The Phase Rule and Its Applications

Wat

rpose of obtaining examples by which we may illustrate the general behaviour of systems, we shall in thi

epresented 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 e

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 temper

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rements of the Phase Rule, and at any given temperature the system w

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

ressure

re. Press

perature. Pr

rc

213 120

8[29] 1

752 150

16 200°

932 250

54 270°

0 364.3°

) 14790.4 (

cal pr

e 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 correspondin

dentical; 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

ditions 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

s of the vapour pressure of

Pressur

re. Press

perature. Pr

rc

.050 -

.121 -

.312 -

.806 -

.279 0

° 1

es 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 w

e 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.

areful measurements of the influence of pressure on the melting point of ice have been made more especially by Ta

Pressur

Pressure in

ange of meltin

e of pr

m. per

0

2

5

7

0.

2.

5.

7.

0.

2.

3

1

9

1

4

6

8

0

0.0

00

00

00

01

01

01

01

01

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 var

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out 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 equilibriu

an 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

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 abstra

e fusion curve. The fusion curve, as we have seen, is the curve of equilibrium between ice and water; and since at the triple point i

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se 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,

ed.[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

e 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,

w, 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 tabl

of Ice and of Su

Pressure in

Ice. Di

4.602 0

95 3.92

50 3.33

58 2.37

197 1.9

492 1.2

005 0.8

ature water and ice have the same vapour pressure, the vapour pressure

ve 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

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, t

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 val

Sul

morphism. 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 re

e 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

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 pris

the help of the Phase Rule, we see that the f

tems: One compon

ombic

oclinic

lphur

quid s

stems: One compon

c sulphur

nic sulphur

c sulphur

nic sulphur

and monocl

uid and

tems: One componen

monoclinic sul

ulphur, liqui

sulphur, liqu

monoclinic sul

g.

olid 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 dia

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

nic sulphur was found by Reicher[51] to lie at 95.5°. Below this tem

re 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 ph

efinite 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 transformat

n 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] Th

ure, 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

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 condit

ng 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

f 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 m

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-mo

esented according to scale in Fig. 6,[57] a being the curve for monoclinic sulphur and liq

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 o

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xistence of other systems of the one-component sulphur besides those alr

T

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 met

oes 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,

llotropic modification occurs, and to the reason of the change. Under the guidance of the Phase Rule, however, the confusion which o

mperature, 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 sm

s is increased, and Cohen and van Eyk found that the temperature of maximum velocity is

antities 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 temper

th 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

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d that these forms exhibit a definite transition point at which their relative stability is reversed. Each form, therefore, possesses a definite r

stance 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 phenom

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enantiotropic substances the transition point lies below the melting point, while in the case of monotropic

respectively. From Fig. 9 we see that the crystalline form I. at all temperatures up to its melting point is met

ting 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

in organic chemistry, that the substance first melts, then solidifies, and remelts at a higher temperature.

e 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 allo

hosp

ed 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

s of White and

uid white phosphorus.

sph

ture. P

mperature

emperatur

a

360° 3.2

440° 7.5

494° 18.

503° 21.

511° 26.

.0 - -

- 57

so represented grap

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resence 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 e

n the case of phosphorus can be best r

s at O1, the melting point of red phosphorus. By heating in capillary tubes of hard glass, Chapman[73] found that red phos

liquid phosphorus, O1A, and the fusion curve of red phosphorus, O1F. Although these have not been determined, the latter curve must, fro

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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 T

ion 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 phosph

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

ave the molecular weight represented by P4,[79] and the same molecular weight has been found for phosphorus in solution.[80] On the other

ve 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 m

ur 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 vap

he case of cyanogen and paracyanogen, which have been studied by Chappuis,[84] Troost

e 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 rel

uid Cr

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

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

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, w

paration 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 hi

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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,

c representation of the relationships f

st 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

for the stable existence of the four single phases, sol

ces hitherto found to form

ce. Tra

t. M

in

benzoate 1

sole 118

etole 134

product fro

zidine

yethylbenzald

product from

nzidin

innamic ac