point D was found to be 27.7deg. The equilibrium curve DE was determined by
heating the liquid mixtures at different temperatures until equilibrium was
attained, and then rapidly cooling the liquid. In all cases the freezing
point was practically that of the point D. From this it is seen that the
equilibrium curve must be a straight line parallel to the temperature axis;
and, therefore, isomeric transformation in the case of the two
benzaldoximes is not accompanied by any heat effect (p. 197). This
behaviour has also been found in the case of acetaldoxime.[287]
The isomeric benzaldoximes are also of interest from the fact that the
stable modification has the _lower_ melting point (_v._ p. 202).
_Acetaldehyde and Paraldehyde._--As a second example of the equilibria
between two isomerides, we shall take the two isomeric (polymeric) forms of
acetaldehyde, which have recently been exhaustively studied.[288]
{205}
In the case of these two substances the reaction
3CH_{3}.CHO <--> (CH_{3}.CHO)_{3}
takes place at the ordinary temperature with very great slowness. For this
reason it is possible to determine the freezing point curves of
acetaldehyde and paraldehyde. The three chief points on these curves,
represented graphically in Fig. 62, are:--
m.p. of acetaldehyde - 118.45deg
m.p. of paraldehyde + 12.55deg
eutectic point - 119.9deg
[Illustration: FIG. 62.]
In order to determine the position of the natural melting point, it was
necessary, on account of the slowness of transformation, to employ a
catalytic agent in order to increase the velocity with which the
equilibrium was established. A drop of concentrated sulphuric acid served
the purpose. In presence of a trace of this substance, isomeric
transformation very speedily occurs, and leads to the condition of
equilibrium. Starting in the one case with fused paraldehyde, and in the
other case with acetaldehyde, the same freezing point, viz. 6.75deg, was
obtained, the solid phase being paraldehyde. This temperature, 6.75deg, is
therefore the natural freezing point, and paraldehyde, the solid in
equilibrium with the liquid phase at this point, is the stable form.
With regard to the change of equilibrium with the temperature, it was found
that whereas the liquid phase contained 11.7 molecules per cent. of
acetaldehyde at the natural freezing point, the liquid at the temperature
of 41.6deg contains 46.6 molecules per cent. of acetaldehyde. As the
temperature {206} rises, therefore, there is increased formation of
acetaldehyde, or a decreasing amount of polymerisation. This is in harmony
with the fact that the polymerisation of acetaldehyde is accompanied by
evolution of heat.
While speaking of these isomerides, it may be mentioned that at the
temperature 41.6deg the equilibrium mixture has a vapour pressure equal to
the atmospheric pressure. At this temperature, therefore, the equilibrium
mixture (obtained quickly with the help of a trace of sulphuric acid)
boils.[289]
* * * * *
{207}
CHAPTER XII
SUMMARY.--APPLICATION OF THE PHASE RULE TO THE STUDY OF SYSTEMS OF TWO
COMPONENTS
In this concluding chapter on two-component systems, it is proposed to
indicate briefly how the Phase Rule has been applied to the elucidation of
a number of problems connected with the equilibria between two components,
and how it has been employed for the interpretation of the data obtained by
experiment. It is hoped that the practical value of the Phase Rule may
thereby become more apparent, and its application to other cases be
rendered easier.
The interest and importance of investigations into the conditions of
equilibrium between two substances, lie in the determination not only of
the conditions for the stable existence of the participating substances,
but also of whether or not chemical action takes place between these two
components; and if combination occurs, in the determination of the nature
of the compounds formed and the range of their existence. In all such
investigations, the Phase Rule becomes of conspicuous value on account of
the fact that its principles afford, as it were, a touchstone by which the
character of the system can be determined, and that from the form of the
equilibrium curves obtained, conclusions can be drawn as to the nature of
the interaction between the two substances. In order to exemplify the
application of the principles of the Phase Rule more fully than has already
been done, illustrations will be drawn from investigations on the
interaction of organic compounds; on the equilibria between optically
active compounds; and on alloys. {208}
Summary of the Different Systems of Two Components.--Before passing to the
consideration of the application of the Phase Rule to the investigation of
particular problems, it will be well to collect together the different
types of equilibrium curves with which we are already acquainted; to
compare them with one another, in order that we may then employ these
characteristic curves for the interpretation of the curves obtained as the
result of experiment.
In investigating the equilibria between two components, three chief classes
of curves will be obtained according as--
I. No combination takes place between the two components.
II. The components can form definite compounds.
III. The components separate out in the form of mixed crystals.
The different types of curves which are obtained in these three cases are
represented in Figs. 63, 64, 65. These different diagrams represent the
whole series of equilibria, from the melting point of the one component (A)
to that of the other component (B). The curves represent, in all cases, the
composition of the solution, or phase of variable composition; the
temperature being measured along one axis, and the composition along the
other.
We shall now recapitulate very briefly the characteristics of the different
curves.
[Illustration: FIG. 63.]
If no compound is formed between the two components, {209} the general form
of the equilibrium curve will be that of curve I. or II., Fig. 63. Type I.
is the simplest form of curve found, and consists, as the diagram shows, of
only two branches, AC and BC, meeting at the point C, _which lies below the
melting point of either component_. The solid phase which is in equilibrium
with the solutions AC is pure A; that in equilibrium with BC, pure B. C is
the eutectic point. Although at the eutectic point the solution solidifies
entirely without change of temperature, the solid which is deposited is not
a homogeneous solid phase, but a mixture, or conglomerate of the two
components. _The eutectic point, therefore, represents the melting or
freezing point, not of a compound, but of a mixture_ (p. 119).
Curve II., Fig. 63, is obtained when two liquid phases are formed. C is an
eutectic point, D and F are transition points at which there can co-exist
the four phases--solid, two liquid phases, vapour. DEF represents the
change in the composition of the two liquid phases with rise of
temperature; the curve might also have the reversed form with the critical
solution point below the transition points D and F.
[Illustration: FIG. 64.]
In the second class of systems (Fig. 64), that in which combination between
the components occurs, there are again two types according as the compound
formed has a definite melting point (_i.e._ can exist in equilibrium with a
solution of the same composition), or undergoes only partial fusion; that
is, exhibits a transition point.
If a compound possessing a definite melting point is formed, the
equilibrium curve will have the general form shown by curve I., Fig. 64. A,
B, and D are the melting points of pure A, pure B, and of the compound
A_{x}B_{y} respectively. AC {210} is the freezing point curve of A in
presence of B; BE that of B in presence of A; and DC and DE the freezing
point curves of the compound in presence of a solution containing excess of
one of the components. C and E are eutectic points at which mixtures of A
and A_{x}B_{y}, or B and A_{x}B_{y} can co-exist in contact with solution.
The curve CDE may be large or small, and the melting point of the compound,
D, may lie above or below that of each of the components, or may have an
intermediate position. If more than one compound can be formed, a series of
curves similar to CDE will be obtained (_cf._ p. 152).
On the other hand, if the compound undergoes transition to another solid
phase at a temperature below its melting point, a curve of the form II.,
Fig. 64, will be found. This corresponds to the case where a compound can
exist only in contact with solutions containing excess of one of the
components. The metastable continuation of the equilibrium curve for the
compound is indicated by the dotted line, the summit of which would be the
melting point of the compound. Before this temperature is reached, however,
the solid compound ceases to be able to exist in contact with solution, and
transition to a different solid phase occurs at the point E (_cf._ p. 134).
This point, therefore, represents the limit of the existence of the
compound AB. If a series of compounds can be formed none of which possess a
definite melting point, then a series of curves will be obtained which do
not exhibit a temperature-maximum, and there will be only one eutectic
point. The limits of existence of each compound will be marked by a break
in the curve (_cf._ p. 143).
[Illustration: FIG. 65.]
Turning, lastly, to the third class of systems, in which formation of mixed
crystals can occur, five different types of curves can be obtained, as
shown in Fig. 65. With regard to the first three types, curves I., II., and
III., {211} these differ entirely from those of the previous classes, in
that they are continuous; they exhibit no eutectic point, and no transition
point. Curve II. bears some resemblance to the melting-point curve of a
compound (_e.g._ CDE, Fig. 64, I.), but differs markedly from it in not
ending in eutectic points.
Further, in the case of the formation of a compound, the composition of the
solid phase remains unchanged throughout the whole curve between the
eutectic points; whereas, when mixed crystals are produced, the composition
of the solid phase varies with the composition of the liquid solution. On
passing through the maximum, the relative proportions of A and B in the
solid and the liquid phase undergo change; on the one side of the maximum,
the solid phase contains relatively more A, and on the other side of the
maximum, relatively more B than the liquid phase. Lastly, when mixed
crystals are formed, the temperature at which complete solidification
occurs changes as the composition of the solution changes, whereas in the
case of the formation of compounds, the temperature of complete
solidification for all solutions is a eutectic point.
The third type of curve, Fig. 65, can be distinguished in a similar manner
from the ordinary eutectic curve, Fig. 63, I., to which it bears a certain
resemblance. Whereas in the case of the latter, the eutectic point is the
temperature of complete solidification of all solutions, the point of
minimum temperature in the case of the formation of mixed crystals, is the
solidification point only of solutions having one particular composition;
that, namely, of the minimum point. For all other solutions, the
temperature of complete solidification is different. Whereas, also, in the
case of the simple eutectic curve, the solid which separates out from the
solutions represented by either curve remains the same throughout the whole
extent of that curve, the composition of the mixed crystal varies with
variation of the composition of the liquid phase, and the relative
proportions of the two components in the solid and the liquid phase are
reversed on passing through the minimum.[290]
In a similar manner, type IV., Fig. 65, can be distinguished from type II.,
Fig. 64, by the fact that it does not exhibit a {212} eutectic point, and
that the composition of the solid phase undergoes continuous variation with
variation of the liquid phase on either side of the transition point.
Lastly, type V., which does exhibit a eutectic point, differs from the
eutectic curve of Fig. 63, in that the eutectic point does not constitute
the point of complete solidification for all solutions, and that the
composition of the solid phase varies with the composition of the liquid
phase.
Such, then, are the chief general types of equilibrium curves for
two-components; they are the pattern curves with which other curves,
experimentally determined, can be compared; and from the comparison it will
be possible to draw conclusions as to the nature of the equilibria between
the two components under investigation.
1. _Organic Compounds._
[Illustration: FIG. 66.]
The principles of the Phase Rule have been applied to the investigation of
the equilibria between organic compounds, and Figs. 66-69 reproduce some of
the results which have been obtained.[291]
{213}
Fig. 66, the freezing point curve (curve of equilibrium) for
_o_-nitrophenol and _p_-toluidine, shows a curve of the simplest type[292]
(type I., Fig. 63), in which two branches meet at an eutectic point. The
solid phase in equilibrium with solutions represented by the left-hand
branch of the curve was _o_-nitrophenol (m.p. 44.1deg); that in equilibrium
with the solutions represented by the right-hand branch, was _p_-toluidine
(m.p. 43.3deg). At the eutectic point (15.6deg), these two solid phases
could co-exist with the liquid phase. This equilibrium curve, therefore,
shows that _o_-nitrophenol and _p_-toluidine do not combine with one
another.
In connection with this curve, attention may be called to the interesting
fact that although the solid produced by cooling the liquid phase at the
eutectic point has a composition approximating to that of a compound of
equimolecular proportions of the phenol and toluidine, and a constant
melting point, it is nevertheless a _mixture_. Although, as a rule, the
constituents of the eutectic mixture are not present in simple molecular
proportions, there is no reason why they should not be so; and it is
therefore necessary to beware of assuming the formation of compounds in
such cases.[293]
Fig. 67, on the other hand, indicates with perfect certainty the formation
of a compound between phenol and [alpha]-naphthylamine.[294] (_Cf._ curve
I., Fig. 64.)
Phenol freezes at 40.4deg, but the addition of [alpha]-naphthylamine lowers
the freezing point as represented by the curve AC. At C (16.0deg) the
compound C_{6}H_{5}OH,C_{10}H_{7}NH_{2} is formed, and the system becomes
invariant. On increasing the amount of the amine, the temperature of
equilibrium rises, the solid phase now being the compound. At D, the curve
passes through a maximum (28.8deg), at which the solid and liquid phases
have the same composition. This is the melting point of the compound.
Further addition of the amine lowers the temperature of equilibrium, until
at E solid [alpha]-naphthylamine separates out, and a second eutectic point
(24.0deg) is obtained. BE is the {214} freezing-point curve of
[alpha]-naphthylamine in presence of phenol, the freezing point of the pure
amine being 48.3deg.
On account of the great sluggishness with which the compound of phenol and
[alpha]-naphthylamine crystallizes, it was found possible to follow the
freezing point curves of phenol and the amine to temperatures considerably
below the eutectic points, as shown by the curves CF and EG.
[Illustration: FIG. 67.]
Phenol can also combine with _p_-toluidine in equimolecular proportions;
and this compound is of interest, from the fact that it exists in two
crystalline forms melting at 28.5deg and 30deg. Each of these forms now
must have its own equilibrium curve, and it was found that the intermediate
portion of the freezing point curve was duplicated, as shown in Fig.
68.[295]
{215}
[Illustration: FIG. 68.]
[Illustration: FIG. 69.]
{216}
Lastly, a curve is given, Fig. 69,[296] which corresponds with curve II.,
Fig. 64. Picric acid and benzene can form a compound, which, however, can
exist only in contact with solutions _containing excess of benzene_. When
the temperature is raised, a point (K) is reached at which the compound
melts with separation of solid picric acid. The point, K, is, therefore, a
_transition point_; analysis, however, showed that the composition of the
solution at this point is very nearly that of the compound
C_{6}H_{2}(NO_{2})_{3}OH,C_{6}H_{6}, so that the melting point of the
compound can almost be reached. The fusion of the compound of benzene and
picric acid with separation of the latter is analogous to the (partial)
fusion of Glauber's salt with separation of anhydrous sodium sulphate.
2. _Optically Active Substances._
The question as to whether a resolvable inactive body is a mixture of the
two oppositely active constituents (a _dl_-mixture), or a racemic compound,
is one which has given rise to considerable discussion during the past
decade; and several investigators have endeavoured to establish general
rules by which the question could be decided. In the case of inactive
liquids it is a matter of great difficulty to arrive at a certain
conclusion as to whether one is dealing with a mixture or a compound, for
in this case the usual physical methods give but a dubious answer; and
although the existence of a racemate in the liquid state (in the case of
conine) has been asserted,[297] most chemists incline to the belief that
such a thing is improbable.
Even in the case of crystalline substances, where the differences between
the various forms is greater, it was not always easy to discriminate
between the _dl_-mixture and the racemic compound. The occurrence of
hemihedral faces was considered by Pasteur to be a sufficient criterion for
an optically active substance. It has, however, been found that hemihedry
in crystals, although a frequent accompaniment of {217} optical activity,
is by no means a necessary or constant expression of this property. Other
rules, also, which were given, although in some cases reliable, were in
other cases insufficient; and all were in so far unsatisfactory that they
lacked a theoretical basis.
With the help of the Phase Rule, however, it is possible from a study of
the solubility or fusion curves of the optically active and inactive
substances, to decide the nature of the inactive substance, at least under
certain conditions. On account of the interest and importance which these
compounds possess, a brief description of the application of the Phase Rule
to the study of such substances will be given here;[298] the two optical
antipodes being regarded as the two components.
In the present chapter we shall consider only the fusion curves, the
solubility curves being discussed in the next section on three-component
systems. The rules which are hereby obtained, have reference only to the
nature of the inactive substance in the neighbourhood of the melting
points.
I. _The inactive substance is a _dl_-mixture._
In this case the fusion curves will have the simple form shown in type I,
Fig. 63. A and B are the melting points of the two optical isomerides, and
C the eutectic point at which the inactive mixture consisting of equal
amounts of d- and l-form melts. Owing to the similar effect of the one form
on the freezing point of the other, the figure is symmetrical. No example
of this simple case has been investigated.
II. _The two components form a racemic compound._
In this case there will be three melting point curves as in Fig. 64, type
I. In this case also the figure must be symmetrical.
Examples.--As examples of this, may be taken dimethyl tartrate and mandelic
acid, the freezing point curves of which are given in Figs. 70 and 71.[299]
As can be seen, the curve for the racemic tartrate occupies a large part of
the diagram, {218} while that for racemic mandelic acid is much smaller. In
the case of dimethyldiacetyl tartrate, this middle portion is still less.
[Illustration: FIG. 70.]
[Illustration: FIG. 71.]
[Illustration: FIG. 72.]
Active dimethyl tartrate melts at 43.3deg; racemic dimethyl tartrate at
89.4deg. Active mandelic acid melts at 132.8deg; the racemic acid at
118.0deg. In the one case, therefore, the racemic compound has a higher, in
the other a lower melting point than the active forms. {219}
In the case of partially racemic compounds (_i.e._ the compound of a
racemate with an optically active substance) the type of curve will be the
same, but the figure will no longer be symmetrical. Such a curve has been
found in the case of the l-menthyl esters of d- and l-mandelic acid (Fig.
72).[300] The freezing point of l-menthyl d-mandelate is 97.2deg, of
l-menthyl l-mandelate 77.6deg, and of l-menthyl r-mandelate 83.7.deg It
will be observed that the summit of the curve for the partially racemic
mandelate is very flat, indicating that the compound is largely dissociated
into its components at the temperature of fusion.
III. _The inactive substance is a pseudo-racemic mixed crystal._
In cases where the active components can form mixed crystals, the
freezing-point curve will exhibit one of the forms given in Fig. 65. The
inactive mixed crystal containing 50 per cent. of the dextro and laevo
compound, is known as a pseudo-racemic mixed crystal.[301] So far, only
curves of the types I. and II. have been obtained.
Examples.--The two active camphor oximes are of interest from the fact that
they form a continuous series of mixed crystals, _all of which have the
same melting point_. The curve which is obtained in this case is,
therefore, a straight line joining the melting points of the pure active
components; the melting point of the active isomerides and of the whole
series of mixed crystals being 118.8deg.
[Illustration: FIG. 73.]
In the case of the carvoximes mixed crystals are also formed, but the
equilibrium curve in this case exhibits a maximum (Fig. 73). At this
maximum point the composition of the solid and of the liquid solution is
the same. Since the curve must be symmetrical, this maximum point must
occur in the case of the solution containing 50 per cent. {220} of each
component, which will therefore be inactive. Further, this inactive mixed
crystal will melt and solidify at the same temperature, and behave,
therefore, like a chemical compound (p. 187). The melting point of the
active compounds is 72deg; that of the inactive pseudo-racemic mixed
crystal is 91.4deg.
Transformations.--As has already been remarked, the conclusions which can
be drawn from the fusion curves regarding the nature of the inactive
substances formed hold only for temperatures in the neighbourhood of the
melting points. At temperatures below the melting point transformation may
occur; _e.g._ a racemate may break up into a _dl_-mixture, or a
pseudo-racemic mixed crystal may form a racemic compound. We shall at a
later point meet with examples of a racemic compound changing into a
_dl_-mixture at a definite transition point; and the pseudo-racemic mixed
crystal of camphoroxime is an example of the second transformation.
Although at temperatures in the neighbourhood of the melting point the two
active camphoroximes form only mixed crystals but no compound, a racemic
compound is formed at temperatures below 103deg. At this temperature the
inactive pseudo-racemic mixed crystal changes into a racemic compound; and
in the case of the other mixed crystals transformation to racemate and
(excess of) active component also occurs, although at a lower temperature
than in the case of the inactive mixed crystal. Although this behaviour is
one of considerable importance, this brief reference to it must suffice
here.[302]
3. _Alloys._
One of the most important classes of substances in the study of which the
Phase Rule has been of very considerable importance, is that formed by the
mixtures or compounds of metals with one another known as alloys. Although
in the investigation of the nature of these bodies various methods are
employed, one of the most important is the determination of the character
of the freezing-point curve; for from the form of this, valuable
information can, as we have already learned, be {221} obtained regarding
the nature of the solid substances which separate out from the molten
mixture.
Although it is impossible here to discuss fully the experimental results
and the oftentimes very complicated relationships which the study of the
alloys has brought to light, a brief reference to these bodies will be
advisable on account both of the scientific interest and of the industrial
importance attaching to them.[303]
We have already seen that there are three chief types of freezing-point
curves in systems of two components, viz. those obtained when (1) the pure
components crystallize out from the molten mass; (2) the components form
one or more compounds; (3) the components form mixed crystals. In the case
of the metals, representatives of these three classes are also found.
1. _The components separate out in the pure state._
In this case the freezing-point curve is of the simple type, Fig. 63, I.
Such curves have been obtained in the case of a number of pairs of metals,
_e.g._ zinc--cadmium, zinc--aluminium, copper--silver (Heycock and
Neville), tin--zinc, bismuth--lead (Gautier), and in other cases. From
molten mixtures represented by one branch of the freezing-point curve one
of the metals will be deposited; while from mixtures represented by the
other branch, the other metal will separate out. At the eutectic point the
molten mass will solidify to a _heterogeneous mixture_ of the two metals,
forming what is known as the _eutectic alloy_. Such an alloy, therefore,
will melt at a definite temperature lower than the melting point of either
of the pure metals.
{222}
In the following table are given the temperature and the composition of the
liquid at the eutectic point, for three pairs of metals:--
-------------------------------------------------------------------
| Temperature. | Composition of liquid.
-------------------------------------------------------------------
Zinc--cadmium | 264.5deg | 73.5 atoms per cent. of cadmium.
Zinc--aluminium | 380.5deg | 11 " " aluminium.
Copper--silver | 778deg | 40 " " copper.
-------------------------------------------------------------------
The melting points of the pure metals are, zinc, 419deg; cadmium, 322deg;
silver, 960deg; copper, 1081deg; aluminium, 650deg.
2. _The two metals can form one or more compounds._
In this case there will be obtained not only the freezing-point curves of
the pure metals, but each compound formed will have its own freezing-point
curve, exhibiting a point of maximum temperature, and ending on either side
in an eutectic point. The simplest curve of this type will be obtained when
only one compound is formed, as is the case with mercury and thallium.[304]
This curve is represented in Fig. 74, where the summit of the intermediate
curve corresponds with a composition TlHg_{2}. Similar curves are also
given by nickel and tin, by aluminium and silver, and by other metals, the
formation of definite compounds between these pairs of metals being thereby
indicated.[305]
[Illustration: FIG. 74.]
{223}
A curve belonging to the same type, but more complicated, is obtained with
gold and aluminium;[306] in this case, several compounds are formed, some
of which have a definite melting point, while others exhibit only a
transition point. The chief compound is AuAl_{2}, which has practically the
same melting point as pure gold.
3. _The two metals form mixed crystals (solid solutions)._
The simplest case in which the metals crystallize out together is found in
silver and gold.[307] The freezing-point curve in this case is an almost
straight line joining the freezing points of the pure metals (_cf._ curve
I., Fig. 65, p. 210). These two metals, therefore, can form an unbroken
series of mixed crystals.
In some cases, however, the two metals do not form an unbroken series of
mixed crystals. In the case of zinc and silver,[308] for example, the
addition of silver _raises_ the freezing point of the mixture, until a
transition point is reached. This corresponds with curve IV., Fig. 65.
Silver and copper, and gold and copper, on the other hand, do not form
unbroken series of mixed crystals, but the freezing-point curve exhibits an
eutectic point, as in curve V., Fig. 65.
Not only may there be these three different types of curves, but there may
also be combinations of these. Thus the two metals may not only form
compounds, but one of the metals may not separate out in the pure state at
all, but form mixed crystals. In this case the freezing point may rise (as
in the case of silver and zinc), and one of the eutectic points will be
absent.
Iron-Carbon Alloys.--Of all the different binary alloys, probably the most
important are those formed by iron and carbon: alloys consisting not of two
metals, but of a metal and a non-metal. On account of the importance of
these alloys, an attempt will be made to describe in brief some of the most
important relationships met with.
Before proceeding to discuss the applications of the Phase Rule to the
study of the iron-carbon alloys, however, the main {224} facts with which
we have to deal may be stated very briefly. With regard to the metal
itself, it is known to exist in three different allotropic modifications,
called [alpha]-, [beta]-, and [gamma]-ferrite respectively. Like the two
modifications of sulphur and of tin, these different forms exhibit
transition points at which the relative stability of the forms changes.
Thus the transition point for [alpha]- and [beta]-ferrite is about 780deg;
and below this temperature the [alpha]- form, above it the [beta]- form is
stable. For [beta]- and [gamma]-ferrite, the transition point is about
870deg, the [gamma]- form being the stable modification above this
temperature.
The different modifications of iron also possess different properties.
Thus, [alpha]-ferrite is magnetic, but does not possess the power of
dissolving carbon; [beta]-ferrite is non-magnetic, and likewise does not
dissolve carbon; [gamma]-ferrite is also non-magnetic, but possesses the
power of dissolving carbon, and of thus giving rise to solid solutions of
carbon in iron.
Various alloys of iron and carbon, also, have to be distinguished. First of
all, there is _hard steel_, which contains varying amounts of carbon up to
2 per cent. Microscopic examination shows that these mixtures are all
homogeneous; and they are therefore to be regarded as solid solutions of
carbon in iron ([gamma]-ferrite). To these solutions the name _martensite_
has been given. _Pearlite_ contains about 0.8 per cent. of carbon, and, on
microscopic examination, is found to be a heterogeneous mixture. If heated
above 670deg, pearlite becomes homogeneous, and forms martensite. Lastly,
there is a definite compound of iron and carbon, iron carbide or
_cementite_, having the formula Fe_{3}C.
A short description may now be given of the application of the Phase Rule
to the two-component system iron--carbon; and of the diagram showing how
the different systems are related, and with the help of which the behaviour
of the different mixtures under given conditions can be predicted.
Although, with regard to the main features of this diagram, the different
areas to be mapped and the position of the frontier lines, there is general
agreement; a final decision has not yet been reached with regard to the
interpretation to be put on all the curves.
[Illustration: FIG. 75.]
The chief relationships met with in the case of the {225} iron-carbon
alloys are represented graphically in Fig. 75.[309] The curve AC is the
freezing-point curve for iron,[310] BC the unknown freezing-point curve for
graphite. C is an eutectic point. Suppose, now, that we start with a
mixture of iron and carbon, represented by the point _x_. On lowering the
temperature, a point, _y_, will be reached at which solid begins to
separate out. This solid phase, however, is not pure iron, but a solid
solution of carbon in iron, having the composition represented by _y'_ (cf.
p. 185). As the temperature continues to fall, the {226} composition of the
liquid phase changes in the direction of _y_C, while the composition of the
solid which separates out changes in the direction _y'_D; and, finally,
when the composition of the molten mass is that of the point C (4.3 per
cent. of carbon), the whole mass solidifies to a heterogeneous mixture of
two solid solutions, one of which is represented by D (containing 2 per
cent. of carbon), while the other will consist practically of pure
graphite, and is not shown in the figure. The temperature of the eutectic
point is 1130deg.
Even below the solidification point, however, changes can take place. As
has been said, the solid phase which finally separates out from the molten
mass is a solid solution represented by the point D; and the curve DE
represents the change in the composition of this solid solution with the
temperature. As indicated in the figure, DE forms a part of a curve
representing the mutual solubility of graphite in iron and iron in
graphite; the latter solutions, however, not being shown, as they would lie
far outside the diagram. As the temperature falls below 1130deg, more and
more graphite separates out, until at E, when the temperature is 1000deg,
the solid solution contains only 1.8 per cent. of carbon. At this
temperature cementite also begins to be formed, so that as the temperature
continues to fall, separation of cementite (represented by the line E'F')
occurs, and the composition of the solid solution undergoes alteration, as
represented by the curve EF. Below the temperature of the point F (670deg)
the martensite becomes heterogeneous, and forms pearlite.
From the above description, therefore, it follows that if we start with a
molten mixture of iron and carbon, the composition of which is represented
by any point between D and C (from 2 to 4.3 per cent. of carbon), we shall
obtain, on cooling the mass, first of all solid solutions, the composition
of which will be represented by points on the line AD; that then, after the
mass has completely solidified at 1130deg, further cooling will lead to a
separation of graphite and a change in the composition of the martensite
(from 2 to 1.8 per cent. of carbon). On cooling below 1000deg, however, the
martensite and graphite will give rise to cementite and solid solutions
{227} containing less carbon than before, until, at temperatures below
670deg, we are left with a mixture of pearlite and cementite.
We have already said that iron consists in three allotropic modifications,
the regions of stability of which are separated by definite transition
points. The transition point for [alpha]- and [beta]-ferrite (780deg) is
represented in Fig. 75 by the point H; and the transition point for [beta]-
and [gamma]-ferrite (870deg) by the point I. Since neither the [alpha]- nor
the [beta]-ferrite dissolves carbon, the transition point will be
unaffected by addition of carbon, and we therefore obtain the horizontal
transition curve HG. In the case of the [beta]- and [gamma]-ferrite,
however, the latter dissolves carbon, and the transition point is
consequently affected by the amount of carbon present. This is shown by the
line IG.
If a martensite containing less carbon than that represented by the point G
is cooled down from a temperature of, say, 900deg, then when the
temperature has fallen to that, represented by a point on the curve IG,
[beta]-ferrite will separate out, and, as the temperature falls, the
composition of the solid solution will alter as represented by IG. On
passing below the temperature of HG, the [beta]-ferrite will be converted
into [alpha]-ferrite, and, as the temperature falls, the latter will
separate out more and more, while the composition of the solid solution
alters in the direction GF. On passing to still lower temperatures, the
solid solution at F (0.8 per cent. of carbon) breaks up into pearlite. If
the percentage of carbon in the original solid solution was between that
represented by the points G and F, then, on cooling down, no
[beta]-ferrite, but only [alpha]-ferrite would separate out.
We see, therefore, that when martensite is allowed to cool _slowly_, it
yields a heterogeneous mixture either of ferrite and pearlite (when the
original mixture contained up to 0.8 per cent. of carbon), or pearlite and
cementite (when the original mixture contained between 0.8 and 2 per cent.
of carbon). These heterogeneous mixtures constitute soft steels, or, when
the carbon content is low, wrought iron.
The case, however, is different if the solid solution of carbon in iron is
_rapidly_ cooled (quenched) from a temperature above the curve IGFE to a
temperature below this {228} curve. In this case, the rapid cooling does
not allow time for the various changes which have been described to take
place; so that the homogeneous solid solution, on being rapidly cooled,
remains homogeneous. In this way hard steel is obtained. By varying the
rapidity of cooling, as is done in the tempering of steel, varying degrees
of hardness can be obtained.
The interpretation of the curves given above is that due essentially to
Roozeboom, who concluded from the experimental data that at temperatures
below 1000deg the stable systems are martensite and cementite, or ferrite
and cementite, graphite being labile. It has, however, been pointed out,
more especially by E. Heyn,[311] that this is not in harmony with the facts
of metallurgy, which show that graphite is undoubtedly formed on slow
cooling, and more especially when small quantities of silicon are present
in the iron.[312] While, therefore, the relationships represented by Fig.
75 are obtained under certain conditions (especially when manganese is
present), Heyn considers that all the curves in that figure, except ACB,
represent _metastable_ systems--systems, therefore, akin to supercooled
liquids. Rapid cooling will favour the production of the metastable systems
containing cementite, and therefore give rise to relationships represented
by Fig. 75; whereas slow cooling will lead to the stable system ferrite and
graphite. Presence of silicon tends to prevent, presence of manganese tends
to assist, the production of the metastable systems.
Although this view put forward by Heyn has not been conclusively proved, it
must be said that there is much evidence in its favour. Further
investigation is, however, required before a final decision as to the
interpretation of the curves can be reached.
Determination of the Composition of Compounds, without Analysis.--Since the
equilibrium between a solid and a liquid phase depends not only on the
composition of the liquid (solution) but also on that of the solid, it is
necessary {229} to determine the composition of the latter. In some cases
this is easily effected by separating the solid from the liquid phase and
analyzing it. In other cases, however, this method is inapplicable, or is
accompanied by difficulties, due either to the fact that the solid phase
undergoes decomposition (_e.g._ when it contains a volatile constituent),
or to the difficulty of completely separating the mother liquor; as, for
example, in the case of alloys. In all such cases, therefore, recourse must
be had to other methods.
In the first place, synthetic methods may be employed.[313] In this case we
start with a solution of the two components, to which a third substance is
added, which, however, does not enter into the solid phase.[314] We will
assume that the initial solution contains _x_ gm. of A and _y_ gm. of B to
1 gm. of C. After the solution has been cooled down to such a temperature
that solid substance separates out, a portion of the liquid phase is
removed with a pipette and analyzed. If, now, the composition of the
solution is such that there are _x'_ gm. of A and _y'_ gm. of B to 1 gm. of
C., then the composition of the solid phase is _x_ - _x'_ gm. of A and _y_
- _y'_ gm. of B. When _x_ = _x'_, the solid phase is pure B; when _y_ =
_y'_, the solid phase is pure A.
We have assumed here that there is only one solid phase present, containing
A and B. To make sure that the solid phase is not a solid solution in which
A and B are present in the same ratio as in the liquid solution, a second
determination of the composition must be made, with different initial and
end concentrations. If the solid phase is a solid solution, the composition
will now be found different from that found previously.
The composition of the solid phase can, however, be determined in another
manner, viz. by studying the fusion curve and the curve of cooling. From
the form of the fusion curve alone, it is possible to decide whether the
two components {230} form a compound or not; and if the compounds which may
be formed have a definite melting point, the position of the latter gives
at once the composition of the compounds (cf. p. 231).
This method, however, cannot be applied when the compounds undergo
decomposition before the melting point is reached. In such cases, however,
the form of the cooling curve enables one to decide the composition of the
solid phase.[315] If a solution is allowed to cool slowly, and the
temperature noted at definite times, the graphic representation of the rate
of cooling will give a continuous curve; _e.g._ _ab_ in Fig. 76. So soon,
however, as a solid phase begins to be formed, the rate of cooling alters
abruptly, and the cooling curve then exhibits a break, or change in
direction (point _b_). When the eutectic point is reached, the temperature
remains constant, until all the liquid has solidified. This is represented
by the line _cd_. When complete solidification has occurred, the fall of
temperature again becomes uniform (_de_).
[Illustration: FIG. 76.]
[Illustration: FIG. 77.]
[Illustration: FIG. 78.]
The length of time during which the temperature remains constant at the
point _c_, depends, of course, on the eutectic solution. If, therefore, we
take equal amounts of solution having a different initial composition, the
period of constant temperature in the cooling curve will evidently be
greatest in the case of the solution having the composition of the eutectic
point; and the period will become less and less as we increase the amount
of one of the components. The relationship between initial composition of
solution and the duration of constant temperature at the eutectic point is
represented by the curve _a'c'b'_ (Fig. 77). When a compound possessing a
definite melting point is formed, it behaves as a pure substance. If,
therefore, the initial composition of the {231} solution is the same as
that of the compound, no eutectic solution will be obtained; and therefore
no line of constant temperature, such as _cd_ (Fig. 76). In such a case, if
we represent graphically the relation between the initial composition of
the solution and the duration of constant temperature, a diagram is
obtained such as shown in Fig. 78. The two maxima on the time-composition
curve represent eutectic points, and the minima, _a'_, _b'_, _e'_, pure
substances. The position of _e'_ gives the composition of the compound.
When a series of compounds is formed, then for each compound a minimum is
found on the time-composition curve.
[Illustration: FIG. 79.]
If the compound formed has no definite melting point, the diagram obtained
is like that shown in Fig. 79. If we start with a solution, the composition
of which is represented by a point between _d_ and _b_, then, on cooling,
_b_ will separate out first, and the temperature will fall until the point
_d_ is reached. The temperature then remains constant until the component
_b_, which has separated out, is converted into the compound. After this
the temperature again falls, until it again remains constant at the
eutectic point c. In the case of the first halt, the period of constant
temperature is greatest when the initial composition of the solution is the
same as that of the compound; and it becomes shorter and shorter with {232}
increase in the amount of either component. In this way we obtain the
time-composition curve _b'e"d'_, of which the maximum point _e"_ gives the
composition of the compound.
On the other hand, the period of constant temperature for the eutectic
point _c_ is greatest in the case of solutions having the same initial
_composition_ as that corresponding with the eutectic point; and it
decreases the more the initial composition approaches that of the pure
component _a_ or the component e. In this way we obtain the
time-composition curve _a'c'e'_. Here also the point _e'_ represents the
composition of the compound. We see, therefore, that from the graphic
representation of the freezing-point curve, and from the duration of the
temperature-arrests on the cooling curve, for solutions of different
initial composition, it is possible, without having recourse to analysis,
to decide what solid phases are formed, and what is their composition.
Formation of Minerals.--Important and interesting as is the application of
the Phase Rule to the study of alloys, its application to the study of the
conditions regulating the formation of minerals is no less so; and although
we do not propose to consider different cases in detail here, still
attention must be drawn to certain points connected with this interesting
subject.
In the first place, it will be evident from what has already been said,
that that mineral which first crystallizes out from a molten magma is not
necessarily the one with the highest melting point. The _composition_ of
the fused mass must be taken into account. When the system consists of two
components which do not form a compound, one or other of these will
separate out in a pure state, according as the composition of the molten
mass lies on one or other side of the eutectic composition; and the
separation of the one component will continue until the composition of the
eutectic point is reached. Further cooling will then lead to the
simultaneous separation of the two components.
If, however, the two components form a stable compound (_e.g._ orthoclase,
from a fused mixture of silica and potassium aluminate), then the
freezing-point curve will resemble that {233} shown in Fig. 64; _i.e._
there will be a middle curve possessing a dystectic point, and ending on
either side at a eutectic point. This curve would represent the conditions
under which orthoclase is in equilibrium with the molten magma. If the
initial composition of the magma is represented by a point between the two
eutectic points, orthoclase will separate first. The composition of the
magma will thereby change, and the mass will finally solidify to a mixture
of orthoclase and silica, or orthoclase and potassium aluminate, according
to the initial composition.
What has just been said holds, however, only for stable equilibria, and it
must not be forgotten that complications can arise owing to suspended
transformation (when, for example, the magma is rapidly cooled) and the
production of metastable equilibria. These conditions occur very frequently
in nature.
The study of the formation of minerals from the point of view of the Phase
Rule is still in its initial stages, but the results which have already
been obtained give promise of a rich harvest in the future.[316]
* * * * *
{234}
CHAPTER XIII
SYSTEMS OF THREE COMPONENTS
General.--It has already been made evident that an increase in the number
of the components from one to two gives rise to a considerable increase in
the possible number of systems, and introduces not a few complications into
the equilibrium relations of these. No less is this the case when the
number of components increases from two to three; and although examples of
all the possible types of systems of three components have not been
investigated, nor, indeed, any one type fully, nevertheless, among the
systems which have been studied experimentally, cases occur which not only
possess a high scientific interest, but are also of great industrial
importance. On account not only of the number, but more especially of the
complexity of the systems constituted of three components, no attempt will
be made to give a full account, or, indeed, even a survey of all the cases
which have been subjected to a more or less complete experimental
investigation; on the contrary, only a few of the more important classes
will be selected, and the most important points in connection with the
behaviour of these described.
On applying the Phase Rule
P + F = C + 2
to the systems of three components, we see that in order that the system
shall be invariant, no fewer than five phases must be present together, and
an invariant system will therefore exist at a _quintuple_ point. Since the
number of liquid phases can never exceed the number of the components, and
since there can be only one vapour phase, it is evident that in this case,
{235} as in others, there must always be at least one solid phase present
at the quintuple point. As the number of phases diminishes, the variability
of the system can increase from one to four, so that in the last case the
condition of the system will not be completely defined until not only the
temperature and the total pressure of the system, but also the
concentrations of two of the components have been fixed. Or, instead of the
concentrations, the partial pressures of the components may also be taken
as independent variables.
Graphic Representation.--Hitherto the concentrations of the components have
been represented by means of rectangular co-ordinates, although the
numerical relationships have been expressed in two different ways. In the
one case, the concentration of the one component was expressed in terms of
a fixed amount of the other component. Thus, the solubility of a salt was
expressed by the number of grams of salt dissolved by 100 grams of water or
other solvent; and the numbers so obtained were measured along one of the
co-ordinates. The second co-ordinate was then employed to indicate the
change of another independent variable, _e.g._ temperature. In the other
case, the combined weights of the two components A and B were put equal to
unity, and the concentration of the one expressed as a fraction of the
whole amount. This method allows of the representation of the complete
series of concentrations, from pure A to pure B, and was employed, for
example, in the graphic representation of the freezing point curves.
Even in the case of three components rectangular co-ordinates can also be
employed, and, indeed, are the most convenient in those cases where the
behaviour of two of the components to one another is very different from
their behaviour to the third component; as, for example, in the case of two
salts and water. In these cases, the composition of the system can be
represented by measuring the amounts of each of the two components in a
given weight of the third, along two co-ordinates at right angles to one
another; and the change of the system with the temperature can then be
represented by a third axis at right angles to the first two. In those
cases, {236} however, where the three components behave in much the same
manner towards one another, the rectangular co-ordinates are not at all
suitable, and instead of these a _triangular diagram_ is employed. Various
methods have been proposed for the graphic representation of systems of
three components by means of a triangle, but only two of these have been
employed to any considerable extent; and a short description of these two
methods will therefore suffice.[317]
[Illustration: FIG. 80.]
In the method proposed by Gibbs an equilateral triangle of unit height is
used (Fig 80).[318] The quantities of the different components are
expressed as fractional parts of the whole, and the sum of their
concentrations is therefore equal to unity, and can be represented by the
height of the triangle. The corners {237} of the triangle represent the
pure substances A, B, and C respectively. A point on one of the sides of
the triangle will give the composition of a mixture in which only two
components are present, while a point within the triangle will represent
the composition of a ternary mixture. Since every point within the triangle
has the property that the sum of the perpendiculars from that point on the
sides of the triangle is equal to unity (the height of the triangle), it is
evident that the composition of a ternary mixture can be represented by
fixing a point within the triangle such that the lengths of the
_perpendiculars_ from the point to the sides of the triangle are equal
respectively to the fractional amounts of the three components present; the
fractional amount of A, B, or C being represented by the perpendicular
distance from the side of the triangle _opposite_ the corners A, B, and C
respectively.
The location of this point is simplified by dividing the normals from each
of the corners on the opposite side into ten or one hundred parts, and
drawing through these divisions lines at right angles to the normal and
parallel to the side of the triangle. A network of rhombohedra is thus
obtained, and the position of any point can be read off in practically the
same manner as in the case of rectangular co-ordinates. Thus the point P in
Fig. 80 represents a ternary mixture of the composition A = 0.5, B = 0.3, C
= 0.2; the perpendiculars P_a_, P_b_, and P_c_ being equal respectively to
0.5, 0.2, and 0.3 of the height of the triangle.
Another method of representation, due to Roozeboom, consists in employing
an equilateral triangle, the length of whose _side_ is made equal to unity,
or one hundred; the sum of the fractional or percentage amounts of the
three components being represented therefore by a side of the triangle. In
this case the composition of a ternary mixture is obtained by determining,
not the _perpendicular_ distance of a point P from the three sides of the
triangle, but the distance in a direction _parallel_ to the sides of the
triangle (Fig. 81). Conversely, in order to represent a mixture consisting
of _a_, _b_, and _c_ parts of the components A, B, and C respectively, one
side of the triangle, say AB, is first of all divided into ten or one {238}
hundred parts; a portion, B_x_ = _a_, is then measured off, and represents
the amount of A present. Similarly, a portion, A_x'_ = _b_, is measured off
and represents the fractional amount of B, while the remainder, _xx'_ =
_c_, represents the amount of C. From _x_ and _x'_ lines are drawn parallel
to the sides of the triangle, and the point of intersection, P, represents
the composition of the ternary mixture of given composition; for, as is
evident from the figure, the distance of the point P from the three sides
of the triangle, when measured in directions _parallel_ to the sides, is
equal to _a_, _b_, and _c_ respectively. From the division marks on the
side AB, it is seen that the point P in this figure also represents a
mixture of 0.5 parts of A, 0.2 parts of B, and 0.3 parts of C. This gives
exactly the same result as the previous method. The employment of a
right-angled isosceles triangle has also been suggested,[319] but is not in
general use.
[Illustration: FIG. 81.]
In employing the triangular diagram, it will be of use to note a property
of the equilateral triangle. A line drawn from one corner of the triangle
to the opposite side, represents the composition of all mixtures in which
the _relative_ amounts of two of the components remain unchanged. Thus, as
Fig. 82 shows, if the component C is added to a mixture x, in which A and B
are present in the proportions of _a_ : _b_, a mixture _x'_, which is
thereby obtained, also contains A and B in the ratio _a_ : b. For the two
triangles AC_x_ and BC_x_ are similar to the two triangles HC_x'_ and
KC_x'_; and, {239} therefore, A_x_ : B_x_ = H_x'_ : K_x'_. But A_x_ = D_x_
and B_x_ = E_x_; further H_x'_ = F_x'_ and K_x'_ = G_x'_. Therefore, D_x_ :
E_x_ = F_x'_ : G_x'_ = _b_ : a. At all points on the line C_x_, therefore,
the ratio of A to B is the same.
[Illustration: FIG. 82.]
[Illustration: FIG. 83.]
If it is desired to represent at the same time the change of another
independent variable, _e.g._ temperature, this can be done by measuring the
latter along axes drawn perpendicular to the corners of the triangle. In
this way a right prism (Fig. 83) is obtained, and each section of this cut
parallel to the base represents therefore an _isothermal surface_.
* * * * *
{240}
CHAPTER XIV
SOLUTIONS OF LIQUIDS IN LIQUIDS
We have already seen (p. 95) that when two liquids are brought together,
they may mix in all proportions and form one homogeneous liquid phase; or,
only partial miscibility may occur, and two phases be formed consisting of
two mutually saturated solutions. In the latter case, the concentration of
the components in either phase and also the vapour pressure of the system
had, at a given temperature, perfectly definite values. In the case of
three liquid components, a similar behaviour may be found, although
complete miscibility of three components with the formation of only one
liquid phase is of much rarer occurrence than in the case of two
components. When only partial miscibility occurs, various cases are met
with according as the three components form one, two, or three pairs of
partially miscible liquids. Further, when two of the components are only
partially miscible, the addition of the third may cause either an increase
or a diminution in the mutual solubility of these. An increase in the
mutual solubility is generally found when the third component dissolves
readily in each of the other two; but when the third component dissolves
only sparingly in the other two, its addition diminishes the mutual
solubility of the latter.
We shall consider here only a few examples illustrating the three chief
cases which can occur, viz. (1) A and B, and also B and C are miscible in
all proportions, while A and C are only partially miscible. (2) A and B are
miscible in all proportions, but A and C and B and C are only partially
miscible. (3) A and B, B and C, and A and C are only partially miscible. A,
B, and C here represent the three components.
1.--_The three components form only one pair of partially miscible
liquids._ {241}
An example of this is found in the three substances: chloroform, water, and
acetic acid.[320] Chloroform and acetic acid, and water and acetic acid,
are miscible with one another in all proportions, but chloroform and water
are only partially miscible with one another. If, therefore, chloroform is
shaken with a larger quantity of water than it can dissolve, two layers
will be formed consisting one of a saturated solution of water in
chloroform, the other of a saturated solution of chloroform in water. The
composition of these two solutions at a temperature of about 18deg, will be
represented by the points _a_ and _b_ in Fig. 84; _a_ representing a
solution of the composition: chloroform, 99 per cent.; water, 1 per cent.;
and _b_ a solution of the composition: chloroform, 0.8 per cent.; water,
99.2 per cent. When acetic acid is added, it distributes itself between the
two liquid layers, and two conjugate _ternary_ solutions, consisting of
chloroform, water, and acetic acid are thereby produced which are in
equilibrium with one another, and the composition of which will be
represented by two points inside the triangle. In this way a series of
pairs of ternary solutions will be obtained by the addition of acetic acid
to the mixture of chloroform and water. By this addition, also, not only do
the two liquid phases become increasingly rich in acetic acid, but the
mutual solubility of the chloroform and water increases; so that the layer
_a_ becomes relatively richer in water, and layer _b_ relatively richer in
chloroform. This is seen from the following table, which gives the
percentage composition of different conjugate ternary solutions at 18deg.
-------------------------------------------------------------------------
Heavier layer. | Lighter layer.
-------------------------------------------------------------------------
Chloroform. | Water. | Acetic acid. | Chloroform. | Water. | Acetic acid.
-------------------------------------------------------------------------
99.01 | 0.99 | 0 | 0.84 | 99.16 | 0
91.85 | 1.38 | 6.77 | 1.21 | 73.69 | 25.10
80.00 | 2.28 | 17.72 | 7.30 | 48.58 | 44.12
70.13 | 4.12 | 25.75 | 15.11 | 34.71 | 50.18
67.15 | 5.20 | 27.65 | 18.33 | 31.11 | 50.56
59.99 | 7.93 | 32.08 | 25.20 | 25.39 | 49.41
55.81 | 9.58 | 34.61 | 28.85 | 23.28 | 47.87
-------------------------------------------------------------------------
{242}
By the continued addition of acetic acid, the composition of the successive
conjugate solutions in equilibrium with one another becomes, as the table
shows, more nearly the same, and a point is at length reached at which the
two solutions become identical. This will therefore be a _critical point_
(p. 98). Increased addition of acetic acid beyond this point will lead to a
single homogeneous solution.
These relationships are represented graphically by the curve _a_K_b_, Fig.
84. The points on the branch _a_K represent the composition of the
solutions relatively rich in chloroform (heavier layer), those on the curve
_b_K the composition of solutions relatively rich in water (lighter layer);
and the points on these two branches representing conjugate solutions are
joined together by "tie-lines." Thus, the points _a'b'_ represent conjugate
solutions, and the line _a'b'_ is a tie-line.
[Illustration: FIG. 84.]
Since, now, acetic acid when added to a heterogeneous mixture of chloroform
and water does not enter in equal amounts into the two layers, but in
amounts depending on its coefficient of distribution between chloroform and
water,[321] the {243} tie-lines will not be parallel to AB, but will be
inclined at an angle. As the solutions become more nearly the same, the
tie-lines diminish in length, and at last, when the conjugate solutions
become identical, shrink to a point. For the reason that the tie-lines are,
in general, not parallel to the side of the triangle, the critical point at
which the tie-line vanishes will not be at the summit of the curve, but
somewhere below this, as represented by the point K.
The curve _a_K_b_, further, forms the boundary between the heterogeneous
and homogeneous systems. A mixture of chloroform, water, and acetic acid
represented by any point outside the curve _a_K_b_, will form only one
homogeneous phase; while any mixture represented by a point within the
curve, will separate into two layers having the composition represented by
the ends of the tie-line passing through that point. Thus, a mixture of the
total composition _x_, will separate into two layers having the composition
_a'_ and _b'_ respectively.
Since three components existing in three phases (two liquid and a vapour
phase) constitute a bivariant system, the final result, _i.e._ the
composition of the two layers and the total vapour pressure, will not
depend merely on the temperature, as in the case of two-component systems
(p. 102), but also on the composition of the mixture with which we start.
At constant temperature, however, all mixtures, the composition of which is
represented by a point on one and the same tie-line, will separate into the
same two liquid phases, although the relative _amounts_ of the two phases
will vary. If we omit the vapour phase, the condition of the system will
depend on the pressure as well as on the temperature and composition of the
initial mixture. By keeping the pressure constant, _e.g._ at atmospheric
pressure (by working with open vessels), the system again becomes
bivariant. We see, therefore, that the position of the curve _a_K_b_, or,
in other words, the composition of the different conjugate ternary
solutions, will vary with the temperature, and only with the temperature,
if we assume either constancy of pressure or the presence of the vapour
phase. Since at the critical point the condition is imposed that the two
liquid phases become identical, one degree of freedom is thereby {244}
lost, and therefore only one degree of freedom remains. The critical point,
therefore, depends on the temperature, and only on the temperature; always
on the assumption, of course, that the pressure is constant, or that a
vapour phase is present. Fig. 84, therefore, represents an isothermal
(p. 239).
It is of importance to note that the composition of the different ternary
solutions obtained by the addition of acetic acid to a heterogeneous
mixture of chloroform and water, will depend not only on the amount of
acetic acid added, but also on the relative amounts of chloroform and water
at the commencement. Suppose, for example, that we start with chloroform
and water in the proportions represented by the point _c'_ (Fig. 84). On
mixing these, two liquid layers having the composition _a_ and _b_
respectively will be formed. Since by the addition of acetic acid the
relative amounts of these two substances in the system as a whole cannot
undergo alteration, the total composition of the different ternary systems
which will be obtained must be represented by a point on the line C_c'_
(p. 238). Thus, for example, by the addition of acetic acid a system may be
obtained, the total composition of which is represented by the point _c"_.
Such a system, however, will separate into two conjugate ternary solutions,
the composition of which will be represented by the ends of the tie-line
passing through the point _c"_. So long as the total composition of the
system lies below the point S, _i.e._ the point of intersection of the line
C_c'_ with the boundary curve, two liquid layers will be formed; while all
systems having a total composition represented by a point on the line
C_c'_, above S, will form only one homogeneous solution.
From the figure, also, it is evident that as the amount of acetic acid is
increased, the relative amounts of the two liquid layers formed differ more
and more until at S a limiting position is reached, when the amount of the
one liquid layer dwindles to nought, and only one solution remains.
The same reasoning can be carried through for different initial amounts of
chloroform and water, but it would be fruitless to discuss all the
different systems which can be obtained. The reason for the preceding
discussion was to show that {245} although the addition of acetic acid to a
mixture of chloroform and water will, in all cases, lead ultimately to a
limiting system, beyond which homogeneity occurs, that point is not
necessarily the critical point. On the contrary, in order that addition of
acetic acid shall lead to the critical mixture, it is necessary to start
with a binary mixture of chloroform and water in the proportions
represented by the point _c'_. In this case, addition of acetic acid will
give rise to a series of conjugate ternary solutions, the composition of
which will gradually approach to one another, and at last become identical.
From the foregoing it will be evident that the amount of acetic acid
required to produce a homogenous solution, will depend on the relative
amounts of chloroform and water from which we start, and can be ascertained
by joining the corner C with the point on the line AB representing the
total composition of the initial binary system. The point where this line
intersects the boundary curve _a_K_b_ will indicate the minimum amount of
acetic acid which, under these particular conditions, is necessary to give
one homogeneous solution.
Retrograde Solubility.--As a consequence of the fact that acetic acid
distributes itself unequally between chloroform and water, and the critical
point K, therefore, does not lie at the summit of the curve, it is possible
to start with a homogeneous solution in which the percentage amount of
acetic acid is greater than at the critical point, and to pass from this
first to a heterogenous and then again to a homogenous system merely by
altering the relative amounts of chloroform and water. This phenomenon, to
which the term _retrograde solubility_ is applied, will be observed not
only in the case of chloroform, water, and acetic acid, but in all other
systems in which the critical point lies below the highest point of the
boundary curve for heterogeneous systems. This will be seen from the
diagram, Fig. 85. Starting with the homogeneous system represented by _x_,
in which, therefore, the concentration of C is greater than in the critical
mixture (K), if the relative amounts of A and B are altered in the
direction _xx'_, while the amount of C is maintained constant, the system
will become heterogeneous when the composition reaches the point _y_, and
will remain {246} heterogeneous with changing composition until the point
_y'_ is passed, when it will again become homogeneous. If the relative
concentration of C is increased above that represented by the line SS, this
phenomenon will, of course, no longer be observed.
[Illustration: FIG. 85.]
Relationships similar to those described for chloroform, water, and acetic
acid are also found in the case of a number of other trios, _e.g._ ether,
water, and alcohol; chloroform, water, and alcohol.[322] They have also
been observed in the case of a considerable number of molten metals.[323]
Thus, molten lead and silver, as well as molten zinc and silver, mix in all
proportions; but molten lead and zinc are only partially miscible with one
another. When melted together, therefore, the last two metals will separate
into two liquid layers, one rich in lead, the other rich in zinc. If silver
is now added, and the temperature maintained above the freezing point of
the mixture, the silver passes for the most part, in accordance with the
law of distribution, into the upper layer, which is rich in zinc; silver
being more soluble in molten zinc than in molten lead. This is clearly
shown by the following figures:--[324]
{247}
--------------------------------------------------
Heavier alloy. | Lighter alloy.
--------------------------------------------------
Percentage amount of | Percentage amount of
Silver. | Lead. | Zinc. | Silver. | Lead. | Zinc.
--------------------------------------------------
1.25 | 96.69 | 2.06 | 38.91 | 3.12 | 57.97
1.71 | 96.43 | 1.86 | 45.01 | 3.37 | 51.62
5.55 | 93.16 | 1.29 | 54.93 | 4.21 | 40.86
--------------------------------------------------
The numbers in the same horizontal row give the composition of the
conjugate alloys, and it is evident that the upper layer consists almost
entirely of silver and zinc. On allowing the mixture to cool slightly, the
upper layer solidifies first, and can be separated from the still molten
lead layer. It is on this behaviour of silver towards a mixture of molten
lead and zinc that the Parkes's method for the desilverization of lead
depends.[325] If aluminium is also added, a still larger proportion of
silver passes into the lighter layer, and the desilverization of the lead
is more complete.[326]
[Illustration: FIG. 86.]
[Illustration: FIG. 87.]
The Influence of Temperature.--As has already been said, a ternary system
existing in three phases possesses two degrees of freedom; and the state of
the system is therefore dependent not only on the relative concentration of
the components, but also on the temperature. As the temperature changes,
therefore, the boundary curve of the heterogeneous system will also alter;
and in order to represent this alteration we shall make use of the right
prism, in which the temperature is measured upwards. In this way the
boundary curve passes into a boundary surface (called a dineric surface),
as shown in Fig. 86. In this figure the curve _akb_ is the isothermal for
the ternary system; the curve _a_K_b_ shows the change in the _binary_
system AB with the temperature, with {248} a critical point at K. This
curve has the same meaning as those given in Chapter VI. The curve _k_K is
a critical curve joining together the critical points of the different
isothermals. In such a case as is shown in Fig. 86, there does not exist
any real critical temperature for the ternary system, for as the
temperature is raised, the amount of C in the "critical" solution becomes
less and less, and at K only two components, A and B, are present. In the
case, however, represented in Fig. 87, a real ternary critical point is
found. In this figure _ak'b_ is an isothermal, _ak"_ is the curve for the
binary system, and K is the ternary critical point. All points outside the
helmet-shaped boundary surface represent homogeneous ternary solutions,
while all points within the surface belong to heterogeneous systems. Above
the temperature of the point K, the three components are miscible in all
proportions. An example of a ternary system yielding such a boundary
surface is that consisting of phenol, water, and acetone.[327] In this case
the critical temperature K is 92deg, and the composition at this ternary
critical point is--
Water 59 per cent.
Acetone 12 "
Phenol 29 "
[Illustration: FIG. 88.]
The difference between the two classes of systems just mentioned, is seen
very clearly by a glance at the Figs. 88 and 89, which show the projection
of the isothermals on the base of the prism. In Fig. 88, the projections
yield paraboloid curves, the two branches of which are cut by one side of
the triangle; and the critical point is represented by a point on {249}
this side. In the second case (Fig. 89), however, the projections of the
isothermals form ellipsoidal curves surrounding the supreme critical point,
which now lies _inside the triangle_. At lower temperatures, these
isothermal boundary curves are cut by a side of the triangle; at the
critical temperature, _k"_, of the binary system AB, the boundary curve
_touches_ the side AB, while at still higher temperatures the boundary
curve comes to lie entirely within the triangle. At any given temperature,
therefore, between the critical point of the binary system (_k"_), and the
supreme critical point of the ternary system (K), each pair of the three
components are miscible with one another in all proportions; for the region
of heterogeneous systems is now bounded by a closed curve lying entirely
within the triangle. Outside this curve only homogeneous systems are found.
Binary mixtures, therefore, represented by any point on one of the sides of
the triangle must be homogeneous, for they all lie outside the boundary
curve for heterogeneous states.
[Illustration: FIG. 89.]
2. _The three components can form two pairs of partially miscible liquids._
In the case of the three components water, alcohol, and succinic nitrile,
water and alcohol are miscible in all proportions, but not so water and
succinic nitrile, or alcohol and succinic nitrile.
[Illustration: FIG. 90.]
[Illustration: FIG. 91.]
As we have already seen (p. 122), water and succinic nitrile can form two
liquid layers between the temperatures 18.5deg and 55.5deg; while alcohol
and nitrile can form two liquid layers between 13deg and 31deg. If, then,
between these two temperature limits, alcohol is added to a heterogeneous
mixture of water and nitrile, or water is added to a mixture of alcohol and
nitrile, two heterogeneous ternary systems will be formed, {250} and two
boundary curves will be obtained in the triangular diagram, as shown in
Fig. 90.[328] On changing the temperature, the boundary curves will also
undergo alteration, in a manner similar to that just discussed. As the
temperature falls, the two curves will spread out more and more into the
centre of the triangle, and might at last meet one another; while at still
lower temperatures we may imagine the curves still further expanding so
that the two heterogeneous regions flow into one another and form a _band_
on the triangular diagram (Fig. 91). This, certainly, has not been realized
in the case of the three components mentioned, because at a temperature
higher than that at which the two heterogeneous regions could fuse
together, solid separates out.
[Illustration: FIG. 92.]
The gradual expansion of a paraboloid into a band-like area of
heterogeneous ternary systems, has, however, been observed in the case of
water, phenol, and aniline.[329] In Fig. 92 are shown three isothermals,
viz. those for 148deg, 95deg, and 50deg. At 148deg, water and aniline form
two layers having the composition--
Water, 83.5 per cent. } { water, 20 per cent.
} and {
Aniline, 16.5 " } { aniline, 80 "
{251}
and the critical point _k'_ has the composition--
Water, 65; phenol, 13.2; aniline, 21.8 per cent.
At 95deg, the composition of the two binary solutions is--
Water, 93 per cent. } { water 8 per cent.
} and {
Aniline, 7 " } { aniline, 92 "
while the point _k"_ has the composition
Water, 69.9; phenol, 26.6; aniline, 3.5 per cent.
At 50deg, the region of heterogeneous states now forms a band, and the two
layers formed by water and aniline have the composition--
Water, 96.5 per cent. } { water, 5.5 per cent.
} and {
Aniline, 3.5 " } { aniline, 94.5 "
while the two layers formed by water and phenol have the composition--
Water, 89 per cent.} { water, 38 per cent.
} and {
Phenol, 11 " } { phenol, 62 "
All mixtures of water, phenol, and aniline, therefore, the composition of
which is represented by any point within the band _abcd_, will form two
ternary solutions; while if the composition is represented by a point
outside the band, only one homogeneous solution will be produced.
3. _The three components form three pairs of partially miscible liquids._
[Illustration: FIG. 93.]
The third chief case which can occur is that no two of the components are
completely miscible with one another. In this case, therefore, we shall
obtain three paraboloid boundary curves, as shown in Fig. 93. If, now, we
imagine these three curves to expand in towards the centre of the triangle,
as might happen, for example, by lowering the temperature, a point will
{252} be reached at which the curves partly overlap, and we shall get the
appearance shown in Fig. 94.
The points _a_, _b_, and _c_ represent the points where the three curves
cut, and the triangle _abc_ is a region where the curves overlap. From this
diagram we can see that any mixture having a composition represented by a
point in one of the clear spaces at the corners of the larger triangle,
will form a homogeneous solution; if the composition corresponds to any
point lying in one of the quadrilateral regions _x__{1}, _x__{2} or
_x__{3}, two ternary solutions will be formed; while, if the composition is
represented by any point in the inner triangle, separation into three
layers will occur.
[Illustration: FIG. 94.]
Since in the clear regions at the corners of the triangle we have three
components in two phases, liquid and vapour, the systems have three degrees
of freedom. At constant temperature, therefore, the condition of the system
is not defined until the concentrations of two of the components are fixed.
A system belonging to one of the quadrilateral spaces has, as we have seen,
two degrees of freedom; besides the temperature, one concentration must be
fixed. Lastly, a system the composition of which falls within the inner
triangle _abc_, will form three layers, and will therefore possess only one
degree of freedom. If the temperature is fixed, the composition of the
three layers is also determined, viz. that of the points _a_, _b_, and _c_
respectively; and a change in the composition of the original mixture can
lead only to a difference in the relative amounts of the three layers, not
to a difference in their composition.
An example of a system which can form three liquid phases is found in
water, ether, and succinic nitrile.[330]
* * * * *
{253}
CHAPTER XV
PRESENCE OF SOLID PHASES
A. The Ternary Eutectic Point.--In passing to the consideration of those
ternary systems in which one or more solid phases can exist together with
one liquid phase, we shall first discuss not the solubility curves, as in
the case of two-component systems, but the simpler relationships met with
at the freezing point. That is, we shall first of all examine the freezing
point curves of ternary systems.
[Illustration: FIG. 95.]
Since it is necessary to take into account not only the changing
composition of the liquid phase, but also the variation of the temperature,
we shall employ the right prism for the graphic representation of the
systems, as shown in Fig. 95. A, B, and C in this figure, therefore, denote
the melting points of the pure components. If we start with the component A
at its melting point, and add B, which is capable of dissolving in liquid
A, the freezing point of A will be lowered; and, similarly, the freezing
point of B by addition of A. In this way we get the freezing point curve
A_k__{1}B for the binary system; _k__{1}; being an eutectic point. This
curve will of course lie in the plane formed by one face of the prism. In a
similar manner we obtain the freezing point curves A_k__{2}C and B_k__{3}C.
These curves give the composition of the binary liquid phases in
equilibrium {254} with one of the pure components, or at the eutectic
points, with a mixture of two solid components. If, now, to the system
represented say by the point _k__{1}, a small quantity of the third
component, C, is added, the temperature at which the two solid phases A and
B can exist in equilibrium with the liquid phase is lowered; and this
depression of the eutectic point is all the greater the larger the addition
of C. In this way we obtain the curve _k__{1}K, which slopes inwards and
downwards, and indicates the varying composition of the ternary liquid
phase with which a mixture of solid A and B are in equilibrium. Similarly,
the curves _k__{2}K and _k__{3}K are the corresponding eutectic curves for
A and C, and B and C in equilibrium with ternary solutions. At the point K,
the three solid components are in equilibrium with the liquid phase; and
this point, therefore, represents _the lowest temperature attainable with
the three components given_. Each of the ternary eutectic curves, as they
may be called, is produced by the intersection of two surfaces, while at
the ternary eutectic point, three surfaces, viz. A_k__{1}K_k__{2},
B_k__{1}K_k__{3}, and C_k__{1}K_k__{3} intersect. Any point on one of these
surfaces represents a ternary solution in equilibrium with only one
component in the solid state; the lines or curves of intersection of these
represent equilibria with two solid phases, while at the point K, the
ternary eutectic point, there are three solid phases in equilibrium with a
liquid and a vapour phase. The surfaces just mentioned represent bivariant
systems. One component in the solid state can exist in equilibrium with a
ternary liquid phase under varying conditions of temperature and
concentration of the components in the solution; and before the state of
the system is defined, these two variables, temperature and composition of
the liquid phase, must be fixed. On the other hand, the curves formed by
the intersection of these planes represent univariant systems; at a given
temperature two solid phases can exist in equilibrium with a ternary
solution, only when the latter has a definite composition. Lastly, the
ternary eutectic point, K, represents an invariant system; three solid
phases can exist in equilibrium with a ternary solution, only when the
latter has one fixed composition and when the temperature has a definite
value. This eutectic point, therefore, {255} has a perfectly definite
position, depending only on the nature of the three components.
Instead of employing the prism, the change in the composition of the
ternary solutions can also be indicated by means of the _projections_ of
the curves _k__{1}K, _k__{2}K, and _k__{3}K on the base of the prism, the
particular temperature being written beside the different eutectic points
and curves. This is shown in Fig. 96.
[Illustration: FIG. 96.]
The numbers which are given in this diagram refer to the eutectic points
for the system bismuth--lead--tin, the data for which are as
follows:--[331]
--------------------------------------------------------------------
Melting point of | Percentage composition of | Temperature of binary
pure metal. | binary eutectic mixture. | eutectic point.
--------------------------------------------------------------------
| Bi Pb Sn |
Bismuth, 268deg | 55 45 -- | Bi--Pb, 127deg
Lead, 325deg | 58 -- 42 | Bi--Sn, 133deg
Tin, 232deg | -- 37 63 | Pb--Sn, 182deg
--------------------------------------------------------------------
--------------------------------------------------
Percentage composition of | Temperature of ternary
ternary eutectic mixture. | eutectic point.
--------------------------------------------------
Bi Pb Sn |
52 32 16 | 96deg
--------------------------------------------------
Formation of Compounds.--In the case just discussed, the components
crystallized out from solution in the pure state. If, however, combination
can take place between two of the components, the relationships will be
somewhat different; the curves which are obtained in such a case being
represented in Fig. 97. From the figure, we see that the two components B
{256} and C form a compound, and the freezing point curve of the binary
system has therefore the form shown in Fig. 64 (p. 209). Further, there are
two _ternary_ eutectic points, K_{1} and K_{2}, the solid phases present
being A, B, and compound, and A, C, and compound respectively.
[Illustration: FIG. 97.]
The particular point, now, to which it is desired to draw attention is
this. Suppose the ternary eutectic curves projected on a plane parallel to
the face of the prism containing B and C, _i.e._ suppose the concentrations
of the two components B and C, between which interaction can occur,
expressed in terms of a constant amount of the third component A,[332]
curves will then be obtained which are in every respect analogous to the
freezing point curves of binary systems. Thus, suppose the eutectic curves
_k__{1}K and _k__{2}K in Fig. 95 projected on the face BC of the prism,
then evidently a curve will be obtained consisting of two branches
meeting in an eutectic point. On the other hand, the projection of the
ternary eutectic curves in Fig. 97 on the face BC of the prism, will
give a curve consisting of three portions, as shown by the outline
_k__{1}K_{1}K_{2}_k__{2} in Fig. 97.
Various examples of this have been studied, and the following table
contains some of the data for the system ethylene bromide (A), picric acid
(B), and [beta]-naphthol (C), obtained by Bruni.[333]
{257}
-------------------------------------------------------------------------
| Temperature | Solid phases present.
-------------------------------------------------------------------------
Point _k__{1} | 9.41deg | Ethylene bromide, picric acid.
Curve _k__{1}K_{1} | -- | " "
Point K_{1} | 9.32deg | Ethylene bromide, picric acid, and
| | [beta]-naphthol picrate.
Curve K_{1}D'K_{2} | -- | Ethylene bromide,
| | [beta]-naphthol picrate.
Point D' | 9.75deg | " " " "
Point K_{2} | 8.89deg | " " [beta]-naphthol,
| | and picrate.
Curve K_{2}_k__{2} | -- | " " [beta]-naphthol.
Point _k__{2} | 9.04deg | " " "
-------------------------------------------------------------------------
From what has been said, it will be apparent that if the ternary eutectic
curve of a three-component system (in which one of the components is
present in constant amount) is determined, it will be possible to state,
from the form of curve obtained, whether or not the two components present
in varying amount crystallize out pure or combine with one another to form
a compound. It may be left to the reader to work out the curves for the
other possible systems; but it will be apparent, that the projections of
the ternary eutectic curves in the manner given will yield a series of
curves alike in all points to the binary curves given in Figs. 63-65,
pp. 208-210.
Since, from the method of investigation, the temperatures of the eutectic
curves will depend on the melting point of the third component (A), it is
possible, by employing substances with widely differing melting points, to
investigate the interaction of the two components (_e.g._ two optical
antipodes) B and C over a range of temperature; and thus determine the
range of stability of the compound, if one is formed. Since, in some cases,
two substances which at one temperature form mixed crystals combine at
another temperature to form a definite compound, the relationships which
have just been described can be employed, and indeed, have been employed,
to determine the temperature at which this change occurs.[334] By means of
this method, Adriani found that below 103deg _i_-camphoroxime exists as a
racemic compound, while above {258} that temperature it occurs as a racemic
mixed crystal[335] (_cf._ p. 219).
B. Equilibria at Higher Temperatures. Formation of Double Salts.--After
having studied the relationships which are found in the neighbourhood of
the freezing points of the components, we now pass to the discussion of the
equilibria which are met with at higher temperatures. In this connection we
shall confine the discussion entirely to the systems formed of two salts
and water, dealing more particularly with those cases in which the water is
present in relatively large amount and acts as solvent. Further, in
studying these systems, one restriction must be made, viz. that the single
salts are salts either of the same base or of the same acid; or are, in
other words, capable of yielding a common ion in solution. Such a
restriction is necessary, because otherwise the system would be one not of
three but of four components.[336]
Transition Point.--As is very well known, there exist a number of hydrated
salts which, on being heated, undergo apparent partial fusion; and in
Chapter V. the behaviour of such hydrates was more fully studied in the
light of the Phase Rule. Glauber's salt, or sodium sulphate decahydrate,
for example, on being heated to a temperature of about 32.5deg, partially
liquefies, owing to the fact that the water of crystallization is split off
and anhydrous sodium sulphate formed, as shown by the equation--
Na_{2}SO_{4},10H_{2}O = Na_{2}SO_{4} + 10H_{2}O
The temperature of 32.5deg, it was learned, constituted a _transition
point_ for the decahydrate and anhydrous salt plus water; decomposition of
the hydrated salt occurring above this temperature, combination of the
anhydrous salt and water below it.
Analogous phenomena are met with in systems constituted of two salts and
water in which the formation of double salts can take place. Thus, for
example, if _d_-sodium potassium {259} tartrate is heated to above 55deg,
apparent partial fusion occurs, and the two single salts, _d_-sodium
tartrate and _d_-potassium tartrate, are deposited, the change which occurs
being represented by the equation--
4NaKC_{4}O_{6}H_{4},4H_{2}O = 2Na_{2}C_{4}O_{6}H_{4},2H_{2}O
+ 2K_{2}C_{4}O_{6}H_{4},1/2H_{2}O + 11H_{2}O
On the other hand, if sodium and potassium tartrates are mixed with water
in the proportions shown on the right side of the equation, the system will
remain partially liquid so long as the temperature is maintained above
55deg (in a closed vessel to prevent loss of water), but on allowing the
temperature to fall below this point, complete solidification will ensue,
owing to the formation of the hydrated double salt. Below 55deg, therefore,
the hydrated double salt is the stable system, while above this temperature
the two single salts plus saturated solution are stable.[337]
A similar behaviour is found in the case of the double salt copper
dipotassium chloride (CuCl_{2},2KCl,2H_{2}O or CuK_{2}Cl_{4},2H_{2}O).[338]
When this salt is heated to 92deg, partial liquefaction occurs, and the
original blue plate-shaped crystals give place to brown crystalline needles
and white cubes; while on allowing the temperature to fall, re-formation of
the blue double salt ensues. The temperature 92deg is, therefore, a
transition point at which the reversible reaction--
CuK_{2}Cl_{4},2H_{2}O <--> CuKCl_{3} + KCl + 2H_{2}O
takes place.
The decomposition of sodium potassium tartrate, or of copper dipotassium
chloride, differs in so far from that of Glauber's salt that _two_ new
solid phases are formed; and in the case of copper dipotassium chloride,
one of the decomposition products is itself a double salt.
In the two examples of double salt decomposition which have just been
mentioned, sufficient water was yielded to cause a partial liquefaction;
but other cases are known where this is not so. Thus, when copper calcium
acetate is heated to a {260} temperature of 75deg, although decomposition
of the double salt into the two single salts occurs as represented by the
equation[339]--
CuCa(C_{2}H_{3}O_{2})_{4},8H_{2}O = Cu(C_{2}H_{3}O_{2})_{2},H_{2}O
+ Ca(C_{2}H_{3}O_{2})_{2},H_{2}O
+ 6H_{2}O
the amount of water split off is insufficient to give the appearance of
partial fusion, and, therefore, only a change in the crystals is observed.
The preceding examples, in which decomposition of the double salt was
effected by a rise of temperature, were chosen for first consideration as
being more analogous to the case of Glauber's salt; but not a few examples
are known where the reverse change takes place, formation of the double
salt occurring _above_ the transition point, and decomposition into the
constituent salts below it. Instances of this behaviour are found in the
case of the formation of astracanite from sodium and magnesium sulphates,
and of sodium ammonium racemate from the two sodium ammonium tartrates, to
which reference will be made later. Between these various systems, however,
there is no essential difference; and whether decomposition or formation of
the double salt occurs at temperatures above the transition point, will of
course depend on the heat of change at that point. For, in accordance with
van't Hoff's law of movable equilibrium (p. 58), that change will take
place at the higher temperature which is accompanied by an absorption of
heat. If, therefore, the formation of the double salt from the single salts
is accompanied by an absorption of heat, the double salt will be formed
from the single salts on raising the temperature; but if the reverse is the
case, then the double salt on being heated will decompose into the
constituent salts.[340]
In those cases, now, which have so far been studied, the change at the
transition point is accompanied by a taking up or a splitting off of water;
and _in such cases the general rule can be given, that if the water of
crystallization of the two constituent {261} salts together is greater than
that of the double salt, the latter will be produced from the former on
raising the temperature_ (_e.g._ astracanite from sodium and magnesium
sulphates); _but if the double salt contains more water of crystallization
than the two single salts, increase of temperature will effect the
decomposition of the double salt_. When we seek for the connection between
this rule and the law of van't Hoff, it is found in the fact that the heat
effect involved in the hydration or dehydration of the salts is much
greater than that of the other changes which occur, and determines,
therefore, the sign of the total heat effect.[341]
Vapour Pressure. Quintuple Point.--In the case of Glauber's salt, we saw
that at a certain temperature the vapour pressure curve of the hydrated
salt cut that of the saturated solution of anhydrous sodium sulphate. That
point, it will be remembered, was a quadruple point at which the four
phases sodium sulphate decahydrate, anhydrous sodium sulphate, solution,
and vapour, could co-exist; and was also the point of intersection of the
curves for four univariant systems. In the case of the formation of double
salts, similar relationships are met with; and also certain differences,
due to the fact that we are now dealing with systems of three components.
Two cases will be chosen here for brief description, one in which
formation, the other in which decomposition of the double salt occurs with
rise of temperature.
On heating a mixture of sodium sulphate decahydrate and magnesium sulphate
heptahydrate, it is found that at 22deg partial liquefaction occurs with
formation of astracanite. At this temperature, therefore, there can coexist
the five phases--
Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O;
solution; vapour.
This constitutes, therefore, a _quintuple point_; and since there are three
components present in five phases, the system is invariant. This point,
also, will be the point of intersection of curves for five univariant
systems, which, in this case, must each be composed of four phases. These
systems are--
{262}
I. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O;
Na_{2}Mg(SO_{4})_{2},4H_{2}O; vapour.
II. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O; solution; vapour.
III. MgSO_{4},7H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution; vapour.
IV. Na_{2}SO_{4},10H_{2}O; Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution;
vapour.
V. Na_{2}SO_{4},10H_{2}O; MgSO_{4},7H_{2}O;
Na_{2}Mg(SO_{4})_{2},4H_{2}O; solution.
[Illustration: FIG. 98.]
On representing the vapour pressures of these different systems
graphically, a diagram is obtained such as is shown in Fig. 98,[342] the
curves being numbered in accordance with the above list. When the system I.
is heated, the vapour pressure increases until at the quintuple point the
liquid phase (solution) is formed, and it will then depend on the relative
amounts of the different phases whether on further heating there is formed
system III., IV., or V. If either of the first two is produced, we shall
obtain the vapour pressure of the solutions saturated with respect to both
double salt and one of the single salts; while if the vapour phase
disappears, there will be obtained the pressure of the condensed systems
formed of double salt, two single salts and solution. This curve,
therefore, indicates the _change of the transition point with pressure_;
and since in the ordinary determinations of the transition point in open
vessels, we are in reality dealing with condensed systems under the
pressure of 1 atm., it will be evident that the transition point does not
accurately coincide with the quintuple point (at which the system is under
the pressure of its own vapour). As in the case of other condensed systems,
however, pressure has only a slight influence on the temperature of the
transition point. Whether or not pressure raises or lowers the transition
point will depend on whether transformation is accompanied by an increase
or {263} diminution of volume (theorem of Le Chatelier, p. 58). In the case
of the formation of astracanite, expansion occurs, and the transition point
will therefore be raised by increase of pressure. Although measurements
have not been made in the case of this system, the existence of such a
curve has been experimentally verified in the case of copper and calcium
acetates and water (v. _infra_).[343]
[Illustration: FIG. 99.]
The vapour pressure diagram in the case of copper calcium acetate and water
(Fig. 99), is almost the reverse of that already discussed. In this case,
the double salt decomposes on heating, and the decomposition is accompanied
by a contraction. Curve I. is the vapour pressure curve for double salt,
two single salts (p. 260), and vapour; curves II. and III. give the vapour
pressures of solutions saturated with respect to double salt and one of the
single salts; curve IV. is the curve of pressures for the solutions
saturated with respect to the two single salts; while curve V. again
represents the change of the transition point with pressure. On examining
this diagram, it is seen that whereas {264} astracanite could exist both
above and below the quintuple point, copper calcium acetate can exist only
_below_ the quintuple point. This behaviour is found only in those cases in
which the double salt is decomposed by rise of temperature, and where the
decomposition is accompanied by a diminution of volume.[344]
As already mentioned, the decomposition of copper calcium acetate into the
single salts and saturated solution is accompanied by a contraction, and it
was therefore to be expected that increase of pressure would _lower_ the
transition point. This expectation of theory was confirmed by experiment,
for van't Hoff and Spring found that although the transition point under
atmospheric pressure is about 75deg, decomposition of the double salt took
place even at the ordinary temperature when the pressure was increased to
6000 atm.[345]
Solubility Curves at the Transition Point.--At the transition point, as has
already been shown, the double salt and the two constituent salts can exist
in equilibrium with the same solution. The transition point, therefore,
must be the point of intersection of two solubility curves; the solubility
curve of the double salt and the solubility curve of the mixtures of the
two constituent salts. It should be noted here that we are not dealing with
the solubility curves of the single salts separately, for since the systems
are composed of three components, a single solid phase can, at a given
temperature, be in equilibrium with solutions of different composition, and
two solid phases in contact with solution (and vapour) are therefore
necessary to give an univariant system. The same applies, of course, to the
solubility of the double salt; for a double salt also constitutes a single
phase, and can therefore exist in equilibrium with solutions of varying
composition. If, however, we make the restriction (which we do for the
present) that the double salt is not decomposed by water, then the solution
will contain the constituent salts in the same relative proportions as they
are contained in the double salt, and the system may therefore be regarded
as one of _two_ components, viz. double salt and water. In this case one
solid phase is sufficient, with solution and {265} vapour, to give an
univariant system; and at a given temperature, therefore, the solubility
will have a perfectly definite value.
Since in almost all cases the solubility is determined in open vessels, we
shall in the following discussion consider that the vapour phase is absent,
and that the system is under a constant pressure, that of the atmosphere.
With this restriction, therefore, four phases will constitute an invariant
system, three phases an univariant, and two phases a bivariant system.
It has already been learned that in the case of sodium sulphate and water,
the solubility curve of the salt undergoes a sudden change in direction at
the transition point, and that this is accompanied by a change in the solid
phase in equilibrium with the solution. The same behaviour is also found in
the case of double salts. To illustrate this, we shall briefly discuss the
solubility relations of a few double salts, beginning with one of the
simplest cases, that of the formation of rubidium racemate from rubidium
_d_- and _l_-tartrates. The solubilities are represented diagrammatically
in Fig. 100, the numerical data being contained in the following table, in
which the solubility is expressed as the number of gram-molecules
Rb_{2}C_{4}H_{4}O_{6} in 100 gm.-molecules of water.[346]
---------------------------------------------------------------
Temperature. | Solubility of tartrate | Solubility of racemate.
| mixture. |
---------------------------------------------------------------
25deg | 13.03 | 10.91
35deg | -- | 12.63
40.4deg | -- | 13.48
40.7deg | 13.46 | --
54deg | 13.83 | --
---------------------------------------------------------------
In Fig. 100 the curve AB represents the solubility of the racemate, while
A'BC represents the solubility of the mixed tartrates. Below the transition
point, therefore, the solubility of the racemate is less than that of the
mixed tartrates. The solution, saturated with respect to the latter, will
be supersaturated with respect to the racemate; and if a nucleus of this is
present, racemate will be deposited, and the mixed tartrates, if present in
equimolecular amounts, will ultimately {266} entirely disappear, and only
racemate will be left as solid phase. The solution will then have the
composition represented by a point on the curve AB. Conversely, above the
transition point, the saturated solution of the racemate would be
supersaturated with respect to the two tartrates, and transformation into
the latter would ensue. If, therefore, a solution of equimolecular
proportions of rubidium _d_- and _l_-tartrates is allowed to evaporate at a
temperature above 40deg, a mixture of the two tartrates will be deposited;
while at temperatures below 40deg the racemate will separate out.
[Illustration: FIG. 100.]
Similar relationships are met with in the case of sodium ammonium _d_- and
_l_-tartrate and sodium ammonium racemate; but in this case the racemate is
the stable form in contact with solution above the transition point
(27deg).[347] Below the transition point, therefore, the solubility curve
of the mixed tartrates will lie below the solubility curve of the racemate.
Below the transition point, therefore, sodium ammonium racemate will break
up in contact with solution into a mixture of sodium ammonium _d_- and
_l_-tartrates. At a higher temperature, 35deg, sodium ammonium racemate
undergoes decomposition into sodium racemate and ammonium racemate.[348]
The behaviour of sodium ammonium racemate is of interest from the fact that
it was the first racemic substance to be resolved into its optically active
forms by a process of crystallization. On neutralizing a solution of
racemic tartaric acid, half with soda and half with ammonia, and allowing
the solution to evaporate, Pasteur[349] obtained a mixture of sodium
ammonium {267} _d_- and _l_-tartrates. Since Pasteur was unaware of the
existence of a transition point, the success of his experiment was due to
the happy chance that he allowed the solution to evaporate at a temperature
below 27deg; for had he employed a temperature above this, separation of
the racemate into the two enantiomorphous forms would not have occurred.
For this reason the attempt of Staedel to perform the same resolution met
only with failure.[350]
Decomposition of the Double Salt by Water.--In the two cases just
described, the solubility relationships at the transition point are of a
simpler character than in the case of most double salts. If, at a
temperature above the transition point, a mixture of rubidium _d_- and
_l_-tartrates in equimolecular proportions is brought in contact with water
a solution will be obtained, which is saturated with respect to both
enantiomorphous forms; and since the solubility of the two optical
antipodes is identical, and the effect of one on the solubility of the
other also the same, the solution will contain equimolecular amounts of the
_d_- and _l_-salt. If, now, the solution is cooled down in contact with the
solid salts to just below the transition point, it becomes supersaturated
with respect to the racemate, and this will be deposited. The solution
thereby becomes unsaturated with respect to the mixture of the active
salts, and these must therefore pass into solution. As the latter are
equally soluble, equal amounts of each will dissolve, and a further
quantity of the racemate will be deposited. These processes of solution and
deposition will continue until the single tartrates have completely
disappeared, and only racemate is left as solid phase. As a consequence of
the identical solubility of the two tartrates, therefore, no excess of
either form will be left on passing through the transition point. From this
it will be evident that the racemate can exist as single solid phase in
contact with its saturated solution at the transition point; or, in other
words, the racemate is not decomposed by water at the transition point. The
same behaviour will evidently be exhibited by sodium ammonium racemate at
27deg, for the two enantiomorphous sodium ammonium tartrates have also
identical solubility.
{268}
Very different, however, is the behaviour of, say, astracanite, or of the
majority of double salts; for the solubility of the constituent salts is
now no longer the same. If, for example, excess of a mixture of sodium
sulphate and magnesium sulphate, in equimolecular proportions, is brought
in contact with water below the transition point (22deg), more magnesium
sulphate than sodium sulphate will dissolve, the solubility of these two
salts in a common solution being given by the following figures, which
express number of molecules of the salt in 100 molecules of water.[351]
COMPOSITION OF SOLUTIONS SATURATED WITH RESPECT TO
Na_{2}SO_{4},10H_{2}O AND MgSO_{4},7H_{2}O.
----------------------------------------
Temperature. | Na_{2}SO_{4}. | MgSO_{4}.
----------------------------------------
18.5deg | 2.16 | 4.57
24.5deg | 3.43 | 4.68
----------------------------------------
At the transition point, then, it is evident that the solution contains
more magnesium sulphate than sodium sulphate: and this must still be the
case when astracanite, which contains sodium sulphate and magnesium
sulphate in equimolecular proportions, separates out. If, therefore, the
temperature is raised slightly above the transition point, magnesium
sulphate and sodium sulphate will pass into solution, the former, however,
in larger quantities than the latter, and astracanite will be deposited;
and this will go on until all the magnesium sulphate has disappeared, and a
mixture of astracanite and sodium sulphate decahydrate is left as solid
phases. Since there are now three phases present, the system is univariant
(by reason of the restriction previously made that the vapour phase is
absent), and at a given temperature the solution will have a definite
composition; as given in the following table:--
COMPOSITION OF SOLUTIONS SATURATED WITH RESPECT TO
Na_{2}Mg(SO_{4})_{2},4H_{2}O AND Na_{2}SO_{4},10H_{2}O.
----------------------------------------
Temperature. | Na_{2}SO_{4}. | MgSO_{4}.
----------------------------------------
22deg | 2.95 | 4.70
24.5deg | 3.45 | 3.62
----------------------------------------
{269}
From the above figures, therefore, it will be seen that at a temperature
just above the transition point a solution in contact with the two solid
phases, astracanite and Glauber's salt, contains a relatively smaller
amount of sodium sulphate than a pure solution of astracanite would; for in
this case there would be equal molecular amounts of Na_{2}SO_{4} and
MgSO_{4}. A solution which is saturated with respect to astracanite alone,
will contain more sodium sulphate than the solution saturated with respect
to astracanite plus Glauber's salt, and the latter will therefore be
deposited. From this, therefore, it is clear that if astracanite is brought
in contact with water at about the transition point, it will undergo
decomposition with separation of Glauber's salt (supersaturation being
excluded).
[Illustration: FIG. 101.]
This will perhaps be made clearer by considering Fig. 101. In this diagram
the ordinates represent the ratio of sodium sulphate to magnesium sulphate
in the solutions, and the abscissae represent the temperatures. The line AB
represents solutions saturated with respect to a mixture of the single
salts (p. 268); BC refers to solutions in equilibrium with astracanite and
magnesium sulphate; while BX represents the composition of solutions in
contact with the solid phases astracanite and Glauber's salt. The values of
the solubility are contained in the following table, and in that on p. 268,
and are, as before, expressed in gm.-molecules of salt in 100 gm.-molecules
of water.[352]
{270}
-------------------------------------------------------------------------
| Astracanite | Astracanite
Temperature. | + sodium sulphate. | + magnesium sulphate.
|----------------------------|------------------------------
| Na_{2}SO_{4}. | MgSO_{4}. | Na_{2}SO_{4}. | MgSO_{4}.
-------------------------------------------------------------------------
18.5deg | -- | -- | 3.41 | 4.27
22deg | 2.95 | 4.70 | 2.85 | 4.63
24.5deg | 3.45 | 3.62 | 2.68 | 4.76
30deg | 4.58 | 2.91 | 2.30 | 5.31
35deg | 4.30 | 2.76 | 1.73 | 5.88
-------------------------------------------------------------------------
At the transition point the ratio of sodium sulphate to magnesium sulphate
is approximately 1 : 1.6. In the case of solutions saturated with respect
to both astracanite and Glauber's salt, the relative amount of sodium
sulphate increases as the temperature rises, while in the solutions
saturated for astracanite and magnesium sulphate, the ratio of sodium
sulphate to magnesium sulphate decreases.
If, now, we consider only the temperatures above the transition point, we
see from the figure that solutions represented by points above the line BX
contain relatively more sodium sulphate than solutions in contact with
astracanite and Glauber's salt; and solutions lying below the line BC
contain relatively more magnesium sulphate than solutions saturated with
this salt and astracanite. These solutions will therefore not be stable,
but will deposit in the one case, astracanite and Glauber's salt, and in
the other case, astracanite and magnesium sulphate, until a point on BX or
BC is reached. All solutions, however, lying to the right of CBX, will be
_unsaturated_ with respect to these two pairs of salts, and only the
solutions represented by the line XY (and which contain equimolecular
amounts of sodium and magnesium sulphates) will be saturated with respect
to the pure double salt.
Transition Interval.--Fig. 101 will also render intelligible a point of
great importance in connection with astracanite, and of double salts
generally. At temperatures between those represented by the points B and X,
the double salt when brought in contact with water will be decomposed with
separation of sodium sulphate. Above the temperature of the point {271} X,
however, the solution of the pure double salt is stable, because it can
still take up a little of either of the components. At temperatures, then,
above that at which the solution in contact with the double salt and the
less soluble single salt, contains the single salts in the ratio in which
they are present in the double salt, solution of the latter will take place
without decomposition. _The range of temperature between that at which
double salt can begin to be formed (the transition point) and that at which
it ceases to be decomposed by water is called the transition
interval._[353] If the two single salts have identical solubility at the
transition point, the transition interval diminishes to nought.
In those cases where the double salt is the stable form below the
transition point, the transition interval will extend downwards to a lower
temperature. Fig. 101 will then have the reverse form.
Summary.--With regard to double salts we have learned that their formation
from and their decomposition into the single salts, is connected with a
definite temperature, the _transition temperature_. At this transition
temperature two vapour pressure curves cut, viz. a curve of dehydration of
a mixture of the single salts and the solubility curve of the double salt;
or the dehydration curve of the double salt and the solubility curve of the
mixed single salts. The solubility curves, also, of these two systems
intersect at the transition point, but although the formation of the double
salt commences at the transition point, complete stability in contact with
water may not be attained till some temperature above (or below) that
point. _Only when the temperature is beyond the transition interval, will a
double salt dissolve in water without decomposition (_e.g._ the alums)._
* * * * *
{272}
CHAPTER XVI
ISOTHERMAL CURVES AND THE SPACE MODEL
In the preceding chapter we considered the changes in the solubility of
double salts and of mixtures of their constituent salts with the
temperature; noting, more especially, the relationships between the two
systems at the transition point. It is now proposed to conclude the study
of the three-component systems by discussing very briefly the solubility
relations at constant temperature, or the isothermal solubility curves. In
this way fresh light will be thrown on the change in the solubility of one
component by the addition of another component, and also on the conditions
of formation and stable existence of double salts in solution. With the
help of these isothermal curves, also, the phenomena of crystallization at
constant temperature--phenomena which have not only a scientific interest
but also an important bearing on the industrial preparation of double
salts--will be more clearly understood.[354]
A brief description will also be given of the method of representing the
variation of the concentration of the two salts in the solution with the
temperature.
Non-formation of Double Salts.--In Fig. 102 are shown the solubility curves
of two salts, A and B, which at the given temperature do not form a double
salt.[355] The ordinates represent the amount of A, the abscissae the
amount of B in a _constant amount_ of the third component, the solvent. The
{273} point A, therefore, represents the solubility of the salt A at the
given temperature; and similarly, point B represents the solubility of B.
Since we are dealing with a three-component system, one solid phase in
contact with solution will constitute a bivariant system (in the absence of
the vapour phase and under a constant pressure). At any given temperature,
therefore, the concentration of the solution in equilibrium with the solid
can undergo change. If, now, to a pure solution of A a small quantity of B
is added, the solubility of A will in general be altered; as a rule it is
diminished, but sometimes it is increased.[356] The curve AC represents the
varying composition of the solution in equilibrium with the solid component
A. Similarly, the curve BC represents the composition of the solutions in
contact with pure B as solid phase. At the point, C, where these two curves
intersect, there are two solid phases, viz. pure A and pure B, in
equilibrium with solution, and the system becomes invariant. At this point
the solution is saturated with respect to both A and B, and at a given
temperature must have a perfectly definite composition. To take an example,
if we suppose A to represent sodium sulphate decahydrate, and B, magnesium
sulphate heptahydrate, and the temperature to be 18.5deg (_i.e._ below the
transition point), the point C would represent a solution containing 2.16
gm.-molecules Na_{2}SO_{4} and 4.57 gm.-molecules MgSO_{4} per 100
gm.-molecules of water (p. 268). The curve ACB is the boundary curve for
saturated solutions; solutions lying outside this curve are supersaturated,
those lying within the area ACBO, are unsaturated.
[Illustration: FIG. 102.]
[Illustration: FIG. 103.]
[Illustration: FIG. 104.]
Formation of Double Salt.--We have already learned in the preceding chapter
that if the temperature is outside[357] the {274} transition interval, it
is possible to prepare a pure saturated solution of the double salt. If,
now, we suppose the double salt to contain the two constituent salts in
equimolecular proportions, its saturated solution must be represented by a
point lying on the line which bisects the angle AOB; _e.g._ point D, Fig.
103. But a double salt constitutes only a single phase, and can exist,
therefore, in contact with solutions of varying concentration, as
represented by EDF.
Let us compare, now, the relations between the solubility curve for the
double salt, and those for the two constituent salts. We shall suppose that
the double salt is formed from the single salts when the temperature is
raised above a certain point (as in the formation of astracanite). At a
temperature below the transition point, as we have already seen, the
solubility of the double salt is greater than that of a mixture of the
single salts. The curve EDF, therefore, must lie above the point C, in the
region representing solutions supersaturated with respect to the single
salts (Fig. 104). Such a solution, however, would be metastable, and on
being brought in contact with the single salts would deposit these and
yield a solution represented by the point C. At this particular
temperature, therefore, the isothermal solubility curve will consist of
only two branches.
[Illustration: FIG. 105.]
Suppose, now, that the temperature is that of the transition point. At this
point, the double salt can exist together with the single salts in contact
with solution. The solubility curve {275} of the double salt must,
therefore, pass through the point C, as shown in Fig. 105.
From this figure, now, it is seen that a solution saturated with respect to
double salt alone (point D), is supersaturated with respect to the
component A. If, then, at the temperature of the transition point, excess
of the double salt is brought in contact with water,[358] and if
supersaturation is excluded, _the double salt will undergo decomposition
and the component A will be deposited_. The relative concentration of the
component B in the solution will, therefore, increase, and the composition
of the solution will be thereby altered in the direction DC. When the
solution has the composition of C, the single salt ceases to be deposited,
for at this point the solution is saturated for both double and single
salt; and the system becomes invariant.