SYSTEMS OF FOUR COMPONENTS 312
Reciprocal salt-pairs. Choice of components, 313. Transition
point, 314. Formation of double salts, 315. Transition
interval, 315. Graphic representation, 316. Example, 317.
Ammonia-soda process, 320. Preparation of barium nitrite, 327.
Barium carbonate and potassium sulphate, 328.
APPENDIX
EXPERIMENTAL DETERMINATION OF THE TRANSITION POINT 331
I. The dilatometric method, 331. II. Measurement of
the vapour pressure, 334. III. Solubility measurements, 335.
IV. Thermometric method, 337. V. Optical method, 338.
VI. Electrical methods, 338.
NAME INDEX 341
SUBJECT INDEX 345
* * * * *
{1}
THE PHASE RULE
CHAPTER I
INTRODUCTION
General.--Before proceeding to the more systematic treatment of the Phase
Rule, it may, perhaps, be not amiss to give first a brief forecast of the
nature of the subject we are about to study, in order that we may gain some
idea of what the Phase Rule is, of the kind of problem which it enables us
to solve, and of the scope of its application.
It has long been known that if water is placed in a closed, exhausted
space, vapour is given off and a certain pressure is created in the
enclosing vessel. Thus, when water is placed in the Torricellian vacuum of
the barometer, the mercury is depressed, and the amount of depression
increases as the temperature is raised. But, although the pressure of the
vapour increases as the temperature rises, its value at any given
temperature is constant, no matter whether the amount of water present or
the volume of the vapour is great or small; if the pressure on the vapour
is altered while the temperature is maintained constant, either the water
or the vapour will ultimately disappear; the former by evaporation, the
latter by condensation. At any given temperature within certain limits,
therefore, water and vapour can exist permanently in contact with one
another--or, as it is said, be in equilibrium with one another--only when
the pressure has a certain definite value. The same law of constancy of
vapour pressure at a given {2} temperature, quite irrespective of the
volumes of liquid and vapour,[1] holds good also in the case of alcohol,
ether, benzene, and other pure liquids. It is, therefore, not unnatural to
ask the question, Does it hold good for all liquids? Is it valid, for
example, in the case of solutions?
We can find the answer to these questions by studying the behaviour of a
solution--say, a solution of common salt in water--when placed in the
Torricellian vacuum. In this case, also, it is observed that the pressure
of the vapour increases as the temperature is raised, but the pressure is
no longer independent of the volume; as the volume increases, the pressure
slowly diminishes. If, however, solid salt is present in contact with the
solution, then the pressure again becomes constant at constant temperature,
even when the volume of the vapour is altered. As we see, therefore,
solutions do not behave in the same way as pure liquids.
Moreover, on lowering the temperature of water, a point is reached at which
ice begins to separate out; and if heat be now added to the system or
withdrawn from it, no change will take place in the temperature or vapour
pressure of the latter until either the ice or the water has
disappeared.[2] Ice, water, and vapour, therefore, can be in equilibrium
with one another only at one definite temperature and one definite
pressure.
In the case of a solution of common salt, however, we may have ice in
contact with the solution at different temperatures and pressures. Further,
it is possible to have a solution in equilibrium not only with anhydrous
salt (NaCl), but also with the hydrated salt (NaCl, 2H_{2}O), as well as
with ice, and the question, therefore, arises: Is it possible to state in a
general manner the conditions under which such different systems can exist
in equilibrium; or to obtain some insight {3} into the relations which
exist between pure liquids and solutions? As we shall learn, the Phase Rule
enables us to give an answer to this question.
The preceding examples belong to the class of so-called "physical"
equilibria, or equilibria depending on changes in the physical state. More
than a hundred years ago, however, it was shown by Wenzel and Berthollet
that "chemical" equilibria can also exist; that chemical reactions do not
always take place completely in one direction as indicated by the usual
chemical equation, but that before the reacting substances are all used up
the reaction ceases, and there is a condition of equilibrium between the
reacting substances and the products of reaction. As an example of this,
there may be taken the process of lime-burning, which depends on the fact
that when calcium carbonate is heated, carbon dioxide is given off and
quicklime is produced. If the carbonate is heated in a closed vessel it
will be found, however, not to undergo entire decomposition. When the
pressure of the carbon dioxide reaches a certain value (which is found to
depend on the temperature), decomposition ceases, and calcium carbonate
exists side by side with calcium oxide and carbon dioxide. Moreover, at any
given temperature the pressure is constant and independent of the amount of
carbonate or oxide present, or of the volume of the gas; _nor does the
addition of either of the products of dissociation, carbon dioxide or
calcium oxide, cause any change in the equilibrium_. Here, then, we see
that, although there are three different substances present, and although
the equilibrium is no longer due to physical, but to chemical change, it
nevertheless obeys the same law as the vapour pressure of a pure volatile
liquid, such as water.
It might be supposed, now, that this behaviour would be shown by other
dissociating substances, _e.g._ ammonium chloride. When this substance is
heated it dissociates into ammonia and hydrogen chloride, and at any given
temperature the pressure of these gases is constant,[3] and is independent
of the amounts of solid and gas present. So far, therefore, ammonium
chloride behaves like calcium carbonate. If, however, one of the {4}
products of dissociation be added to the system, it is found that the
pressure is no longer constant at a given temperature, but varies with the
amount of gas, ammonia or hydrogen chloride, which is added. In the case of
certain dissociating substances, therefore, addition of one of the products
of dissociation alters the equilibrium, while in other cases it does not.
With the help of the Phase Rule, however, a general interpretation of this
difference of behaviour can be given--an interpretation which can be
applied not only to the two cases cited, but to all cases of dissociation.
Again, it is well known that sulphur exists in two different crystalline
forms, octahedral and prismatic, each of which melts at a different
temperature. The problem here is, therefore, more complicated than in the
case of ice, for there is now a possibility not only of one solid form, but
of two different forms of the same substance existing in contact with
liquid. What are the conditions under which these two forms can exist in
contact with liquid, either singly or together, and under what conditions
can the two solid forms exist together without the presence of liquid
sulphur? To these questions an answer can also be given with the help of
the Phase Rule.
These cases are, however, comparatively simple; but when we come, for
instance, to study the conditions under which solutions are formed, and
especially when we inquire into the solubility relations of salts capable
of forming, perhaps, a series of crystalline hydrates; and when we seek to
determine the conditions under which these different forms can exist in
contact with the solution, the problem becomes more complicated, and the
necessity of some general guide to the elucidation of the behaviour of
these different systems becomes more urgent.
It is, now, to the study of such physical and chemical equilibria as those
above-mentioned that the Phase Rule finds application; to the study, also,
of the conditions regulating, for example, the formation of alloys from
mixtures of the fused metals, or of the various salts of the Stassfurt
deposits; the behaviour of iron and carbon in the formation of steel and
the {5} separation of different minerals from a fused rock-mass.[4] With
the help of the Phase Rule we can group together into classes the large
number of different isolated cases of systems in equilibrium; with its aid
we are able to state, in a general manner at least, the conditions under
which a system can be in equilibrium, and by its means we can gain some
insight into the relations existing between different kinds of systems.
Homogeneous and Heterogeneous Equilibrium.--Before passing to the
consideration of this generalization, it will be well to first make mention
of certain restrictions which must be placed on its treatment, and also of
the limitations to which it is subject. If a system is uniform throughout
its whole extent, and possesses in every part identical physical properties
and chemical composition, it is called _homogeneous_. Such is, for example,
a solution of sodium chloride in water. An equilibrium occurring in such a
homogeneous system (such as the equilibrium occurring in the formation of
an ester in alcoholic solution) is called _homogeneous equilibrium_. If,
however, the system consists of parts which have different physical
properties, perhaps also different chemical properties, and which are
marked off and separated from one another by bounding surfaces, the system
is said to be _heterogeneous_. Such a system is formed by ice, water, and
vapour, in which the three portions, each in itself homogeneous, can be
mechanically separated from one another. When equilibrium exists between
different, physically distinct parts, it is known as _heterogeneous
equilibrium_. It is, now, with heterogeneous equilibria, with the
conditions under which a heterogeneous system can exist, that we shall deal
here.
Further, we shall not take into account changes of equilibrium due to the
action of electrical, magnetic, or capillary forces, or of gravity; but
shall discuss only those which are due to changes of pressure, temperature,
and volume (or concentration).
Real and Apparent Equilibrium.--In discussing equilibria, also, a
distinction must be drawn between real and {6} apparent equilibria. In the
former case there is a state of rest which undergoes continuous change with
change of the conditions (_e.g._ change of temperature or of pressure), and
for which the chief criterion is that _the same condition of equilibrium is
reached from whichever side it is approached_. Thus in the case of a
solution, if the temperature is maintained constant, the same concentration
will be obtained, no matter whether we start with an unsaturated solution
to which we add more solid, or with a supersaturated solution from which we
allow solid to crystallize out; or, in the case of water in contact with
vapour, the same vapour pressure will be obtained, no matter whether we
heat the water up to the given temperature or cool it down from a higher
temperature. In this case, water and vapour are in _real_ equilibrium. On
the other hand, water in contact with hydrogen and oxygen at the ordinary
temperature is a case only of _apparent_ equilibrium; on changing the
pressure and temperature continuously within certain limits there is no
continuous change observed in the relative amounts of the two gases. On
heating beyond these limits there is a sudden and not a continuous change,
and the system no longer regains its former condition on being cooled to
the ordinary temperature. In all such cases the system may be regarded as
undergoing change and as tending towards a state of true or real
equilibrium, but with such slowness that no change is observed.
Although the case of water in contact with hydrogen and oxygen is an
extreme one, it must be borne in mind that the condition of true
equilibrium may not be reached instantaneously or even with measurable
velocity, and in all cases it is necessary to be on one's guard against
mistaking apparent (or false) for real (or true) equilibrium. The
importance of this will be fully illustrated in the sequel.
* * * * *
{7}
CHAPTER II
THE PHASE RULE
Although the fact that chemical reactions do not take place completely in
one direction, but proceed only to a certain point and there make a halt,
was known in the last quarter of the eighteenth century (Wenzel, 1777;
Berthollet, 1799); and although the opening and subsequent decades of the
following century brought many further examples of such equilibria to our
knowledge, it was not until the last quarter of the nineteenth century that
a theorem, general in its application and with foundations weakened by no
hypothetical assumptions as to the nature or constitution of matter, was
put forward by Willard Gibbs;[5] a generalization which serves at once as a
golden rule by which the condition of equilibrium of a system can be
tested, and as a guide to the similarities and dissimilarities existing in
different systems.
Before that time, certainly, attempts had been made to bring the different
known cases of equilibria--chemical and physical--under general laws. From
the very first, both Wenzel[6] and Berthollet[7] recognized the influence
exercised by the _mass_ of the substances on the equilibrium of the system.
It was reserved, however, for Guldberg and Waage, by their more general
statement and mathematical treatment of the Law of Mass Action,[8] to
inaugurate the period of quantitative study of equilibria. The law which
these investigators enunciated {8} served satisfactorily to summarize the
conditions of equilibrium in many cases both of homogeneous and, with the
help of certain assumptions and additions, of heterogeneous equilibrium. By
reason, however, of the fact that it was developed on the basis of the
kinetic and molecular theories, and involved, therefore, certain
hypothetical assumptions as to the nature and condition of the substances
taking part in the equilibrium, the law of mass action failed, as it
necessarily must, when applied to those systems in which neither the number
of different molecular aggregates nor the degree of their molecular
complexity was known.
Ten years after the law of mass action was propounded by Guldberg and
Waage, Willard Gibbs,[9] Professor of Physics in Yale University, showed
how, in a perfectly general manner, free from all hypothetical assumptions
as to the molecular condition of the participating substances, all cases of
equilibrium could be surveyed and grouped into classes, and how
similarities in the behaviour of apparently different kinds of systems, and
differences in apparently similar systems, could be explained.
As the basis of his theory of equilibria, Gibbs adopted the laws of
thermodynamics,[10] a method of treatment which had first been employed by
Horstmann.[11] In deducing the law of equilibrium, Gibbs regarded a system
as possessing only three independently variable factors[12]--temperature,
pressure, and the concentration of the components of the system--and he
enunciated the general theorem now usually known as the _Phase Rule_, by
which he defined the conditions of equilibrium as a relationship between
the number of what are called the phases and the components of the system.
Phases.--Before proceeding farther we shall first consider what exactly is
meant by the terms _phase_ and _component_. We have already seen (p. 5)
that a heterogeneous system is made {9} up of different portions, each in
itself homogeneous, but marked off in space and separated from the other
portions by bounding surfaces. These homogeneous, physically distinct and
mechanically separable portions are called _phases_. Thus ice, water, and
vapour, are three phases of the same chemical substance--water. A phase,
however, whilst it must be physically and chemically homogeneous, need not
necessarily be chemically simple. Thus, a gaseous mixture or a solution may
form a phase; but a heterogeneous mixture of solid substances constitutes
as many phases as there are substances present. Thus when calcium carbonate
dissociates under the influence of heat, calcium oxide and carbon dioxide
are formed. There are then _two_ solid phases present, viz. calcium
carbonate and oxide, and one gas phase, carbon dioxide.
The _number of phases_ which can exist side by side may vary greatly in
different systems. In all cases, however, there can be but one gas or
vapour phase on the account of the fact that all gases are miscible with
one another in all proportions. In the case of liquid and solid phases the
number is indefinite, since the above property does not apply to them. The
number of phases which can be formed by any given substance or group of
substances also differs greatly, and in general increases with the number
of participating substances. Even in the case of a single substance,
however, the number may be considerable; in the case of sulphur, for
example, at least eight different solid phases are known (_v._ Chap. III.).
It is of importance to bear in mind that equilibrium is _independent of the
amounts_ of the phases present.[13] Thus it is a familiar fact that the
pressure of a vapour in contact with a {10} liquid (_i.e._ the pressure of
the saturated vapour) is unaffected by the amounts, whether relative or
absolute, of the liquid and vapour; also the amount of a substance
dissolved by a liquid is independent of the amount of solid in contact with
the solution. It is true that deviations from this general law occur when
the amount of liquid or the size of the solid particles is reduced beyond a
certain point,[14] owing to the influence of surface energy; but we have
already (p. 5) excluded such cases from consideration.
Components.--Although the conception of phases is one which is readily
understood, somewhat greater difficulty is experienced when we come to
consider what is meant by the term _component_; for the components of a
system are not synonymous with the chemical elements or compounds present,
_i.e._ with the _constituents_ of the system, although both elements and
compounds may be components. By the latter term there are meant only those
constituents the concentration of which can undergo _independent_ variation
in the different phases, and it is only with these that we are concerned
here.[15]
To understand the meaning of this term we shall consider briefly some cases
with which the reader will be familiar, and at the outset it must be
emphasized that the Phase Rule is concerned merely with those constituents
which take part in the state of real equilibrium (p. 5); for it is only to
the final state, not to the processes by which that state is reached, that
the Phase Rule applies.
Consider now the case of the system water--vapour or ice--water--vapour.
The number of constituents taking part in the equilibrium here is only one,
viz. the chemical substance, water. Hydrogen and oxygen, the constituents
of water, are not to be regarded as components, because, in the first
place, they are {11} not present in the system in a state of real
equilibrium (p. 6); in the second place, they are combined in definite
proportions to form water, and their amounts, therefore, cannot be varied
independently. A variation in the amount of hydrogen necessitates a
definite variation in the amount of oxygen.
In the case, already referred to, in which hydrogen and oxygen are present
along with water at the ordinary temperature, we are not dealing with a
condition of true equilibrium. If, however, the temperature is raised to a
certain point, a state of true equilibrium between hydrogen, oxygen, and
water-vapour will be possible. In this case hydrogen and oxygen will be
components, because now they do take part in the equilibrium; also, they
need no longer be present in definite proportions, but excess of one or the
other may be added. Of course, if the restriction be arbitrarily made that
the free hydrogen and oxygen shall be present always and only in the
proportions in which they are combined to form water, there will be, as
before, only one component, water. From this, then, we see that a change in
the conditions of the experiment (in the present case a rise of
temperature) may necessitate a change in the number of the components.
It is, however, only in the case of systems of more than one component that
any difficulty will be found; for only in this case will a choice of
components be possible. Take, for instance, the dissociation of calcium
carbonate into calcium oxide and carbon dioxide. At each temperature, as we
have seen, there is a definite state of equilibrium. When equilibrium has
been established, there are three different substances present--calcium
carbonate, calcium oxide, and carbon dioxide; and these are the
constituents of the system between which equilibrium exists. Now, although
these constituents take part in the equilibrium, they are not all to be
regarded as components, for they are not mutually independent. On the
contrary, the different phases are related to one another, and if two of
these are taken, the composition of the third is defined by the equation
CaCO_{3} = CaO + CO_{2}
{12} Now, in deciding the number of components in any given system, not
only must the constituents chosen be capable of independent variation, but
a further restriction is imposed, and we obtain the following rule: _As the
components of a system there are to be chosen the_ smallest number _of
independently variable constituents by means of which the composition of
each phase participating in the state of equilibrium can be expressed in
the form of a chemical equation._
Applying this rule to the case under consideration, we see that of the
three constituents present when the system is in a state of equilibrium,
only two, as already stated, are independently variable. It will further be
seen that in order to express the composition of each phase present, two of
these constituents are necessary. The system is, therefore, one of _two
components_, or a system of the second order.
When, now, we proceed to the actual choice of components, it is evident
that any two of the constituents can be selected. Thus, if we choose as
components CaCO_{3} and CaO, the composition of each phase can be expressed
by the following equations:--
CaCO_{3} = CaCO_{3} + 0CaO
CaO = CaO + 0CaCO_{3}
CO_{2} = CaCO_{3} - CaO
As we see, then, both zero and negative quantities of the components have
been introduced; and similar expressions would be obtained if CaCO_{3} and
CO_{2} were chosen as components. The matter can, however, be simplified
and the use of negative quantities avoided if CaO and CO_{2} are chosen;
and it is, therefore, customary to select these as the components.
While it is possible in the case of systems of the second order to choose
the two components in such a way that the composition of each phase can be
expressed by positive quantities of these, such a choice is not always
possible when dealing with systems of a higher order (containing three or
four components).
From the example which has just been discussed, it might {13} appear as if
the choice of the components was rather arbitrary. On examining the point,
however, it will be seen that the arbitrariness affects only the _nature_,
not the _number_, of the components; a choice could be made with respect to
which, not to how many, constituents were to be regarded as components. As
we shall see presently, however, it is only the number, not the nature of
the components that is of importance.
After the discussion of the conditions which the substances chosen as
components must satisfy, another method may be given by which the number of
components present in a system can be determined. Suppose a system
consisting of several phases in equilibrium, and the composition of each
phase determined by analysis. If each phase present, regarded as a whole,
has the same composition, the system contains only one component, or is of
the first order. If two phases must be mixed in suitable quantities in
order that the composition of a third phase may be obtained, the system is
one of two components or of the second order; and if three phases are
necessary to give the composition of a fourth coexisting phase, the system
is one of three components, or of the third order.[16]
Although the examples to be considered in the sequel will afford sufficient
illustration of the application of the rules given above, one case may
perhaps be discussed to show the application of the method just given for
determining the number of components.
Consider the system consisting of Glauber's salt in equilibrium with
solution and vapour. If these three phases are analyzed, the composition of
the solid will be expressed by Na_{2}SO_{4}, 10H_{2}O; that of the solution
by Na_{2}SO_{4} + _x_H_{2}O, while the vapour phase will be H_{2}O. The
system evidently cannot be a one-component system, for the phases have not
all the same composition. By varying the amounts of two phases, however
(_e.g._ Na_{2}SO_{4}, 10H_{2}O and H_{2}O), the composition of the third
phase--the solution--can be obtained. The system is, therefore, one of _two
components_.
But sodium sulphate can also exist in the anhydrous form and as the hydrate
Na_{2}SO_{4}, 7H_{2}O. In these cases there may {14} be chosen as
components Na_{2}SO_{4} and H_{2}O, and Na_{2}SO_{4}, 7H_{2}O and H_{2}O
respectively. In both cases, therefore, there are two components. But the
two systems (Na_{2}SO_{4}, 10H_{2}O--H_{2}O, and Na_{2}SO_{4},
7H_{2}O--H_{2}O) can be regarded as special cases of the system
Na_{2}SO_{4}--H_{2}O, and these two components will apply to all systems
made up of sodium sulphate and water, no matter whether the solid phase is
anhydrous salt or one of the hydrates. In all three cases, of course, the
_number_ of components is the same; but by choosing Na_{2}SO_{4} and H_{2}O
as components, the possible occurrence of negative quantities of components
in expressing the composition of the phases is avoided; and, further, these
components apply over a much larger range of experimental conditions.
Again, therefore, we see that, although the number of the components of a
system is definite, a certain amount of liberty is allowed in the choice of
the substances; and we also see that the choice will be influenced by the
conditions of experiment.
Summing up, now, we may say--
(1) The components are to be chosen from among the constituents which are
present when the system is in a state of true equilibrium, and which take
part in that equilibrium.
(2) As components are to be chosen the _smallest number_ of such
constituents necessary to express the composition of each phase
participating in the equilibrium, zero and negative quantities of the
components being permissible.
(3) In any given system the _number_ of the components is definite, but may
alter with alteration of the conditions of experiment. A certain freedom of
choice, however, is allowed in the (qualitative, not quantitative)
selection of the components, the choice being influenced by considerations
of simplicity, suitability, or generality of application.[17]
Degree of Freedom. Variability of a System.--It is well known that in
dealing with a certain mass of gas or vapour, _e.g._ water vapour, if only
one of the independently variable factors--temperature, pressure, and
concentration (or volume)--is fixed, the state of the gas or vapour is
undefined; while occupying the same volume (the concentration, therefore,
remaining {15} unchanged), the temperature and the pressure may be altered;
at a given temperature, a gas can exist under different pressures and
occupy different volumes, and under any given pressure the temperature and
volume may vary. If, however, two of the factors are arbitrarily fixed,
then the third factor can only have a certain definite value; at any given
values of temperature and pressure a given mass of gas can occupy only a
definite volume.
Suppose, however, that the system consists of water in contact with vapour.
The condition of the system then becomes perfectly defined on arbitrarily
giving one of the variables a certain value. If the temperature is fixed,
the pressure under which water and water vapour can coexist is also
determined; and conversely, if a definite pressure is chosen, the
temperature is also defined. Water and vapour can coexist under a given
pressure only at a definite temperature.
Finally, let the water and vapour be cooled down until ice begins to
separate out. So soon as the third phase, ice, appears, the state of the
system as regards temperature and pressure of the vapour is perfectly
defined, and none of the variables can be arbitrarily changed without
causing the disappearance of one of the phases, ice, water, or vapour.
We see, therefore, that in the case of some systems two, in other cases,
only one of the independent variables (temperature, pressure,
concentration) can be altered without destroying the nature of the system;
while in other systems, again, these variables have all fixed and definite
values. We shall therefore define the number of degrees of freedom[18] of a
system as the _number of the variable factors, temperature, pressure, and
concentration of the components, which must be arbitrarily fixed in order
that the condition of the system may be perfectly defined_. From what has
been said, therefore, we shall describe a gas or vapour as having two
degrees of freedom; the system water--vapour as having only one; and the
system ice--water--vapour as having no degrees of freedom. We may also
speak of the {16} _variability_ or _variance_ of a system, and describe a
system as being invariant, univariant, bivariant, multivariant,[19]
according as the number of degrees of freedom is nought, one, two, or more
than two.
A knowledge of its variability is, therefore, of essential importance in
studying the condition and behaviour of a system, and it is the great merit
of the Phase Rule that _the state of a system is defined entirely by the
relation existing between the number of the components and the phases
present_, no account being taken of the molecular complexity of the
participating substances, nor any assumption made with regard to the
constitution of matter. It is, further, as we see, quite immaterial whether
we are dealing with "physical" or "chemical" equilibrium; in principle,
indeed, no distinction need be drawn between the two classes, although it
is nevertheless often convenient to make use of the terms, in spite of a
certain amount of indefiniteness which attaches to them--an indefiniteness,
indeed, which attaches equally to the terms "physical" and "chemical"
process.[20]
The Phase Rule.--The Phase Rule of Gibbs, which defines the condition of
equilibrium by the relation between the number of coexisting phases and the
components, may be stated as follows: A system consisting of n components
can exist in _n_ + 2 phases only when the temperature, pressure, and
concentration have fixed and definite values; if there are _n_ components
in _n_ + 1 phases, equilibrium can exist while one of the factors varies,
and if there are only _n_ phases, two of the varying factors may be
arbitrarily fixed. This rule, the application of which, it is hoped, will
become clear in the sequel, may be very concisely and conveniently
summarized in the form of the equation--
P + F = C + 2, or F = C + 2 - P
where P denotes the number of the phases, F the degrees of freedom, and C
the number of components. From the second form of the equation it can be
readily seen that the greater the number of the phases, the fewer are the
degrees of freedom. With increase in the number of the phases, therefore,
the {17} condition of the system becomes more and more defined, or less and
less variable.
Classification of Systems according to the Phase Rule.--We have already
learned in the introductory chapter that systems which are apparently quite
different in character may behave in a very similar manner. Thus it was
stated that the laws which govern the equilibrium between water and its
vapour are quite analogous to those which are obeyed by the dissociation of
calcium carbonate into carbon dioxide and calcium oxide; in each case a
certain temperature is associated with a definite pressure, no matter what
the relative or absolute amounts of the respective substances are. And
other examples were given of systems which were apparently similar in
character, but which nevertheless behaved in a different manner. The
relations between the various systems, however, become perfectly clear and
intelligible in the light of the Phase Rule. In the case first mentioned,
that of water in equilibrium with its vapour, we have one
component--water--present in two phases, _i.e._ in two physically distinct
forms, viz. liquid and vapour. According to the Phase Rule, therefore,
since C = 1, and P = 2, the degree of freedom F is equal to 1 + 2 - 2 = 1;
the system possesses one degree of freedom, as has already been stated. But
in the case of the second system mentioned above there are two components,
viz. calcium oxide and carbon dioxide (p. 12), and three phases, viz. two
solid phases, CaO and CaCO_{3}, and the gaseous phase, CO_{2}. The number
of degrees of freedom of the system, therefore, is 2 + 2 - 3 = 1; this
system, therefore, also possesses one degree of freedom. We can now
understand why these two systems behave in a similar manner; both are
univariant, or possess only one degree of freedom. We shall therefore
expect a similar behaviour in the case of all univariant systems, no matter
how dissimilar the systems may outwardly appear. Similarly, all bivariant
systems will exhibit analogous behaviour; and generally, systems possessing
the same degree of freedom will show a like behaviour. In accordance with
the Phase Rule, therefore, we may classify the different systems which may
be found into invariant, univariant, bivariant, multivariant, {18}
according to the relation which obtains between the number of the
components and the number of coexisting phases; and we shall expect that in
each case the members of any particular group will exhibit a uniform
behaviour. By this means we are enabled to obtain an insight into the
general behaviour of any system, so soon as we have determined the number
of the components and the number of the coexisting phases.
The adoption of the Phase Rule for the purposes of classification has been
of great importance in studying changes in the equilibrium existing between
different substances; for not only does it render possible the grouping
together of a large number of isolated phenomena, but the guidance it
affords has led to the discovery of new substances, has given the clue to
the conditions under which these substances can exist, and has led to the
recognition of otherwise unobserved resemblances existing between different
systems.
Deduction of the Phase Rule.--In the preceding pages we have restricted
ourselves to the statement of the Phase Rule, without giving any indication
of how it has been deduced. At the close of this chapter, therefore, the
mathematical deduction of the generalization will be given, but in brief
outline only, the reader being referred to works on Thermodynamics for a
fuller treatment of the subject.[21]
All forms of energy can be resolved into two factors, the _capacity_ factor
and the _intensity_ factor; but for the production of equilibrium, only the
intensity factor is of importance. Thus, if two bodies having the same
temperature are brought in contact with each other, they will be in
equilibrium as regards heat energy, no matter what may be the amounts of
heat (capacity factor) contained in either, because the intensity
factor--the temperature--is the same. But if the temperature of the two
bodies is different, _i.e._ if the intensity factor of heat energy is
different, the two bodies will no longer be in equilibrium; but heat will
pass from the hotter to the colder until both have the same temperature.
As with heat energy, so with chemical energy. If we have a substance
existing in two different states, or in two different {19} phases of a
system, equilibrium can occur only when the intensity factor of chemical
energy is the same. This intensity factor may be called the _chemical
potential_; and we can therefore say that a system will be in equilibrium
when the chemical potential of each component is the same in all the phases
in which the component occurs. Thus, for example, ice, water, and vapour
have, at the triple point, the same chemical potential.
The potential of a component in any phase depends not only on the
composition of the phase, but also on the temperature and the pressure (or
volume). If, therefore, we have a system of C components existing in P
phases, then, in order to fix the composition of unit mass of each phase,
it is necessary to know the masses of (C - 1) components in each of the
phases. As regards the composition, therefore, each phase possesses (C - 1)
variables. Since there are P phases, it follows that, as regards
composition, the whole system possesses P(C - 1) variables. Besides these
there are, however, two other variables, viz. temperature and pressure, so
that altogether a system of C components in P phases possesses P(C - 1) + 2
variables.
In order to define the state of the system completely, it will be necessary
to have as many equations as there are variables. If, therefore, there are
fewer equations than there are variables, then, according to the deficiency
in the number of the equations, one or more of the variables will have an
undefined value; and values must be assigned to these variables before the
system is entirely defined. The number of these undefined values gives us
the variability or the degree of freedom of the system.
The equations by which the system is to be defined are obtained from the
relationship between the potential of a component and the composition of
the phase, the temperature and the pressure. Further, as has already been
stated, equilibrium occurs when the potential of each component is the same
in the different phases in which it is present. If, therefore, we choose as
standard one of the phases in which all the components occur, then in any
other phase in equilibrium with {20} it, the potential of each component
must be the same as in the standard phase. For each phase in equilibrium
with the standard phase, therefore, there will be a definite equation of
state for each component in the phase; so that, if there are P phases, we
obtain for each component (P - 1) equations; and for C components,
therefore, we obtain C(P - 1) equations.
But we have seen above that there are P(C - 1) + 2 variables, and as we
have only C(P - 1) equations, there must be P(C - 1) + 2 - C(P - 1) = C + 2
- P variables undefined. That is to say, the degree of freedom (F) of a
system consisting of C components in P phases is--
F = C + 2 - P
* * * * *
{21}
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 H_{2}O. 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
abscissae 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 {22} 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.
[Illustration: FIG. 1.]
Now, the results of experiment are quite in agreement with the requirements
of the Phase Rule, and at any given temperature the system water--vapour
can exist in equilibrium only under a definite pressure.
The vapour pressure of water at different temperatures has been subjected
to careful measurement by Magnus,[22] Regnault,[23] Ramsay and Young,[24]
Juhlin,[25] Thiesen and Scheel,[26] and others. In the following table the
values of the vapour pressure from -10deg to +100deg are those calculated
from the measurements of Regnault, corrected by the measurements of Wiebe
and Thiesen and Scheel;[27] those from 120deg to 270deg were determined
{23} 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. | Temperature. | Pressure in cm.
| mercury. | | mercury.
-------------+-----------------+--------------+--------------------
| | |
-10deg | 0.213 | 120deg | 148.4
0deg | 0.458[29] | 130deg | 201.9
+20deg | 1.752 | 150deg | 356.8
40deg | 5.516 | 200deg | 1162.5
60deg | 14.932 | 250deg | 2973.4
80deg | 35.54 | 270deg | 4110.1
100deg | 76.00 | 364.3deg | 14790.4 (194.6 atm.)
| | (critical | (critical pressure).
| | temperature) |
-------------+-----------------+--------------+--------------------
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 {24} 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 364.3deg, 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 0deg, 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 0deg (_vide infra_), the sublimation curve must
end at that temperature.
The following are the values of the vapour pressure of ice between 0deg and
-50deg.[30]
{25}
VAPOUR PRESSURE OF ICE.
---------------------------------------------------------------
Temperature. | Pressure in mm. | Temperature. | Pressure in mm.
| mercury. | | mercury.
-------------+-----------------+--------------+----------------
-50deg | 0.050 | -8deg | 2.379
-40deg | 0.121 | -6deg | 2.821
-30deg | 0.312 | -4deg | 3.334
-20deg | 0.806 | -2deg | 3.925
-15deg | 1.279 | 0deg | 4.602
-10deg | 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 {26} of 8.1 atm. the melting point
of ice was -0.059deg; under a pressure of 16.8 atm. the melting point was
-0.129deg.
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.
---------------------------------------------------------------------
| Pressure in kilogms. per | Change of melting point for
Temperature. | sq. cm.[34] | an increase of pressure of
| | 1 kilogm. per sq. cm.
---------------------------------------------------------------------
-0deg | 1 |
-2.5deg | 336 | 0.0074deg
-5deg | 615 | 0.0090deg
-7.5deg | 890 | 0.0091deg
-10.0deg | 1155 | 0.0094deg
-12.5deg | 1410 | 0.0100deg
-15.0deg | 1625 | 0.0116deg
-17.5deg | 1835 | 0.0119deg
-20.0deg | 2042 | 0.0121deg
-22.1deg | 2200 | 0.0133deg
---------------------------------------------------------------------
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 0.0076deg,[35] or an increase of pressure of 135 atm. is required
to produce a lowering of the melting point of 1deg. We see further that the
fusion curve bends slightly as the pressure is increased, which signifies
that the variation of {27} the melting point with the pressure changes; at
-15deg, when the pressure is 1625 kilogm. per sq. cm., increase of pressure
by 1 kilogm. per sq. cm. lowers the melting point by 0.012deg. This
curvature of the fusion curve we shall later (Chap. IV.) see to be an
almost universal phenomenon.
[Illustration: FIG. 2.]
[Illustration: FIG. 3.]
Equilibrium between Ice, Water, and Vapour. The Triple Point.--On examining
the vapour-pressure curves of ice and water (Fig. 3), we see that at a
temperature of about 0deg 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 {28} 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 0deg 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 0.0076deg; the melting
point of ice, therefore, when under the pressure of its own vapour, will be
very nearly +0.0076deg, and the pressure of the vapour will be very
slightly greater than 4.579 mm., which is the pressure at 0deg (p. 21). The
difference is, however, slight, and may be neglected here. At the
temperature, then, of +0.0076deg, 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 {29} vapour of the same pressure, they must, of course,
also be in equilibrium with one another.
[Illustration: FIG. 4.]
Bivariant Systems of Water.--If we examine Fig. 4, we see that the curves
OA, OB, OC, which represent diagrammatically the conditions under which
water and vapour, ice and vapour, and water and ice are in equilibrium,
form the boundaries of three "fields," or areas, I., II., III. These areas,
now, represent the conditions for the existence of the single phases,
solid, liquid, and vapour respectively. At temperatures and pressures
represented by any point in the field I., solid only can exist as a stable
phase. Since we have here one component in only one phase, the system is
bivariant, and at any given temperature, therefore, ice can exist under a
series of pressures; and under any given pressure, at a series of
temperatures, these pressures and temperatures being limited only by the
curves OB, OC. Similarly also with the areas II. and III.
We see, further, that the different areas are the regions of stability of
the phase common to the two curves by which the area is enclosed.[37] Thus,
the phase common to the two systems {30} 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 0deg, 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 -9.4deg 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 0deg--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, {31} 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.
---------------------------------------------------------------------
| Pressure in mm. mercury.
------------------------------------------------------
Temperature. | | |
| Water. | Ice. | Difference.
---------------------------------------------------------------------
0deg | 4.618 | 4.602 | 0.016[40]
-2deg | 3.995 | 3.925 | 0.070
-4deg | 3.450 | 3.334 | 0.116
-8deg | 2.558 | 2.379 | 0.179
-10deg | 2.197 | 1.999 | 0.198
-15deg | 1.492 | 1.279 | 0.213
-20deg | 1.005 | 0.806 | 0.199
---------------------------------------------------------------------
At all temperatures below 0deg (more correctly +0.0076deg), 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 0deg, 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 0deg 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 {32} 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 0deg. 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 {33} 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 -22deg, 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.
{34}
Sulphur exists in two well-known crystalline forms--rhombic, or octahedral,
and monoclinic, or prismatic sulphur. Of these, the former melts at
114.5deg; the latter at 120deg.[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 96deg, 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.
[Illustration: FIG. 5.]
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 0deg 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 {35} 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._ {36}
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 95.5deg. 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. {37} 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 0.04deg-0.05deg.[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).
{38}
Triple Point--Monoclinic Sulphur, Liquid, and Vapour. Melting Point of
Monoclinic Sulphur.--Above 95.5deg, monoclinic sulphur is, as we have seen,
the stable form. On being heated to 120deg, 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 151deg, 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 151deg 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 {39} 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
(114.5deg). 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 120deg, 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 151deg 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, {40}
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.
[Illustration: FIG. 6.]
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).
{41}
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 {42} 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 20deg. 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 20deg,
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 -50deg. 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 0deg. 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 {43} 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."
[Illustration: FIG. 7.]
{44}
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 13.9deg and 27.2deg,[63]
and those of benzophenone at 26deg and 48deg.[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.
[Illustration: FIG. 8.]
[Illustration: FIG. 9.]
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, O_{3} is the transition point, O_{1} and O_{2} the
melting points of the metastable and stable forms {45} 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 O_{1}, 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 {46} 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 | Temperature.| Pressure | Temperature.| Pressure
| in cm. | | in atm. | | in atm.
------------+----------+-------------+-----------+-------------+---------
165deg | 12 | 360deg | 3.2 | 360deg | 0.1
180deg | 20.4 | 440deg | 7.5 | 440deg | 1.75
200deg | 26.6 | 494deg | 18.0 | 487deg | 6.8
219deg | 35.9 | 503deg | 21.9 | 510deg | 10.8
230deg | 51.4 | 511deg | 26.2 | 531deg | 16.0
290deg | 76.0 | -- | -- | 550deg | 31.0
-- | -- | -- | -- | 577deg | 56.0
-------------------------------------------------------------------------
These values are also represented graphically in Fig. 10.
[Illustration: FIG. 10.]
At all temperatures above about 260deg, 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 {47} 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, BO_{1} represents the conditions of equilibrium of the
univariant system red phosphorus and vapour, which ends at O_{1}, 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 630deg,[74] but the pressure at this
temperature is unknown.
At O_{1}, then, we have the triple point, red phosphorus, liquid, and
vapour, and starting from it, we should have the {48} vaporization curve of
liquid phosphorus, O_{1}A, and the fusion curve of red phosphorus, O_{1}F.
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.
[Illustration: FIG. 11.]
When white phosphorus is heated to 44deg, 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, O_{2}E, 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 0.029deg. 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 500deg; 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 {49} 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 O_{1}, the melting point of red
phosphorus, and must be continuous with the curve O_{1}A, 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 BO_{1} and O_{1}A 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 500deg and more, show it to have the
molecular weight represented by P_{4},[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 P_{8}, and very possibly higher.
In the case of phosphorus, therefore, it is more than possible that we are
dealing, not simply with two polymorphic {50} 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 630deg under high pressure also exists in a polymeric form, greater than
P_{4}, 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 350deg is less than the vapour pressure of
white phosphorus at 200deg, 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 324deg
and 350deg.[83]
Relationships similar to those found in the case of phosphorus are also met
with in the case of cyanogen and {51} 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 {52} 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 {53} 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]
[Illustration: FIG. 12.]
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).
{54}
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 118.27deg, and the change of
the transition point with the pressure is +0.032deg 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 135.85deg, and the
melting point is raised 0.0485deg 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 | Melting
| point. | point.
----------------------------------+------------+--------
| |
Cholesteryl benzoate | 145.5deg | 178.5deg
Azoxyanisole | 118.3deg | 135.9deg
Azoxyphenetole | 134.5deg | 168.1deg
Condensation product from | |
benzaldehyde and benzidine | 234deg | 260deg
Azine of _p_-oxyethylbenzaldehyde | 172deg | 196deg
Condensation product from | |
_p_-tolylaldehyde and benzidine | 231deg | --
_p_-Methoxycinnamic acid | 169deg | 185deg
----------------------------------+------------+--------
* * * * *
{55}
CHAPTER IV
GENERAL SUMMARY
In the preceding pages we have learned how the principles of the Phase Rule
can be applied to the elucidation of various systems consisting of one
component. In the present chapter it is proposed to give a short summary of
the relationships we have met with, and also to discuss more generally how
the Phase Rule applies to other one-component systems. On account of the
fact that beginners are sometimes inclined to expect too much of the Phase
Rule; to expect, for example, that it will inform them as to the exact
behaviour of a substance, it may here be emphasized that the Phase Rule is
a general rule; it informs us only as to the general conditions of
equilibrium, and leaves the determination of the definite, numerical data
to experiment.
Triple Point.--We have already (p. 28) defined a triple point in a
one-component system, as being that pressure and temperature at which three
phases coexist in equilibrium; it represents, therefore, an invariant
system (p. 16). At the triple point also, three curves cut, viz. the curves
representing the conditions of equilibrium of the three univariant systems
formed by the combination of the three phases in pairs. The most common
triple point of a one-component system is, of course, the triple point,
solid, liquid, vapour (S-L-V), but other triple points[99] are also
possible when, as in the case of {56} sulphur or benzophenone, polymorphic
forms occur. Whether or not all the triple points can be experimentally
realized will, of course, depend on circumstances. We shall, in the first
place, consider only the triple point S-L-V.
As to the general arrangement of the three univariant curves around the
triple point, the following rules may be given. (1) The prolongation of
each of the curves beyond the triple point must lie between the other two
curves. (2) The middle position at one and the same temperature in the
neighbourhood of the triple point is taken by that curve (or its metastable
prolongation) which represents the two phases of most widely differing
specific volume.[100] That is to say, if a line of constant temperature is
drawn immediately above or below the triple point so as to cut the three
curves--two stable curves and the metastable prolongation of the third--the
position of the curves at that temperature will be such that the middle
position is occupied by that curve (or its metastable prolongation) which
represents the two phases of most widely differing specific volume.
Now, although these rules admit of a considerable variety of possible
arrangements of curves around the triple point,[101] only two of these have
been experimentally obtained in the case of the triple point
solid--liquid--vapour. At present, therefore, we shall consider only these
two cases (Figs. 13 and 14).
[Illustration: FIG. 13.]
[Illustration: FIG. 14.]
An examination of these two figures shows that they satisfy the rules laid
down. Each of the curves on being prolonged passes between the other two
curves. In the case of substances of the first type (Fig. 13), the specific
volume of the solid is greater than that of the liquid (the substance
contracts on fusion); the difference of specific volume will, therefore, be
greatest between liquid and vapour. The curve, therefore, for liquid and
vapour (or its prolongation) must lie between the other two curves; this is
seen from the figure to be the case. Similarly, the rule is satisfied by
the arrangement of curves in Fig. 14, where the difference of specific
volumes is {57} greatest between the solid and vapour. In this case the
curve S-V occupies the intermediate position.
As we see, the two figures differ from one another only in that the fusion
curve OC in one case slopes to the right away from the pressure axis, thus
indicating that the melting point is raised by increase of pressure; in the
other case, to the left, indicating a lowering of the melting point with
the pressure. These conditions are found exemplified in the case of sulphur
and ice (pp. 29 and 35). We see further from the two figures, that O in
Fig. 13 gives the highest temperature at which the solid can exist, for the
curve for solid--liquid slopes back to regions of lower temperature; in
Fig. 14, O gives the lowest temperature at which the liquid phase can exist
as stable phase.[102]
Theorems of van't Hoff and of Le Chatelier.--So far we have studied only
the conditions under which various systems exist in equilibrium; and we now
pass to a consideration of the changes which take place in a system when
the external conditions of temperature and pressure are altered. For all
such changes there exist two theorems, based on the laws of thermodynamics,
by means of which the alterations in a system can be qualitatively
predicted.[103] The first of these, usually {58} known as van't Hoff's _law
of movable equilibrium_,[104] states: When the temperature of a system in
equilibrium is raised, that reaction takes place which is accompanied by
absorption of heat; and, conversely, when the temperature is lowered, that
reaction occurs which is accompanied by an evolution of heat.
The second of the two theorems refers to the effect of change of pressure,
and states:[105] When the pressure on a system in equilibrium is increased,
that reaction takes place which is accompanied by a diminution of volume;
and when the pressure is diminished, a reaction ensues which is accompanied
by an increase of volume.
The demonstration of the universal applicability of these two theorems is
due chiefly to Le Chatelier, who showed that they may be regarded as
consequences of the general law of action and reaction. For this reason
they are generally regarded as special cases of the more general law, known
as the _theorem of Le Chatelier_, which may be stated in the words of
Ostwald, as follows:[106] _If a system in equilibrium is subjected to a
constraint by which the equilibrium is shifted, a reaction takes place
which opposes the constraint, _i.e._ one by which its effect is partially
destroyed._
This theorem of Le Chatelier is of very great importance, for it applies to
all systems and changes of the condition of equilibrium, whether physical
or chemical; to vaporization and fusion; to solution and chemical action.
In all cases, whenever changes in the external condition of a system in
equilibrium are produced, processes also occur within the system which tend
to counteract the effect of the external changes.
_Changes at the Triple Point._--If now we apply this theorem to equilibria
at the triple point S-L-V, and ask what changes will occur in such a system
when the external conditions of pressure and temperature are altered, the
general answer to the question will be: So long as the three phases are
present, no {59} change in the temperature or pressure of the system can
occur, but _only changes in the relative amounts of the phases_; that is to
say, the effect on the system of change in the external conditions is
opposed by the reactions or changes which take place within the system
(according to the theorems of van't Hoff and Le Chatelier). We now proceed
to discuss what these changes are, and shall consider first the effect of
alteration of the temperature at constant volume and constant pressure, and
then the effect of alteration of the pressure both when the temperature
remains constant and when it varies.
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 passes 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 pass into liquid, _i.e._ there occurs the
change S + V --> L. It will, therefore, depend on the relative quantities
of solid and vapour, which will disappear first. If the solid disappears
first, then we shall pass 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 quantity of the solid must also
pass 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 passes along the line OD into the
region of vapour; on heat being withdrawn, the system passes along OD' into
the area of solid.
[Illustration: 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 pass 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).
[Illustration: FIG. 16.]
[Illustration: FIG. 17.]
Lastly, if the temperature is maintained constant, _i.e._ if heat can pass
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 passing 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 passing 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.
{63}
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 | 35deg
[alpha]-rhombic --> rhombohedral | 83deg
Rhombohedral --> regular | 125deg
Mercuric iodide | 126deg
Potassium nitrate | 129deg
Silver iodide | 145deg
Silver nitrate | 160deg
Sulphur | 95.5deg
Tetrabrommethane | 46.8deg
Thallium nitrate-- |
Rhombic --> rhombohedral | 80deg
Rhombohedral --> regular | 142.5deg
Thallium picrate | 46deg
Tin | 20deg
------------------------------------+-------------
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 assumed vapour to be given off by the
solid substance, although the pressure of the vapour has not hitherto been
measured. The assumption, 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.
{64}
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
quantitative 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 participating 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 constitute 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 passing 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.
{65}
If transference of heat to or from the system is prevented, increase of
volume (diminution of pressure) will cause the system L-V to pass along the
curve BO; liquid will pass 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 pass along the curve AO to higher temperatures and pressures; at O
the solid will melt, and the system will ultimately pass 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 pass 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 relationship 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 relationship 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 passing 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 passage 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
_quantitatively_ 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 0.03deg,
but in the case of water the change is much less (0.0076deg), and in the
case of camphor much more (0.13deg). 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 1deg.
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 passage 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.
-------------+----------
|
85.85deg | 1 atm.
84.38deg | 100 "
83.03deg | 200 "
82.29deg | 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 passes through a point of maximum
temperature, and exhibits, therefore, a form similar to that assumed by
Tammann for the fusion curve.[124]
{69}
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
passes into such a condition that the existence of that phase is possible_;
but rather, instead of the system undergoing transformation so as to pass
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, quantitative 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 x
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.