the quantum theory is fully borne out. From measurements of the
positions of the successive sub- maxima it is found in the case of water
vapour that the largest value for \ Y is of the order 4oo/x, i.e. water
vapour should exhibit absorption at various positions down to this
region, a result in agreement with the observations of Rubens. On the
other hand, the experimental results obtained by E. von Bahr indicate
a lack of symmetry with respect to the principal maximum of a given
band, which indicates in turn that a single frequency difference is in-
sufficient to account for the whole series of band heads.
The question of the rotational spectrum of water vapour, i.e. the
infra-red spectrum beyond IO/A, hasmlso been investigated by Eucken
(Verh. d. D. phys. Ges., 15, 1159 (1913)), who adopts Bjerrum's expres-
sion for the series of frequencies to be expected, viz.
v = n . /2ir 2 . I.
This expression is based upon the assumption that the molecule is
symmetrical, and possesses therefore but one moment of inertia. Eucken,
however, emphasises the lack of symmetry of the water molecule and
ascribes to it at least two principal moments of inertia, which would lead
us to expect two different series in the spectrum. Sackur (Ann. physik,
[iv.], 40, 95 (1913)) has arrived at the conclusion on quite other
88
A SYSTEM OF PHYSICAL CHEMISTRY
grounds that the water molecule possesses two main moments of inertia,
which are in the ratio of i : 2. Eucken finds, on taking two values for
I, one of which is approximately twice the other, and calculating the
corresponding series as given by the Bjerrum expression, that satisfactory
agreement is obtained between the calculated and observed band
maxima. The following table contains Eucken's results :
INFRA-RED ABSORPTION BANDS OF WATER VAPOUR.
A Observed (in jx).
A Calculated (in /*).
Direct.l
i
Indirect.^
First Series.
I = 0-96 x io- 40 . n.
Second Series.
1= 2'2i x to" 40 , n.
j
IO'9/A
iQ'SfjL 16
ir6
n-6 15
12-4
12-4 14
i3'4
I3'3 13
i4'3
14-4 12
157
I5'8 ii
J 7'5
17*3 10
i9'3 9
21-6 8
__
2 5M
24-8 7
25-0,11 16
11
28-9 6
267 15
28-5 14
30
307 13
33
347 5
33*3 12
36
36-5 ii
39
40-0 io
42
43*3 4
45
44'5 9
50
47-5
50-0 8
ca. 58
54-56
57*8 3
57'2 7
66-67
64-69
66-6 6
78-80
79-81
80-0 5
91-92
8 7 2
ca. 103
109
100 4
124-128
133 3
170-172
173 I
240-250
2OO 2
385-398
40O I
E. von Bahr (ibid., 15, 1154) has examined very carefully a simpler
case than that of water vapour, namely, the infra-red spectrum of HC1
gas. The subdivision of each part of the double band (found by Bur-
meister and shown by the dotted line in the figure) into twelve separ-
ate maxima is very clearly shown in Fig. 9.
The data corresponding to Fig. 9 are given in the following table.
The wave-lengths Aj and A 2 of the right and left parts of the double
absorption band, as well as the corresponding frequencies, are given.
1 Rubens and Aschkinass, Wied. Ann., 64, 584, 1898; Rubens, Ber. Berliner^
Akad. t 1913, p. 513.
2 E. von. Bahr, Verh. D. phys. Ges., 15, 731, 1913.
ABSORPTION SPECTRUM OF HYDROGEN CHLORIDE 89
The first column contains the value of n t i.e. the order of that maximum
reckoned from the middle of the absorption band. The middle of the
3-4 /A 3-3 f* 3-2/t,
80
FIG. g. Hydrogen chloride gas (E. von Bahr). S denotes slit width.
Dotted curve due to Burmeister. Upper full curve corresponds to
760 mm. ; lower full curve corresponds to 380 mm.
absorption band is characterised by the wave-length
the frequency v = 8-636 x io 13 .
HYDROGEN CHLORIDE GAS.
3'474/x, and
ft.
A! in /x.
\z in IJL.
1 x io- 13 -
f 2 X jo" 13 '
I
3*444
3*504
8711
8-562
2
3-416
3-528
8-782
8-503
3
3*394
3-556
8-839
8-436
4
3'372
3'583
8-897
8-373
5
3'352
3'6lo
8-950
8-310
6
3-332
9-004
7
33I7
~
9-044
In the following table are given the values of (i/j - v ) and (i/ - v 2 ),
which, according to Bjerrum, should be equal to the rotational fre-
quencies of the molecule of HC1 :
.
n^-.
-11
VQ v>2 x io
-11
v xio mean.
\ in /*.
v' 10 " 11 '
I
7-5
7'4
7*45
403
7*45
2
14-6
13-3
I3-95
215
6-98
3
20-3
2O-O
20-15
149
6-72
4
26-1
26-3
26-20
"5
6-55
5
3 1 '4
3 2-6
32-0
94
6-40
6
36*8
36-8
82
6-13
7
40-8
*"*"_
40-8
74
5*83
9 o A SYSTEM OF PHYSICAL CHEMISTRY
On Bjerrum's theory we would expect vrjn to be constant. As
a matter of fact it is moderately constant over a considerable range but
shows a definite tendency to diminish as n increases. E. von Bahr re-
gards the variation as outside the limit of experimental error, although
Eucken (loc. tit.) considers the question as not yet definitely settled.
By employing the Bjerrum theory it is seen that HC1 should absorb at
several positions down to A. = 400^. Rubens and von Wartenberg
(Verh. d. D. phys. Ges., 13, 796 (191 1)) have found considerable absorp-
tion exhibited by this substance in the residual-ray range, the absorption
being especially strong at 150^.
Bjerrum (Verh. d. D. phys. Ges., 16, 640 (1914)) recalculates the
rotation frequencies of HC1 in a manner somewhat different from that
employed by E. von Bahr, and concludes that the agreement is as good
as can be expected in view of the unavoidable experimental error in de-
termining the position of the successive sub-maxima.
Whilst we are dealing with inter-relations between different portions
of the infra-red spectrum, it is convenient to refer briefly to certain ap-
plications of Bjerrum's theory of molecular rotations which have been
made by Baly (Phil. Mag., 27, 632 (1914) ; ibid., 29, 223 (1915) ; ibid., 30,
510 (1915)) in connection with the structure of bands exhibited by sub-
stances in the ultra-violet region. According to Baly the frequency of
the central position of such a band must be a multiple of the principal
frequency exhibited in the infra-red region by the same substance.
Further, the breadth which such ultra-violet bands possess is due to
a number of lines lying close together, and the distances apart of
these lines, when expressed in terms of frequencies, are found to have
constant values, and are to be ascribed, according to Baly, to the exist-
ence of certain basis constants characteristic of the molecule, these
basis constants being also capable of accounting for the infra-red ab-
sorption of the molecule. This method of linking up the infra-red and
ultra-violet absorption spectra has found considerable support in the
spectra of benzene and other organic compounds, and particularly so in
connection with the absorption of gaseous sulphur dioxide which has
been examined by Garrett (Phil. Mag., 31, 505 (1916)) and by Miss
Lowater (As trophy s. Journ. , 31, 311 (1910)). Thus Garrett has found
that the optical centre of the marked absorption band of SO 2 in the
near ultra-violet occurs at A, = 296'i/A/x, the corresponding wave-number
(i.e. reciprocal of wave-length) being 3378. On applying the quantum
hypothesis as Baly has done in this case we would expect that this wave-
number should be an even multiple of the principal infra-red band of
SO 2 . The infra-red spectrum of SO 2 has been examined by Coblentz
(Pub. Carnegie Inst., Washington, 1905) and has found a pronounced
band at 7 '4/n or wave-number 135*1. If this wave-number be multiplied
by 25 we obtain the number 33 7 7 '5, which is almost identical with the
centre of the ultra-violet band as determined by Garrett. For details of
the method of determining the basis constants the papers referred to
must be consulted. At the present time the mechanical significance of
these constants is not clear. The accuracy, however, with which they
KRUGERS THEORY 9 t
permit of even very complex spectra being calculated in detail indicates
quite clearly that there is a very close connection between the infra-red
and the ultra-violet absorption of any substance. This question is taken
up briefly at a later stage (in Chap. VI.) in connection with the selective
photo-electric effect.
KRUGER'S THEORY.
(Cf. Kriiger, Ann. Physik, [iv.], 50, 346; ibid., 51, 45 (iQ 1 ^).)
Kriiger assumes on the basis of the now generally accepted Ruther-
ford-Bohr structure of atoms and molecules (if. Chap. V.) that gaseous
molecules must be gyroscopic in nature; that is, such molecules are
quite incapable of rotations, but are capable of carrying out precessio nal
vibrations (cf. Chap. I.). These vibrations are totally distinct from the
ordinary vi oration of two atoms along the line joining their centres, to
which we have already referred in dealing with Bjerrum's treatment.
As a result of collisions with other molecules Kriiger considers that the
ring of rotating electrons in a molecule, which serves to unite the two
atoms together (in the case of a diatomic molecule), suffers displace-
ments perpendicular to its direction of motion, with the result that the
atoms themselves describe small circular orbits and the molecule as a
whole exhibits motion of the precessional type. Such precession has
already been shown diagram matically in Chapter I. Kriiger points out
that such precessional motion is entirely kinetic and necessarily involves
two degrees of freedom. In the temperature range in which the
principle of equipartition holds good the energy term corresponding to-
these two degrees of freedom will be RT, which is identical in magni-
tude with that postulated by Bjerrum for a diatomic molecule on the
basis of rotation of the molecule as a whole over the same temperature
region. Over this range therefore the numerical values for the energy
term is the same on either view in the case of a diatomic molecule.
The chief advantage of Kriiger's theory lies however in the explana-
tion which it offers of the behaviour of monatomic gases. Monatomic
molecules, such as that of argon, possess energy which is completely
taken account of by the free translation alone. An atom is therefore
incapable of rotation. If molecular rotations were possible in general
we would expect monatomic molecular rotations to be possible also.
The theory of molecular rotations is not very satisfactory in this con-
nection. On the other hand, Kriiger shows that precessional vibrations
in the case of a spherical monatomic molecule become practically
infinitely rapid, and may therefore be left out of account in regard ta
the energy content. This conclusion becomes clearer when we intro-
duce the quantum hypothesis.
At low temperatures it is necessary to treat all kinds of constrained
motion, such as ordinary atomic vibrations, precessional vibrations or
rotations from the standpoint of the quantum theory. The molecular
heat in virtue of any of these types of motion decreases with falling
92 A SYSTEM OF PHYSICAL CHEMISTRY
temperature and converges to zero at absolute zero of temperature. In
the special case of a monatomic molecule in which the precessional
vibrations are exceedingly rapid the corresponding quantum must be
exceedingly great, and consequently the likelihood of any molecule
possessing even one such quantum is negligibly small. In other words,
the observed energy content can only be due to the free translational
motion which is always directly proportional to the absolute temperature,
i.e. the molecular heat in such a case is 3/2R, in complete agreement
with experiment. In the case of diatomic gases Kriiger has shown
that the vibration frequencies of precession correspond to the farther
infra-red region, about 30^ in the case of hydrogen, and that the so-
called rotation spectrum is really due to precessional vibrations. It is
only at high temperatures that the precessional vibration becomes so
violent as to merge into a rotation (cf. Chap. I.). It must be clearly
understood that the conflicting views of Bjerrum and Kriiger have
nothing to do with the ordinary atomic to-and-fro vibration inside the
molecule, which Bjerrum has shown gives rise to bands in the short
infra-red region, i.e. at wave-lengths shorter than io//
On the whole, Kriiger's view serves to bring the behaviour of mon-
atomic and diatomic gases into much closer accord than had hitherto
been the case. Considerable experimental work is still necessary, how-
ever, before we can decide between the theory of molecular rotations
and the theory of precessional vibrations.
A translation of the major part of Kriiger's papers will be found in
Appendix III. for those desirous of following Kriiger's treatment in
greater detail.
THE MOLECULAR HEAT OF HYDROGEN GAS AT Low TEMPERATURES.
(Cf. A. Eucken, Sitzungsber. kon. preuss. Akad. Wissenschaft ,
p. 141, 1912.)
The molecular heat of hydrogen is of particular interest because
hydrogen is the simplest type of diatomic molecule. The most striking
result obtained by Eucken is that at low temperatures hydrogen behaves
as a monatomic gas, i.e. it exhibits no energy of a rotational or of a
gyroscopic kind. The experimental details will be found in the paper
Teferred to. It is sufficient in the present instance to quote certain of
the results. Eucken employed different quantities of hydrogen in the
calorimeter (internal volume, 39 c.cs.), and his experiments are suf-
ficiently exact to show that the molecular heat varied with the concen-
tration of the gas. The following table gives the value of the molecular
heat observed corresponding to a particular temperature, 1 and the mass
of hydrogen employed :
1 The molecular heat values are " instantaneous," or true molecular heats at a
tgiven temperature; not mean values over a temperature range.
MOLECULAR HEAT OF HYDROGEN GAS
TAbs.
Gram-moles of HJ.
Molecular Heat.
273-1
0-0966
4*94
194
0-1710
4'39
91
0-3550
3 '44
91
82
0-1315
0-1268
332
3'30
The data obtained at still lower temperatures are given in the follow-
ing table :
TAbs.
Gram-moles
H a .
Molecular
Heat.
Gram-moles
H a .
Molecular
Heat.
Gram-moles
H 2 .
Molecular
Heat.
35
0-1909
3'42
0-1794
0-1040
3*20
40
3-28
3-18
45
3*30
3*28
3-13
50
3*13
60
3-09
65
3-21
3*14
70
3-26
80
0-2585
3-29
0-1295
3-23
3*20
85
3*34
3-27
3*32
90
3*42
, 3*31
IOO
3*64
3'37
no
3-82
3-62
It will be seen that the lower the concentration of the gas the lower
the molecular heat. 1 This effect is taken account of by the thermo-
dynamic relation
?
To make use of this expression Eucken employs a relation of D. Ber-
thelot which connects the molecular heat at any given concentration
with that of the substance in the ideal gas state. 2 In this way Eucken
arrives at the following values for the molecular heat of hydrogen in the
ideal gas state :
1 This is not true universally for hydrogen, i.e. at very high pressures there is a
tendency for Cv to diminish.
2 Berthelot's expression involves the critical temperature and critical pressure of
the gas.
94 A SYSTEM OF PHYSICAL CHEMISTRY
MOLECULAR HEAT OF HYDROGEN IN THE IDEAL GAS STATE.
T Abs. Molecular Heat.
35 2-g8
40 2-98
45 3'oo
50 3-01
60 2-99
65 3'4
70 3-10
80 3-14
85 3'2i
90 3-26
ioo 3 '42
no 3*62
I96-5 439
273-1 4-84
These values show that the molecular heat of hydrogen falls rapidly
with a fall in temperature, finally (at about 60 abs.) attaining a value
2-98, identical with that of a monatomic gas. Eucken finds that the
molecular heat curve shows even a more rapid fall than that given by
the Einstein formula for monatomic solids. The reason of the rapid
fall is not clear ; Eucken discusses the question in the paper referred to.
The most striking fact, as already stated, is that at low temperatures the
molecule of hydrogen possesses only energy of translation. Possibly
other diatomic gases behave similarly ; nothing is known on this point.
Eucken (Ber. Deutsch. phys. Ges., 18, 4 (1916)) has measured the
molecular heats of hydrogen and helium at high pressures and at low
temperatures, and has found that when the gases are strongly compressed,
the molecular heat falls below the value 3/2R(2'98). This is discussed
by Nernst (Zeitsch. Elektrochem., 22, 185 (1916)), who regards this
" degradation of the ideal gaseous state " as due to the substitution of
circular rotational motion of the molecules at very low temperatures in
place of free translation. Thus, instead of the energy of the gas being
expressed by the relation : E = 3/2 RT (due to translation), we have to
substitute the expression
F - 3 R ft*
- 7* ^O"/T - j
where v is the frequency of rotation. This expression can give rise to a
molecular heat which is less than 3/2R. Experiment shows, however,
that even at fairly low temperatures the relation E = 3/2 RT holds good,
so that the second expression is only of importance at very low tem-
peratures.
As a result of this modified expression for the energy of the gas
which is functioning as a monatomic one, it follows that, under the same
extreme conditions, in place of the perfect gas law P = RT/V we must
write
p = R &v
V ' i - e
MOLECULAR HEAT OF HYDROGEN 95
The most striking conclusion arrived at by Nernst (loc. at. ; also
Verh. Deutsch. phys. Ges., p. 99 (1916)) is that this "degradation phe-
nomenon " stands in close causal connection with the chemical con-
stant of the gas, i.e. the quantity ; where / is the integration
constant of the Clapeyron equation for the vapour pressure of the
liquified substance. We have already discussed the significance of the
chemical constant in connection with the application of Nernst J s Heat
Theorem to gaseous reactions.
Nernst finds theoretically that the chemical constant of a monatomic
substance, or a substance which functions as such, can be expressed by
a relation of the form
C = C + i'5 log 10 M,
where M is the gram- molecular weight of the substance, C is the
chemical constant (identical with the C of Chap. III., Vol. II.), and C
in the present instance is a constant independent of the nature of the
substance considered. Nernst shows from an examination of the most
reliable data existing at present that an expression of this form is in
agreement with the experimental values of C. Nernst's theoretical expres-
sion is likewise in agreement with those 01 Sackur (Nernst Festschrift,
p. 405 ; Ann. Physik, 40, 67 (1913)), and of Tetrode (Ann. Physik, 38,
434 j 39, 255 (1912); Proc. Acad. Amsterdam (1915)).
By introducing the quantum theory into a vapour pressure relation
based upon his Heat Theorem, Nernst calculates Goto be 1*590, the
unit of pressure being the atmosphere, which is in good agreement with
the mean observed value, namely, 1*62 obtained from a consideration
of the data available in connection with hydrogen, argon, monatomic
iodine gas, and mercury vapour. Tetrode's (calculated) value for C is
- i -608.
CHAPTER V.
(Physical equilibrium, continued) Distribution of electrons in atoms Structure
of the atom from the standpoint of the quantum theory The Rutherford-
Bohr atom-model Parson's magneton and the structure of the atom High
frequency spectra of the elements Moseley's relation.
The Rutherford Atom-model.
(Rutherford, Phil. Mag., [vi.], 21, 669 (1911).)
THE underlying idea in this atom-model is that the atom consists of a
central charge, concentrated on a nucleus, which is surrounded by an
" atmosphere " of electrons, rotating in certain orbits. Practically the
whole mass of the atom is to be ascribed to the nucleus. The number
of "atmosphere " electrons is not large. The dimensions of the nucleus
are considerably smaller than the dimensions of the atom as a whole.
Rutherford estimates the diameter of the nucleus to be of the order
io~~ 12 cm., whilst the diameter of the atom as a whole is of the order
io~ 8 cm. The nucleus itself possesses a structure about which, how-
ever, nothing is known. It, the nucleus, contains probably quite a
large number of electrons, but these are bound firmly, except in the
case of radioactive materials which are capable of expelling one of these
bound electrons in the form of y3-rays. The outermost ring of the
"atmosphere'' electrons contains those which give the property of
valency to the atom. The innermost electron of the atmosphere is the
source of X-rays, the shortest wave-length which the atom is capable
of emitting. The necessary disturbance of the innermost electron is
brought about by collision with a /2-ray (cathode ray) which jerks the
innermost electron out ; on its return journey to the innermost ring it
causes the emission of the extremely short waves which we call X-rays.
Rutherford was led to this planetary view of the atom in order to ex-
plain the scattering of a and (3 particles. When such particles pass
very close to the centre of an atom they are violently deflected, an
effect which could be produced by a nuclefls of the kind referred to.
The angle of scattering has been measured by Geiger and Marsden in
the case of a-rays, and by Crowther in the case of /3-rays. The outer
" atmosphere " of electrons rotating in rings do not appreciably affect
the direction of the motion of an a particle travelling through the atom.
It is only when the a particle comes into close contact with the nucleus
that its path is abruptly altered, the path becoming hyperbolic.
RUTHERFORD-BOHR ATOM-MODEL 97
Bohr's Application of the Quantum Theory to the Rutherford Atom-model.
(Bohr, Phil. Mag., [vi.], 26, i., 4?6, 857 (1913) ; ibid., 27, 56 (1914) ;
30, 394 (iQ^)-)
The following is a brief account of the line of thought pursued by
Bohr, given to a large extent in his own words.
Bohr first of all points out that the Rutherford atom-model, which
has just been referred to, suffers from the serious drawback that the
system of "atmosphere" electrons is unstable; that is, unstable from
the standpoint of classical electro-dynamics. If, however, we introduce
Planck's concept, the instability may no longer exist from the theoretical
point of view. The problem of atomic structure affords therefore a
further instance of the necessity of introducing some new concept, such
as that of Planck, into electro-dynamics in order to account for the
observed facts. Bohr first attempts to apply the quantum theory to
the process whereby a free electron such for example as exists in
a vacuum tube when a discharge is passed may be conceived of as
attaching itself to a positively charged nucleus. It will be shown
that it is possible from the point of view taken to account, in a simple
way, for the Balmer law of line* spectra of hydrogen and helium, and
possibly the theory will eventually be capable of dealing with the spectra
of more complicated atomic structures.
According to Rutherford, the hydrogen atom consists of a nucleus
with a single electron describing a closed orbit around it The first as-
sumption is that the mass of the electron is negligible compared with the
mass of the nucleus, and that the velocity of the electron is small compared
with that of light. Bohr considers that whilst the electron is rotating in
this orbit or stationary state it is neither radiating or absorbing energy.
Such rotation represents an equilibrium condition of the system as a
whole. The production of spectra must be due therefore to some kind
of departure from an equilibrium state. In general Bohr considers the
possibility of a series of such stationary states corresponding to electron
orbits of different radius, and emission or absorption of radiation is due
to the electron passing from one stationary or equilibrium state to* another*
The quantum idea is introduced in the further hypothesis that during
the passage of the electron from one equilibrium state to another,
homogeneous radiation of a certain frequency v is emitted or absorbed,
the amount of radiant energy so emitted or absorbed being hv, where
h is Planck's constant. It will be realised at once how very different
these considerations are from those based on classical electrodynamics.
On the classical view a charged particle rotating steadily in a closed