Indifferent line

If we express the state of a system in terms of intensive variables, then the representative points of a system in equilibrium form a continuum in w dimensions. The equilibrium indifferent states, or static indifferent states, will also be represented in this continuum but they must satisfy in addition, the further ( - 1) conditions of indifference (29.19). These ( - 1) relations among the w variables leave only one independent variable the static indifferent states of the system hence fall on a line, called the indifferent line. 

We note that in a non-reacting two-phase system, the indifferent line is simply the line of uniform composition. 

We may now proceed to calculate the equation of the indifferent line assuming the gas phase to be ideal. 

The indifferent line is thus given approximately by a simple equation analogous to the integrated form of the Clausius-Clapeyron equation. 

Comparison of (29.77) with (29.63) shows immediately the close similarity between the indifferent line of a polyvariant system, and the equilibrium line of a monovariant system. This similarity is perhaps less surprising when we remember that all equilibrium states of a monovariant system are indifferent states. Thus (29.63) can be regarded as a particular case of (29.77), for in this case, since w = l,... 

This indifferent state must lie on the indifferent line of the subsystem considered. 

If 8sP is the increase in pressure which accompanies an increase in temperature 8T in the sub-system along the indifferent line, then we have at the point considered (c/. 29.77)... 

When in moving along the monovariant line, a monovariant system of phases passes through a point at which (j)s of these phases form an indifferent subsystem, then the projection on the T, p) plane of the monovariant line, is at this point tangential to the projection of the indifferent line of the subsystem," ... 

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. 

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. 

We see, therefore, that for all systems in the first category, T and p may vary along the indifferent line, but the composition of the system remains constant. 

In systems of the second tyjpe, the composition varies in general from point to point along the indifferent line. 

In both cases, unless the equations exhibit a singularity, a closed system of the second kind cannot move along the indifferent line. 

Within certain limits of compositions e.g. if the composition along the indifferent line varies between certain limiting values only) and in the absence of singularities in the equations e.g. if the ratios of certain weight fractions remain constant along the indifferent line as in the example (c) of 7). 

In this case the composition corresponding to the indifferent state which is here the azeotropic composition, depends upon the temperature. Thus if certain values of mj and m are given, it is in general possible to find a temperature such that the azeotropic composition satisfies (29.121). A closed system of this kind, chosen at random, can in general reach an indifferent state if one exists, but it cannot move along the indifferent line, for the other values of the composition along the line cannot be reached from the initial state. 

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