After the discussion of the conditions which the substances chosen as components must satisfy, another method may be given by which the number of components present in a system can be determined. Suppose a system consisting of several phases in equilibrium, and 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 composition of a third phase may be obtained, the system is one of two components or of the second order; and if three phases are necessary to give the composition of a fourth coexisting phase, the system is one of three components, or of the third order.[16]
Although the examples to be considered in the sequel will afford sufficient illustration of the application of the rules given above, one case may perhaps be discussed to show the application of the method just given for determining the number of components.
Consider the system consisting of Glauber's salt in equilibrium with solution and vapour. If these three phases are analyzed, the composition of the solid will be expressed by Na2SO4, 10H2O; 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. By varying the amounts of two phases, however (e.g.Na2SO4, 10H2O and H2O), the composition of the third phase—the solution—can be obtained. The system is, therefore, one oftwo components.
But sodium sulphate can also exist in the anhydrous form and as the hydrate Na2SO4, 7H2O. In these cases there maybe chosen as components Na2SO4and H2O, and Na2SO4, 7H2O and H2O respectively. In both cases, therefore, there are two components. But the two systems (Na2SO4, 10H2O—H2O, and Na2SO4, 7H2O—H2O) can be regarded as special cases of the system Na2SO4—H2O, 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, thenumberof components is the same; but by choosing Na2SO4and H2O as components, the possible occurrence of negative quantities of components in expressing the composition of the phases is avoided; and, further, these components apply over a much larger range of experimental conditions. Again, therefore, we see that, although the number of the components of a system is definite, a certain amount of liberty is allowed in the choice of the substances; and we also see that the choice will be influenced by the conditions of experiment.
Summing up, now, we may say—
(1) The components are to be chosen from among the constituents which are present when the system is in a state of true equilibrium, and which take part in that equilibrium.
(2) As components are to be chosen thesmallest numberof such constituents necessary to express the composition of each phase participating in the equilibrium, zero and negative quantities of the components being permissible.
(3) In any given system thenumberof the components is definite, but may alter with alteration of the conditions of experiment. A certain freedom of choice, however, is allowed in the (qualitative, not quantitative) selection of the components, the choice being influenced by considerations of simplicity, suitability, or generality of application.[17]
Degree of Freedom. Variability of a System.—It is well known that in dealing with a certain mass of gas or vapour,e.g.water vapour, if only one of the independently variable factors—temperature, pressure, and concentration (or volume)—is fixed, the state of the gas or vapour is undefined; while occupying the same volume (the concentration, therefore, remainingunchanged), 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 given pressure the temperature and volume may vary. If, however, two of the factors are arbitrarily fixed, then the third factor can only have a certain definite value; at any given values of temperature and pressure a given mass of gas can occupy only a definite volume.
Suppose, however, that the system consists of water in contact with vapour. The condition of the system then becomes perfectly defined on arbitrarily giving 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 conversely, if a definite pressure is chosen, the temperature is also defined. Water and vapour can coexist under a given pressure only at a definite temperature.
Finally, let the water and vapour be cooled down until ice begins to separate out. So soon as the third phase, ice, appears, the state of the system as regards temperature and pressure of the vapour is perfectly defined, and none of the variables can be arbitrarily changed without causing the disappearance of one of the phases, ice, water, or vapour.
We see, therefore, that in the case of some systems two, in other cases, only one of the independent variables (temperature, pressure, concentration) can be altered without destroying the nature of the system; while in other systems, again, these variables have all fixed and definite values. We shall therefore define the number of degrees of freedom[18]of a system as thenumber 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; the system water—vapour as having only one; and the system ice—water—vapour as having no degrees of freedom. We may also speak of thevariabilityorvarianceof a system, and describe a system as being invariant, univariant, bivariant, multivariant,[19]according as the number of degrees of freedom is nought, one, two, or more than two.
A knowledge of its variability is, therefore, of essential importance in studying the condition and behaviour of a system, and it is the great merit of the Phase Rule thatthe state of a system is defined entirely by the relation existing between the number of the components and the phases present, 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 distinction need be drawn between the two classes, although it is nevertheless often convenient to make use of the terms, in spite of a certain amount of indefiniteness which attaches to them—an indefiniteness, indeed, which attaches equally to the terms "physical" and "chemical" process.[20]
The Phase Rule.—The Phase Rule of Gibbs, which defines the condition of equilibrium by the relation between the number of coexisting phases and the components, may be stated as follows: A system consisting of n components can exist inn+ 2 phases only when the temperature, pressure, and concentration have fixed and definite values; if there arencomponents inn+ 1 phases, equilibrium can exist while one of the factors varies, and if there are onlynphases, two of the varying factors may be arbitrarily fixed. This rule, the application of which, it is hoped, will become clear in the sequel, may be very concisely and conveniently summarized in the form of the equation—
P + F = C + 2, or F = C + 2 - P
where P denotes the number of the phases, F the degrees of freedom, and C the number of components. From the second form of the equation it can be readily seen that the greater the number of the phases, the fewer are the degrees of freedom. With increase in the number of the phases, therefore, thecondition of the system becomes more and more defined, or less and less variable.
Classification of Systems according to the Phase Rule.—We have already learned in the introductory chapter that systems which are apparently quite different in character may behave in a very similar manner. Thus it was stated that the laws which govern the equilibrium between water and its vapour are quite analogous to those which are obeyed by the dissociation of calcium carbonate into carbon dioxide and calcium oxide; in each case a certain temperature is associated with a definite pressure, no matter what the relative or absolute amounts of the respective substances are. And other examples were given of systems which were apparently similar in character, but which nevertheless behaved in a different manner. The relations between the various systems, however, become perfectly clear and intelligible in the light of the Phase Rule. In the case 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 therefore expect a similar behaviour in the case of all univariant systems, no matter how dissimilar the systems may outwardly appear. Similarly, all bivariant systems will exhibit analogous behaviour; and generally, systems possessing the same degree of freedom will show a like behaviour. In accordance with the Phase Rule, therefore, we may classify the different systems which may be found into invariant, univariant, bivariant, multivariant,according to the relation which obtains between the number of the components and the number of coexisting phases; and we shall expect that in each case the members of any particular group will exhibit a uniform behaviour. By this means we are enabled to obtain an insight into the general behaviour of any system, so soon as we have determined the number of the components and the number of the coexisting phases.
The adoption of the Phase Rule for the purposes of classification has been of great importance in studying changes in the equilibrium existing between different substances; for 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 given the clue to the conditions under which these substances can exist, and has led to the recognition of otherwise unobserved resemblances existing between different systems.
Deduction of the Phase Rule.—In the preceding pages we have restricted ourselves to the statement of the Phase Rule, without giving any indication of how it has been deduced. At the close of this chapter, therefore, the mathematical deduction of the generalization will be given, but in brief outline only, the reader being referred to works on Thermodynamics for a fuller treatment of the subject.[21]
All forms of energy can be resolved into two factors, thecapacityfactor and theintensityfactor; but for the production of equilibrium, only the intensity factor is of importance. Thus, if two bodies having the same temperature are brought 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 temperature of the two bodies is different,i.e.if the intensity factor of heat energy is different, the two bodies will no longer be in equilibrium; but heat will pass from the hotter to the colder until both have the same temperature.
As with heat energy, so with chemical energy. If we have a substance existing in two different states, or in two differentphases of a system, equilibrium can occur only when the intensity factor of chemical energy is the same. This intensity factor may be called thechemical potential; and we can therefore say that a system will be in equilibrium when the chemical potential of each component is the same in all the phases in which the component occurs. Thus, for example, ice, water, and vapour have, at the triple point, the same chemical potential.
The potential of a component in any phase depends not only on the composition of the phase, but also on the temperature and the pressure (or volume). If, therefore, we have a system of C components existing in P phases, then, in order to fix the 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, as regards composition, the whole system possesses P(C - 1) variables. Besides these there are, however, two other variables, viz. temperature and pressure, so that altogether a system of C components in P phases possesses P(C - 1) + 2 variables.
In order to define the state of the system completely, it will be necessary to have as many equations as there are variables. If, therefore, there are fewer equations than 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 assigned to these variables before the system is entirely defined. The number of these undefined values gives us the variability or the degree of freedom of the system.
The equations by which the system is to be defined are obtained from the relationship between the potential of a component and the composition of the phase, the temperature and the pressure. Further, as has already been stated, equilibrium occurs when the potential of each component 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 withit, the potential of each component must be the same as in the standard phase. For each phase in equilibrium with the standard phase, therefore, there will be a definite equation of state for each component in the phase; so that, if there are P phases, we obtain for each component (P - 1) equations; and for C components, therefore, we obtain C(P - 1) equations.
But we have seen above that there are P(C - 1) + 2 variables, and as we have only C(P - 1) equations, there must be P(C - 1) + 2 - C(P - 1) = C + 2 - P variables undefined. That is to say, the degree of freedom (F) of a system consisting of C components in P phases is—
F = C + 2 - P
TYPICAL SYSTEMS OF ONE COMPONENT
A.Water.
For the sake of rendering the Phase Rule more readily intelligible, and at the same time also for the purpose of obtaining examples by which we may illustrate the general behaviour of systems, we shall in this chapter examine in detail the behaviour of several well-known systems consisting of only one component.
The most familiar examples of equilibria in a one-component system are those furnished by the three phases of water, viz. ice, water, water vapour. The system consists of one component, because all three phases have the same chemical composition, represented by the formula H2O. As the criterion of equilibrium we shall choose a definite pressure, and shall study the variation of the pressure with the temperature; and for the purpose of representing the relationships which we obtain we shall employ a temperature-pressure diagram, in which the temperatures are measured as abscissæ and the pressures as ordinates. In such a diagram invariant systems will be represented by points; univariant systems by lines, and bivariant systems by areas.
Equilibrium between Liquid and Vapour. Vaporization Curve.—Consider in the first place the conditions for the coexistence of liquid and vapour. According to the Phase Rule (p.16), a system consisting of one component in two phases has one degree of freedom, or is univariant. We should therefore expect that it will be possible for liquid water to coexist with water vapour at different values of temperature andpressure, but that if we arbitrarily fix one of the variable factors, pressure, temperature, or volume (in the case of a given mass of substance), the state of the system will then be defined. If we fix, say, the temperature, then the pressure will have a definite value; or if we adopt a certain pressure, the liquid and vapour can coexist only at a certain definite temperature. Each temperature, therefore, will correspond to a definite pressure; and if in our diagram we join by a continuous line all the points indicating the values of the pressure corresponding to the different temperatures, we shall obtain a curve (Fig. 1) representing the variation of the pressure with the temperature. This is the curve of vapour pressure, or thevaporization curveof water.
Fig. 1Fig. 1.
Now, the results of experiment are quite in agreement with the requirements of the Phase Rule, and at any given temperature the system water—vapour can exist in equilibrium only under a definite pressure.
The vapour pressure of water at different temperatures has been subjected to careful measurement by Magnus,[22]Regnault,[23]Ramsay and Young,[24]Juhlin,[25]Thiesen and Scheel,[26]and others. In the following table the values of the vapour pressure from -10° to +100° are those calculated from the measurements of Regnault, corrected by the measurements of Wiebe and Thiesen and Scheel;[27]those from 120° to 270° were determinedby Ramsay and Young, while the values of the critical pressure and temperature are those determined by Battelli.[28]
Vapour Pressure of Water.
The pressure is, of course, independent of the relative or absolute volumes of the liquid and vapour; on increasing the volume at constant temperature, a certain amount of the liquid will pass into vapour, and the pressure will regain its former value. If, however, the pressure be permanently maintained at a value different from that corresponding to the temperature employed, then either all the liquid will pass into vapour, or all the vapour will pass into liquid, and we shall have either vapour alone or liquid alone.
Upper Limit of Vaporization Curve.—On continuing to add heat to water contained in a closed vessel, the pressure of the vapour will gradually increase. Since with increase of pressure the density of the vapour must increase, and since with rise of temperature the density of the liquid must decrease, a point will be reached at which the density of liquid and vapour become identical; the system ceases to be heterogeneous, and passes into one homogeneous phase. The temperature at which this occurs is called thecritical temperature. To this temperature there will, of course, correspond a certain definite pressure, called thecritical pressure. The curve representing theequilibrium between liquid and vapour must, therefore, end abruptly at the critical point. At temperatures above this point no pressure, however great, can cause the formation of the liquid phase; at temperatures above the critical point the vapour becomes a gas. In the case of water, the critical temperature is 364.3°, and the critical pressure 194.6 atm.; at the point representing these conditions the vapour-pressure curve of water must cease.
Sublimation Curve of Ice.—Vapour is given off not only by liquid water, but also by solid water, or ice. That this is so is familiar to every one through the fact that ice or snow, even at temperatures below the melting point, gradually disappears in the form of vapour. Even at temperatures considerably lower than 0°, the vapour pressure of ice, although small, is quite appreciable; and it is possible, therefore, to have ice and vapour coexisting in equilibrium. When we inquire into the conditions under which such a system can exist, we see again that we are dealing with a univariant system—one component existing in two phases—and that, therefore, just as in the case of the system water and vapour, there will be for each temperature a certain definite pressure of the vapour, and this pressure will be independent of the relative or absolute amounts of the solid or vapour present, and will depend solely on the temperature. Further, just as in the case of the vapour pressure of water, the condition of equilibrium between ice and water vapour will be represented by a line or curve showing the change of pressure with the temperature. Such a curve, representing the conditions of equilibrium between a solid and its vapour, is called asublimation curve. At temperatures represented by any point on this curve, the solid (ice) will sublime or pass into vapour without previously fusing. Since ice melts at 0° (vide infra), the sublimation curve must end at that temperature.
The following are the values of the vapour pressure of ice between 0° and -50°.[30]
Vapour Pressure of Ice.
Equilibrium between Ice and Water. Curve of Fusion.—There is still another univariant system of the one component water, the existence of which, at definite values of temperature and pressure, the Phase Rule allows us to predict. This is the system solid—liquid. Ice on being heated to a certain temperature melts and passes into the liquid state; and since this system solid—liquid is univariant, there will be for each temperature a certain definite pressure at which ice and water can coexist or be in equilibrium, independently of the amounts of the two phases present. Since now the temperature at which the solid phase is in equilibrium with the liquid phase is known as the melting point or point of fusion of the solid, the curve representing the temperatures and pressures at which the solid and liquid are in equilibrium will represent the change of the melting point with the pressure. Such a curve is called thecurve of fusion, or the melting-point curve.
It was not until the middle of the nineteenth century that this connection between the pressure and the melting point, or the change of the melting point with the pressure, was observed. The first to recognize the existence of such a relationship was James Thomson,[31]who in 1849 showed that from theoretical considerations such a relationship must exist, and predicted that in the case of ice the melting point would be lowered by pressure. This prediction was fully confirmed by his brother, W. Thomson[32](Lord Kelvin), who found that under a pressureof 8.1 atm. the melting point of ice was -0.059°; under a pressure of 16.8 atm. the melting point was -0.129°.
The experiments which were first made in this connection were more of a qualitative nature, but in recent years careful measurements of the influence of pressure on the melting point of ice have been made more especially by Tammann,[33]and the results obtained by him are given in the following table and represented graphically in Fig. 2.
Fusion Pressure of Ice.
From the numbers in the table and from the figure we see that as the pressure is increased the melting point of ice is lowered; but we also observe that a very large change of pressure is required in order to produce a very small change in the melting point. The curve, therefore, is very steep. Increase of pressure by one atmosphere lowers the melting point by only 0.0076°,[35]or an increase of pressure of 135 atm. is required to produce a lowering of the melting point of 1°. We see further that the fusion curve bends slightly as the pressure is increased, which signifies that the variation ofthe melting point with the pressure changes; at -15°, when the pressure is 1625 kilogm. per sq. cm., increase of pressure by 1 kilogm. per sq. cm. lowers the melting point by 0.012°. This curvature of the fusion curve we shall later (Chap. IV.) see to be an almost universal phenomenon.
Fig. 2Fig. 2.
Fig. 3Fig. 3.
Equilibrium between Ice, Water, and Vapour. The Triple Point.—On examining the vapour-pressure curves of ice and water (Fig. 3), we see that at a temperature of about 0° and under a pressure of about 4.6 mm. mercury, the two curves cut. At this point liquid water and solid ice are each in equilibrium with vapour at the same pressure. Since this is so, they must, of course, be in equilibriumwith one another, as experiment also shows. At this point, therefore, ice, water, and vapour can be in equilibrium, and as there are three phases present, the point is called atriple point.[36]
The triple point, however, does not lie exactly at 0° C., for this temperature is defined as the melting point of ice under atmospheric pressure. At the triple point, however, the pressure is equal to the vapour pressure of ice and water, and this pressure, as we see from the tables on pp. 21 and 23, is very nearly 4.6 mm., or almost 1 atm. less than in the previous case. Now, we have just seen that a change of pressure of 1 atm. corresponds to a change of the melting point of 0.0076°; the melting point of ice, therefore, when under the pressure of its own vapour, will be very nearly +0.0076°, and the pressure of the vapour will be very slightly greater than 4.579 mm., which is the pressure at 0° (p.21). The difference is, however, slight, and may be neglected here. At the temperature, then, of +0.0076°, and under a pressure of 4.6 mm. of mercury, ice, water, and vapour will be in equilibrium; the point in our diagram representing this particular temperature and pressure is, therefore, the triple point of the system ice—water—vapour.
Since at the triple point we have three phases of one component, the system at this point is invariant—it possesses no degrees of freedom. If the temperature is changed, the system will undergo alteration in such a way that one of the phases will disappear, and a univariant system will result; if heat be added, ice will melt, and we shall have left water and vapour; if heat be abstracted, water will freeze, and we shall have left ice and vapour; if, when the temperature is altered, the pressure is kept constant, then we shall ultimately obtain only one phase (see Chap. IV.).
The triple point is not only the point of intersection of the vaporization and sublimation curves, but it is also the end-point of the fusion curve. The fusion curve, as we have seen, is the curve of equilibrium between ice and water; and since at the triple point ice and water are each in equilibrium withvapour of the same pressure, they must, of course, also be in equilibrium with one another.
Fig. 4Fig. 4.
Bivariant Systems of Water.—If we examine Fig. 4, we see that the curves OA, OB, OC, which represent diagrammatically the conditions under which water and vapour, ice and vapour, and water and ice are in equilibrium, form the boundaries of three "fields," or areas, I., II., III. These areas, now, represent the conditions for the existence of the single phases, solid, liquid, and vapour respectively. At temperatures and pressures represented by any point in the field I., solid only can exist as a stable phase. Since we have here one component in only one phase, the system is bivariant, and at any given temperature, therefore, ice can exist under a series of pressures; and under any given pressure, at a series of temperatures, these pressures and temperatures being limited only by the curves OB, OC. Similarly also with the areas II. and III.
We see, further, that the different areas are the regions of stability of the phase common to the two curves by which the area is enclosed.[37]Thus, the phase common to the two systemsrepresented by BO (ice and vapour), and OA (water and vapour) is the vapour phase; and the area BOA is therefore the area of the vapour phase. Similarly, BOC is the area of the ice phase, and COA the area of the water phase.
Supercooled Water. Metastable State.—When heated under the ordinary atmospheric pressure, ice melts when the temperature reaches 0°, and it has so far not been found possible to raise the temperature of ice above this point without liquefaction taking place. On the other hand, it has long been known that water can be cooled below zero without solidification occurring. This was first discovered in 1724 by Fahrenheit,[38]who found that water could be exposed to a temperature of -9.4° without solidifying; so soon, however, as a small particle of ice was brought in contact with the water, crystallization commenced. Superfused or supercooled water—i.e.water cooled below 0°—is unstable only in respect of the solid phase; so long as the presence of the solid phase is carefully avoided, the water can be kept for any length of time without solidifying, and the system supercooled water and vapour behaves in every way like a stable system. A system, now, which in itself is stable, and which becomes instable only in contact with a particular phase, is said to bemetastable, and the region throughout which this condition exists is called the metastable region. Supercooled water, therefore, is in a metastable condition. If the supercooling be carried below a certain temperature, solidification takes place spontaneously without the addition of the solid phase; the system then ceases to be metastable, and becomesinstable.
Not only has water been cooled to temperatures considerably below the melting point of ice, but the vapour pressure of the supercooled water has been measured. It is of interest and importance, now, to see what relationship exists between the vapour pressure of ice and that of supercooled water at the same temperature. This relationship is clearly shown by the numbers in the following table,[39]and is represented in Fig. 3,p.27., and diagrammatically in Fig. 4, the vapour pressures of supercooled water being represented by the curve OA′, which is the unbroken continuation of AO.
Vapour Pressure of Ice and of Supercooled Water.
At all temperatures below 0° (more correctly +0.0076°), at which temperature water and ice have the same vapour pressure, the vapour pressure of supercooled water isgreaterthan that of ice at the same temperature.
From the relative positions of the curves OB and OA (Fig. 4) we see that at all temperatures above 0°, the (metastable) sublimation curve of ice, if it could be obtained, would be higher than the vaporization curve of water. This shows, therefore, that at 0° a "break" must occur in the curve of states, and that in the neighbourhood of this break the curve above that point must ascend less rapidly than the curve below the break. Since, however, the differences in the vapour pressures of supercooled water and of ice are very small, the change in the direction of the vapour-pressure curve on passing from ice to water was at first not observed, and Regnault regarded the sublimation curve as passing continuously intothe vaporization curve. The existence of a break was, however, shown by James Thomson[41]and by Kirchhoff[42]to be demanded by thermo-dynamical considerations, and the prediction of theory was afterwards realized experimentally by Ramsay and Young in their determinations of the vapour pressure of water and ice, as well as in the case of other substances.[43]
From what has just been said, we can readily understand why ice and water cannot exist in equilibrium below 0°. For, suppose we have ice and water in the same closed space, but not in contact with one another, then since the vapour pressure of the supercooled water is higher than that of ice, the vapour of the former must be supersaturated in contact with the latter; vapour must, therefore, condense on the ice; and in this way there will be a slow distillation from the water to the ice, until at last all the water will have disappeared, and only ice and vapour remain.[44]
Other Systems of the Substance Water.—We have thus far discussed only those systems which are constituted by the three phases—ice, water, and water vapour. It has, however, been recently found that at a low temperature and under a high pressure ordinary ice can pass into two other crystalline varieties, called by Tammann[45]ice II. and ice III., ordinary ice being ice I. According to the Phase Rule, now, since each of these solid forms constitutes a separate phase (p.9), it will be possible to have the following (and more) systems of water, in addition to those already studied, viz. water, ice I., ice II.; water, ice I., ice III.; water, ice II., ice III., forming invariant systems and existing in equilibrium only at a definite triple point; further, water, ice II.; water, ice III.; ice I., ice II.; ice I., ice III.; ice II., ice III., forming univariant systems, existing, therefore, at definite corresponding values oftemperature and pressure; and lastly, the bivariant systems, ice II. and ice III. Several of these systems have been investigated by Tammann. The triple point for water, ice I., ice III., lies at -22°, and a pressure of 2200 kilogms. per sq. cm. (2130 atm.), as indicated in Fig. 2,p.27.[46]In contrast with the behaviour of ordinary ice, the temperature of equilibrium in the case of water—ice II., and water—ice III., israisedby increase of pressure.
B.Sulphur.
Polymorphism.—Reference has just been made to the fact that ice can exist not only in the ordinary form, but in at least two other crystalline varieties. This phenomenon, the existence of a substance in two or more different crystalline forms, is calledpolymorphism. Polymorphism was first observed by Mitscherlich[47]in the case of sodium phosphate, and later in the case of sulphur. To these two cases others were soon added, at first of inorganic, and later of organic substances, so that polymorphism is now recognized as of very frequent occurrence indeed.[48]These various forms of a substance differ not only in crystalline shape, but also in melting point, specific gravity, and other physical properties. In the liquid state, however, the differences do not exist.
According to our definition of phases (p.9), each of these polymorphic forms constitutes a separate phase of the particular substance. As is readily apparent, the number of possible systems formed of one component may be considerably increased when that component is capable of existing in different crystalline forms. We have, therefore, to inquire what are the conditions under which different polymorphic forms can coexist, either alone or in presence of the liquid and vapour phase. For the purpose of illustrating the general behaviour of such systems, we shall study the systems formed by the different crystalline forms of sulphur, tin, and benzophenone.
Sulphurexists in two well-known crystalline forms—rhombic, or octahedral, and monoclinic, or prismatic sulphur. Of these, the former melts at 114.5°; the latter at 120°.[49]Further, at the ordinary temperature, rhombic sulphur can exist unchanged, whereas, on being heated to temperatures somewhat below the melting point, it passes into the prismatic variety. On the other hand, at temperatures above 96°, prismatic sulphur can remain unchanged, whereas at the ordinary temperature it passes slowly into the rhombic form.
If, now, we examine the case of sulphur with the help of the Phase Rule, we see that the following systems are theoretically possible:—