Joseph William Mellor.

A comprehensive treatise on inorganic and theoretical chemistry (Volume 1) online

. (page 136 of 177)
Online LibraryJoseph William MellorA comprehensive treatise on inorganic and theoretical chemistry (Volume 1) → online text (page 136 of 177)
Font size
QR-code for this ebook


when free. The number and kind of other atoms present and their mode of
combination seem to have no influence on the numerical value of this property.
The observations of Neumann and Joule indicate that the constituent atoms of a
sob'd compound behave as if the solid were a mechanical mixture of its component
atoms, and each atom were free to vibrate independently of the others. 1

According to the kinetic theory, temperature is proportional to the kinetic
energy of the molecules ; and consequently, Dulong and Petit's rule points to a
similar relation. It must be added that we can form no real conception of the
" temperature of an atom "or of the " temperature of a molecule." All our con-
ceptions of temperature are based on the properties of atoms and of molecules en
masse. R. Clausius 2 supposes the specific heat of an element to be made up of two
magnitudes (i) the heat c v required to raise the kinetic energy of the molecules ;
and (ii) the heat e required to perform internal work. Clausius calls c v the true
specific heat of the solid. Hence if M be the atomic weight of an element, the
observed atomic heat MC is equal to M(c v -\-). It is often stated that at the
absolute zero of temperature, 273, all atomic motion must cease. This is a mere
assertion, of no intrinsic value, and probably wrong. The statement might be
true of the translatory motion of the molecules such that B. Clausius' Mc v is zero
because of the convergence of the specific heats of the elements to zero as the
temperature approaches absolute zero. The same fact also shows that the internal
work c becomes very small at absolute zero, and the fact that Dulong and Petit's
rule is so nearly exact at ordinary temperatures coupled with the assumption that
at the same temperature the kinetic energy of the molecules is the same, leads to
the inference that when the temperature of elementary solids is raised from absolute
zero, the internal work per atom is approximately the same.

W. Jankowsky 3 argued that the heat of a chemical reaction is developed for the most
part by the conversion of potential energy into heat, and that at the absolute zero, the
energy content of a substance is entirely potential, and that there must be an absolute
upper limit or maximum temperature where the energy content of a substance consists
entirely of heat.

Among the evidence which indicates that the atoms of a solid^even at absolute
zero, probably oscillate about a position of equilibrium, the following may be cited :
(i) The low coefficient of thermal expansion of solids shows that tne volume would
be very little changed if the solids were cooled to absolute zero ; (ii) it is not pro-
bable that solids would lose their compressibility at absolute zero ; (iii) the natural
frequency of the vibrations of the atoms of a solid calculated by different methods
shows no signs of ceasing at absolute zero, (iv) P. Debye's effect, in which the
intensity of the higher orders of the X-ray spectrum of crystalline solids increases
as the temperature of the crystal is lowered, points in the same direction.

L. Boltzmann, in a paper Ueber die Natur der Gasmolekule (1876), has shown that



THE KINETIC THEORY OF ATOMS AND MOLECULES 809

the kinetic and potential energies of the molecules of a monatomic solid vibrating
about a position of equilibrium are equal in magnitude. Consequently, the total
energy of a vibrating system called for convenience an oscillator is shared equally
between the average kinetic and potential energies, and is twice the value of either
alone. 4

This interesting result follows by considering the motion of a particle under the influence
of a central attractive force moving on an orbit about its position of equilibrium. If the
particle were at rest in any part of its orbit, it would tend to move to its centre of attraction,
and in so doing, would acquire such a velocity that its kinetic energy would be the same as it
possessed when oscillating in its former orbit. Hence, a particle oscillating about a centre
of rest possesses both kinetic and potential energy, and on the average, the one is equal to
the other, provided that the time average of its kinetic energy is equal to that of the
potential energy. This is the case if the potential energy is zero as the particle passes through
its position of equilibrium. In reality, the equipartition theorem applies only to the kinetic
energy, but if the average kinetic and potential energies are equal, each will make the same
contribution to the specific heat.

L. Boltzmann assumed that the atoms of a solid have natural periods of vibration,
so that if a monatomic gas be in contact with a solid, the bombardment of the
gaseous molecules produces a state of thermal equilibrium when the mean kinetic
energy of vibratory motion of the atoms of the solid is equal to the mean kinetic
energy of the translatory motions of the molecules of the gas. With a solid, the
average kinetic energy of the atoms in each state will be the same ; but the average
kinetic energy per atom of a monatomic gas is f KT per atom, hence, the sum of the
kinetic and potential energy of the solid will be 2 X^KT, or 3KT per atom ; and if
there are N atoms in a gram-atom of the solid, the total kinetic and potential energy
will be 3NKT or 3RT per gram-atom, where NK=R is nearly equivalent to two
calories per gram-atom per degree. Accordingly, the atomic heat, wC v , must be
3R=6 nearly, or with a more exact value of R, 5'95. This interesting argument
shows how the atomic heats of the monatomic solids are nearly twice the molecular
heats of the monatomic gases ; and it furnishes a brilliant deduction of Dulong and
Petit's rule for solids.

The agreement between the result of Boltzmann's assumption and Dulong and
Petit's observation, shows that the atoms of a monatomic solid probably vibrate
so that their energy is equally divided between the kinetic and potential energy.
If the oscillations of the atoms are not harmonic in character, the time averages of
the kinetic and potential energies will not generally be equal. The agreement in
question also shows that the opposing forces attraction and repulsion between
the atoms just balance one another so that as two atoms approach one another the
attractive forces gradually diminish, and the repulsive forces gradually increase
until the latter predominate.

The discrepancies between Boltzmann's 3R and Dulong and Petit's constant.
There must be a flaw somewhere, because the theory does not explain (i) how the
solid elements with a low atomic heat carbon, silicon, and boron have normal
atomic heats at high enough temperatures ; nor (ii) how all solids give abnormally
low values at low temperatures. Many attempts have been made to explain the
discrepancy between theory and fact. It may be necessary to consider :

(a) The time required for the atoms to adjust themselves to a change of temperature.
L. Boltzmann assumed that the atoms take a long time to adjust themselves
to the temperature but no corresponding variation of specific heat with tempera-
ture has been detected ; and the specific heats of solids are so related with the
melting points that if the specific heat changed with time, the melting point
ought likewise to change. Such a phenomenon has not been observed even in
the case of artificial minerals and natural minerals formed aeons ago. While the
translational energy may be rapidly distributed between the internal motions of
a molecule during a collision, yet, if the distribution is slow, so that it becomes
appreciable only after millions of collisions, the number of collisions per second



810 INORGANIC AND THEORETICAL CHEMISTRY

is so great a million occurs in about one-seven-thousandth of a second according
to G. J. Stoney that even when the exchange is slow, a second of time is a com-
paratively long interval.

(b) The oscillations of the atoms are not harmonic. L. Boltzmann assumes that the
vibrations of the atoms is harmonic ; and this assumption is probably valid for most
metals far from their melting points ; but if the amplitudes of the vibrations of the
atoms are large, oscillations may be no longer harmonic. I. Langmuir 5 has
emphasized the fact that if the oscillations of the atoms are not harmonic, the time
averages of the kinetic and potential energies will not be equal. The average
kinetic and potential energies will be equal, only when the motion is harmonic,
in which case, the restoring force acting on the atom is proportional to the dis-
placement from the position of equilibrium. If the restoring force increases more
slowly than the displacement, the potential energy will be greater than the kinetic,
and from the principle of equipartition, the atomic heat will be greater than 2>R ;
conversely, if the restoring force increases more rapidly than the displacement, the
atomic heat will be less than 3/2. The remarkable closeness of the atomic heats of
the elements to the value 3R, is taken to show that the forces to which the atoms of
a solid are subjected vary approximately with the displacement of the atoms from
their position of equilibrium.

There must then be both attractive and repulsive forces acting between the atoms. On
the average, these opposing forces must just balance each other. As one atom approaches
another the repulsive force must gradually increase and the attractive force decrease until
the repulsive force greatly predominates. We cannot consider that the repulsive forces
in solids are exerted only during collisions between atoms, for under these conditions there
would be no potential energy and the atomic heat would be %R.

(c) The congealing of molecules to more rigid systems. R. A. Millikan (1912) 6
considers that it may possibly be assumed that as the temperature is reduced, the
atoms of the solid are frozen, so to speak, into rigid systems of continually increas-
ing size where each system is endowed with the kinetic energy of agitation appro-
priate to its temperature before absolute zero is attained, it might be possible for
the total energy of the whole mass to become that of a single molecule of the sur-
rounding gas. C. Benedicks (1913) has also shown that the equipartition law is
avoided by assuming that the solids are not always monatomic, but at low tempera-
ture form atomic complexes, which change the number of degrees of freedom. The
equipartition law applies only to free atoms. However, from Joule's law, it appears
probable that the rule for atomic heats applies to atoms in combination as well as
free.

(d) Another explanation of the reduction in the atomic heats below 3R when the
temperature is low was suggested by A. Einstein (1907) . It is based on the so-called
quantum theory of energy ; and has been remarkably successful.

REFERENCES.

1 W. Sutherland, Phil. Mag., (5), 32. 550, 1891.

2 R. Clausius, Pogg. Ann., 116. 100, 1862 ; W. Sutherland, Phil. Mag., (5), 32. 550, 1891 ;
W. Jankowsky, Zeit. Ehktrochem., 23. 368, 1917.

3 W. Jankowsky, Zeit. Eleklrochem., 25. 325, 1919.

4 L. Boltzmann, Sitzber. Alcad. Wien, 74. 555, 1876 ; H. Petrini, Zeit. phys. Chem., 16. 97,
1895 ; E. J. Routh, The Dynamics of a System of Rigid Bodies, London, 2. 54, 1892.

5 I. Langmuir, Journ. Amer. Chem. Soc., 38. 2236, 1916.

6 R. A. Millikan, Science, 37. 119, 1913 ; C. Benedicks, Ann. Physik, (4), 42, 1333, 1913 ;
F. Richarz, Zeit. anorg. Chem., 58. 356, 1908 ; 59, 146, 1908 ; J. J. van Laar, Proc. Acad.
Amsterdam, 11. 765, 1909 ; 12. 120, 133, 1909 ; 13. 454, 636, 1910 ; J. Duclaux, Cmn.pt. Rend.,
155. 1015, 1912; A. H. Compton, Phys. Rev. (2), 6 377, 1915; F. Schwers, ib., (2), 8. 117, 1916.



THE KINETIC THEOKY OF ATOMS AND MOLECULES 811
15. The Quantum Theory of Energy and Dulong and Petit's Rule

An observer who does not allow himself to be led in his work by any hypothesis, how-
ever cautious and provisional, renounces beforehand all deeper understanding of his own
results. M. PLANCK (1914).

An attempt to imagine a universe in which action is atomic leaves the mind in a hopeless
state of confusion.' J. H. JEANS (1914).

J. H. Jeans, in his The Dynamical Theory of Gases (Cambridge, 1904), shows that
Maxwell-Boltzmann's theorem of the equipartition of energy is based upon the
assumption that there is no interaction between matter and aether, whereas every
ray of light which reaches the eye is evidence against the truth of the assumption.
With ordinary diatomic transparent gases two (rotational) degrees of freedom
appear to be directly affected by the translational motions during a collision ; with
the coloured gases there appear to be motions which consume energy in the pro-
duction of sethereal vibrations. In 1906, M. Planck, in his Vorlesungen uber Theorie
der Warmestrahlung (Leipzig, 1906), assumed that the interchange of energy between
the aether and a vibrating atom is not a continuous process, but takes place per
saltum that is, discontinuously by jumps in definite amounts hv, where v represents
t he Schwingungszahl or the frequency of the atomic vibrations, and h is a universal
constant in the same sense that e, the unit of electrical change, is a universal constant.
The constant h called Planck's constant seems to be a fundamental unit which
regulates and controls the ceaseless ebb and flow of energy in the world of matter.
For brevity, write ehv. This means that for any given temperature, a certain
amount of energy is associated with the vibrating atom, and that this amount is
a function of the vibration-frequency v of the atom ; and energy can be absorbed
or emitted by a vibrating system spasmodically, and only in amounts e or in integral
multiples of this magnitude such as e,2e, 3c, . . ., but not in intermediate quan-
tities, say, Je, e, f e, . . . This virtually means, said H. Poincare (1911) , l that a
physical system can exist only in a finite number of states, it leaps from one of these
states to another without passing through a continuous series of intermediate states ;
and, adds M. Planck : 2

The continuity of all dynamical effects was formerly taken for granted as the basis
of all physical theories and in close correspondence with Aristotle, was condensed in the
well-known dogma- -natura nonfacit saltus- nature makes no leaps. However, present-day
investigation has made a considerable breach even in this venerable stronghold of physical
science. This time it is the principle of thermodynamics with which that theorem has
been brought into collision by new facts, and unless all signs are misleading, the days of
its validity are numbered. Nature does indeed seem to make jumps- and very extra-
ordinary ones.

The ration or unit of energy e is called a quantum, and hence this hypothesis is
called the quantum theory of energy. According to this remarkable hypothesis,
the vibrating atoms radiate definite loads hv of energy which, for any given
vibration frequency, v, are indivisible. M. Planck inferred that the average energy
possessed by an oscillating unit, with two degrees of freedom,

Average energy = , ; or, Average energy = . . (12)

J.

per degree of freedom ; and three times this value for a monatomic oscillator with
three degrees of freedom instead of the average value 2>kT per atom deduced by an
application of Maxwell-Boltzmann's theorem which assumes that the evolution
or absorption of energy is a continuous process. Here u is written in place of the
fraction hv/kT.

M. Planck follows the theory of probability in deducing his formula ; D. L. Chapman
starts from J. H. van't Hoff's well-known expression Q/RT z = (d log k)/dT. From the
quantum law, if the resonators of vibration period v are attached to the molecules of a gas,




812



INORGANIC AND THEORETICAL CHEMISTRY



then there will be vibrators possessing amounts of energy 0, hv, 2kv, . . . , but no
vibrators with intermediate amounts of energy. Let the number of vibrators with 0. //r,
2hv, . . . amounts of energy be respectively n Q , w lt 2 , . . . n m . Then, from J. H. van't
Hoff's rule, rnhvdT/kT 2 =d log (W^WQ), and by integration between T and oo, and writing
u in place of hv/kT, it follows that w OT = e ~" mM - Consequently, the mean energy of a
vibrator is (hve- u + 2hve- 2u + 3hve~ 3u + . . .)'(! +e~ M + e~ 2M + . . .) which, by division,
reduces to (30) above. It might also be added that F. R. von Bichowsky 3 has shown that
(i) Planck's radiation law, (ii) the quantum theory, and (iii) the equipartition law are not
independent, because if any two be assumed the third will follow ; and further, S. Ratnowsky
has shown that if J. W. Gibb's assumption (that the free energy of a system cannot be
generated until the magnitude of the co-ordinates fixing the energy of the system has
reached a certain value, and is thereafter given off continuously) be made, Planck's
radiation law follows directly without the assumption of the quantum hypothesis. Other
attempts to establish a theory of radiation without quanta have been made by M. Brillouiti,
A. Byk, H. L. Callendar, and R. C. Tolman.

In one modification of the hypothesis, the oscillator is supposed to absorb energy
continuously until an amount hv has been absorbed, when it has a chance of emit 1 i ng
the whole of this unit. Otherwise, energy will continue being absorbed until it
reaches 2hv, 3hv, . . . Only when the amount of energy reaches an exact mull iple
of hv is the oscillator in a condition to emit the whole of its energy. 4

It is an open question what are the receptacles of energy in a solid. As H. A. Lorentz
(1913) 6 has shown, the phenomena of light make it highly probable that energy quanta

can have no individual and permanent existence in the
aether, and cannot be regarded as accumulations of energy
in minute spaces travelling about with the velocity of
light. It seems more probable that the energy of solids
is localized in the elastic vibrations of the solid, and that
the mean energy of an oscillator is equal to the mean
energy of an aether vibration of the same frequency. No
reason can be assigned why the electric charge e always
acts as if it were atomic, or why electrons, each with a
fractional charge say, e do not exist ; so also no
reason can be assigned why energy can change only by
complete quanta.



1000




200 400 600 800 1000 K
Fia. 9. Values of the Function
f}v/(e u l) at Different Tem-
peratures.



In 1907, A. Einstein, 6 in his paper Die Plancksche
Theorie der Strahlung und die Theorie der spezifischen
Wdrme, extended Planck's atomic theory of radia-
tion to the radiation of heat. He assumed that the longer heat waves emitted and
absorbed by solids are due to vibrations of the constituent atoms about a mean posi-
tion of rest. A. Einstein further assumed that the energy of the solid does not reside
solely in the kinetic energy of the atoms, but the vibration-frequency v of each atom
has three degrees of freedom, and the energy of these vibrations is governed by
M. Planck's law, and A. Einstein thus deduced a formula analogous with that of
M.Planck:



Bv
Average energy 3R ^

& JL



(13)



for the energy of the vibrating atoms of a solid. If fiv is very small, the function is
approximately 3RT, and the expression corresponds with Dulong and Petit's law,
which requires the atomic heat of monatomic solids to be proportional to the
temperature. For all other values of f$v the function is less than 3RT. The values
of the function are plotted in Fig. 9. At any given temperature, the value of
the function differs more and more from the value of T as the value of fiv is
increased. Differentiating for dEjdT, the atomic heat, C v , he obtained :



Atomic heat =3R



(14)



where u, for convenience, has been written in place of fiv/T, and j8 is written in



THE KINETIC THEORY OF ATOMS AND MOLECULES 813

place of h/k ; k is the atomic gas constant represented by R/N, when R is the
ordinary gas constant, and N (approximately 6'OGxlO 23 ) denotes the number of
atoms per gram-atom. It will be evident that when the fraction u=f$v/T is very
large, v will either be very large, or T very small, and C v will be virtually zero ;
and when u is small, C V =3R. For example, if $v\T be greater than 10, C V =3R
X0'004 ; and if it be less than unity, C v will be less than 3.RxO'92. In the former
case, the specific heat approaches unity, and in the latter case, C v is nearly normal.

The numerical values of the constants h and ft. Seven different lines of argument 7
show that Planck's constant h, is equivalent to (6'5543i0'0025) X 10~ 27 ergs per second, and is
the same for all substances. For the yellow )-sodium line with a wave-length 0'5896ju, it
follows that v is 3 x 10 10 /0'5896 x 10~ 4 or 5'088 x 10 14 , so that hv for this radiation is
6-62 x!0~ 27 x 5-088 xlO 14 33-7 x 10~ 13 ergs per second. The numerical value of )3 is
4-865 xlO- 11 .

W. Nernst and F. A. Lindemann (1911) 8 have shown that Einstein's equation
is in fair agreement with their observations of specific heats at low temperatures,
although discrepancies appear as the temperatures approach absolute zero ; and
they tried to rectify Einstein's equation by introducing a new term. So that the
atomic heat C c becomes



Atomic heat = - R



(15)



on the assumption that the solid is a mixture of oscillating atoms half of which
have the vibration frequency v and half the frequency \v. M. Planck and
A. Einstein assumed that all the oscillating atoms had a frequency v ; and W. Nernst
and F. A. Lindemann's assumption is a first approximation to a summation ex-
tending over an infinite number of values of v. W. Nernst and F. A. Lindemann's
equation represents the observed atomic heats of solids aluminium, copper, silver,
lead, mercury, zinc, iodine, and the diamond down to the lowest temperatures.
A few numbers selected from Nernst's tables for silver and the diamond are indicated
in Table XXIV.



TABLE XXIV.- THE ATOMIC HEATS OF SILVER AND THE DIAMOND AT DIFFERENT

TEMPERATURES.



Silver.


Diamond.


PI/ = 221.


/3v=1940.


Temperatures.


C P (calc.).


C P (obs.).


Temperatures.


C P (calc.).


C P (obs.).


35-0


1-59


1-58


30


o-oo


o-oo


53-8


2-98


2-90


92


o-oi


0-03


100


4.77


4-86


205


0-62


0-62


200


5-77


5-78


243


0-97


0-95


273


6-02


6-00


306


1-59


1-58


331


6-12


6-01


358


2-08


2-12


535


6-45


6-46


413 2-55


2-66


589


6-57


6-64


1169


5-41


5-45



According to the form of quantum hypothesis now under consideration, oscillat-
ing atoms cannot absorb energy unless it comes to them with a certain degree of
intensity equal to hv, or some whole multiple thereof. As the temperature rises,
the number of molecules which take up loads of energy from the low intensity heat
waves increases rapidly in accord with the equipartition law, and the need for ab-
sorbing energy in integral multiples of hv. Molecules of chlorine and bromine begin
to absorb this energy at a lower temperature than the transparent diatomic gases
because (i) the bond of union between the respective atoms is weak, and their



INORGANIC AND THEORETICAL CHEMISTRY



frequency v and consequently also their quantum hv is small ; hence (ii) the
quanta or loads of energy hv absorbed by these oscillators are correspondingly small ;
and (iii) the temperature at which the kinetic energy of the diatomic oscillators
attain the value hv is correspondingly low. Diatomic hydrogen molecules at a low
temperature act like monatomic molecules because the rotatory motions at any
given temperature correspond with a definite frequency v, and when the energy
of impact falls below this value of hv, no energy can go into these rotations, and
energy is solely distributed among the three degrees of freedom corresponding with
translatory motion.

The quantum hypothesis gives a qualitative explanation : (i) how the atomic
heats of the elements approach zero as the temperature falls, and (ii) how abnormally
low values appear at a higher temperature with the elements of low atomic
weight. From measurements of atomic heats, it seems as if, as the temperature
rises, different kinds of atoms can take on their normal load at different stages,
the heaviest atoms take it on first, the lighter atoms last. With a given rise of
temperature there is a corresponding increase in the vibratory energy of the atoms
of an element, and at a sufficiently low temeprature, only a definite fraction of the
atoms can take on the normal quota hv. The higher the vibration frequency v,
the higher the temperature at which energy can be absorbed. Again, other things
being equal, with a falling temperature, the greater the vibration frequency v, the



Online LibraryJoseph William MellorA comprehensive treatise on inorganic and theoretical chemistry (Volume 1) → online text (page 136 of 177)