The Phase Rule and Its Applications Part 5

When the volume is kept constant, the effect of the addition of heat to a system at the triple point S-L-V differs somewhat according as there is an increase or diminution of volume when the solid pa.s.ses into the liquid state. In the former and most general case (Fig. 14), addition of heat will cause a certain amount of the solid phase to melt, whereby the heat which is added becomes latent; the temperature of the system therefore does not rise. Since, however, the melting of the solid is accompanied by an increase of volume, whereby an increase of pressure would result, a certain portion of the vapour must condense to liquid, in order that the pressure may remain constant. The total effect of addition of heat, therefore, is to cause both solid and vapour to pa.s.s into liquid, _i.e._ there occurs the change S + V --> L. It will, therefore, depend on the relative quant.i.ties of solid and vapour, which will disappear first. If the solid disappears first, then we shall pa.s.s to the system L-V; if vapour disappears first, we shall obtain the system S-L. Withdrawal of heat causes the reverse change, L --> S + V; at all temperatures below the triple point the liquid is unstable or metastable (p. 30).

When fusion is accompanied by a diminution of volume (_e.g._ ice, Fig. 13), then, since the melting of the solid phase would decrease the total volume, _i.e._ would lower the pressure, a certain quant.i.ty of the solid must also pa.s.s into vapour in order that the pressure may be maintained constant. On addition of heat, therefore, there occurs the reaction S --> L + V; withdrawal of heat causes the reverse change L + V --> S. Above the temperature of the triple point the {60} solid cannot exist; below the triple point both systems, S-L and S-V, can exist, and it will therefore depend on the relative amounts of liquid and vapour which of these two systems is obtained on withdrawing heat from the system at constant volume.

The same changes in the phases occur when heat is added or withdrawn at constant pressure, so long as the three phases are present. Continued addition of heat, however, at constant pressure will ultimately cause the formation of the bivariant system vapour alone; continued withdrawal of heat will ultimately cause the formation of solid alone. This will be readily understood from Fig. 15. The dotted line D'OD is a line of constant pressure; on adding heat, the system pa.s.ses along the line OD into the region of vapour; on heat being withdrawn, the system pa.s.ses along OD' into the area of solid.

[Ill.u.s.tration: FIG. 15.]

Similar changes are produced when the volume of the system is altered.

Alteration of volume may take place either while transference of heat to or from the system is cut off (adiabatic change), or while such transference may occur (isothermal change). In the latter case, the temperature of the system will remain constant; in the former case, since at the triple point the pressure must be constant so long as the three phases are present, increase of volume must be compensated by the evaporation of liquid. This, however, would cause the temperature to fall (since communication of heat from the outside is supposed to be cut off), and a portion of the liquid must therefore freeze. In this way the latent heat of evaporation is counterbalanced by the latent heat of fusion. As the result of increase of volume, therefore, the process occurs L --> S + V. Diminution of volume, without transference of heat, will bring about the opposite change, S + V --> L. In the former case there is ultimately obtained the univariant system S-V; in the latter case there will be {61} obtained either S-L or L-V according as the vapour or solid phase disappears first.

This argument holds good for both types of triple point shown in Figs. 13 and 14 (p. 57). A glance at these figures will show that increase of volume (diminution of pressure) will lead ultimately to the system S-V, for at pressures lower than that of the triple point, the liquid phase cannot exist. Decrease of volume (increase of pressure), on the other hand, will lead either to the system S-L or L-V, because these systems can exist at pressures higher than that of the triple point. If the vapour phase disappears and we pa.s.s to the curve S-L, continued diminution of volume will be accompanied by a fall in temperature in the case of systems of the first type (Fig. 13), and by a rise in temperature in the case of systems of the second type (Fig. 14).

[Ill.u.s.tration: FIG. 16.]

[Ill.u.s.tration: FIG. 17.]

Lastly, if the temperature is maintained constant, _i.e._ if heat can pa.s.s into or out of the system, then on changing the volume the same changes in the phases will take place as described above until one of the phases has disappeared. Continued increase of volume (decrease of pressure) will then cause the disappearance of a second phase, the system pa.s.sing along the dotted line OE' (Figs. 16, 17), so that ultimately there remains only the vapour phase. Conversely, diminution of volume (increase of pressure) will ultimately lead either to solid (Fig. 16) or to liquid alone (Fig. 17), the system pa.s.sing along the dotted line OE. {62}

In discussing the alterations which may take place at the triple point with change of temperature and pressure, we have considered only the triple point S-L-V. The same reasoning, however, applies, _mutatis mutandis_, to all other triple points, so that if the specific volumes of the phases are known, and the sign of the heat effects which accompany the transformation of one phase into the other, it is possible to predict (by means of the theorem of Le Chatelier) the changes which will be produced in the system by alteration of the pressure and temperature.

In all cases of transformation at the triple point, it should be noted that all _three phases are involved in the change_,[107] and not two only; the fact that in the case, say, of the transformation from solid to liquid, or liquid to solid, at the melting point with change of temperature, only these two phases appear to be affected, is due to there generally being a large excess of the vapour phase present and to the prior disappearance therefore of the solid or liquid phase.

In the case of triple points at which two solid phases are in equilibrium with liquid, other arrangements of the curves around the triple point are found. It is, however, unnecessary to give a general treatment of these here, since the principles which have been applied to the triple point S-L-V can also be applied to the other triple points.[108]

Triple Point Solid--Solid--Vapour.--The triple point solid--solid--vapour is one which is of considerable importance. Examples of such a triple point have already been given in sulphur and tin, and a list of other substances capable of yielding two solid phases is given below. The triple point S-S-V is not precisely the same as the transition point, but is very nearly so.

The transition point is the temperature at which the relative stability of the two solid phases undergoes change, when the vapour phase is absent and the pressure is 1 atm.; whereas at the triple point the pressure is that of the system itself. The transition point, therefore, bears the same relation to the triple point S-S-V as the melting point to the triple point S-L-V.

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In the following table is given a list of the most important polymorphous substances, and the temperatures of the transition point.[109]

------------------------------------+------------- Substance. Transition temperature.

------------------------------------+------------- Ammonium nitrate-- [beta]-rhombic --> [alpha]-rhombic 35 [alpha]-rhombic --> rhombohedral 83 Rhombohedral --> regular 125 Mercuric iodide 126 Pota.s.sium nitrate 129 Silver iodide 145 Silver nitrate 160 Sulphur 95.5 Tetrabrommethane 46.8 Thallium nitrate-- Rhombic --> rhombohedral 80 Rhombohedral --> regular 142.5 Thallium picrate 46 Tin 20 ------------------------------------+-------------

Sublimation and Vaporization Curves.--We have already seen, in the case of ice and liquid water, that the vapour pressure increases as the temperature rises, the increase of pressure per degree being greater the higher the temperature. The sublimation and vaporization curves, therefore, are not straight lines, but are bent, the convex side of the curve being towards the temperature axis in the ordinary _pt_-diagram.

In the case of sulphur and of tin, we a.s.sumed vapour to be given off by the solid substance, although the pressure of the vapour has not hitherto been measured. The a.s.sumption, however, is entirely justified, not only on theoretical grounds, but also because the existence of a vapour pressure has been observed in the case of many solid substances at temperatures much below the melting point,[110] and in some cases, _e.g._ camphor,[111] the vapour pressure is considerable.

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As the result of a large number of determinations, it has been found that all vapour pressure curves have the same general form alluded to above.

Attempts have also been made to obtain a general expression for the quant.i.tative changes in the vapour pressure with change of temperature, but without success. Nevertheless, the _qualitative_ changes, or the general direction of the curves, can be predicted by means of the theorem of Le Chatelier.

As we have already learned (p. 16), the Phase Rule takes no account of the molecular complexity of the substances partic.i.p.ating in an equilibrium. A dissociating substance, therefore, in contact with its vaporous products of dissociation (_e.g._ ammonium chloride in contact with ammonia and hydrogen chloride), will likewise const.i.tute a univariant system of one component, provided the composition of the vapour phase as a whole is the same as that of the solid or liquid phase (p. 13). For all such substances, therefore, the conditions of equilibrium will be represented by a curve of the same general form as the vapour pressure curve of a non-dissociating substance.[112] The same behaviour is also found in the case of substances which polymerize on pa.s.sing into the solid or liquid state (_e.g._ red phosphorus). Where such changes in the molecular state occur, however, the time required for equilibrium to be established is, as a rule, greater than when the molecular state is the same in both phases.

From an examination of Figs. 13 and 14, it will be easy to predict the effect of change of pressure and temperature on the univariant systems S-V or L-V. If the volume is kept constant, addition of heat will cause an increase of pressure, the system S-V moving along the curve AO until at the triple point the liquid phase is formed, and the system L-V moving along the curve OB; so long as two phases are present, the condition of the system must be represented by these two curves. Conversely, withdrawal of heat will cause condensation of vapour, and therefore diminution of pressure; the system will therefore move along the vaporization or sublimation curve to lower temperatures and pressures, so long as the system remains univariant.

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If transference of heat to or from the system is prevented, increase of volume (diminution of pressure) will cause the system L-V to pa.s.s along the curve BO; liquid will pa.s.s into vapour and the temperature will fall.[113]

At O solid may appear, and the temperature of the system will then remain constant until the liquid phase has disappeared (p. 57); the system will then follow the curve OA until the solid phase disappears, and we are ultimately left with vapour. On the other hand, diminution of volume (increase of pressure) will cause condensation of vapour, and the system S-V will pa.s.s along the curve AO to higher temperatures and pressures; at O the solid will melt, and the system will ultimately pa.s.s to the curve OB or to OC (p. 57).

Addition or withdrawal of heat at constant pressure, and increase or diminution of the pressure at constant temperature, will cause the system to pa.s.s along lines parallel to the temperature and the pressure axis respectively; the working out of these changes may be left to the reader, guided by what has been said on pp. 60 and 61.

The sublimation curve of all substances, so far as yet found, has its upper limit at the melting point (triple point), although the possibility of the existence of a superheated solid is not excluded. The lower limit is, theoretically at least, at the absolute zero, provided no new phase, _e.g._ a different crystalline modification, is formed. If the sublimation pressure of a substance is greater than the atmospheric pressure at any temperature below the point of fusion, then the substance will _sublime without melting_ when heated in an open vessel; and fusion will be possible only at a pressure higher than the atmospheric. This is found, for example, in the case of red phosphorus (p. 47). If, however, the sublimation pressure of a substance at its triple point S-L-V is less than one atmosphere, then the substance will melt when heated in an open vessel.

In the case of the vaporization curve, the upper limit lies at the critical point where the liquid ceases to exist;[114] the {66} lower limit is determined by the range of the metastable state of the supercooled liquid.

The interpolation and extrapolation of vapour-pressure curves is rendered very easy by means of a relations.h.i.+p which Ramsay and Young[115] found to exist between the vapour-pressure curves of different substances. It was observed that in the case of closely related substances, the ratio of the absolute temperatures corresponding to equal vapour pressures is constant, _i.e._ T_{1}/T'_{1} = T_{2}/T'_{2}. When the two substances are not closely related, it was found that the relations.h.i.+p could be expressed by the equation T_{1}/T'_{1} = T_{2}/T'_{2} + _c_(_t_' - _t_) where _c_ is a constant having a small positive or negative value, and _t_' and _t_ are the temperatures at which one of the substances has the two values of the vapour pressure in question. By means of this equation, if the vapour-pressure curve of one substance is known, the vapour-pressure curve of any other substance can be calculated from the values at any two temperatures of the vapour pressure of that substance.

Fusion Curve--Transition Curve.--The fusion curve represents the conditions of equilibrium between the solid and liquid phase; it shows the change of the melting point of a substance with change of pressure.

As shown in Figs. 13 and 14, the fusion curve is inclined either towards the pressure axis or away from it; that is, increase of pressure can either lower or raise the melting point. It is easy to predict in a qualitative manner the different effect of pressure on the melting point in the two cases mentioned, if we consider the matter in the light of the theorem of Le Chatelier (p. 58). Water, on pa.s.sing into ice, expands; therefore, if the pressure on the system ice--water be increased, a reaction will take place which is accompanied by a diminution in volume, _i.e._ the ice will melt. Consequently, a lower temperature will be required in order to counteract the effect of increase of pressure; or, in other words, the melting point will {67} be lowered by pressure.[116] In the second case, the pa.s.sage of the liquid to the solid state is accompanied by a diminution of volume; the effect of increase of pressure will therefore be the reverse of that in the previous case.

If the value of the heat of fusion and the alteration of volume accompanying the change of state are known, it is possible to calculate _quant.i.tatively_ the effect of pressure.[117]

We have already seen (p. 25) that the effect of pressure on the melting point of a substance was predicted as the result of theoretical considerations, and was first proved experimentally in the case of ice.

Soon after, Bunsen[118] showed that the melting point of other substances is also affected by pressure; and in more recent years, ample experimental proof of the change of the melting point with the pressure has been obtained. The change of the melting point is, however, small; as a rule, increase of pressure by 1 atm. changes the melting point by about 0.03, but in the case of water the change is much less (0.0076), and in the case of camphor much more (0.13). In other words, if we take the mean case, an increase of pressure of more than 30 atm. is required to produce a change in the melting point of 1.

Investigations which were made of the influence of pressure on the melting-point, showed that up to pressures of several hundred atmospheres the fusion curve is a straight line.[119] Tammann[120] has, however, found that on increasing the pressure the fusion curve no longer remains straight, but bends towards the pressure axis, so that, on sufficiently increasing the pressure, a maximum temperature might at length be reached.

This maximum has, so far, however, not been attained, although the melting point curves of various substances have been studied up to pressures of 4500 atm. This is to be accounted for partly {68} by the fact that the probable maximum temperature in the case of most substances lies at very great pressures, and also by the fact that other solid phases make their appearance, as, for example, in the case of ice (p. 32).

As to the upper limit of the fusion curve, the view has been expressed[121]

that just as in the case of liquid and vapour, so also in the case of solid and liquid, there exists a critical point at which the solid and the liquid phase become identical. Experimental evidence, however, does not appear to favour this view.[122]

The _transition point_, like the melting point, is also influenced by the pressure, and in this case also it is found that pressure may either raise or lower the transition point, so that the transition curve may be inclined either away from or towards the pressure axis. The direction of the transition curve can also be predicted if the change of volume accompanying the pa.s.sage of one form into the other is known. In the case of sulphur, we saw that the transition point is raised by increase of pressure; in the case of the transition of rhombohedral into [alpha]-rhombic form of ammonium nitrate, however, the transition point is lowered by pressure, as shown by the following table.[123]

-------------+---------- Temperature. Pressure.

-------------+---------- 85.85 1 atm.

84.38 100 "

83.03 200 "

82.29 250 "

So far as investigations have been carried out, it appears that in most cases the transition curve is practically a straight line.

It has, however, been found in the case of Glauber's salt, that with increase of pressure the transition curve pa.s.ses through a point of maximum temperature, and exhibits, therefore, a form similar to that a.s.sumed by Tammann for the fusion curve.[124]

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Suspended Transformation. Metastable Equilibria.--Hitherto we have considered only systems in stable equilibrium. We have, however, already seen, in the case of water, that on cooling the liquid down to the triple point, solidification did not necessarily take place, although the conditions were such as to allow of its formation. Similarly, we saw that rhombic sulphur can be heated above the transition point, and monoclinic sulphur can be obtained at temperatures below the transition point, although in both cases transformation into a more stable form is possible; the system becomes metastable.

The same reluctance to form a new phase is observed also in the phenomena of superheating of liquids, and in the "hanging" of mercury in barometers, in which case the vapour phase is not formed. In general, then, we may say that _a new phase will not necessarily be formed immediately the system pa.s.ses into such a condition that the existence of that phase is possible_; but rather, instead of the system undergoing transformation so as to pa.s.s into the most stable condition under the existing pressure and temperature, this transformation will be "suspended" or delayed, and the system will become metastable. Only in the case of the formation of the liquid from the solid phase, in a one-component system, has this reluctance to form a new phase not been observed.

_To ensure the formation of the new phase, it is necessary to have that phase present._ The presence of the solid phase will prevent the supercooling of the liquid; and the presence of the vapour phase will prevent the superheating of the liquid. However, even in the presence of the more stable phase, transformation of the metastable phase occurs with very varying velocity; in some cases so quickly as to appear almost instantaneous; while in other cases, the change takes place so slowly as to require hundreds of years for its achievement. It is this slow rate of transformation that renders the existence of metastable forms possible, when in contact with the more stable phase. Thus, for example, although calcite is the most stable form of calcium carbonate at the ordinary temperature,[125] the less stable {70} modification, aragonite, nevertheless exists under the ordinary conditions in an apparently very stable state.

As to the amount of the new phase required to bring about the transformation of the metastable phase, quant.i.tative measurements have been carried out only in the case of the initiation of crystallization in a supercooled liquid.[126] As the result of these investigations, it was found that, in the case of superfused salol, the very small amount of 1 10^{-7} gm. of the solid phase was sufficient to induce crystallization.

Crystallization of a supercooled liquid, however, can be initiated only by a "nucleus" of the same substance in the solid state, or, as has also been found, by a nucleus of an isomorphous solid phase; it is not brought about by the presence of any chance solid.