to reach a maximum. But on purely thermodynamical grounds we know
that in spontaneous processes the entropy of a system tends towards
a maximum value consistent with the total energy of the system. It
follows therefore that there must be some close relation between the
thermodynamical probability of a state and the entropy of the state.
We can express this by writing
where S is the entropy of the system in any state, not necessarily the
equilibrium state, w the thermodynamic probability of the same state,
and F is some function still to be determined. To determine the nature
of F, let us suppose that we have two independent systems, each one
in a definite state, the entropy of the first being denoted by Si, the
probability of the state or arrangement of the first system being w^, the
entropy of the second system being S 2 , and the probability of the state
of the second system being ze/ 2 . We then have the relations
51 = F(w l )
5 2 = F(w z ).
The total entropy S of the two systems taken together is the sum of
the separate entropies. That is
S = Sj + S 2 = F(w 1 ) + F(w 2 ).
Since the particular state or arrangement of the first system can be
realised by selecting any one of the w l complexions (contained in or
characteristic of that arrangement or state) and similarly for the second
system, it follows that the state or arrangement of the combined
system can be realised by selecting any one of the w\ complexions of
the first and combining them with the w 2 complexions of the second.
That is, the compound arrangement is obtained by selecting any one
of the w l and zv 2 complexions. That is the probability w of the
compound state is w : x w z .
But for the compound system we have the relation : S = F(w).
Hence from the above we get : S = F(z/j . w 2 ). But we have already
seen that S = F(w 1 ) + F(w z ). Hence, F(w l . w 2 ) = Ffa) + F(o; 2 ).
The only function which will satisfy this relation is the logarithmic
one, i.e. log xy = log x + log y.
Hence, the connection between the thermodynamic probability and
the entropy of a system is given by the relation :
Entropy oc log e Probability
or S = k log, W,
where k is a constant independent of the chemical nature of the system
1 6 A SYSTEM OF PHYSICAL CHEMISTRY
and independent of the conditions under which the system is examined.
A more general form of the above expression is
S = k log W + constant.
We have now to find out the significance of the universal constant k.
To do this we make use of a statistical expression arrived at by
Boltzmann for the entropy of a perfect monatomic gas. A monatomic
gas is one in which the total or internal energy U is due entirely to the
kinetic energy of the molecules. Boltzmann's expression is
S = 3/2N log U + m log v + K
where U and k are defined above, N is the number of molecules in the
system, v the volume of the system, and K is a constant independent of
the energy and volume but involving the number and mass of the
molecules. It has already been shown in Volume II. that the following
purely thermodynamical relation holds good :
*S J W
yr ~ T Vr
Differentiating Boltzmann's expression for the entropy of a perfect
monatomic gas we obtain :
as , ,i au
Hence U = 3/2/&NT.
If N be taken as denoting the number of molecules in one gram-mole-
cule, then U denotes the total energy of one gram-molecule of monatomic
But we have already seen (Vol. I.) that in the case of a perfect gas,
the total kinetic energy of all the molecules forming one gram-molecule
is 3/2RT, where R is the gas constant per gram -molecule. Further
in the case of a monatomic gas the internal energy is entirely kinetic.
Hence for one gram-molecule of a monatomic gas : U = 3/2 RT.
It follows therefore that
N = R,
or k is the gas constant per single molecule.
Further, in the case of a perfect gas, 3/2 R = C v where C w is the
gram-molecular heat at constant volume. Hence the equation of
Boltzmann for the monatomic gas becomes
S = C w log T + R log v + K!
where S now denotes the entropy of one gram-molecule. This expres-
sion is in complete agreement with that already deduced in Volume II.
on thermodynamical grounds, viz. :
S = C Jog T + R log v + Si
if we identify Si with Kj. This constant represents the value of the
entropy under certain conditions. On purely thermodynamical grounds
ENTROPY AND THERMODYNAMIC PROBABILITY 17
it is legitimate to consider that the entropy may assume in general any
value whatsoever, positive or negative, and that therefore all that we can
measure is the change in entropy resulting from a physical or chemical
process. It will be pointed out later in dealing with Nernst's Heat
Theorem, that according to Planck, the Heat Theorem itself is equivalent
to regarding the entropy of all substances as zero at the absolute zero
of temperature, and possessing therefore a positive value at all other
temperatures. 1 This gives us a starting point from which to calculate
not only change in entropy but its absolute value under given conditions.
This likewise agrees with the simplified expression : S = k log W in
which W has been defined as a quantity greater than unity, and con-
sequently S is essentially positive. Of course if we retain the constant
in the expression : S = k log W + constant, the value of S may be
positive or negative depending upon the magnitude and sign of the
integration constant whether VV itself is greater than unity or not.
Classical statistical mechanics, which did not attempt to assign any
particular limit to the value of the entropy, is represented by the above
expression. If we assume with Planck that the integration constant is
zero, and remembering that W as defined above is greater than unity, it
follows that S is a positive term becoming zero at absolute zero. This
is equivalent to assuming the quantum hypothesis.
The general position which we have now reached as a result of the
considerations dealt with in this section may be summarised as follows :
The second law of thermodynamics, regarded as a law of experience,
states that, whilst work may always be completely converted into heat,
heat on the other hand cannot be completely converted into work. In
other words, all natural spontaneous processes are thermodynamically
irreversible. In mechanics we deal only with reversible processes, and
from the standpoint of mechanics alone we would expect heat to be as
readily convertible into work as work into heat. Since this is not the
case there must be something characteristic of molecular systems to
which the irreversibility is due. This " something " is discovered in the
fact that heat consists of a chaotic motion of the molecules, and that as
a result of collisions this motion tends to become as chaotic or disordered
as possible. In other words, the irreversibility which finds expression in
the second law of thermodynamics is due essentially to the fact that
ordered motion always tends, of its own accord, to become disordered,
and chaotic motion never tends, of its own accord, to become ordered.
This statement is a statement of the second law of thermodynamics
not expressed simply as a result of experience but in terms of statistical
mechanics. We have therefore found a mechanical basis for the second
It is obvious at the same time why Gibbs gave the significance
1 According to Planck this assumption is "the very quintessence of the hypo-
thesis of quanta ". It must be pointed out that whilst this assumption makes the
theory of quanta and Nernst's Heat Theorem agree, it is not essential to the deduc-
tion of the heat theorem itself, which only requires that the entropy of all substances
at absolute zero shall be the same, but not necessarily zero.
VOL. IIL 2
1 8 A SYSTEM OF PHYSICAL CHEMISTRY
" mixed- up-ness" to the concept of entropy; for maximum mixed-up-
ness means on the mechanical view maximum stability or equilibrium,
and hence when equilibrium exists the entropy is a maximum. The
same idea is involved in the term "run-down-ness" employed by
Tolman as a descriptive mechanical term for entropy. The latter term
appeals to the mind as being particularly applicable to a chemical re-
action which proceeds in order to attain an equilibrium state at which
the system will, chemically speaking, have completely " run down ".
THE PRINCIPLE OF EQUIPARTITION OF KINETIC ENERGY.
This principle, which has been deduced on the basis of statistical
mechanics by Maxwell and by Boltzmann (cf. Jeans, Dynamical Theory
of Gases), has already been employed in Chap. I. of Vol. I. in dealing
with Perrin's method of determining the Avogadro Constant from
measurements made upon emulsions. The principle states that in a
system consisting of a large number of particles (e.g. molecules) the
kinetic energy is on the average equally distributed amongst the various
degrees of freedom possessed by these particles. The first point with
which we have to deal is the term degree of freedom.
A degree of freedom is represented by a co-ordinate. We may
take the term as meaning an independent mode in which a body may
be displaced or a possible mode or direction of motion. We shall con-
sider briefly the problem of the number of degrees of freedom possessed
by bodies in the gaseous and solid states of matter respectively.
First of all a word about energy in general. We are familiar with the
two kinds of energy which material systems may possess, namely, kinetic
and potential. These " kinds " of energy are not to be confused with
the "types" of energy of which we are about to speak. A "type"
may consist of kinetic and potential energy together, or simply kinetic
alone. A gas molecule can possess theoretically three types of energy,
each of which is a function of the temperature : (i) Energy of Transla-
tion ; (2) Energy of Vibration, and (3) Energy of Rotation.
1. Energy of Translation. This type of energy is possessed in virtue
of the free translational motion of the molecules along free paths
throughout the whole of the system. The energy in this case is entirely
kinetic. It has been represented diagrammatically in Chap. I. of Vol. I.
in connection with Perrin's work on Brownian movement. Every gas
molecule possesses translational motion and consequently translational
energy. Since the direction of any movement of this kind can be repre-
sented by the three space co-ordinates, we conclude that translational
energy corresponds to three degrees of freedom. The energy of transla-
tion is the same for any moving molecule at a given temperature.
Further, such a molecule possesses just three degrees of freedom in re-
spect of translation whether the molecule be monatomic or polyatomic ;
the constitution of a molecule does not enter into the question of its
2. Energy of Vibration. This type of energy exists in virtue of the
EQUIPARTITION OF KINETIC ENERG Y 19
oscillations of particles with respect to a fixed centre of gravity. Vibra-
tion is only possible when there is a restoring force acting upon the
particle which tends to make it take up a mean position. The existence
of the restoring force is the factor which distinguishes vibration from
translation and also from rotation to be mentioned later. We meet
with vibration in the case of the atoms inside a molecule of a gas, and
likewise in the case of the atoms composing a solid. Vibration is
always a constrained movement. We can conceive of vibration as
corresponding either to one or to two or to three degrees of freedom.
When we speak of vibration in connection with molecules it is always
to be understood that we are referring to the vibrations of the atoms
inside the molecule. One such atom can vibrate with respect to the
other in the case of a diatomic molecule. In this case the vibration
is linear, that is, it is along the line joining the two atom centres.
Since the vibration is linear there is one degree of freedom in this case.
Linear vibration is represented in diagram (a) Fig. 2.
FIG. 2 (a). Linear vibration of atoms in a diatomic FIG. 2 (6). Circular
molecule. One degree of freedom. Energy, libration or spin of
kinetic + potential. an atom round a
centre of gravity.
Two degrees of free-
dom. Energy, ki-
netic + potential.
It is also conceivable that the vibration of the particle, an atom
or an electron, may be circular. That is, the particle may describe a
circular path about a centre of gravity, the orbit being traced out on a
surface. Hence in such a case there are two degrees of freedom to be
attributed to the vibration. This is represented by the spinning motion
shown in diagrams (b) and (c), Fig. 2. Further, in the case of a mon-
atomic solid (such as a metal) the only type of motion which can be
ascribed to the atom is vibration. Free translation cannot exist, for
if it did the solid would not retain its crystalline form. In this case
the vibration of every atorn can take place in three directions in space
with respect to the centre of gravity or mean position of the atom.
There are therefore three degrees of freedom to be attributed to the
vibrations of the ultimate particles in a solid. This is represented in
diagram (d), Fig. 2. The characteristic feature of all vibrations is the
existence of potential as well as kinetic energy. As will be shown later
20 A SYSTEM OF PHYSICAL CHEMISTRY
each complete vibration on the average contains just as much potential
energy as it does kinetic.
To return to the vibrations of the atoms in a gas molecule. In the
case of a diatomic molecule the molecule is said to possess one degree
of freedom in respect to the to-and-fro vibration of one atom with
respect to the other. The molecule also possesses three degrees of
freedom in respect of translation of the molecule as a whole. In a
triatomic molecule the atoms ABC probably vibrate in pairs, AB,
BC, CA, each pair functioning like a diatomic molecule, so that the
molecule as a whole has three degrees of freedom in respect of atomic
vibrations. The molecule possesses in addition three degrees of
freedom in respect of translation. Hence, in the case of a triatomic
molecule there are three degrees of freedom in respect of translation
and three in respect of vibration.
It is obvious that no atomic vibration is possible in the case of a
monatomic molecule in a gas. If such an atom is displaced there is no
restoring force ; the atom is not connected to any other as in the case
also Axis of Spin"
^' (Circular Vibration.)
FIG. 2 (c). * Spinning molecule." Two degrees of freedom. Energy, FIG. 2 (d). Trilinear
kinetic +' potential. (Motion not maintained by collisions.) vibration of an atom
in a solid. Three
degrees of freedom.
Energy, kinetic +
of diatomic molecules, and any displacement would simply be identical
with tree translation. Contrast this with the behaviour of monatomic
molecules in solids in which any displacement can only take place
against a restoring force, with the result, as already mentioned, that- the
vibration in the case of a solid possesses three degrees of freedom. In
the case of a monatomic gas vibration is impossible, and, so far as we
have gone, we can only ascribe to this kind of molecule energy of the
translational kind. A difficulty crops up when we come to consider
a monatomic gas molecule in the light of the third possible type of
motion, viz. molecular rotation.
The circular vibration represented by diagram (c), Fig. 2, requires
a little more consideration. The amount of energy represented by a
circular vibration or spin depends upon the square of the rate of spin
and upon the moment of inertia of the spinning particle. 1 If a di-
1 If a particle of mass m moving in a circle of radius r round a fixed position
with an angular velocity o> (u> being the number of radians swept out by the
particle per second), then the speed of the particle is <ar and its kinetic energy
E = 1/2 m . a r a . This expression can also be written : E = 1 . 1/2 . w 2 , where i = mr*.
The quantity I is called the moment of inertia of the particle. The dimensions of
EQ UIPARTITION OF KINETIC ENERG Y 2 1
atomic molecule were really represented by the diagram (<:), Fig. 2,
i.e. if the atoms were massive spheres extending a sensible distance
from the axis, the energy of the motion represented in the diagram
would be sufficiently large to make its presence felt in the molecular
heat of the gas. As a matter of fact this type of circular vibration does
not affect the molecular heat sensibly. This arises from the now
accepted conclusion that practically the entire mass of any atom is
concentrated in a nucleus situated at the centre of the atom, the di-
mensions of the nucleus being small even compared with those of the
atom (cf. Chap. V., the Rutherford-Bohr atomic model). Hence, in
the motion represented in diagram (c\ Fig. 2, the mass of the two atoms,
i.e. the mass of the whole molecule is practically all concentrated upon
the axis of circular vibration (the figure axis in this case), and the
moment of inertia of the atoms, and consequently the energy in respect
of this motion is negligible, because the r term referred to in the foot-
note is practically zero. Further, as shown in Chap. V., each molecule
possesses a number of electrons likewise spinning, as in diagram (c),
Fig. 2. In the case of the electrons the distance r is not negligible,
but on the other hand the mass of the electron is so small a fraction of
the total mass of the molecule that again the moment of inertia is
small and the energy of electronic spin does not enter sensibly into the
" ordinary" energy content of the molecule, the variation of which
(energy) with temperature is given by the molecular heat. The mole-
cular heat term is due to energy of translation, of linear vibration and
of molecular rotation or its equivalent, precessional vibration (cf. infra).
The electron spin enters into the question of the ultra-violet spectrum
of the gas, cf. Chap. V. It must be assumed, of course, that such a
spin as that represented in the diagram (c) t Fig. 2, takes place in all
cases. It is scarcely affected by temperature, however, and conse-
quently does not enter into molecular heat values, except in the limit
when the temperature is very high.
3. Energy of Molecular Rotation. If a molecule resemble a solid
sphere we would expect it to rotate in the manner indicated in Fig. 3,
diagram (a}. The rotation of a sphere can be referred to three axes
of rotation, i.e. there are three degrees of freedom. The energy is
entirely kinetic. Molecules, however, are not necessarily spherical
unless they contain a number of atoms. It is believed that at least
three atoms must be present in a molecule before we can possibly
ascribe to the molecule as a whole the limiting number (3) of degrees
of freedom in respect of rotation.
It is a remarkable fact that monatomic gas molecules do not appear
to possess rotational energy. This conclusion rests upon the experi-
mental fact that the molecular heat of argon and other monatomic
gases and metallic vapours can be accounted for by simply assuming
translational energy. This point will be dealt with later. A monatomic
the particle in the above case are supposed to be small compared with r. In the
case of a solid sphere the moment of inertia of the sphere can be shown to be
2/5 . M . r 2 , the axis of rotation passing through the centre of the sphere.
A SYSTEM OF PHYSICAL CHEMISTRY
gas molecule appears to function simply as a massive point, and not
as a massive sphere. This is a remarkable conclusion. It is self-
evident that vibration of the atom cannot occur in the case of a mon-
FIG. 3 (a). Sphere with three com- FIG. 3 (6). Molecular rotation of diatomic molecule
ponent rotations. Three degrees (due to collisions). Two degrees of freedom,
of freedom. Energy, kinetic. Energy, kinetic.
c '>cular Orbit.
FIG. 3 (c).Precessional
vibration (due to col-
lisions). The molecule
properties. Two de-
grees of freedom. En-
FIG. 3 (d). Extreme precessional vi~
bration (due to collisions at high
temperatures). Merging into or-
dinary molecular rotation.
atomic molecule. It is by no means clear why rotation of the monatomic
molecule as a whole apparently does not occur. It is possible that
rotation is actually occurring but that the energy term corresponding
thereto is negligibly small for the same reasons as those advanced in
EQ UIPARTITION OF KINETIC ENERG Y 23
considering the circular vibration of diagram (c\ Fig. 2. This point
has not yet been settled.
Molecular rotations are ascribed to collisions with other molecules.
This distinguishes molecular rotation from atomic vibration as far as
origin is concerned. Atomic vibration can indeed be affected by
collisions, but the origin of atomic vibration is more deeply seated, so
to speak, than that of molecular rotations. Presumably atomic vibra-
tion is set up as a result of absorption of radiant energy on the part of
the molecule. This might occur at a collision, but not necessarily so ;
unless the collision be very inelastic (cf. Chap. VI., the section dealing
with resonance and ionisation potentials).
When a molecule consists of more than one atom it seems reason-
able at first sight to ascribe to the molecule as a whole a certain amount
of rotational energy. The reason for the qualifying clause will not be
given now ; the point is taken up in Chap. IV. in connection with the
theory of the molecular heats of gases. As Kriiger has shown, in place
of true molecular rotations we may have to substitute another kind of
motion, namely, precessional vibrations. 1 The possible kinds of rota-
tions and precessional vibrations in the case of a diatomic molecule
are illustrated in Fig. 3, diagrams (b) to (d). It must be understood
that rotation or precessional vibration always refers to the molecule as
a whole. This is in contrast with the view taken of true atomic vibra-
tions which have been discussed.
As regards the rotation of a diatomic molecule the diagram (),
Fig. 3, shows us that there are two degrees of freedom, i.e. there are two
co-ordinates at right angles defining the surface over which the rota-
tion of such a molecule can take place. The axis of rotation is at right
angles to the plane indicated in the diagram.
From the point of view of the internal molecular energy the signifi-
cance of the rotation of the molecule depends upon its moment of in-
ertia, i.e. the moment of inertia of each of the atoms with respect to the
axis of rotation and the number of revolutions v which the molecule
makes per second (as a result of collisions). The rotational energy is
given by the expression 1/2 . I . (27rr)' 2 , where I is the moment of
inertia and v has already been defined.
1 The term " precessional vibration " requires perhaps a word of explanation.
The type of motion represented by the term is shown in diagrams (c) and (c/), Fig. 3.
It is similar to the true precession of a gyroscope, but with this difference that a di-
atomic gas molecule does not precess in a definite field of force. It is assumed, in
fact, that there is no field of force. Precessional vibration (which must not be con-
fused with nutation) is brought about in the case considered by the collisions of
molecules with one another, each molecule being assumed to possess gyroscopic
properties in virtue of the electrons which it contains, and which spin [circular
vibration] around the figure axis as already indicated in diagram (c), Fig. 2. If no
collision occurred a diatomic molecule would spin with a fixed axis. As a result of
collision the spin is disturbed and the figure axis itself describes the circular motion
represented in diagram (c), Fig. 3. It maintains this type until a further collision
occurs which sets up a new vibrational precession, the figure axis now precessing
round a circle which has a greater or less circumference than before. Such pre-
cessional vibrations -might be more accurately described as Poinsot movement, dealt
with in rigid dynamics. The introduction of the idea into molecular motion and
molecular heat is due to Kruger, cf. Chap. IV.
A SYSTEM OF PHYSICAL CHEMISTRY
In the case of a diatomic molecule we have therefore three degrees
of freedom in respect of translation, one degree of freedom in respect
of linear vibration of the atoms, and two degrees of freedom in respect
of molecular rotation ; six degrees in all. If we regard rotation as im-