CHAPTER III
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 and
pressure, 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 the vaporization curve of water.
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 determined
by Ramsay and Young, while the values of the critical pressure and temperature are those determined by Battelli.[[28]]
Vapour Pressure of Water.
| Temperature. | Pressure in cm. mercury. | Temperature. | Pressure in cm. mercury. |
| -10° | 0.213 | 120° | 148.4 |
| 0° | 0.458[[29]] | 130° | 201.9 |
| +20° | 1.752 | 150° | 356.8 |
| 40° | 5.516 | 200° | 1162.5 |
| 60° | 14.932 | 250° | 2973.4 |
| 80° | 35.54 | 270° | 4110.1 |
| 100° | 76.00 | 364.3° (critical temperature) | 14790.4 (194.6 atm.) (critical pressure). |
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 the critical temperature. To this temperature there will, of course, correspond a certain definite pressure, called the critical pressure. The curve representing the
equilibrium 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 a sublimation 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.
| Temperature. | Pressure in mm. mercury. | Temperature. | Pressure in mm. mercury. |
| -50° | 0.050 | -8° | 2.379 |
| -40° | 0.121 | -6° | 2.821 |
| -30° | 0.312 | -4° | 3.334 |
| -20° | 0.806 | -2° | 3.925 |
| -15° | 1.279 | 0° | 4.602 |
| -10° | 1.999 |
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 the curve 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 pressure
of 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.
| Temperature. | Pressure in kilogms. per sq. cm.[[34]] | Change of melting point for an increase of pressure of 1 kilogm. per sq. cm. |
| -0° -2.5° -5° -7.5° -10.0° -12.5° -15.0° -17.5° -20.0° -22.1° | 1 336 615 890 1155 1410 1625 1835 2042 2200 | 0.0074° 0.0090° 0.0091° 0.0094° 0.0100° 0.0116° 0.0119° 0.0121° 0.0133° |
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 of
the 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.
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 equilibrium
with 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 a triple 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 with
vapour of the same pressure, they must, of course, also be in equilibrium with one another.
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 systems
represented 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 be metastable, 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 becomes instable.
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.
| Temperature. | Pressure in mm. mercury. | ||
| Water. | Ice. | Difference. | |
| 0° | 4.618 | 4.602 | 0.016[[40]] |
| -2° | 3.995 | 3.925 | 0.070 |
| -4° | 3.450 | 3.334 | 0.116 |
| -8° | 2.558 | 2.379 | 0.179 |
| -10° | 2.197 | 1.999 | 0.198 |
| -15° | 1.492 | 1.279 | 0.213 |
| -20° | 1.005 | 0.806 | 0.199 |
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 is greater than 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 into
the 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 of
temperature 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., is raised by 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 called polymorphism. 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.
Sulphur exists 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:—
I. Bivariant Systems: One component in one phase.
(a) Rhombic sulphur.
(b) Monoclinic sulphur.
(c) Sulphur vapour.
(d) Liquid sulphur.
II. Univariant Systems: One component in two phases.
(a) Rhombic sulphur and vapour.
(b) Monoclinic sulphur and vapour.
(c) Rhombic sulphur and liquid.
(d) Monoclinic sulphur and liquid.
(e) Rhombic and monoclinic sulphur.
(f) Liquid and vapour.
III. Invariant Systems: One component in three phases.
(a) Rhombic and monoclinic sulphur and vapour.
(b) Rhombic sulphur, liquid and vapour.
(c) Monoclinic sulphur, liquid and vapour.
(d) Rhombic and monoclinic sulphur and liquid.
Triple Point—Rhombic and Monoclinic Sulphur and Vapour. Transition Point.—In the case of ice, water and vapour, we saw that at the triple point the vapour pressures of ice and water are equal; below this point, ice is stable; above this point, water is stable. We saw, further, that below 0° the vapour pressure of the stable system is lower than that of the metastable, and therefore that at the triple point there is a break in the vapour pressure curve of such a kind that above
the triple point the vapour-pressure curve ascends more slowly than below it. Now, although the vapour pressure of solid sulphur has not been determined, we can nevertheless consider that it does possess a certain, even if very small, vapour pressure,[[50]] and that at the temperature at which the vapour pressures of rhombic and monoclinic sulphur become equal, we can have these two solid forms existing in equilibrium with the vapour. Below that point only one form, that with the lower vapour pressure, will be stable; above that point only the other form will be stable. On passing through the triple point, therefore, there will be a change of the one form into the other. This point is represented in our diagram (Fig. 5) by the point O, the two curves AO and OB representing diagrammatically the vapour pressures of rhombic and monoclinic sulphur respectively. If the vapour phase is absent and the system maintained under a constant pressure, e.g.
atmospheric pressure, there will also be a definite temperature at which the two solid forms are in equilibrium, and on passing through which complete and reversible transformation of one form into the other occurs. This temperature, which refers to equilibrium in absence of the vapour phase, is known as the transition temperature or inversion temperature.
Were we dependent on measurements of pressure and temperature, the determination of the transition point might be a matter of great difficulty. When we consider, however, that the other physical properties of the solid phases, e.g. the density, undergo an abrupt change on passing through the transition point, owing to the transformation of one form into the other, then any method by which this abrupt change in the physical properties can be detected may be employed for determining the transition point. A considerable number of such methods have been devised, and a description of the most important of these is given in the Appendix.
In the case of sulphur, the transition point of rhombic into monoclinic sulphur was found by Reicher[[51]] to lie at 95.5°. Below this temperature the octahedral, above it the monoclinic, is the stable form.
Condensed Systems.—We have already seen that in the change of the melting point of water with the pressure, a very great increase of the latter was necessary in order to produce a comparatively small change in the temperature of equilibrium. This is a characteristic of all systems from which the vapour phase is absent, and which are composed only of solid and liquid phases. Such systems are called condensed systems,[[52]] and in determining the temperature of equilibrium of such systems, practically the same point will be obtained whether the measurements are carried out under atmospheric pressure or under the pressure of the vapour of the solid or liquid phases. The transition point, therefore, as determined in open vessels at atmospheric pressure, will differ only by a very slight amount from the triple point, or point at which the two solid or liquid phases are in equilibrium under the pressure of their vapour.
The determination of the transition point is thereby greatly simplified.
Suspended Transformation.—In many respects the transition point of two solid phases is analogous to the melting point of a solid, or point at which the solid passes into a liquid. In both cases the change of phase is associated with a definite temperature and pressure in such a way that below the point the one phase, above the point the other phase, is stable. The transition point, however, differs in so far from a point of fusion, that while it is possible to supercool a liquid, no definite case is known where the solid has been heated above the triple point without passing into the liquid state. Transformation, therefore, is suspended only on one side of the melting point. In the case of two solid phases, however, the transition point can be overstepped in both directions, so that each phase can be obtained in the metastable condition. In the case of supercooled water, further, we saw that the introduction of the stable, solid phase caused the speedy transformation of the metastable to the stable condition of equilibrium; but in the case of two solid phases the change from the metastable to the stable modification may occur with great slowness, even in presence of the stable form. This tardiness with which the stable condition of equilibrium is reached greatly increases in many cases the difficulty of accurately determining the transition point. The phenomena of suspended transformation will, however, receive a fuller discussion later (p. [68]).
Transition Curve—Rhombic and Monoclinic Sulphur.—Just as we found the melting point of ice to vary with the pressure, so also do we find that change of pressure causes an alteration in the transition point. In the case of the transition point of rhombic into monoclinic sulphur, increase of pressure by 1 atm. raises the transition point by 0.04°-0.05°.[[53]] The transition curve, or curve representing the change of the transition point with pressure, will therefore slope to the right away from the pressure axis. This is curve OC (Fig. 5).
Triple Point—Monoclinic Sulphur, Liquid, and Vapour. Melting Point of Monoclinic Sulphur.—Above 95.5°, monoclinic sulphur is, as we have seen, the stable form. On being heated to 120°, under atmospheric pressure, it melts. This temperature is, therefore, the point of equilibrium between monoclinic sulphur and liquid sulphur under atmospheric pressure. Since we are dealing with a condensed system, this temperature may be regarded as very nearly that at which the solid and liquid are in equilibrium with their vapour, i.e. the triple point, solid (monoclinic)—liquid—vapour. This point is represented in the diagram by B.
Triple Point—Rhombic and Monoclinic Sulphur and Liquid.—In contrast with that of ice, the fusion point of monoclinic sulphur is raised by increase of pressure, and the fusion curve, therefore, slopes to the right. The transition curve of rhombic and monoclinic sulphur, as we have seen, also slopes to the right, and more so than the fusion curve of monoclinic sulphur. There will, therefore, be a certain pressure and temperature at which the two curves will cut. This point lies at 151°, and a pressure of 1320 kilogm. per sq. cm., or about 1288 atm.[[54]] It, therefore, forms another triple point, the existence of which had been predicted by Roozeboom,[[55]] at which rhombic and monoclinic sulphur are in equilibrium with liquid sulphur. It is represented in our diagram by the point C. Beyond this point monoclinic sulphur ceases to exist in a stable condition. At temperatures and pressures above this triple point, rhombic sulphur will be the stable modification, and this fact is of mineralogical interest, because it explains the occurrence in nature of well-formed rhombic crystals. Under ordinary conditions, prismatic sulphur separates out on cooling fused sulphur, but at temperatures above 151° and under pressures greater than 1288 atm., the rhombic form would be produced.[[56]]
Triple Point—Rhombic Sulphur, Liquid, and Vapour. Metastable Triple Point.—On account of the slowness with
which transformation of one form into the other takes place on passing the transition point, it has been found possible to heat rhombic sulphur up to its melting point (114.5°). At this temperature, not only is rhombic sulphur in a metastable condition, but the liquid is also metastable, its vapour pressure being greater than that of solid monoclinic sulphur. This point is represented in our diagram by the point b.
From the relative positions of the metastable melting point of rhombic sulphur and the stable melting point of monoclinic sulphur at 120°, we see that, of the two forms, the metastable form has the lower melting point. This, of course, is valid only for the relative stability in the neighbourhood of the melting point; for we have already learned that at lower temperatures rhombic sulphur is the stable, monoclinic sulphur the metastable (or unstable) form.
Fusion Curve of Rhombic Sulphur.—Like any other melting point, that of rhombic sulphur will be displaced by increase of pressure; increase of pressure raises the melting point, and we can therefore obtain a metastable fusion curve representing the conditions under which rhombic sulphur is in equilibrium with liquid sulphur. This metastable fusion curve must pass through the triple point for rhombic sulphur—monoclinic sulphur—liquid sulphur, and on passing this point it becomes a stable fusion curve. The continuation of this curve, therefore, above 151° forms the stable fusion curve of rhombic sulphur (curve CD).
These curves have been investigated at high pressures by Tammann, and the results are represented according to scale in Fig. 6,[[57]] a being the curve for monoclinic sulphur and liquid; b, that for rhombic sulphur and liquid; and c, that for rhombic and monoclinic sulphur.
Bivariant Systems.—Just as in the case of the diagram of states of water, the areas in Fig. 5 represent the conditions for the stable existence of the single phases: rhombic sulphur in the area to the left of AOCD; monoclinic sulphur in the area OBC; liquid sulphur in the area EBCD; sulphur vapour below the curves AOBE. As can be seen from the diagram,
the existence of monoclinic sulphur is limited on all sides, its area being bounded by the curves OB, OC, BC. At any point outside this area, monoclinic sulphur can exist only in a metastable condition.
Other crystalline forms of sulphur have been obtained,[[58]] so that the existence of other systems of the one-component sulphur besides those already described is possible. Reference will be made to these later (p. [51]).
C. Tin.
Another substance capable of existing in more than one crystalline form, is the metal tin, and although the general behaviour, so far as studied, is analogous to that of sulphur, a short account of the two varieties of tin may be given here, not only on account of their metallurgical interest, but also on account of the importance which the phenomena possess for the employment of this metal in everyday life.
After a winter of extreme severity in Russia (1867-1868), the somewhat unpleasant discovery was made that a number of blocks of tin, which had been stored in the Customs House at St. Petersburg, had undergone disintegration and crumbled to a grey powder.[[59]] That tin undergoes change on exposure to extreme cold was known, however, before that time, even as far back as the time of Aristotle, who spoke of the tin as "melting."[[60]] Ludicrous as that term may now appear, Aristotle nevertheless unconsciously employed a strikingly accurate analogy, for the conditions under which ordinary white tin passes into the grey modification are, in many ways, quite analogous to those under which a substance passes from the solid to the liquid state. The knowledge of this was, however, beyond the wisdom of the Greek philosopher.
For many years there existed considerable confusion both as to the conditions under which the transformation of white tin into its allotropic modification occurs, and to the reason of the change. Under the guidance of the Phase Rule, however, the confusion which obtained has been cleared away, and the "mysterious" behaviour of tin brought into accord with other phenomena of transformation.[[61]]
Transition Point.—Just as in the case of sulphur, so also in the case of tin, there is a transition point above which the
one form, ordinary white tin, and below which the other form, grey tin, is the stable variety. In the case of this metal, the transition point was found by Cohen and van Eyk, who employed both the dilatometric and the electrical methods (Appendix) to be 20°. Below this temperature, grey tin is the stable form. But, as we have seen in the case of sulphur, the change of the metastable into the stable solid phase occurs with considerable slowness, and this behaviour is found also in the case of tin. Were it not so, we should not be able to use this metal for the many purposes to which it is applied in everyday life; for, with the exception of a comparatively small number of days in the year, the temperature of our climate is below 20°, and white tin is, therefore, at the ordinary temperature, in a metastable condition. The change, however, into the stable form at the ordinary temperature, although slow, nevertheless takes place, as is shown by the partial or entire conversion of articles of tin which have lain buried for several hundreds of years.
On lowering the temperature, the velocity with which the transformation of the tin occurs is increased, and Cohen and van Eyk found that the temperature of maximum velocity is about -50°. Contact with the stable form will, of course, facilitate the transformation.
The change of white tin into grey takes place also with increased velocity in presence of a solution of tin ammonium chloride (pink salt), which is able to dissolve small quantities of tin. In presence of such a solution also, it was found that the temperature at which the velocity of transformation was greatest was raised to 0°. At this temperature, white tin in contact with a solution of tin ammonium chloride, and the grey modification, undergoes transformation to an appreciable extent in the course of a few days.
Fig. 7 is a photograph of a piece of white tin undergoing transformation into the grey variety.[[62]] The bright surface of the tin becomes covered with a number of warty masses, formed of the less dense grey form, and the number and size of these continue to grow until the whole of the white tin has passed
into a grey powder. On account of the appearance which is here seen, this transformation of tin has been called by Cohen the "tin plague."
Enantiotropy and Monotropy.—In the case of sulphur and tin, we have met with two substances existing in polymorphic forms, and we have also learned that these forms exhibit a definite transition point at which their relative stability is reversed. Each form, therefore, possesses a definite range of stable existence, and is capable of undergoing transformation into the other, at temperatures above or below that of the transition point.
Another class of dimorphous substances is, however, met with as, for instance, in the case of the well-known compounds iodine monochloride and benzophenone. Each crystalline form has its own melting point, the dimorphous forms of iodine monochloride melting at 13.9° and 27.2°,[[63]] and those of benzophenone at 26° and 48°.[[64]] This class of substance differs from that which we have already studied (e.g. sulphur and tin), in that at all temperatures up to the melting point, only one of the forms is stable, the other being metastable. There is, therefore, no transition point, and transformation of the crystalline forms can be observed only in one direction. These two classes of phenomena are distinguished by the names enantiotropy and monotropy; enantiotropic substances being such that the change of one form into the other is a reversible process (e.g. rhombic sulphur into monoclinic, and monoclinic sulphur into rhombic), and monotropic substances, those in which the transformation of the crystalline forms is irreversible.
These differences in the behaviour can be explained very well in many cases by supposing that in the case of enantiotropic substances the transition point lies below the melting point, while in the case of monotropic substances, it lies above the melting point.[[65]] These conditions would be represented by the Figs. 8 and 9.
In these two figures, O3 is the transition point, O1 and O2 the melting points of the metastable and stable forms
respectively. From Fig. 9 we see that the crystalline form I. at all temperatures up to its melting point is metastable with respect to the form II. In such cases the transition point could be reached only at higher pressures.
Although, as already stated, this explanation suffices for many cases, it does not prove that in all cases of monotropy the transition point is above the melting point of the two forms. It is also quite possible that the transition point may lie below the melting points;[[66]] in this case we have what is known as pseudomonotropy. It is possible that graphite and diamond,[[67]] perhaps also the two forms of phosphorus, stand in the relation of pseudomonotropy (v. p. [49]).
The disposition of the curves in Figs. 8 and 9 also explains the phenomenon sometimes met with, especially in organic chemistry, that the substance first melts, then solidifies, and remelts at a higher temperature. On again determining the melting point after re-solidification, only the higher melting point is obtained.
The explanation of such a behaviour is, that if the determination of the melting point is carried out rapidly, the point O1, the melting point of the metastable solid form, may be realized. At this temperature, however, the liquid is metastable with respect to the stable solid form, and if the temperature is
not allowed to rise above the melting point of the latter, the liquid may solidify. The stable solid modification thus obtained will melt only at a higher temperature.
D. Phosphorus.
An interesting case of a monotropic dimorphous substance is found in phosphorus, which occurs in two crystalline forms; white phosphorus belonging to the regular system, and red phosphorus belonging to the hexagonal system. From determinations of the vapour pressures of liquid white phosphorus, and of solid red phosphorus,[[68]] it was found that the vapour pressure of red phosphorus was considerably lower than that of liquid white phosphorus at the same temperature, the values obtained being given in the following table.
Vapour Pressures of White and Red Phosphorus.
| Vapour pressure of liquid white phosphorus. | Vapour pressure of red phosphorus. | ||||
| Temperature. | Pressure in cm. | Temperature. | Pressure in atm. | Temperature. | Pressure in atm. |
| 165° | 12 | 360° | 3.2 | 360° | 0.1 |
| 180° | 20.4 | 440° | 7.5 | 440° | 1.75 |
| 200° | 26.6 | 494° | 18.0 | 487° | 6.8 |
| 219° | 35.9 | 503° | 21.9 | 510° | 10.8 |
| 230° | 51.4 | 511° | 26.2 | 531° | 16.0 |
| 290° | 76.0 | — | — | 550° | 31.0 |
| — | — | — | — | 577° | 56.0 |
These values are also represented graphically in Fig. 10.
At all temperatures above about 260°, transformation of the white into the red modification takes place with appreciable velocity, and this velocity increases as the temperature is raised. Even at lower temperatures, e.g. at the ordinary temperature, the velocity of transformation is increased under the influence
of light,[[69]] or by the presence of certain substances, e.g. iodine,[[70]] just as the velocity of transformation of white tin into the grey modification was increased by the presence of a solution of tin ammonium chloride (p. [40]). At the ordinary temperature, therefore, white phosphorus must be considered as the less stable (metastable) form, for although it can exist in contact with red phosphorus for a long period, its vapour pressure, as we have seen, is greater than that of the red modification, and also, its solubility in different solvents is greater[[71]] than that of the red modification; as we shall find later, the solubility of the metastable form is always greater than that of the stable.
The relationships which are met with in the case of phosphorus can be best represented by the diagram, Fig. 11.[[72]]
In this figure, BO1 represents the conditions of equilibrium of the univariant system red phosphorus and vapour, which ends at O1, the melting point of red phosphorus. By heating in capillary tubes of hard glass, Chapman[[73]] found that red phosphorus melts at the melting point of potassium iodide, i.e. about 630°,[[74]] but the pressure at this temperature is unknown.
At O1, then, we have the triple point, red phosphorus, liquid, and vapour, and starting from it, we should have the
vaporization curve of liquid phosphorus, O1A, and the fusion curve of red phosphorus, O1F. Although these have not been determined, the latter curve must, from theoretical considerations (v. p. [58]), slope slightly to the right; i.e. increase of pressure raises the melting point of red phosphorus.
When white phosphorus is heated to 44°, it melts. At this point, therefore, we shall have another triple point, white phosphorus—liquid—vapour; the pressure at this point has been calculated to be 3 mm.[[75]] This point is the intersection of three curves, viz. sublimation curve, vaporization curve, and the fusion curve of white phosphorus. The fusion curve, O2E, has been determined by Tammann[[76]] and by G. A. Hulett,[[77]] and it was found that increase of pressure by 1 atm. raises the melting point by 0.029°. The sublimation curve of white phosphorus has not yet been determined.
As can be seen from the table of vapour pressures (p. [46]), the vapour pressure of white phosphorus has been determined up to 500°; at temperatures above this, however, the velocity with which transformation into red phosphorus takes place is so great as to render the determination of the vapour pressure
at higher temperatures impossible. Since, however, the difference between white phosphorus and red phosphorus disappears in the liquid state, the vapour pressure curve of white phosphorus must pass through the point O1, the melting point of red phosphorus, and must be continuous with the curve O1A, the vapour pressure curve of liquid phosphorus (vide infra). Since, as Fig. 10 shows, the vapour pressure curve of white phosphorus ascends very rapidly at higher temperatures, the "break" between BO1 and O1A must be very slight.
As compared with monotropic substances like benzophenone, phosphorus exhibits the peculiarity that transformation of the metastable into the stable modification takes place with great slowness; and further, the time required for the production of equilibrium between red phosphorus and phosphorus vapour is great compared with that required for establishing the same equilibrium in the case of white phosphorus. This behaviour can be best explained by the assumption that change in the molecular complexity (polymerization) occurs in the conversion of white into red phosphorus, and when red phosphorus passes into vapour (depolymerization).[[78]]
This is borne out by the fact that measurements of the vapour density of phosphorus vapour at temperatures of 500° and more, show it to have the molecular weight represented by P4,[[79]] and the same molecular weight has been found for phosphorus in solution.[[80]] On the other hand, it has recently been shown by R. Schenck,[[81]] that the molecular weight of red phosphorus is at least P8, and very possibly higher.
In the case of phosphorus, therefore, it is more than possible that we are dealing, not simply with two polymorphic
forms of the same substance, but with polymeric forms, and that there is no transition point at temperatures above the absolute zero, unless we assume the molecular complexity of the two forms to become the same. The curve for red phosphorus would therefore lie below that of white phosphorus, for the vapour pressure of the polymeric form, if produced from the simpler form with evolution of heat, must be lower than that of the latter. A transition point would, of course, become possible if the sign of the heat effect in the transformation of the one modification into the other should change. If, further, the liquid which is produced by the fusion of red phosphorus at 630° under high pressure also exists in a polymeric form, greater than P4, then the metastable vaporization curve of white phosphorus would not pass through the melting point of red phosphorus, as was assumed above.[[82]]
We have already seen in the case of water (p. [31]) that the vapour pressure of supercooled water is greater than that of ice, and that therefore it is possible, theoretically at least, by a process of distillation, to transfer the water from one end of a closed tube to the other, and to there condense it as ice. On account of the very small difference between the vapour pressure of supercooled water and ice, this distillation process has not been experimentally realized. In the case of phosphorus, however, where the difference in the vapour pressures is comparatively great, it has been found possible to distil white phosphorus from one part of a closed tube to another, and to there condense it as red phosphorus; and since the vapour pressure of red phosphorus at 350° is less than the vapour pressure of white phosphorus at 200°, it is possible to carry out the distillation from a colder part of the tube to a hotter, by having white phosphorus at the former and red phosphorus at the latter. Such a process of distillation has been carried out by Troost and Hautefeuille between 324° and 350°.[[83]]
Relationships similar to those found in the case of phosphorus are also met with in the case of cyanogen and
paracyanogen, which have been studied by Chappuis,[[84]] Troost and Hautefeuille,[[85]] and Dewar,[[86]] and also in the case of other organic substances.
Enantiotropy combined with Monotropy.—Not only can polymorphic substances exhibit enantiotropy or monotropy, but, if the substance is capable of existing in more than two crystalline forms, both relationships may be found, so that some of the forms may be enantiotropic to one another, while the other forms exhibit only monotropy. This behaviour is seen in the case of sulphur, which can exist in as many as eight different crystalline varieties. Of these only monoclinic and rhombic sulphur exhibit the relationship of enantiotropy, i.e. they possess a definite transition point, while the other forms are all metastable with respect to rhombic and monoclinic sulphur, and remain so up to the melting point; that is to say, they are monotropic modifications.[[87]]
E. Liquid Crystals.
Phenomena observed.—In 1888 it was discovered by Reinitzer[[88]] that the two substances, cholesteryl acetate and cholesteryl benzoate, possess the peculiar property of melting sharply at a definite temperature to milky liquids; and that the latter, on being further heated, suddenly become clear, also at a definite temperature. Other substances, more especially p-azoxyanisole and p-azoxyphenetole, were, later, found to possess the same property of having apparently a double melting point.[[89]] On cooling the clear liquids, the reverse series of changes occurred.
The turbid liquids which were thus obtained were found to possess not only the usual properties of liquids (such as the
property of flowing and of assuming a perfectly spherical shape when suspended in a liquid of the same density), but also those properties which had hitherto been observed only in the case of solid crystalline substances, viz. the property of double refraction and of giving interference colours when examined by polarized light; the turbid liquids are anisotropic. To such liquids, the optical properties of which were discovered by O. Lehmann,[[90]] the name liquid crystals, or crystalline liquids, was given.
Nature of Liquid Crystals.—During the past ten years the question as to the nature of liquid crystals has been discussed by a number of investigators, several of whom have contended strongly against the idea of the term "liquid" being applied to the crystalline condition; and various attempts have been made to prove that the turbid liquids are in reality heterogeneous and are to be classed along with emulsions.[[91]] This view was no doubt largely suggested by the fact that the anisotropic liquids were turbid, whereas the "solid" crystals were clear. Lehmann found, however, that, when examined under the microscope, the "simple" liquid crystals were also clear,[[92]] the apparent turbidity being due to the aggregation of a number of differently oriented crystals, in the same way as a piece of marble does not appear transparent although composed of transparent crystals.[[93]]
Further, no proof of the heterogeneity of liquid crystals has yet been obtained, but rather all chemical and physical investigations indicate that they are homogeneous.[[94]] No separation
of a solid substance from the milky, anisotropic liquids has been effected; the anisotropic liquid is in some cases less viscous than the isotropic liquid formed at a higher temperature; and the temperature of liquefaction is constant, and is affected by pressure and admixture with foreign substances exactly as in the case of a pure substance.[[95]]
Equilibrium Relations in the Case of Liquid Crystals.—Since, now, we have seen that we are dealing here with substances in two crystalline forms (which we may call the solid and liquid[[96]] crystalline form), which possess a definite transition point, at which, transformation of the one form into the other occurs in both directions, we can represent the conditions of equilibrium by a diagram in all respects similar to that employed in the case of other enantiotropic substances, e.g. sulphur (p. [35]).
In Fig. 12 there is given a diagrammatic representation of the relationships found in the case of p-azoxyanisole.[[97]]
Although the vapour pressure of the substance in the solid, or liquid state, has not been determined, it will be understood from what we have already learned, that the curves AO, OB, BC, representing the vapour pressure of solid crystals, liquid crystals, isotropic liquid, must have the relative positions shown in the diagram. Point O, the transition point of the solid into the liquid crystals, lies at 118.27°, and the change of the transition point with the pressure is +0.032° pro 1 atm. The transition curve OE slopes, therefore, slightly to the right. The point B, the melting point of the liquid crystals, lies at 135.85°, and the melting point is raised 0.0485° pro 1 atm. The curve BD, therefore, also slopes to the right, and more so than the transition curve. In this respect azoxyanisole is different from sulphur.
The areas bounded by the curves represent the conditions for the stable existence of the four single phases, solid crystals, liquid crystals, isotropic liquid and vapour.
The most important substances hitherto found to form liquid crystals are[[98]]:—
| Substance. | Transition point. | Melting point. |
| Cholesteryl benzoate | 145.5° | 178.5° |
| Azoxyanisole | 118.3° | 135.9° |
| Azoxyphenetole | 134.5° | 168.1° |
| Condensation product from benzaldehyde and benzidine | 234° | 260° |
| Azine of p-oxyethylbenzaldehyde | 172° | 196° |
| Condensation product from p-tolylaldehyde and benzidine | 231° | — |
| p-Methoxycinnamic acid | 169° | 185° |