obtained, no matter whether we start with an unsaturated solution to which we add more solid, or with a supersaturated solution from which we allow solid to crystallize out; or, in the case of water in contact with vapour, the same vapour pressure will be obtained, no matter whether we heat the water up to the given temperature or cool it down from a higher temperature. In this case, water and vapour are in real equilibrium. On the other hand, water in contact with hydrogen and oxygen at the ordinary temperature is a case only of apparent equilibrium; on changing the pressure and temperature continuously within certain limits there is no continuous change observed in the relative amounts of the two gases. On heating beyond these limits there is a sudden and not a continuous change, and the system no longer regains its former condition on being cooled to the ordinary temperature. In all such cases the system may be regarded as undergoing change and as tending towards a state of true or real equilibrium, but with such slowness that no change is observed.
Although the case of water in contact with hydrogen and oxygen is an extreme one, it must be borne in mind that the condition of true equilibrium may not be reached instantaneously or even with measurable velocity, and in all cases it is necessary to be on one's guard against mistaking apparent (or false) for real (or true) equilibrium. The importance of this will be fully illustrated in the sequel.
CHAPTER II
THE PHASE RULE
Although the fact that chemical reactions do not take place completely in one direction, but proceed only to a certain point and there make a halt, was known in the last quarter of the eighteenth century (Wenzel, 1777; Berthollet, 1799); and although the opening and subsequent decades of the following century brought many further examples of such equilibria to our knowledge, it was not until the last quarter of the nineteenth century that a theorem, general in its application and with foundations weakened by no hypothetical assumptions as to the nature or constitution of matter, was put forward by Willard Gibbs;[5] a generalization which serves at once as a golden rule by which the condition of equilibrium of a system can be tested, and as a guide to the similarities and dissimilarities existing in different systems.
Before that time, certainly, attempts had been made to bring the different known cases of equilibria—chemical and physical—under general laws. From the very first, both Wenzel[6] and Berthollet[7] recognized the influence exercised by the mass of the substances on the equilibrium of the system. It was reserved, however, for Guldberg and Waage, by their more general statement and mathematical treatment of the Law of Mass Action,[8] to inaugurate the period of quantitative study of equilibria. The law which these investigators enunciated served satisfactorily to summarize the conditions of equilibrium in many cases both of homogeneous and, with the help of certain assumptions and additions, of heterogeneous equilibrium. By reason, however, of the fact that it was developed on the basis of the kinetic and molecular theories, and involved, therefore, certain hypothetical assumptions as to the nature and condition of the substances taking part in the equilibrium, the law of mass action failed, as it necessarily must, when applied to those systems in which neither the number of different molecular aggregates nor the degree of their molecular complexity was known.
Ten years after the law of mass action was propounded by Guldberg and Waage, Willard Gibbs,[9] Professor of Physics in Yale University, showed how, in a perfectly general manner, free from all hypothetical assumptions as to the molecular condition of the participating substances, all cases of equilibrium could be surveyed and grouped into classes, and how similarities in the behaviour of apparently different kinds of systems, and differences in apparently similar systems, could be explained.
As the basis of his theory of equilibria, Gibbs adopted the laws of thermodynamics,[10] a method of treatment which had first been employed by Horstmann.[11] In deducing the law of equilibrium, Gibbs regarded a system as possessing only three independently variable factors[12]—temperature, pressure, and the concentration of the components of the system—and he enunciated the general theorem now usually known as the Phase Rule, by which he defined the conditions of equilibrium as a relationship between the number of what are called the phases and the components of the system.
Phases.—Before proceeding farther we shall first consider what exactly is meant by the terms phase and component. We have already seen (p. 5) that a heterogeneous system is made up of different portions, each in itself homogeneous, but marked off in space and separated from the other portions by bounding surfaces. These homogeneous, physically distinct and mechanically separable portions are called phases. Thus ice, water, and vapour, are three phases of the same chemical substance—water. A phase, however, whilst it must be physically and chemically homogeneous, need not necessarily be chemically simple. Thus, a gaseous mixture or a solution may form a phase; but a heterogeneous mixture of solid substances constitutes as many phases as there are substances present. Thus when calcium carbonate dissociates under the influence of heat, calcium oxide and carbon dioxide are formed. There are then two solid phases present, viz. calcium carbonate and oxide, and one gas phase, carbon dioxide.
The number of phases which can exist side by side may vary greatly in different systems. In all cases, however, there can be but one gas or vapour phase on the account of the fact that all gases are miscible with one another in all proportions. In the case of liquid and solid phases the number is indefinite, since the above property does not apply to them. The number of phases which can be formed by any given substance or group of substances also differs greatly, and in general increases with the number of participating substances. Even in the case of a single substance, however, the number may be considerable; in the case of sulphur, for example, at least eight different solid phases are known (v. Chap. III.).
It is of importance to bear in mind that equilibrium is independent of the amounts of the phases present.[13] Thus it is a familiar fact that the pressure of a vapour in contact with a liquid (i.e. the pressure of the saturated vapour) is unaffected by the amounts, whether relative or absolute, of the liquid and vapour; also the amount of a substance dissolved by a liquid is independent of the amount of solid in contact with the solution. It is true that deviations from this general law occur when the amount of liquid or the size of the solid particles is reduced beyond a certain point,[14] owing to the influence of surface energy; but we have already (p. 5) excluded such cases from consideration.
Components.—Although the conception of phases is one which is readily understood, somewhat greater difficulty is experienced when we come to consider what is meant by the term component; for the components of a system are not synonymous with the chemical elements or compounds present, i.e. with the constituents of the system, although both elements and compounds may be components. By the latter term there are meant only those constituents the concentration of which can undergo independent variation in the different phases, and it is only with these that we are concerned here.[15]
To understand the meaning of this term we shall consider briefly some cases with which the reader will be familiar, and at the outset it must be emphasized that the Phase Rule is concerned merely with those constituents which take part in the state of real equilibrium (p. 5); for it is only to the final state, not to the processes by which that state is reached, that the Phase Rule applies.
Consider now the case of the system water—vapour or ice—water—vapour. The number of constituents taking part in the equilibrium here is only one, viz. the chemical substance, water. Hydrogen and oxygen, the constituents of water, are not to be regarded as components, because, in the first place, they are not present in the system in a state of real equilibrium (p. 6); in the second place, they are combined in definite proportions to form water,