LAW OF VARIATION OF ANGLES IN ISOMORPHOUS SERIES. RELATIVE DIMENSIONS OF
UNIT CELLS. FIXITY OF ATOMS IN CRYSTAL.
We are now in a position to approach the conclusion of the long
controversy as to the constancy or otherwise of crystal angles in
the cases of greatest similarity, those of isomorphous substances,
and to appreciate how the conflicting views of Haüy and Mitscherlich
and their schools of thought have at length been reconciled. As the
result of a comprehensive study, on the part of the author, of the
sulphates and selenates of the rhombic series R_{2}S/SeO_{4}, and of
the double sulphates and selenates of the monoclinic series
R_{2}M(S/SeO_{4})_{2}.6H_{2}O, in which R represents the alkali
metals, potassium, rubidium and cæsium, and in which M may be
magnesium, zinc, iron, nickel, cobalt, manganese, copper or cadmium,
four facts of prime significance have been definitely established.
(1) The crystals of the different members of an isomorphous series
exhibit slight but real differences in their interfacial angles, the
magnitude of the angle changing regularly with the alteration of the
atomic weight of the interchangeable metals or negative elements of the
same family group which give rise to the series, as one metal or
acid-forming element is replaced by another. The amount of the
difference increases as the symmetry of the system diminishes. Thus the
maximum difference for the more symmetrical rhombic series of sulphates
and selenates is 56′, which occurs in the case of one angle between
potassium and cæsium selenates, and it is usually much less than this;
in the case of the less symmetrical monoclinic series of double salts
the maximum angular difference observed was 2° 21′, between potassium
and cæsium magnesium sulphates.
(2) The physical properties of the crystals, such as their optical and
thermal constants, are also functions of the atomic weights of the
elements of the same family group which by their interchange produce the
series.
(3) The dimensions of the elementary parallelepipedon of the
space-lattice, or in other words, the separation of the molecular
centres of gravity, the points or nodes of the space-lattice, along the
three directions of the crystal axes, also vary with the atomic weight
of the interchangeable elements.
(4) Specific chemical replacements are accompanied by clearly defined
changes in the crystal structure along equally specific directions.
Thus, when the metal, say potassium, in an alkali sulphate or selenate
is replaced by another of the same alkali-family group, rubidium or
cæsium, there is a marked alteration in the crystal angles and in the
dimensions of the space-lattice, corresponding to elongation of the
vertical axis; and when the acid-forming element sulphur is replaced by
selenium, its family analogue, a similar very definite change occurs,
but the expansion in this case takes place in the horizontal plane of
the crystals.
Confirmatory results have also been obtained as regards the
morphological constants, the investigations not extending to the optical
or thermal properties, by Muthmann for the permanganates, and by Barker
for the perchlorates, of the alkali metals. Hence, there can be no doubt
whatever that, as regards the various series investigated, which are
such as would be expected to afford the most definite results owing to
the electro-positive nature of metals being at its maximum strength in
the alkali group, the above rules are definite laws of nature.
Thus it is clear that in the cases of isomorphous substances, which were
the only possible exceptions to the generalisation that _to every
chemically distinct solid substance of other than perfect cubic symmetry
there appertains a specific crystalline form, endowed with its own
particular angles and morphological crystal elements, which are
absolutely constant for the same temperature_, the law does really hold,
and isomorphous substances are no exceptions. The law of progression of
the crystal properties according to the atomic weight of the
interchangeable elements affords indeed at the same time both an
amplification of the generalisation and a precise explanation of its
mode of operation in these cases.
The discovery of the local effect produced by the two kinds, positive
and negative, of chemical replacement, has a profound bearing on crystal
structure. For it is thereby rendered certain that the atoms are fixed
in the crystal edifice, and therefore in the molecule in the solid
state. It becomes obvious that the atoms—in their stereometric positions
in the molecule, being thus fixed in the solid crystal when the
molecules set themselves rigidly in the regular organisation of the
space-lattice—form the points of the regular point-system of the crystal
structure, which determines to which of the thirty-two classes of
symmetry the crystal shall belong. Any movement of the atoms in the
crystal, other than that which accompanies change of temperature, and
possibly change of pressure, is thus improbable; and this experimental
proof of their fixity, afforded by the fact that definitely orientated
changes accompany the replacement of particular atoms, also doubtless
indicates that the latter are located in the particular directions along
which the changes of exterior angle and of internal structural
dimensions are observed to occur. Stereo-chemistry, which has made such
enormous advances during the last few years, thus becomes of even
greater importance than Wislicenus and its other originators ever dreamt
of.
Within the atoms in the crystal the constituent electronic corpuscles
may be and probably are in rapid movement, and such physical effects as
have hitherto been ascribed to movement of the atoms within the crystal
are doubtless due to movement of the electronic corpuscles within them,
the sphere of influence of the atom itself being fixed in space in the
solid crystal, and being doubtless defined by the area within which the
corpuscular movements occur.
[Illustration:
FIG. 60.—Diagram illustrating Progressive Change of Crystal Angles in
Isomorphous Series.
]
Three illustrations of the law of change of the crystal properties with
variation of the atomic weight of the determinative elements of an
isomorphous series may be given, and will serve to render the practical
meaning of the generalisation clearer. The first is a diagrammatic
representation, in Fig. 60 (in a very exaggerated manner as the real
change would be inappreciable on the scale drawn), of the change of
angle on replacing the potassium in potassium sulphate, K_{2}SO_{4}, or
selenate, K_{2}SeO_{4}, by rubidium or cæsium. The inner crystal
outline, a vertical section, is that of the potassium salt. The vertical
lines represent the intersections of the two faces of the brachypinakoid
_b_ = {010} with a vertical plane parallel to the macropinakoid _a_ =
{100}; the horizontal lines represent the intersection of the two faces
of the basal plane _c_ = {001} with the same vertical plane; and the
oblique lines represent the intersection of the vertical plane with the
four faces of the dome form _q_ = {011}, which are inclined to both _b_
and _c_ planes. The diagram is thus designed to show the variation of
the inclination of these latter dome faces to the two rectangular axial
plane faces _b_ and _c_. The outer crystal outline represents a similar
section of a crystal of the corresponding cæsium salt, and the middle
outline that of a crystal of the rubidium salt.
The progressive alteration of the angle of the q-face will be obvious,
the direction of the change being correct, but the amount of change, as
already stated, being much exaggerated; in reality it never reaches a
degree between the two extreme (potassium and cæsium) salts. It will be
remembered that the respective atomic weights of potassium, rubidium,
and cæsium are 38·85, 84·9 and 131·9, when hydrogen equals 1, that of
rubidium being almost exactly the mean.
[Illustration:
FIG. 61.—Diagram illustrating Progressive Change of Double Retraction
in Isomorphous Series.
]
The second illustration is taken from the optical properties. Fig. 61
represents graphically the regular diminution of double refraction (the
difference between the two extreme indices of refraction α and γ) which
accompanies increase of the atomic weight of the metal present. The
diagram exhibits the closing up of the two spectra afforded by three
analogously orientated 60°-prisms, one of each of the three salts, such
as was used in determining two of the refractive indices of the salt.
Each prism produces two refracted rays from the single ray furnished by
the collimator of the spectrometer, and consequently two images of the
signal-slit of the collimator when monochromatic light is used, or two
spectra if white light be employed. The Websky signal-slit is narrow at
the centre to enable an accurate allocation to the vertical cross-wire
of the telescope to be made, but wide at its top and bottom ends, in
order to transmit ample light, and Fig. 61 shows four images of this
signal produced by each prism, namely, one R in red C-hydrogen light and
another B in greenish-blue F-hydrogen light belonging to each of the two
spectra, in order to locate the two ends of each of the latter, coloured
monochromatic light of each of the two colours in turn and of the exact
C and F wave-lengths having been fed to the spectrometer from the
spectroscopic illuminator. It will be observed in the case of the top
row that the two spectra, each indicated by the adjacent red and
greenish-blue images, are well apart, the relative distance being about
that actually observed in the case of potassium sulphate. They are
nearer together, however, in the second row, which indicates what is
observed in the case of the analogous rubidium salt, and in the lowest
row representing the relative distances of the two spectra apart in the
case of the cæsium salt, they are so close together as to overlap; for
in this latter case the greenish-blue image of the left-hand spectrum,
corresponding to the a index of refraction, occupies the same position
as the image for yellow sodium light of the right-hand spectrum
corresponding to γ would occupy in the case of cæsium sulphate, the a
refractive index for F-light being 1·5660 and the γ index for Na-light
being 1·5662. The progression of the alteration of the amount of the
double refraction is thus very striking, as the atomic weight of the
metal is varied.
The third illustration of the law of progression with atomic weight is
also an optical one, and is taken from the monoclinic series of double
sulphates and selenates. It indicates the rotation, with increase of the
atomic weight of the metal, of the ellipsoid which graphically
represents the optical properties, about the unique axis of symmetry,
which is likewise an axis of optical symmetry, of the crystal. In the
potassium salt the ellipsoid occupies the position indicated by the
ellipse drawn in continuous line in Fig. 62, the section of the
ellipsoid by the symmetry plane; the outline of a tabular crystal
parallel to the symmetry plane is also given, as well as the axes of the
crystal and of the ellipsoid lying in that plane.
[Illustration:
FIG. 62.—Diagram illustrating Progressive Rotation of Optical
Ellipsoid in Monoclinic Isomorphous Series.
]
In the rubidium salt the ellipsoid has rotated over to the left, as
indicated by the dotted ellipse, for a few degrees, the number of which
varies slightly for the different groups of double salts; while in the
cæsium salt it has swung over much more still, to the place marked by
the ellipse drawn in broken line. In both this and the last illustration
it will be remarked that the optical change is greater between the
rubidium and cæsium salts than it is between the potassium and rubidium
salts, the reason being that the optical properties are usually
functions (of the atomic weight of the interchangeable elements) which
are of an order higher than the first corresponding to simple
proportionality.
These three ocular illustrations may serve to render this interesting
law of progression, according to the atomic weight of the
interchangeable elements which give rise to the isomorphous series,
clearer to the mind, by placing before it concrete instances of the
operation of the law.
The generalisation itself may be very concisely expressed in the
statement that:
_The whole of the properties, morphological and physical, of the
crystals of an isomorphous series of salts are functions of the atomic
weights of the interchangeable chemical elements of the same family
group which give rise to the series._
The fact that this law extends to the structural dimensions, equally
with all other morphological properties, as stated under (3) at the
beginning of this chapter, is of especial interest. For it has actually
been found possible to determine the relations of the dimensions of the
unit parallelepipeda of the space-lattices of the various salts, that
is, the separation of the molecular points of the space-lattice in the
directions of the three crystal axes, for the various salts of the
isomorphous series. This is achieved by combining in suitable formulæ
the volume of the unit cell of the space-lattice with the relative
lengths of the three crystal axes, _a_, _b_, _c_.
The axial ratios _a_ : _b_ : _c_ are calculated from the measurements of
the crystal angles, as explained in Chapter VI., page 68, and the volume
is the physical constant long known as “molecular volume,” but now for
the first time understood as regards its meaning in the case of solid
substances. It is the quotient of the chemical constant molecular weight
(the sum of the atomic weights, taking into account the number of atoms
of each element present) by the specific gravity of the substance, here
the solid crystal. Very great care has been taken to obtain absolutely
accurate determinations of the specific gravities of the salts, as much
depends on this now very valuable physical constant, and all the values
obtained were reduced to the constant reference temperature of 20°, as
the density notoriously alters rapidly with change of temperature.
We have thus arrived at morphological constants of very considerable
importance, which are best termed “_Molecular Distance Ratios_,” as they
express the relative distances apart in the three directions of space of
the centres of gravity or other representative points of contiguous
chemical molecules. They are dependent on three experimental
determinations, atomic weight, specific gravity, and crystal angles, all
of which have now been brought to the highest pitch of refinement and
accuracy; hence the molecular distance ratios are particularly
trustworthy constants. If it were only known how much is matter and how
much is space in the molecular parallelepipedal cell, we should actually
have in these constants a relative measure of the sizes of the
molecules. They do give us, however, the relative directional dimensions
of the molecular unit parallelepipedal cells of the space-lattices of
the various members of the isomorphous series, just as the molecular
volumes give us the relative volumes of these cells. For in an
isomorphous series we are absolutely sure that the plan on which the
space-lattice is constructed, its style of architecture, is identical
for all the members of the isomorphous series. Hence, the molecular
distance ratios are in these cases absolutely valid and strictly
comparable. The ratios are generally expressed by the Greek letters χ :
ψ : ω.
On comparing the molecular distance ratios for a potassium, a rubidium,
and a cæsium salt of any of the series of sulphates, selenates,
permanganates, perchlorates, double sulphates or double selenates
investigated, we invariably find that the values of χ, ψ, and ω for the
rubidium salt (rubidium having the intermediate atomic weight) lie
between the analogous sets of three values for the potassium and cæsium
salts respectively, in complete accordance with the law.
For the generalisation to apply absolutely it is essential that the
interchangeable elements shall belong strictly to the same family group
of the periodic classification of Mendeleéff. Potassium, rubidium, and
cæsium fulfil this condition absolutely, and so the law of progression
of the crystal properties with the atomic weight of the interchangeable
elements applies rigidly to their salts. Now there are two bases, the
metal thallium and the complex radicle group ammonium NH_{4}, which are
not thus related to the group of three alkali metals just mentioned, but
which are yet capable of replacing those metals isomorphously in their
crystals without more change of angle or of structural constants than is
provoked by the replacement of potassium by cæsium; and often indeed the
amount of change has been singularly like the lesser amount observed
when rubidium has been interchanged for potassium. But although this is
so, the directions of the changes are irregular, being sometimes the
same as when rubidium or cæsium is introduced, and sometimes
contrariwise, and in the case of thallium there are also striking
optical differences, the thallium salts being exceptionally highly
refractive. Still, morphologically the ammonium and thallium salts may
legitimately be included in the same isomorphous series with the salts
of potassium, rubidium, and cæsium, and a somewhat wider interpretation
has to be given to the term “isomorphism” in order to admit these cases.
To distinguish the inner group formed by family analogues, that is, the
more exclusive group obeying the law of progression according to the
atomic weight, the term “eutropic” is employed.
Thus the “isomorphous series” of rhombic sulphates, selenates,
permanganates, and perchlorates, and the monoclinic series of double
sulphates and double selenates, comprise the potassium, rubidium,
cæsium, thallium and ammonium salts and double salts of sulphuric,
selenic, permanganic, and perchloric acids, while the inner more
exclusive “eutropic series,” following the law absolutely, comprises in
each case only the salts containing the family analogues, potassium,
rubidium, and cæsium.
In this beautiful manner has the controversy between the schools of Haüy
and Mitscherlich now been settled, the interesting law described in this
chapter having definitely laid down the true nature and limitations of
isomorphism, while at the same time absolutely proving as a law of
nature the constancy and specific character of the crystal angles of
every definitely chemically constituted substance.