State indifferent

The concept of indifferent states of systems is introduced in Section 5.11. There we define indifferent states as those states that required the assignment of a value to at least one extensive variable in addition to the number of moles of the components in order to define the state. Such systems are considered in more detail in this section. 

When we consider a one-component, two-phase system, of constant mass, we find similar relations. Such two-phase systems are those in which a solid-solid, solid-liquid, solid-vapor, or liquid-vapor equilibrium exists. These systems are all univariant. Thus, the temperature is a function of the pressure, or the pressure is a function of the temperature. As a specific example, consider a vapor-liquid equilibrium at some fixed temperature and in a state in which most of the material is in the liquid state and only an insignificant amount in the vapor state. The pressure is fixed, and thus the volume is fixed from a knowledge of an equation of state. If we now add heat to the system under the condition that the temperature (and hence the pressure) is kept constant, the liquid will evaporate but the volume must increase as the number of moles in the vapor phase increases. Similarly, if the volume is increased, heat must be added to the system in order to keep the temperature constant. The change of state that takes place is simply a transfer of matter from one phase to another under conditions of constant temperature and pressure. We also see that only one extensive variable—the entropy, the energy, or the volume—is necessary to define completely the state of the system. 

Univariant systems containing two components are exemplified by the equilibrium at a eutectic or peritectic. In each case a liquid phase is in equilibrium with two solid phases. Since such systems are univariant, the temperature is a function of the pressure, or the pressure is a function of 

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 

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States of closed systems for which it is necessary to assign values to at least one extensive variable in addition to the mole numbers are called indifferent states. Thus, all nonvariant and univariant states are indifferent states. These states are discussed in greater detail in Section 5.13. 

In every example that we have discussed, we have pointed out that it is possible to transfer matter at constant pressure and temperature between the phases present without causing any change in the composition of any phase. This observation permits another definition of an indifferent state of... 

In this discussion of indifferent states we have always used the entropy, energy, and volume as the possible extensive variables that must be used, in addition to the mole numbers of the components, to define the state of the system. The enthalpy or the Helmholtz energy may also be used to define the state of the system, but the Gibbs energy cannot. Each of the systems that we have considered has been a closed system in which it was possible to transfer matter between the phases at constant temperature and pressure. The differentials of the enthalpy and the Helmholtz and Gibbs energies under these conditions are... 

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... 

A very complete discussion of indifferent states of thermodynamic systems is given in Prigogine, I., and DeFay, R., op. cit., pp. 450-509. 

A similar remark may be made relative to the composition of each of the phases of a bivariant system in equilibrium at a given pressure and temperature it is a remark whose importance we shall see while studying in Chap. XI the indifferent states of a bivariant system. 

We must mention however, that in certain special cases—called indifferent states—the choice of variables T and p is not convenient in the sense that for these systems these two variables do not suffice to calculate all the others. We indicate below the way in which this state of affairs arises, but a detailed examination is deferred until chap. XXIX. 

The state of a system for which the intensive variables have values T,p,w. .. w" such that the equations (13.22) are indeterminate, is called an indifferent state. This name was introduced by Duhem.f The point M in fig. 13.12 represents a particular case of an indifferent state. A more extensive study of indifferent states will be given in chap. XXIX. 

The general methods used above may be extended without difficulty to more general problems than have been treated here. We shall use them for example in the study of indifferent states and azeotropy in chapters XXVIII and XXIX. 

Let us now consider a closed system which is at least bivariant. Then, apart from the case of indifferent states, we can, from Duhem s theorem (c/. chap. XIII, 6 and 7), describe all equilibrium states of the system in terms of two variables, T and p. We have... 

Azeotropic transformations in systems in which chemical reactions may take place in addition to transfers from one phase to another, or which have more than two phases, are not necessarily associated with states of uniform composition but with the more general class of indifferent states. A study of systems of uniform composition is a natural introduction to the more general question of indifferent states. 

In the preceding chapter we studied the properties of states of uniform composition in two-phase binary systems. As was demonstrated by the work of Duhem, Saurel and Jouguet, these states of uniform composition are but a special case of a much larger family of thermodynamic states indifferent states. We shall end this book therefore, with a chapter dealing with the properties of such states. 

For such a state to exist it is necessary that the equations (29.12) shall be compatible with the so-called conditions of enclosure (29.11). As we shall prove, this is not possible unless the variables in (29.9) have values satisfying certain conditions. If these conditions are satisfied the state defined by (29.9) is called an indifferent state. 

To say that (29.9) is an indifferent state is in effect to say that we can find a solution to this set of equations giving Am. ... .. such... 

Two corollaries follow from the above discussion. The first is that if the set of homogeneous equations has one non-zero solution, then it has an infinity of such solutions. All closed systems which are in an indifferent state, characterized by the weight fractions w. .., ... 

Hence for this system to be indifferent, it is necessary and sufficient that the two phases shall have the same composition. We thus find, as a special case of indifferent states, the states of uniform composition discussed in the preceding chapter. 

In the case of a solution in the presence of a solid phase consisting of crystals of an addition compound (cf. chap. XXIII, 2 and 3) it can be easily verified that the system is in an indifferent state if the composition of the solution is the same as that of the crystals. 

We now proceed to prove that any polyphase system containing two phases of the same composition is in an indifferent state. 

The matrix (29.19) has two identical columns and thus all determinants of order ( + r ) formed from this matrix must be zero hence the system is in an indifferent state. 

The inverse of this theorem is clearly not true, for in 5 we have already seen examples of indifferent systems in which no pair of phases has the same composition. In two-phase non-reacting systems all indifferent states are states of uniform composition, for in the case of only two phases (29.19) reduces to... 

We see that the concept of indifferent states is much more general than that of states of uniform composition. All states of uniform composition of two or more phases are indifferent, but by no means all indifferent states involve phases of the same composition. The properties discussed in the previous chapter in connection with states of uniform composition are nearly all a consequence of the fact that these states are indifferent states, and we shall find most of these properties exhibited in the more general case. 

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 see therefore, that if a system is in an indifferent state, the variables T and p are no longer sufficient to determine its state completely. 

The name indifferent state seems to have had its origin in the fact that certain equilibrium states, even though completely defined by the temperature, pressure and composition of each phase, are indifferent to the mass of the phases present. Monovariant and invariant systems, which we have already seen to be indifferent, exhibit in all their equilibrium states this indifference to the masses of the phases For example, in the very simplest case of a monovariant system, a pure liquid in the presence of its vapour, we find just these properties. For at constant temperature, and hence at constant pressure, a closed system containing a pure substance can have an infinite range of equilibrium states which depend simply on the volume in which the system is confined. These equilibrium states differ only in the amounts of the two phases. 

We met these theorems in studying the properties of states of uniform composition (chap. XVIII, 6) we shall now examine them in a more general form as a property of all static indifferent states in a poly variant system. 

If in any isothermal equilibrium displacement the system passes through an indifferent state, then the pressure passes through an extreme value. 

If, among the pressures which maintain the system in equilibrium at a constant temperature, there exists an extreme value of then the state corresponding to this extreme value is an indifferent state. 

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