We can readily pass to the second manner of formulating Maxwell's
Law, and in so doing clear away the possibility of misconception referred
to a few lines back. Suppose we wish to find the number of molecules
whose speeds (i.e. merely velocity magnitudes) are at any moment
between the limits c and c + dc, but with no limits placed on their
directions of motion. We must obviously find the number of points in
our velocity diagram which lie between spherical surfaces of radii c and
+ dc. The volume of the elementary region of this diagram is
1 64 A SYSTEM OF PHYSICAL CHEMISTRY
and the point density in the region is given by (3). Hence the number
sought is
4 7rN V27/87T 3 ; 6 . &-&&* . dc . . . (4)
The expression (4) has, for a given value of the differential element of
speed, dc, its maximum value for a value of c which makes
a maximum. By the usual methods of the calculus it appears that this
value of c is : >J\c or o'8i6?; so that there are more molecules at an
instant possessing speeds within, say, one foot per second of this speed
than of any other assigned speed. That is, 0-8 1 6c is the speed with
" maximum probability ".
We can now deal with the fallacious inference mentioned above.
Let us consider a velocity with speed c and a definite direction whose
direction angles are a, (3, and y. The velocity has then the com-
ponents
c cos a, c cos ft, c cos y.
Now, undoubtedly, there are more molecules whose velocities have
components between o, o, o and du, dv, dw than there are molecules with
velocity components between c cos a, c cos ft, c cos y and c cos a 4- du,
c cos ft + dv, c cos y + dw. This is a just inference from equation (3)
but this statement is not true if we remove the condition ol ccefiniteness
from the angles a, ft, and y and allow them to take any possible values.
The removal of this restriction does not alter the speed limits, but it
multiplies the number in the second group enormously. From the
point of view of the velocity diagram, the representative points do un-
doubtedly cluster more densely around the origin of the diagram, and
.less and less densely around a point gradually receding from the origin,
but the actual law of distribution provides for the number of points
which lie in a thin spherical shell of definite thickness, increasing as we
go out to a shell of average radius o'8i6and then decreasing as we
go still further. From the origin to 0*816?, the increasing volume of
the shell more than compensates the decreasing density of the points ;
further on it fails to do so.
We can modify expression (4) so as to deal with velocities having
denned limits of direction. Expression (3) shows us that the density-
in-velocity is a function of speed only ; hence there are no privileged
directions. So of those molecules which at an instant have speeds
between c and c + dc, the number whose directions of motion are re-
stricted to the directions represented by points on a superficial element
w of the unit sphere constitute a fraction w/47r. Multiplying expres-
sion (4) by this fraction, we find that the number of molecules sc*
restricted as to speed and direction is
cuN ^27/87^. c 1 . e-v*!** . dc . (5)
If we wish to find the number of molecules which have speeds within,
limits finitely separated we must integrate (4) or (5) between the limits,
APPENDIX 1
165
Thus the number, unrestricted as to direction, with speeds between ^
and ^ is
. . . (6)
Expression (6) can be much simplified by introducing instead of the
speed c the ratio which this speed bears to the maximum probability
speed ^/f . c.
Call this ratio x so that
and hence
c=
dc =
. c . dx,
On introducing these expressions into (6) we find after a little re-
arrangement that the number of molecules whose speed ratios lie
between x l and x% is
_ r**
4N/V7T. ;
J X:
(7)
[In (7) we might refer to x as the speed, if we took the maximum
probability speed as the unit of speed.]
The properties of the integral in (7) can be most readily exhibited
by means of Fig. 1 4, which is the graph of the curve
y = *V**
The graph begins at the origin O; y rises to a maximum value i/* (or
M, M n
FIG. 14.
'37 approximately) when x = i ; the curve then descends asymptoti-
cally to the axis of x, y reaching as low a value as 0*02 when x = 2-5,
showing that all but a very small fraction of the molecules have speeds
at any moment less than two and a half times the maximum probability
speed (which corresponds to x equal to unity) or practically twice the
r-m-s-speed. In fact, the fraction of the molecules whose speed ratios
1 66
A SYSTEM OF PHYSICAL CHEMISTRY
lie between x l and x 2 is given by : 4/ ^/TT area of PjMiMaPa, where
OMi = x l and OM 2 = x 2 . By methods of analysis or quadrature of a
carefully drawn graph numerical values can be obtained. The following
table from Meyer's Kinetic Theory of Gases (translated by Baynes)
illustrates the point :
Of 1000 molecules of oxygen at o C., whose r-m-s-speed is known
(by methods to be explained presently) to be 461-2 metres per second
13 to 14 molecules have speeds below 100 metres per second.
81 82 from 100 to 200
1 66
214
76
167
215
203
152
92
77
200
300
4 00
500
600
above 700
300
400
500
600
700
The table shows that all but about 10 per cent, of the molecules have
speeds between one-half and double the r-m-s-speed. It thus appears
that despite the possibility of any definite molecule acquiring a speed
enormous in comparison with the average, the probability of it doing so
is exceedingly remote. It should also be observed in illustration of the
point dealt with earlier, that the number possessing quite low speeds is
also very small.
The kinetic theory of gases explains the pressure of a gas as the
lesult of molecular bombardment, and connects the pressure and tem-
perature of the gas with its mean molecular kinetic energy. The actual
relations are as follows :
If p grams per c.c. is the density of the gas, and/ dynes per square
cm. its pressure, it can be shown that
where c, the r-m-s-speed, is measured in cms. per second. If p and
p are known, this enables us to calculate c, as, for example, in the table
just given. If E is the energy of motion of a molecule
E
where m is the mass of a molecule ; and if E is the mean molecular
kinetic energy, then
E = mc 2 .
Hence /
where n is the number of molecules per c.c.
The experimental gas laws are summarised in the statement that
pv = RT
where v c.c. is the volume of one gram-molecule of the gas under the
pressure/ at the temperature T and R is the gram -molecular gas con-
stant.
APPENDIX I 167
Hence we derive
But nv is the number of molecules in the gram-molecule ; so R/#z> is
the gas constant per molecule, usually denoted by k. Thus we obtain,
as the connection between the temperature of the gas and its mean
molecular motion
E = f*T . . . . (8)
We have so far dealt with a gas composed of similar molecules, but
if we consider a gas mixture containing molecules of different masses,
little alteration is required in the formulae. It can be shown that among
each group of molecules of one type, the relative distribution in speed
and direction is similar to that already outlined for a simple gas, and is
unaffected by the presence of other types of molecules. This is a
statistical generalisation of the well-known law of Dalton concerning
gas mixtures. Thus, each group of molecules has a certain r-m-s-speed,
but the value of this varies from group to group ; however, a very simple
relation connects them. If we denote types of molecules by suffixes,
i, 2, 3, etc., then
etc.
where m it m 2 , m& etc., are the masses of each type of molecule. In
terms of average kinetic energies per group of molecules this is
Ej = E 2 = E 3 = etc.
Referring to (8) we see that this dynamical conclusion is the statistical
statement of the equality of temperature which obtains throughout the
gas mixture when a steady state has been attained. On account of this
relation it becomes more convenient to state the distribution law in
terms of limits of kinetic energy. This is easily done. Thus, revert-
ing to the case of a simple gas for a moment, we introduce the following
changes into (4) :
Write E = \mc*
E = fync*
ffE = me . dc
and we obtain after a few steps
47rN
as the number of molecules with kinetic energies between E and
E + dE at an instant.
We may avail ourselves of (8) and obtain instead of this expression
the following one :
E*..-*.* .. . . (9)
as the number of molecules limited in the manner mentioned.
1 68 A SYSTEM OF PHYSICAL CHEMISTRY
Now it can be shown that for a gas mixture consisting of N! mole-
cules of type i, N 2 molecules of type 2, etc., in a steady state, and
therefore with the same mean energy of motion (i.e. the same tempera-
ture) throughout, the number of molecules of type i, of type 2, etc.,
within the defined limits of kinetic energy are given by expression (9),
with N replaced by N lf N 2 , etc., in succession. Briefly, the distribution
of kinetic energy ignores differences in molecular mass.
Hitherto we have considered the gas to be free from all external
forces such as gravity, and subject only to the forces arising from collision
with each other or with the comparatively fixed molecules in the solid
walls of the containing vessel. If we introduce external forces of a
conservative type the previous formulae must be modified as follows :
Denote the co-ordinates of a point in the gas as X, Y, Z ; the poten-
tial energy of a molecule m of the gas at this point is a function of
X, Y, Z ; call it X XYZ or simply x .
We no longer have a uniform density of distribution imposition for
the molecules. The density, in fact, diminishes as we move to places
of greater potential energy. A good illustration of this is the progres-
sive decrease in the density of the atmosphere as the altitude increases.
It can be shown that the molecular density is proportional to
e ~ X/*T.
Hence, the number of molecules in an element of volume dx, dy, dz at
the point XYZ is
ne ~ xl^dxdydz
where n is a constant. As a matter of fact n is the molecular density
in a small volume surrounding a point where the potential energy, x , is
zero.
The distribution in kinetic energy of the molecules in this element
of volume is just as before. Hence we can write in full
. dxdydz (i o)
as the number of molecules in an element of volume dxdydz, where the
potential energy of a molecule is x , and whose kinetic energies lie be-
tween E and E + <E.
The extension of this result to the case of a gaseous mixture is easily
given. Let the potential energies of each type of molecule in the ex-
ternal field of force at the point XYZ be xi X2> xs> etc - tnen tne
numbers of each type of molecule in the volume element dxdydz, at
this point limited as in (10), are
. (n)
and so on.
APPENDIX I 169
Here lt n z are the molecular densities of each type of molecule
around a point where the potential energy is zero.
The formulae developed hitherto have postulated implicitly an
absence of intermolecular forces, except such as arise in a collision
between molecules, an event which, though absolutely very frequent
for a molecule, is relatively rare, since the time which a molecule
spends in its free paths is very much larger than the time spent in
collisions. When we introduce intermolecular forces, i.e. when we
approach the liquid or solid state, the formulae become more complex,
and as they are not required in the text of the chapter to which this
appendix refers, may be laid aside.
A more important case for our purpose is the case of a gas whose
molecules can no longer be considered as simple rigid bodies, but for
which an atomic and even a sub-atomic structure must be postulated.
It would take us too far afield to treat the formulae for this case in all
their generality, but we can deal with a few statements of service later.
As before, we take the number of molecules as N, and consider that
the general position of each molecule is given by the three Cartesian
co-ordinates of its centre of mass. Besides these three co-ordinates there
are other geometrical quantities (or co-ordinates in a more general sense)
necessary to define the structure of a molecule. These internal co-or-
dinates can be most readily thought of as the least number of lengths
and angles necessary to specify the position of every distinct and con-
ceivably separate part of the molecule with regard to its centre of mass.
The number of these co-ordinates constitute the internal " degrees of
freedom " of the molecule. Suppose there are n of them, the neces-
sary n co-ordinates being denoted by the letters
The rates of change of these co-ordinates as the molecular parts move
relatively to one another are referred to as "components of velocity"
or simply "velocities," each corresponding to a definite degree of
freedom. Writing q for dq\dt^ these velocities are
As the parts of the molecule act on each other by means of inter-
atomic and intra-atomic forces (of electric origin), the molecule pos-
sesses a certain amount of internal potential energy, as well as a certain
amount of internal kinetic energy due to the motions of the parts rela-
tive to the centre of the mass. This kinetic energy can be expressed
as a quadratic function of the velocities q^ . . . q n . The case which
has been most extensively treated, and which is very amenable to
mathematical analysis, permits of the potential energy also being ex-
pressed as a quadratic function of the co-ordinates q 1 . . . q n . This
assumption is possible when we have to deal with the small oscillations
of the various parts about positions of relative equilibrium, or about
states of steady relative motion, and these conditions have thrown a
great deal of light, e.g. on the radiating mechanisms of molecules. It
1 70 A SYSTEM OF PHYSICAL CHEMISTRY
is possible with this assumption so to chose the co-ordinates that the
total internal energy, kinetic and potential, involves only squares of the
co-ordinates and velocities, so that we can write the total internal energy
as
where a l . . . a n , b l ... b n are constants.
In this manner and under the conditions laid down we can, as it
were, partition the energy among the various degrees of freedom, a^q^
being, for example, the kinetic energy " belonging to " the first degree of
freedom, and b^q^ the potential energy belonging to the same degree of
freedom, and so on. There is in addition, of course, the kinetic energy
of translation of the molecule as a whole, viz. %mx 2 + %my 2 + ^mz 2 ,
which we will write as K (kappa) involving three more degrees of free-
dom, with the energy as before partitioned between the three degrees.
We can now state some results of the application of statistical
mechanics to a system of such molecules. It must be understood,
however, that these results rest on the validity of the classical dyna-
mics. Of late these results have been impugned by Planck and others,
and the foundations of dynamics are undergoing a revision. It will,
however, enable the reader to grasp the modifications proposed if he
masters the following few statements and accepts their truth at all
events provisionally.
As the molecule moves about and its parts oscillate with regard to
each other, its energy changes, not only in toto but also in its several
terms. It can be shown, however, that the mean energy, kinetic or
potential, belonging to any co-ordinate, averaged over a considerable
period of the molecule's history, is the same for all the degrees of free-
dom and is equal to -T. That is
This is a statement of the famous theorem due to Maxwell and usually
referred to as the " equipartition of the energy among the various de-
grees of freedom ".
Further, it can be shown, as before, that the number of molecules
which have velocities between uvw (i.e. xyz) and u + du, v + dv,
w + dw is
A . N . <r K /* T . dudvdw . . . (12)
where A is a constant.
Of this limited number a still more limited number exists at one
instant whose several total internal energies, kinetic and potential, tie
between c and c + de.
This latter number forms a fraction of the former whose value is
B -i<re/fcT^ e ...... (13)
where B is a constant.
Combining the two expressions, we have the number whose internal
APPENDIX I 17*
energies lie between c and c + dt, and whose general velocities of trans-
lation lie between uvw and u + du, v + dv, w + dw, as
C . N .-'. e-( e + Vl hT didudvdw . . . (14)
where C is a constant.
As an important particular case, expression (13) reduces for one
internal degree of freedom to
Be-*l kT <te ... , (15)
Equation (15) forms the basis of the expression of Marcelin and
Rice for the rate of decomposition of a molecule referred to in Chapter
VI. under the heading " Reaction velocity from the standpoint of the
quantum theory ". The expression for the velocity constant on the
basis of statistical mechanics is given by Rice (Brit. Ass. Report, 1915,
P- 397).
APPENDIX II.
FOUNDATIONS OF THE QUANTUM THEORY.
BY J. RICE, M.A.
(From the Transactions of the Faraday Society xi., p. i, 1915.)
DURING the nineteenth century certain facts had been well established
concerning the radiation contained in an enclosure maintained at con-
stant temperature. Balfour Stewart and Kirchhoff had shown that if
the material of the walls was. not perfectly reflecting for any quality of
radiation, any constituent of the radiation having a definite frequency
was present in an amount depending on that frequency, the temperature
and size of the enclosure, but independent of the nature of the wall-
material. Stefan and Boltzmann had proved that the energy of the
total radiation in one c.c. was proportional to the fourth power of the
absolute temperature. Wien had reached the conclusion that the
-energy-density of those constituents of the radiation, whose wave-lengths
lay between narrow limits X and X + 8X, was 5 8X, where the
function /in the numerator, although undetermined in form, was de-
pendent on the single variable XT, the product of the wave-length and
the absolute temperature. These statements had been arrived at by
reasoning of a purely thermodynamic nature, based on the existence
of a radiation pressure. It was only natural that the statistical methods,
already employed as a successful weapon of attack on problems arising
out of the kinetic theory of gases, should be presently pressed into ser-
vice in this new line of research. There is some resemblance between
a vessel containing molecules, individually uncontrollable but maintain-
ing a certain average condition of energy by encounters, and an en-
closure maintaining a statistically permanent condition of radiation by
the emission and absorption of its walls, while the history of any par-
ticular wave-train of definite frequency cannot be followed in detail.
There is one marked difference between the two cases : molecules can,
by direct encounter with each other, as well as by collision with the
walls, exchange energy; but in a temperature enclosure passage of
energy from one wave-train to another must take place through the
agency of the walls alone or of matter contained within them, since two
such trains can pass through the same element of volume (i.e. cross one
172
AP-PENU2X 11 173,
another without any change in the energy or quality of the trains). For
this reason, if the walls are perfectly reflecting, any arbitrary condition
of radiation supposed existing at one moment would remain permanently
unaffected in the energy and quality of its constituents, and only by the
introduction of a piece of non-reflecting matter into the enclosure could
the radiation be gradually brought to the condition of " full " or " com-
plete " radiation for the temperature, the condition considered by Balfour
Stewart and Kirchhoff.
Wien, himself, had made an attempt to determine the form of the
function in the numerator of his so-called displacement formula. 1 He
considered the enclosure to contain a perfectly absorbing or "black"
body, which he assumed to be a gas with its molecules distributed ac-
cording to the Maxwell Law, and its temperature therefore proportional
to the mean-squared velocity. A further hypothesis (a very doubtful
one) was the assumption that those molecules whose velocities lie within
certain narrow limits at a definite instant are at that instant emitting
light within narrow frequency limits, with an intensity proportional to
the number of those molecules. By these means Wien arrived at the
c
form e~J^ for his function /(AT), c being a constant. Measurements
carried out shortly after by Lummer and Pringsheim, Beckmann, and
Rubens verified the formula as a good representation of the facts for
short wave-lengths, but found it completely at variance with the facts
for long wave-lengths.
It was by the application of statistical principles in another direction
that the next advance towards a correct radiation formula was made.
Wien had considered the molecules of the black body. Lord Rayleigh,
and afterwards Jeans, considered the radiation itself, assigned co-ordin-
ates to it and " degrees of freedom," and applied the results of Max-
well's distribution law directly to these concepts. Such applications
are certainly rather easier to apprehend " physically " in the case of gas
molecules than in that of constituent wave-trains of radiation ; there is
more " substantiality " about a molecule than a wave-train. The repre-
sentation of a molecule as a small, hard sphere with perfect resilience,
which is quite adequate for many purposes, and analogies with billiard-
balls, discs, etc., put one on fairly familiar terms with molecular motion
and exchange of energy. The degrees of freedom of such simple systems-
are easily calculable, being in fact six for a " rigid " molecule of any
shape. Even if we introduce atomic structure into the molecules, the
degrees can still be computed if one knows the parts and their connec-
tions. The energy of the system at a given temperature is then obtained 1
by the principle of equipartition, which is derived from the law of dis-
tribution, and asserts that the kinetic energy can be calculated by as-
signing to each degree of freedom an amount -J^T ergs, where k is the
molecular gas constant (1-35 x io~ 16 ergs/degrees), an equal amount
of potential energy being also assigned to any degree of freedom, if there,
are " elastic " forces of the usual simple harmonic type.
1 Wied. Ann., 58, p. 662, 1896.
i 7 4 A SYSTEM OF PHYSICAL CHEMISTRY
If such ideas are to be generalised and extended to the ether of an
enclosure, a definite notion as to the structure of the ether is imperative
before any headway can be made in calculating its degrees of freedom.
At the outset it is fairly evident that the usual conception of the ether
as a perfectly continuous medium indivisible into an enumerable number
of discrete parts, should lead to an infinite number of degrees of freedom,
with the resulting conclusion that in a state of equilibrium the ether in
the enclosure should contain all the energy and the walls none. The.
well-known illustration of the gradual loss of vibratory energy from the
particles of a sounding body to the surrounding air may serve to make
this point clearer. This conclusion is in fact reached by the Rayleigh-
Jeans analysis, and is very much at variance with the facts as we know
them. The formula arrived at, however, is a close approximation to
the truth for long wave-lengths, and the calculation of the number of
degrees of freedom has proved of signal service in itself. It is impos-
sible to reproduce the analysis here, but an analogy from sound waves
may serve to show the principles on which it is based. It is well known
that an organ-pipe will resound only to notes of definite frequencies, the
fundamental and its overtones. This is due to the fact that any state
of " stationary " wave-motion which will persist in the air of the pipe has
to satisfy certain end conditions e.g. at a closed end there can be no
vibratory motion of the air particles, at an open end no change of pres-
sure. Any text-book on sound shows that from these conditions there
can exist in a very narrow pipe, closed at both ends or open at both
ends, only waves whose wave-lengths are 2/, , ..., etc., where
v v
I is the length of the pipe. The frequencies are, of course, - 2 ,