the short infra-red regions. Hence the values of D are to a consider-
able extent arbitrary, the values not being strictly comparable. Thus
the value obtained for the dielectric constant will depend very largely
on whether there exists an absorption band at the wave-length employed
154 A SYSTEM OF PHYSICAL CHEMISTRY
in the dielectric constant apparatus. In the case of water it is con-
sidered that the high value obtained is partly due to the fact that water-
possesses a band for the electro- magnetic waves usually employed.
Further, Kriiger's assumptions are possibly not correct in all cases.
Thus, it is very surprising that the results quoted should be as satis-
factory as they are, for the electrolyte is strong. Nevertheless the
deduction, so far as it goes, indicates that radiation plays a r61e in
ordinary chemical processes. Kriiger shows further how an empirical
relationship of Walden, according to which " saturated solutions of one
and the same electrolyte in various solvents possess the same degree of
ionisation," can be deduced on the radiation basis. For details regard-
ing this and other matters the original paper of Kriiger must be con-
THE HEAT OF REACTION. HABER'S RELATION.
In Chap. VI. we have dealt briefly with the rate of reaction and the
velocity constant from the point of view of the quantum theory. In
the case of a monomolecular reaction in a dilute gaseous system the
variation of the velocity constant with the temperature is given by the
d log kjdl = N^/RT 2
where v is the characteristic vibration frequency, or head of the absorp-
tion band, of the decomposing substance. If the reaction be a re-
versible one, the resultant being characterised by the frequency v', the
corresponding velocity constant being /', we have
d log k'fdT = NA//RT 2 .
d log klk'
But klk' = K, the equilibrium constant. Hence we obtain the relation
d log K/dT = N*(v - v')/RT 2 .
At the same time the statistical mechanical expression of Marcelin
and Rice gives the relation
d log K/tfT = (E - E')/RT 2
where E is the critical increment of the reactant, E' that of the resultant.
On comparing these expressions with the van 't Hoff isochore, viz.
d log K/rfT = - Q V /RT 2
where - Q w is the heat absorbed per stoicheiometric quantity of the
reactant transformed or decomposed, we obtain the relation
- Q v (heat absorbed) = N>&(v - v) = E - E'
or, heat evolved in the reaction = critical quantum of the resultant
minus the critical quantum of the reactant.
HABEAS HEAT RELATION
In the case of reactions in which catalytic effects are absent we may
generalise the above expression, and write
Heat evolved = 2N/&i/ resu i tants - 2N/&i/ reac tants.
This expression, deduced however by a totally different method, was
first given by Haber (Ber. Deutsch. phys. Ges., 13, m7 iQ 11 )- It
may therefore be called the Haber expression for the heat effect. The
above mode of deduction is that given by the writer (Trans. Chem. Soc.,
Ill, 457> I 9 I 7)-
In the above formulation the heat of reaction at constant volume
has been expressed as the difference of the critical quanta or the dif-
ference of the critical increments of the resultants and the reactants.
The heat effect at constant volume is also thermodynamically defined as
the difference of the mean internal energies of the reactants and re-
sultants. To see that these two definitions are concordant we may
proceed in the following manner.
Let us consider the simplest type of reaction, viz. A l B. The
various energy terms involved in the process are represented in the ac-
companying diagram (Fig. 13). The ordinates denote internal energy,
the length ab corresponding to the mean values Uj of the internal
energy possessed by one gram-molecule of the substance A, and similarly
the length de represents the mean energy U 2 of the substance B. Be-
fore one gram-molecule of A, possessing the average internal energy
U x , can become reactive its internal energy must rise to the point c by
the addition of the critical increment E^ At this stage the gram-
molecule possesses the energy (U l 4- EJ. It may now change into
one gram-molecule of B, with an evolution of energy denoted by the
dotted line ce. The gram-molecule now possesses the mean energy de, or
LJ 2 characteristic of the substance B. In passing from c to e the energy
emitted is (U l + E a ) - U 2 . In passing from the mean state b to
the mean state e the total energy
evolved is (Uj + E x - U 2 ) - E lf
or U x - U 2 . The total energy
evolved in passing from e to b is
(U 2 + E 2 - Uj) - E 2 ,orU 2 - Uj.
This expresses the fact that if the
reaction is exothermic in one direc-
tion it necessarily endothermic in
the reverse direction. 1
When a molecule is in the
critical state it is impossible to say
whether it belongs to the system A
or to the system B. That is, the
critical state is common to both A
and B. If E c denotes the total
critical energy not the critical in-
crement E c will have the same
value for the A and B molecules
1 Note that on the above mechanism the system never passes along the line be.
156 A SYSTEM OF PHYSICAL CHEMISTRY
alike. This is indicated in the diagram by the lines ac and df being of
equal length. From the above considerations it follows that
E c = U x + E! = U 2 + E 2 .
Hence Uj - U 2 = E 2 - E x = Q, = Nh(/ - v).
It will be seen that the above mechanism is somewhat more complicated
than that assumed by Einstein in deducing his law of the photochemical
equivalent. It agrees with Einstein's view, however, in that the energy
evolutions just balance the energy absorptions when both reactions are
taken into account, and therefore the radiation equilibrium is not
altered when mass action equilibrium has once been attained. This
assumes that the evolution from c to e is in the form of a single quantum
if the absorption ef takes place in the form of a single quantum ; and
similarly for the quantities represented by fb and be respectively.
In writing Q w as the difference between the critical quantum of the
resultant and reactant we have really assumed that Q v is a constant
independent of temperature. This, of course, cannot be exact, as it
would mean that d\J/dT = o, or that the sum of the molecular heats
of reactants is equal to those of the resultants. The above radiation
expression is therefore an approximation. In this connection it may
be mentioned that a recent investigation of equilibria in the gaseous
state has led Scheffer (Proc. Akad. Wetenschap Amsterdam, 19, 636, 1 9 1 7)
to the conclusion that the variation in Qj, with temperature is a negligible
quantity in general, i.e. " the experimental error in the determination
of K is always such as to render meaningless any attempt to allow for
the variation of Q with T ". This practical conclusion does not affect
the question, however, of the true variation of Q with T.
The integrated expression for K on the above radiation basis takes
the approximate form
log e K = - ^ " v ' + constant
log. K ~ + constant
where v refers to resultants, v to reactants, and K is so defined that the
equilibrium concentration terms of the resultants appear in the numerator.
An interesting application of the above considerations suggests
itself in connection with the phenomenon of fluorescence or phos-
phorescence. In this phenomenon we are dealing with the absorption
of one type of radiation and the emission of another. The effect is
regarded as due to an emission of energy which accompanies the
return of an electron to its original position in the atom, the initial
disturbance from this position having been brought about by absorption
of radiation. According to Stokes' law the frequency of the emitted
radiation is always less than that of the radiation absorbed. The pro-
cess is therefore analogous to an endothermic change, represented in
Fig. 13 by the absorption ef and the evolution fb. Stoke's law means
that the process is irreversible. On the other hand, Baly (loc. cit^] has
THE HEAT OF REACTION 157
made the very important discovery that the fluorescence (emission)
wave-length exhibited by a substance in one solvent is identical with the
absorption band exhibited by the same substance in a second solvent.
That is, under one set of conditions the substance is capable of ab-
sorbing the wave-length which it emits as fluorescence under other
In the " second " solvent it is conceivable that the conditions
exist for a reversal of the process (i.e. possibly Stokes' law is not
necessarily true in general). No information is available on this point
at the present time.
MAXWELL'S DISTRIBUTION LAW AND THE PRINCIPLE OF EQUIPARTITION
BY JAMES RICE, M.A.
THE aim of mechanics is the description of motion. We seek to
specify the position of every part of a system of bodies at every instant.
The most direct way of doing this would be to express all the necessary
geometrical co-ordinates of the system as known functions of the time.
To this end the laws of motion are applied to the special features and
environment of each system, and a series of differential equations are
obtained which, among other quantities, involve the first and second
differential coefficients of each co-ordinate with respect to the time,
i.e. the velocities and accelerations. If the mathematician can solve
for us the particular differential equations arrived at, we have attained
our object for that system. Many special cases of considerable interest
have proved amenable to mathematical treatment, but, at present, no
solution for the general case exists.
In the phrase, "system of bodies," we must be definite as to the
meaning to be attached to the word " body ". In physics and chemistry,
a single body is a system, i.e. a collection of molecules, which are in
themselves discrete, if minute bodies. Indeed the molecule itself is a
system of atoms, and, on present views, the atom is a system of nuclei
and electrons. Even if we regard a body of fluid or solid material as
a system of molecules alone, without concerning ourselves about its
internal structure, the complexity of description involved in a complete
account of molecular motion is so great that it becomes necessary to
introduce the mathematical theory of probability. We are no longer
concerned with an exact solution of the dynamical problem, by which one
could predict with certainty the position and motion of each molecule
at a given instant ; instead we endeavour to find the law of distribution
of the co-ordinates and velocities of the molecules, so as to be able
to state, with but small possibility of error, that at a given instant such
and such a fraction of the molecules will occupy such and such a portion
of the space filled by the body, and have velocities lying between such
and such limits. It is this feature which characterises a problem as one
of statistical mechanics.
In dealing with a system of molecules, the co-ordinates referred to
above are naturally the Cartesian co-ordinates of the centres of each
APPENDIX I 159
molecule referred to a definite set of axes ; but it should be borne in mind
that in general dynamics the word " co-ordinate " is used to indicate
any geometrical quantity which serves to specify the position and con-
figuration of the parts of a system, the least number of such co-ordinates
which is necessary to specify the system completely being in fact its
number of "degrees of freedom". Thus, if we attempt to treat the
problem of a system of molecules which are themselves systems of
atoms, we should have to introduce further geometrical magnitudes to
those mentioned above, which would define the position of each atom
relative to the centre of its corresponding molecule.
As regards the velocities of the individual molecules, there are two
ways open to us to specify such a velocity, both as regards its magnitude
and its direction. We may give the three components of the velocity
parallel to the three axes of co-ordinates, or we may give the actual
speed combined with some convenient way of indicating its direction.
The following is a simple method of defining direction :
Conceive of a sphere of unit radius. Every direction in space is
parallel to some radius of the sphere. Consequently a point P on the
surface of the sphere can be said to represent a direction in space, viz.
the direction parallel to OP where O is the centre of the sphere. Any
direction and its opposite are of course represented by diametrically
opposite points on the sphere. In statistical work we have to classify
directions into groups, and one convenient way of ' doing this is as
follows : Draw a series of plants perpendicular to a given diameter of
the unit sphere, and meeting the diameter in points which divide the
diameter into an exact number, say m> of equal parts. These planes
divide the surface into a series of zones, and it is well known that these
zones have equal areas, viz. 4Tr/m, 4-0- being in fact the entire area of
the sphere whose radius is unity. Still further subdivide the surface by
a number of planes, say n, each containing the diameter and each one
making, with its neighbouring planes, angles which are all equal to an
aliquot part of 2-n- radians, viz. 2ir/n radian. This new family of planes
cuts the surface of the sphere in great circles which subdivide each
zone in equal parts, each part being four-sided and equal in area to
4-ir/mn. (The parts round the ends of the diameter are three sided but
have the same area.) If we call this area w, then the number of such
elements of area is mn. or 47r/w. It will prove convenient presently to
classify directions into groups such that all the directions in one group
correspond to points in one superficial element of the representative
sphere. In the nature of things no finite limit can be assigned to the
number of directions in one group, no matter how small the elements of
area are made by closer and closer subdivision by the planes of " latitude
and longitude," but of course closer subdivision and an increase in the
number of groups involves closer and closer limits to the angular differ-
ences between any pair of directions in one group.
The problem in statistical mechanics which has received the greatest
attention concerns the statistics of a body of gas enclosed in a fixed
volume and in a steady state with regard to such properties as tempera-
160 A SYSTEM Of PHYSICAL CHEMISTRY
ture pressure and density. Here, among other things, answers have
been found to such questions as refer to distribution of the positions
and velocities of the molecules, the mean number of molecular colli-
sions per second, the mean free path, the connection between these
quantities and the pressure, temperature, viscosity, rate of diffusion,
etc., of the gas. We proceed to quote some known results concerning
the distribution of the molecules in position and in velocity.
As regards the positions of the molecules, we conceive the enclosure
to be subdivided into a great number, say n, "physically small" volumes
or cells. This means that each cell is supposed too small to be dealt
with separately by our experimental apparatus and yet large enough
to contain an enormous number of molecules. As it is known that the
number of molecules in one cubic mm. of a gas at N.T.P. is about
3 x io 16 , this condition is easily complied with. Suppose we have
altogether N molecules in the enclosure, then the average number of
molecules per cell is N/ which is assumed to be a large number.
Suppose we express the actual number of molecules in any cell as
N/ . (i +8) where 8 is the fractional variation of this molecular density
from the mean molecular density N/, and may be positive or negative.
It can be proved that there is an enormous probability against the
possibility of 8 acquiring values of an order of magnitude greater than
the order of i/N. As N is enormous, this practically means that there
is an enormous probability in favour of uniform distribution of the
molecules in position. It should be noted as a feature of this statisti-
cal proposition that the proof of it does not prove the impossibility of
the number of molecules in any cell deviating seriously from the mean
number ; it proves that such a state is extremely improbable, and that
the dynamical conditions which would produce it occur so infrequently
and exist for so brief a time that actual demonstration of its existence
would elude our experimental arrangements.
When we come to deal with the distribution of the molecules in
terms of velocity, we do not find this uniformity of distribution. Tak-
ing the question of speed alone, apart from questions of direction, there
is certainly a theoretical upper limit to the possible speed attainable
by any one molecule ; it is in fact the speed which that molecule would
have if it possessed the entire energy of the gas, the other molecules
being absolutely at rest. Such a speed is, however, far beyond any
practical limit; although there is no dynamical impossibility in the
state of affairs pictured, there is an enormous probability against it.
The application of statistical methods to this problem leads to the
view that certain speeds are more privileged than others. Thus there
is one speed such that there are at a given instant more molecules
possessing velocities within, say, one foot per second of this speed, than
there are molecules possessing speeds within one foot per second of
any other speed ; and if we choose speeds smaller and smaller or larger
and larger than this " maximum probability " speed, these speeds are
less and less privileged, until, when we arrive at zero speed or at speeds
very great compared with the maximum probability speed, the proba-
APPENDIX 1 161
bility of their occurrence is very small indeed. There are, however,
no privileged directions, all directions of motion being equally likely.
This rough statement will serve to introduce the mathematical law of
the distribution of velocities, first formulated by Maxwell and often
stated in the words: "the law of distribution of the velocities among
the molecules of a gas in the steady state is the law of errors ".
As a matter of fact there are two ways of stating Maxwell's Law,
one of which assumes a velocity to be measured by its components
parallel to the Cartesian axes, the other way assumes that a velocity
is to be measured by its magnitude and direction on the plan explained
above (the unit sphere device). In the former case the law of the
distribution is for each component similar to the normal law of errors,
but in the latter case the law of distribution of the speeds is not the
normal law of errors. Let us take the first method of formulation.
Considering an enormous number of molecules, N, it can be shown
that m the steady state those of them which at one instant have veloci-
ties whose X components lie between narrow limits, say u and u + du,
are in number equal to
where A and y are constants to be determined shortly. This number
constitutes a fraction Ke~t lft du of the whole. It should be carefully
observed that the group of velocities here dealt with are not neces-
sarily close to each other in magnitude and direction : it is only their
X components which are close to each other in magnitude ; no restric-
tion is so far imposed on the Y and Z components.
Of this limited number there will be at the same instant a certain^
fraction which have their Y components between v and v + dv, this:
fraction being in fact A.e~y v2 dv ; so that the number whose X and Y
components are between u and u + du, and v and v + dv respectively,
A similar step gives the number of those whose X, Y, and Z components
are at one instant within the limits u to u + du, v to v + dv, and w to
w + dw respectively, as
^&Ze-y(u* + v* + ^)dudvdw . . . (i)
We can calculate the constant A in terms of y ; for if we inte-
grate for all values of u, v, and w from - infinity to + infinity, we must
Hence N = NA*f + ~*r7V . ^ e~^dv . I ^
J oo J oo J
Now it is known that the definite integral
VOL. Ill II
A SYSTEM OF PHYSICAL CHEMISTRY
Hence N = NA 3 ( VV^) 3 .
Hence A = J^fc
We can also calculate y in terms of the total kinetic energy of the
molecular motion. For the fraction of the whole, i.e. fractional number
of the whole number of molecules, which is represented by
N AV-r( 2 + v* + ^ dudvdw
have each kinetic energy %m(u 2 + v z + w 2 ) where m is the mass of a
Hence the total kinetic energy is
J oo J
00 _y( M 2+j;2 + a,2)
dw . (ti 2 + v 2 + w 2 ) . e dudiau,
Leaving aside the constant factor for the moment, we see that the. i.ple
integral is the sum of three triple integrals, one of which, for ins ; i ce,
r+oo r+oo r
u 2 e~y u<i du . e-i^dv .
J oo J oo J
Now it is known that
and the two* remaining integral factors in the above chosen integral havt
already been evaluated. Hence the selected member of the three triple
integrals has the value -J ^/yYy 5 , and the other two have each the same
value. Hence the total kinetic energy is
/S /^/?; /> INw/y; since A
Now we could conceive the molecules all moving with one uniform
velocity as a rigid body, for instance, and that velocity such as to give
the same kinetic energy as that due to the gaseous motion. Denote
the magnitude of this hypothetical velocity by c, and we have
or y = 32.
This particular speed c is not the true average speed, nor is it the
maximum probability speed referred to above (its relation with these
will be given presently) ; it is in fact a speed whose square is the aver-
age of the squares of the molecular speeds at one instant, and on that
account is called the root-mean-square or r-m-s-speed. Assuming that
this r-m-s-speed is known, we formulate Maxwell's Law in one way
Of a great number N of gaseous molecules in a steady state with an
r-m-s-speed c, the number whose velocity components at one instant
lie between the limits u to u + du, v to v + dv t w to w + dw, is
. e-sWW+^l^dudvdw .
APPENDIX I 163
and we see that any one particular component, say u, enters into this
expression in a manner similar to the way the error x enters into the
expression for the normal law of errors, viz. fy-^^dx.
We can employ a geometrical method of representing this law.
Imagine that u, v, w, are chosen to be the Cartesian co-ordinates of a
point in a threp dimensional diagram, which we will refer to as the
velocity diagram. The origin O of the co-ordinates in this diagram
represents absolute rest, while any other point P represents a velocity
whose magnitude is given by the length OP and whose direction is the
direction of OP. Let us write c for the length OP, i.e.
<= J u* + v* + ze/ 2 )
gives the actual speed corresponding to the point P. Now suppose we
represent, as it were, every molecule by a point in this diagram, i.e.
each point represents the velocity not the position of some molecule,
so that we have therefore N points marked in the diagram ; these points
will, of course, move about with lapse of time, because of the changing
velocity of the corresponding molecule. The expression (2) states the
volume distribution of these points at a given instant. Thus, taking
dudvdw to be an element of volume of the velocity diagram the
" density " of the points around the point representing zero speed, i.e.
representing the condition of rest, is
Around any point representing a speed c Cwith no restriction as to
direction), i.e. around a point lying anywhere on a sphere of radius c
with its centre representing rest, the density of the points is
-3iW' .... (3)
So the density of the swarm of points diminishes in this exponential
manner as we recede from the point representing rest. The reader is
warned against drawing the erroneous conclusion that there are at any
moment more molecules in the gas at rest, or very nearly so, than there
are those possessing any other assigned speed or near to it. The fallacy
involved in such an inference will be pointed out presently.
An element of volume in the velocity diagram represents what is
called an " extension-in- velocity," just as an element of volume in actual
space occupied by a gas is an " extension-in-position ". Thus we may
refer to expression (3) as the " density-in-velocity " of the molecules
about any velocity of the magnitude c.