The Phase Rule and Its Applications

ury (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 l

e by which the condition of equilibrium of a system can be tested, and as a

cal 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

neral manner, free from all hypothetical assumptions as to the molecular condition of the participating substances, all cases of equilibrium could be surveyed and gr

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 t

, 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

ons. 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 gener

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

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

et it must be emphasized that the Phase Rule is concerned merely with those constituents which take part in the state of real equil

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 com

e possible. In this case hydrogen and oxygen will be components, because now they do take part in the equilibrium; also, they need no longer be present in definite proportions, but excess of one or the other may be added. Of course, if the restriction be arbitrarily made that the free hydrogen and oxygen shall be

s we have seen, there is a definite state of equilibrium. When equilibrium has been established, there are three different substances present-calcium carbonate, calcium oxide, and carbon dioxide; and these are the constituents of the system between which equilibrium exists. Now, although

= Ca

striction is imposed, and we obtain the following rule: As the components of a system there are to be chosen the smallest number of independently variable

brium, only two, as already stated, are independently variable. It will further be seen that in order to express the composition of each

two of the constituents can be selected. Thus, if we choose as components CaCO3 a

= CaCO

CaO +

CaCO

uld be obtained if CaCO3 and CO2 were chosen as components. The matter can, however, be simplified and the use of neg

way that the composition of each phase can be expressed by positive quantities of these, such a choice

ll be seen that the arbitrariness affects only the nature, not the number, of the components; a choice could be made with respect to which, not to how many, co

d the composition of each phase determined by analysis. If each phase present, regarded as a whole, has the same composition, the system contains only one component, or is of the first order. If two phases must be mixed in suitable quantities in order that the

on of the application of the rules given above, one case may perhaps be discussed to s

H2O; that of the solution by Na2SO4 + xH2O, while the vapour phase will be H2O. The system evidently cannot be a one-component system, for the phases have not all the same composition. B

and these two components will apply to all systems made up of sodium sulphate and water, no matter whether the solid phase is anhydrous salt or one of the hydrates. In all three cases, of course, the number of components is the same; but by choosing Na2SO4 and H2O as components, the possible occurrence of negative quantities of components in expressing the composit

p, now, w

tuents which are present when the system is in a state of

necessary to express the composition of each phase participating in the eq

experiment. A certain freedom of choice, however, is allowed in the (qualitative, not quantitative) selection of the

r vapour is undefined; while occupying the same volume (the concentration, therefore, remaining unchanged), the temperature and the pressure may be altered; at a given temperature, a gas can exist under different pressures and occupy different volumes, and under any gi

one of the variables a certain value. If the temperature is fixed, the pressure under which water and water vapour can coexist is also determined; and conv

ppears, the state of the system as regards temperature and pressure of the vapour is perfectly defined, and none of t

rees of freedom[18] of a system as the number of the variable factors, temperature, pressure, and concentration of the components, which must be arbitrarily fixed in order that the condition of the system may be perfectly defined. From what has been said, therefore, we shall describe a gas or vapour as having two degrees of freedom; t

esent, no account being taken of the molecular complexity of the participating substances, nor any assumption made with regard to the constitution of matter. It is, further, as we see, quite immaterial whether we are dealing with "physical" or "chemical" equilibrium; in principle, indeed, no

exist in n + 2 phases only when the temperature, pressure, and concentration have fixed and definite values; if there are n components in n + 1 phases, equilibrium can exist while one of the factors varies, and if there are on

+ 2, or F

e equation it can be readily seen that the greater the number of the phases, the fewer are the degrees of freedom. With increa

first mentioned, that of water in equilibrium with its vapour, we have one component-water-present in two phases, i.e. in two physically distinct forms, viz. liquid and vapour. According to the Phase Rule, therefore, since C = 1, and P = 2, the degree of freedom F is equal to 1 + 2 - 2 = 1; the system possesses one degree of freedom, as has already been stated. But in the case of the second system mentioned above there are two components, viz. calcium oxide and carbon dioxide (p. 12), and three phases, viz. two solid phases, CaO and CaCO3, and the gaseous phase, CO2. The number of degrees of freedom of the system, therefore, is 2 + 2 - 3 = 1; this system, therefore, also possesses one degree of freedom. We can now understand why these two systems behave in a similar manner; both are univariant, or possess only one degree of freedom. We shall th

r not only does it render possible the grouping together of a large number of isolated phenomena, but the guidance it affords has led to the discovery of new substances, has g

y indication of how it has been deduced. At the close of this chapter, therefore, the mathematical deduction of the generalization wil

rought in contact with each other, they will be in equilibrium as regards heat energy, no matter what may be the amounts of heat (capacity factor) contained in either, because the intensity factor-the temperature-is the same. But if the

sity factor of chemical energy is the same. This intensity factor may be called the chemical potential; and we can therefore say that a system will be in equilibrium when the chemical

e composition of unit mass of each phase, it is necessary to know the masses of (C - 1) components in each of the phases. As regards the composition, therefore, each phase possesses (C - 1) variables. Since there are P phases, it follows that,

han there are variables, then, according to the deficiency in the number of the equations, one or more of the variables will have an undefined value; and values must be

ent is the same in the different phases in which it is present. If, therefore, we choose as standard one of the phases in which all the components occur, then in any other phase in equilibrium with it, the potential of each component must be the same as in the standard phase. For

) equations, there must be P(C - 1) + 2 - C(P - 1) = C + 2 - P variables undefined. That

C +