282 ; fa = 229 (215).
7 8 A SYSTEM OF PHYSICAL CHEMISTRY
AgCl./?v 2 = 179; ^=102(114).
zC p Obs.
2 7 8
The numbers in brackets denote the values of /fri actually obtained
by the melting-point method. They are very close to the values of
ftv l which are arbitrarily chosen.
An examination of the data given shows that at very low tempera-
tures the contribution made by the Debye portion is much greater than
that made by the Einstein portion, as we would expect.
It will be recalled that the Nernst-Lindemann formula applies fairly
well to the case of KC1 and NaCl, the frequency employed being that
obtained by the Reststrahlen method. "This agreement must be
regarded as half accidental, and may be explained on algebraic grounds
from the approximation
In the case of KC1 and NaCl the fundamental [molecular] frequency
[vi in the above table] is in fact about 0-75 times the internal [atomic]
frequency [v 2 in the above case]."
An interesting case is that of mercurous chloride. Taking the
molecular formula to be HgCl and calculating the molecular heat by
the above method, discrepancies are observed which lie outside experi-
mental error. Nernst concludes therefore, that in solid calomel the
molecule Hg 2 Cl 2 (and possibly a still more complex molecule) exists,
which is in agreement with conclusions arrived at concerning the mole-
cular formula of calomel in the dissolved state. "The exact complexity
cannot yet be decided since the melting point of the salt is not open to
observation on account of decomposition."
A further interesting case is that of ice. In order to account for
the molecular heat of ice by the above method it is necessary to regard
the molecular weight as 36 approximately (cf. Nernst, The Theory of
the Solid State). Graphite and sulphur likewise show evidence of
Although the method of treatment which has just been outlined is
very ingenious, it is obviously of a somewhat hypothetical character in
MOLECULAR HEATS OF POLYATOMIC SOLIDS 79
its present formulation. It is not improbable that considerable modi-
fications may eventually be introduced. It represents, however, a
partially successful attempt to solve the problem of the heat capacity of
We shall conclude this chapter by quoting some numerical data
obtained by Eueken (Ber. Deutsch. phys. Ges., 18, 4 (1916)) in con-
nection with the heats of fusion and vaporisation of certain condensed
gases at low temperatures :
Substance. of Fusion in
Carbon monoxide . 224*1
Nitrogen was found to exist in two solid forms, the temperature of
transition being 35 '5 abs. and the molecular heat of transformation
53-8 cals. Oxygen exhibits three solid forms ; transition temperatures,
23 '5 and 42*5 abs., the corresponding molecular heats of transforma-
tion being 17*5 and 167*4 cals. respectively. Carbon monoxide exists
in two solid forms ; transition temperature, 60*4 abs., molecular heat
of transformation, 144*1 cals.
Substance. of Vaporisation
Nitrogen . 1363
. Carbon monoxide 1414
Eucken (loc. tit.} has likewise measured the specific heats of solid
and liquid argon, nitrogen, oxygen, and carbon monoxide ; the specific
heats of solid carbon dioxide and of liquid hydrogen, and likewise the
specific heat of highly compressed gaseous helium.
(Physical equilibrium in gaseous systems) Molecular heats of gases Bjerrum's-
theory Kriiger's theory.
BJER RUM'S THEORY.
IN Chapter I. we have already had occasion to discuss briefly the
problem of the molecular heats of gases. It has been pointed out
that the observed values cannot be accounted for on the basis of the
equipartition principle, especially the fact that the molecular heat varies
with the temperature. In view of the considerable advance which has
been made possible in the analogous case of solids by the application
of the quantum theory, it is of interest to see how far the same con-
siderations can be applied in the present case. This problem was
first taken up by Bjerrum (Zeitsch. Elektrochem., 17, 731 (1911) ; ibid., 18,
In Chapter I. we have given a table showing the number of possible
degrees of freedom, as estimated by Bjerrum, which are possessed by
mono- , di-, tri-, and tetra-atomic gas molecules in respect of translation,
rotation, and vibration. In the case of monatomic gases, which appear
to possess translational energy only, the equipartition principle of
classical statistical mechanics gives a satisfactory explanation of the
observed values, e.g. the case of argon already discussed. So long
as we restrict ourselves to translational movement the equipartition
principle necessarily holds good, whether the molecule be monatomic
or polyatomic; the distribution of energy in terms of the quantum
theory only enters when we deal with vibrations or rotations, i.e. move-
ment with respect to a centre of gravity.
As already pointed out, the result obtained in the case of argon
leads us to regard a mona.tomic gas as possessing no energy other than
that of translation. This is somewhat unexpected, and we shall return
to it later in connection with Kriiger's theory. For the present, how-
ever, we are discussing Bjerrum's treatment.
As regards rotation of the molecule as a whole Bjerrum shows that
the potential energy of rotation is negligible compared with the kinetic
energy. (The case is quite different of course for the vibrations of the
atoms inside the molecules.) In regard to vibration of one atom with
respect to the other in a diatomic gas molecule, according to the
quantum hypothesis the sum of the kinetic and potential energies, in-
stead of being RT, is a fraction < of this quantity. That is the total
vibrational energy is <RT. < is a function of the temperature T and
BJERRUM'S THEORY Si
of the vibration frequency v. According to Nernst and Lindemann we
Bjerrum has investigated the Einstein function as well. It is found to
hold fairly well, but is not so exact as that of Nernst and Lindemann.
At absolute zero <f> is zero, and at high temperatures </> approximates
to unity, i.e. at high temperatures the conclusions based on classical
statistical mechanics ought to hold good.
In the case of a triatomic molecule, containing the atoms A, B, and C,
these may vibrate with respect to one another in pairs, viz. A with respect
to B, B with respect to C, and C with respect to A. There are there-
fore three different vibrations to be taken account of, and the total
vibrational energy is
+ + .
The following tables contain the data obtained in the case of a number
of gases. The wave-length A, is that chosen by Bjerrum for substitution
in the Nernst- Lindemann equation. The temperature T is in degrees
absolute. C v (x, T ,) denotes the mean molecular heat (at constant.
volume) over the temperature range between x abs. and T abs.
Molecular Heat of Hydrogen.
The formula employed by Bjerrum is :
C,(o, T) = (5/2). R+ R<[2- 0y a]
where 5/2R covers the translational and rotational energy * of the mole-
1 Bjerrum, as already pointed out, ascribes two degrees of freedom to a diatomic
molecule, such as H 2 , in respect of rotation of the molecule as a whole. By writing
the corresponding energy term as RT (which then added to the 3/2RT, due to
translation, makes in all 5/2RT), Bjerrum is here assuming that rotational energy may
be treated from the standpoint of the equipartition principle. To get the contribution
which translation and rotation make to the total molecular heat it is of course neces-
sary to differentiate with respect to T, i.e. we obtain the term 5/2R. The same as-
sumption of equipartition is employed in the other gases, the molecular heats of which
are considered. In a later paper, however (Nernst Festschrift, 1912, p. 90), Bjerrum
regards rotational energy of the molecule as a quantity which has to be treated
from the quantum standpoint, the characteristic wave-length of molecular rotation
lying far in the infra-red region (about 30/4 in the case of hydrogen), whilst the
characteristic wave-length of atomic vibration (inside the molecule) lies in the short
infra-red region, between i and lo/m approx. In other words, molecular rotations are
much less violent than atomic vibrations and require a much smaller quantum, i.e.
a much smaller v or greater A. It has been observed that the atomic vibrations,
instead of manifesting themselves as sharp lines in the short infra-red portion of the
spectrum, actually exhibit bands more or less broad. An explanation of this has
been suggested by Bjerrum on the following lines. The broadening is regarded as
due to the presence of a number of " secondary " lines situated on either side of the
principal atomic-vibration lire, the latter being identified with the " head " of the
band. These secondary lines are regarded as multiples or overtones of the funda-
mental rotational frequency or wave-length which occurs in the longer wave region.
As the frequency of rotation is small its consecutive multiples will lie close together
VOL. III. 6
A SYSTEM OF PHYSICAL CHEMISTRY
cule and 2*o//, = X is the characteristic wave-length of atomic vibration,
which is of course connected with the frequency of atomic vibration
by means of the relation c = Xi/, where c is the velocity of light. The
following table indicates the degree of agreement between the observed
and calculated molecular heats of hydrogen :
C v (291, TO,)
C v (291, TO,)
Molecular Heat of Nitrogen. (Oxygen and carbon monoxide have
the same molecular heat as nitrogen.)
Formula employed: Ci(o, T,) = (5/2) . R +
C y (273, TO,)
C tf (273, TO,)
C y (agio, T0))
C v (291, T.)
In the case of hydrogen the characteristic wave-length assumed for
the atomic vibration is X = 2 '0/4 ; in the case of nitrogen, X = 2 '4/4.
One would expect that if the vibrating particles are electrically charged
that an absorption band should occur at these positions in the case of
hydrogen and nitrogen. No bands, however, have been observed in
this region of the spectrum. On the other hand, oxygen and carbon
monoxide have the same molecular heat as nitrogen, and direct
measurement has shown that oxygen possesses bands at 3*2/4 and 47/4,
whilst carbon monoxide exhibits bands at 2*4/4 and 4*6/4. The bands
when they occur at all, and, in fact, by measuring the distance apart of the
"secondary" lines referred to, it is quite possible to calculate the actual funda-
mental frequency of the rotation itself. On the basis of a calculation made by
Rayleigh it can be shown that under certain conditions, if v^ is the vibration fre-
quency of the atomic vibrations, and v 2 is the much smaller frequency of tte mole-
cular rotations, then in the region ot the spectrum in which the atomic vibration
manifests itself we may expect in general three absorption lines occurring at the
positions d efined byv l + v z , v lt and j/ x - v z .
of oxygen are weak, and it is possible that hydrogen would exhibit a
band if its absorption were measured when the gas itself were excited,
i.e. at a fairly high temperature.
Molecular Heat of CO 2 . (SO 2 has the same molecular heat.)
As this is a triatomic molecule we have three atomic vibrations to
take account of. Two of these, however, may be expected to be the
same, as the oxygen atoms are presumably linked symmetrically to the
carbon atom. The translational energy plus the rotational energy are
in this case, according to Bjerrum, represented by the term 3RT, i.e.
3/2 RT translational and 3/2RT rotational. The wave-length of vibra-
tion of each of the oxygen atoms with respect to the carbon is taken
by Bjerrum to be 5'ctyi; the wave-length of vibration of the two oxygen
atoms against each other is taken to be 8 -I/A. The formula employed
by Bjerrum is
C^o , T,) = 3 R + 2R4>[ 5 -o / /.] + R^S-i^].
[Note that the second term on the right-hand side of this expression
contains the number 2, to allow for the fact that there are two
similar kinds of atomic vibrations present. It may also be pointed
out that if CO 2 were a linear molecule, O = C = O, the rotational
energy term would be RT (as in a diatomic gas). The fact that the
value 3/2RT for the rotational energy agrees with experiment (cf. the
following table) means that CO 2 is not a linear molecule, its spatial
constitution being represented approximately by
stitution of the CO 2 molecule is considered in detail by Bjerrum (Verh.
d.D.phys. Ges.,16, 737, 1914).]
C v (273, I ,)
C v (273, T,)
C v (273, T,) Calculated
ing the Wave-Lengths
27, 4 '3.
The agreement between observed and calculated values is satisfactory.
We have now to see what evidence is available from the absorption
spectrum of CO 2 as regards the choice of the wave-lengths employed.
Direct measurement has shown that CO 2 possesses bands at 14*7, 4*3,
and 2 "jfji. These are of the same order of magnitude as those used
in the above formula. As a matter of fact, Bjerrum has also used the
three observed values of X and has obtained values for C v which agree
moderately with the observed. The formula in this case is
C (o, T,) = 3 R
A SYSTEM OF PHYSICAL CHEMISTRY
The resulting values are given in the last column of the preceding
Molecular Heat of Water Valour.
The formula employed by Bjerrum is
C,(o, T ,) = 3 R + 2Ri- 3 rf + R^[3-6rf +
The following values are thereby obtained :
C v Calculated.
C v Observed.
These values are true mole-
cular heats, i.e. instan-
taneous values for the
temperatures referred to,
not mean values over a
wide range of tempera-
C v (383, T ,)
C v (383, T.)
Mean values for the mole-
cular heat over the tem-
perature range indicated.
C v (291, T ,)
C v (291, TO,)
The absorption spectrum of water vapour contains many bands
over the range of X's employed above, i.e. bands at i 'i, i -5, 2 -o, 3-2, 47/4,
etc. Where there are so many bands almost any value might be re-
garded as borne out by the absorption spectrum. It appears from this
and other cases that the calculated molecular heat is not very sensitive
to error in the actual value chosen for X The term
empirical term introduced by Bjerrum to account for the fact that the
molecular heat of water vapour increases with the temperature more
rapidly than would be anticipated on the basis of the quantum theory.
The cause of this is by no means clear : possibly it is connected with a
depolymerisation of any bimolecules which may be present, though at
the higher temperatures this would be a negligible quantity.
The molecular heat of ammonia, a tetra-atomic body, may likewise
be calculated more or less satisfactorily by writing down terms for
BJERRUM' S THEORY 85
translation and rotation, and by applying the quantum theory to the
vibrations of the atoms, using the wave-lengths 2* I/A and 8'9/x, which
actually occur in the absorption spectrum of ammonia.
It is evident from the foregoing that to account for the observed
values of the molecular heats of gases it is essential to introduce the
quantum thebry in some form. Bjerrum's mode of treatment, though
marking a considerable advance, is open to criticism, more particularly
as regards the choice of the number of degrees of freedom.
We shall consider later the views put forward by Kriiger in connection
with the same problem.
One point remains to be emphasised in connection with the mole-
cular heat, or rather the energy content of gases. Owing to the fact that
the true atomic vibrations inside the molecule correspond to relatively
high frequencies (i.e. short infra-red region) such vibrations contribute
a relatively small amount to the total energy content ; for as the fre-
quency is high very few molecules will possess even one quantum of t
this type of energy. The so-called rotational frequencies (obtained
on applying the quantum theory to rotation) are much more important
as they occur in the longer infra-red region.
The Absorption Spectrum of Water Vapour and of Hydrogen Chloride
Gas in the infra-red region^ from the point of view of Molecular
Rotations^ treated on the basis of the Quantum Theory.
As already mentioned, Bjerrum has treated the rotational energy of
the di- or tri-atomic gas molecule from the point of view of the quantum
theory. That is, the rotational spectrum should exhibit a number of
lines related to one another by a constant frequency difference ; this, at
any rate, is the simplest possible statement. The spectrum due to such
rotations would be expected to lie in the farther infra-red (beyond IO/A),
the principle lines in the shorter infra-red region being due to the vibra-
tions of the atoms inside the molecule. Bjerrum assumes that the total
rotational energy varies in terms of quanta, and as a further simplification
he assumes that the moment of inertia of the molecule for all axes
through the centre of gravity is the same ; i.e. we have only to deal with
one such moment. The energy of rotation of a particle round an axis
is given by the expression 1/2 . 1 . (2?rj/) 2 , where v is the frequency of
rotation and I is the moment of inertia. On the older quantum theory
this energy must be represented by hv, 2h\>, etc., or in general by nhv t
where n is a whole number. Hence we have
1/2 I . (27Ti/) 2 = n . hv
or, v = n . /$/27r 2 . I.
According to this expression we would expect a difference series in the
frequencies of the band heads in the spectrum of a gas. That is, har-
monics of the fundamental rotational frequency are to be expected in
the spectrum according to the value ascribed to n. From an examina-
tion of the spectrum of water vapour Bjerrum concludes that the fre-
quency difference in the series of rotational bands is 1*73 x io 12 . He
A SYSTEM OF PHYSICAL CHEMISTRY
then proceeds to consider the spectrum between the limits io/x and 20//,,
ascribing all the band heads therein to rotations, n being given con-
secutive values from 10 to 16. The corresponding wave-length series
(in /A) is given by the expression
3_x io a
n x 173 x 10
The following table indicates the degree of concordance obtained by
this means :
*With increasing wave-length the bands become deeper and tend to
overlap. For wave-lengths greater than iy'5/x. the absorption becomes
general and without structure unless very good dispersion is obtained in
the apparatus. Whilst these results are to be regarded as a considerable
step in our understanding of the problem of the source of bands, the
numerical results just quoted are, according to E. von Bahr, open to a
certain amount of doubt (Verh. d. D. phys. Ges., 15, 731 (1913)).
At sufficiently high temperatures the energy of rotation will be
directly proportional to the absolute temperature. Hence, the fre-
quency of rotation will be directly proportional to the square root of
the temperature, since the rotational energy term involves v 2 . Hence,
the product of the rotational wave-length into */T should be a con-
stant. The rotational wave-length is obtained in the following way. In
the short infra-red region (i.e. the region of the spectrum in which the
bands are due to ordinary vibrations of atoms inside the molecule) it is
to be expected, on the basis of Rayleigh's considerations, already re-
ferred to, that, instead of a single well-defined band head due to vibra-
tion, we should find a triple-headed band, viz. a centre one due to
atomic vibration and two others due to compounded rotational and
vibrational motion. As a matter of fact, what is frequently found with
a certain degree of dispersional power in the spectrometer, is a double-
headed band. E. von Bahr suggests that the middle band is not really
missing, but only apparently so, on account of insufficient dispersion.
The two wave-lengths A^ and X 2 , which in general lie fairly close together,
correspond to two frequencies v and i/ 2 , and their difference (v 2 vi) is
just twice the most probable frequency v r of the rotation. v r is- of course
considerably smaller than either v l or i/ 2 and its normal position is far in
the infra-red. On expressing v r in terms of its wave-length \r we find
that \ r = 2 *' * . The above is a convenient way of measuring the
A! - AS
rotational wave-length without having to investigate the far infra-red
ABSORPTION SPECTRUM OF WATER VAPOUR 87
region. The validity of the expression can of course be tested by com-
parison with actual measurements in the far infra-red where the bands
are due to rotation only. From what has been said above, it follows
that the product
* ' * . JT should be a constant, provided the
- A 2
theoretical considerations are sound. Making use of some data ob-
tained by Paschen in the region 5 to 6/x, in the case of water vapour,
E. von Bahr has found that the two heads of the double band (due to
molecular rotations compounded with atomic vibration) alter their
relative position in accordance with the above expression.
. A! . A 2
V T A! - A 2 ]
In view of the fact that the temperature measurements were not very
exact, the constancy of the expression in the last column is very satis-
E. von Bahr has investigated more closely the constitution of the
bands of water vapour between 5 and 6/x and has found that each ex-
hibits a number of sub-maxima. Similar behaviour is shown by other
gases. That is, the discontinuous nature of the absorption required by