flow analysis, flow injection analysis, sequential injection analysis, multisyringe flow injection analysis, batch injection analysis, mono-segmented flow analysis, flow-batch analysis, multi-pumping flow analysis, all injection analysis and bead injection analysis, all of which have acronyms [176]. In view of the existence of several common features, however, all flow analysers can be broadly classified as either segmented or unsegmented, with the most common example of the later mode being the flow injection analyser. 

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