is 7T/- 2 /// 3 or 7T/- 2 // 2 . If the average velocity is u, the time of a journey
between two successive collisions is Iju. Hence the number of en-
counters per second is (ufl) x chance of collision = u-n-r 2 // 3 . This
6 A SYSTEM OF PHYSICAL CHEMISTRY
quantity is obviously a constant for a particular gas at constant tempera-
ture, and hence is really included in the proportionality factor or velocity
constant k.
Another illustration of the application of probability considerations
is afforded by the phenomenon of coagulation of colloids by electrolytes
(cf. Whetham, Phil Mag., [v.], 48, 474 (1899). It is supposed that
the coagulation is due solely to the neutralisation of the electrical
charge of the colloid, and that one trivalent ion is exactly equivalent to
i *5 divalent ions, or to three univalent ions as far as coagulating efficiency
is concerned. This is, of course, too simple an assumption, as we
have already seen in the criticism of Schultze's rule (Vol. II., Chap. VIII.).
Adopting this simple view of the effect, however, Whetham has shown
how the numerical values obtained experimentally may be accounted
for approximately. The coagulating power of an ion is defined as the
reciprocal of the number of gram-equivalents of the ion which are just
capable of bringing about coagulation of a given quantity of colloid.
Linder and Picton found that negatively charged colloidal arsenic
sulphide could be coagulated by tri-, di-, or uni-valent cations, the
relative coagulating powers being in the ratios
1023 : 35 : i.
Let us suppose that in order to produce coagulation a certain
minimum electric charge has to be brought into contact with the
colloidal particle, and that such conjunctions must occur with a certain
minimum frequency throughout the solution. We shall get equal
charges by the conjunction of 2n trivalent ions, or $n divalent ions,
or 6n univalent ions, where n is any whole number. In a solution
where ions are moving freely the probability that an ion is at any instant
at a certain position is represented by a fraction which is proportional
to the ratio between the volume occupied by the sphere of influence of
the ion and the total volume of the solution. The probability is there-
fore proportional to the concentration of the ion, and may be written
as (Ac), where A is any constant. The probability that two such ions
are present together is (A<:) 2 , and the chance that n such ions are
present simultaneously at the position is (A<r) M . Let us suppose that
three solutions containing respectively tri-, di-, and uni-valent ions at
concentrations C 3 , C 2 , and Cj possess exactly the same coagulating
power upon a given amount of a given colloid. Then, since the coagu-
lating power is the same in these three cases
A 2 Ca 2 = A 3 C 2 3M = A^Cj 6 " = a constant = B.
Hence, the critical concentration of the trivalent ions is given by
and similarly C 2 = r- and Q = r-.
WHETHAM'S RULE OF COLLOID COAGULATION 7
The ratios in which these three concentrations stand to one another are
Cf~* f** "D^W ~D3^ 15 2#
! . U 2 . C 8 = > . -D . &
= i : B^ : B 3 *.
Putting B 6w = i/jc, the ratios can be written
These represent numbers which indicate relative concentrations of equal
coagulating power. Hence the relative coagulating powers P I} P 2 , P 3 of
equal concentrations of these three ions are given by the reciprocal of
the above numbers. That is
P. "D "P ._ T <V T^
-, . Jr 2 -^3 ~~ " **
The value of x, which depends upon a number of unknown factors
characteristic of the colloid considered, cannot be found on a priori
grounds. If we take Linder and Picton's experiments into account and
set x = 32, we get for the relative coagulating powers of univalent,
divalent, and trivalent ions respectively, the values i : 32 : 1024.
It will be seen that these numbers are of quite the same order of magni-
tude as those observed. Whetham predicted on this basis that the
coagulating power of a tetravalent ion on the above colloid should be
a large number, approximately 33000. Recent measurements have
corroborated this result in so far as an extraordinarily large coagulating
power is actually obtained.
The Law of Error. It is a familiar fact in physico-chemical
measurements that repetitions of a certain measurement give rise to a
series of numbers which are not identical. The variations we speak of
as experimental errors. The measurements are as likely to be too high
as too low, that is, the errors are as often positive as negative, provided
we make a very large number of determinations. (It is assumed that
there is no systematic error in the apparatus or in the method of
measurement.) The treatment of such results so as to obtain the most
probable result, i.e. the most accurate determination, is a further illus-
tration of the application of the theory of probability, somewhat more
complex in nature than that hitherto considered.
Thus, it is possible to construct a probability curve, by means of
the probability equation given below, which has been found to agree
closely with the actual results obtained in a series of experimental de-
terminations of a given quantity. Examples will be found in a text-
book of mathematics, e.g. Mellor's Higher Mathematics. The large
majority of the readings will fall very closely together, i.e. they will
not be far removed from the true result, a smaller number of readings
will be farther away on either side of the true result, and only a very
small number will be much to one side or the other. This distribution
of values may be represented by an expression of the form
A SYSTEM OF PHYSICAL CHEMISTRY
This is known as the normal law of errors. The curve is shown in
Fig. i. x denotes the error and y the probability of its occurrence.
As x increases numerically, positively, or negatively, y decreases rapidly,
and when x becomes large y becomes vanishingly small. It will be
observed that the curve is symmetrical.
Maxwell's Law. Maxwell has applied the principle of probability
to the problem of the distribution of velocities among the molecules of
a gas, the gas being in a condition of statistical equilibrium at a uni-
form temperature throughout. A gas is to be regarded as a molecular
chaos, the speed of any molecule varying from zero to infinity, its in-
stantaneous value being the result of chance collisions with its neigh-
bours. Although all values are theoretically possible for the speed of
a molecule, it is found that in a system containing a large number of
molecules, there are very few possessing either very great or very small
Negative Errors. Positive Errors.
FIG. i.
speeds. The majority of the molecules possess speeds which lie within
relatively restricted limits. A numerical illustration taken from Meyer's
Kinetic Theory of Gases is given in Appendix I.
There are, as a matter of fact, two ways of expressing Maxwell's
law of distribution. One of these ways has already been stated in
Chap. I., Vol. I. This way of expressing the law may be put in the
form
dn = constant x N x
dc
where N is the total number of molecules in the system, and dn is the
number whose speeds lie between the limits c and c + dc. It is to
be clearly understood that the speed here referred to is simply a velocity
magnitude, and no restriction has been introduced as to direction of
motion. The term c is known as the root-mean-square speed (or
r-m-s speed). At a given temperature the system is characterised by
a certain mean or average kinetic energy of its molecules, which is
MAXWELL'S LAW 9
maintained as long as the temperature is maintained constant. This
average kinetic energy may be written as -^Nm? 2 , where f is the sum
of the squares of the velocities of all the molecules at any given instant
divided by N. This r-m-s speed is not the same thing as the mean or
average speed, which is simply the sum of all the velocities divided by
N, though the two quantities are not very different numerically. The
above way of expressing Maxwell's law, i.e. the above expression, is of
the general form : y = x 2 e~* 2 , which, it is to be observed, is not identical
with the normal law of errors. The resulting curve is, in fact, not
symmetrical. The alternative mode of expressing Maxwell's law where-
in the curve is symmetrical, and the distinction between the two modes,
is considered in Appendix I.
Maxwell's, law expresses the distribution of velocities as a continuous
function of the number of molecules present. That is, the speed of one
molecule may differ by any amount (down to the infinitesimally small)
from the speed of any other molecule. When we come to consider
the quantum theory we shall find continuous functions replaced by dis-
continuous ones, i.e. abrupt changes in finite small steps in place of
gradual change in infinitely small steps. 1
It will be observed that Maxwell's expression involves the squares of
velocities. Since the kinetic energy of a molecule depends upon the
square of its velocity, it should be possible, on similar lines, to express
the distribution of kinetic energy amongst the molecules constituting
a gas system. If we denote by dn the number of molecules which
possess kinetic energy lying between the limits E and E + <E, it can be
shown that at a given temperature
dn = constant x N x e~ E l kT . E^ . dE
where N is the total number of molecules, and k the gas constant per
molecule (i.e. k = R/N , where R is the gas constant per gram-molecule,
and N is the number of molecules in i gram-molecule).
Further, potential energy of molecules may be considered as dis-
tributed in an analogous manner. We are here considering the
potential energy of a molecule, free from intermolecular forces (as in
the case of a perfect gas), the molecules being placed, however, in a
field of force and the potential energy being due to this field of force,
which might be conceived of as originating in some external body. A
good example is the earth's gravitational field which acts on a column
of gas in such a way that the molecules are more dense in the lower
portion of the column than in the upper. Denoting potential energy
a This statement must not be taken as meaning that the distribution of kinetic
energy of translation takes place in the discontinuous manner postulated by the
quantum theory. The quantum theory applies only to energy of the vibrational or
rotational type, i.e. motion with respect to some fixed centre of gravity. Free
translational motion such as that considered above must be treated in a con-
tinuous manner, the velocity being capable of changing by an infinitely small
amount.
io A SYSTEM OF PHYSICAL CHEMISTRY
by W, we can state that the number of molecules, the potential energy
of which lies between W and W + <AV, is given by
dn = constant x N x <r w /* T . dW.
We can integrate this to obtain the number x of molecules of a
perfect gas (in equilibrium in a field of force at a uniform temperature)
which possess potential energy W 15 that is all values from zero up to W r
If the total number of molecules in the system be N, the expression
is ! = N(i - *- w i/* T ).
If we introduce the Avogadro constant N , i.e. the number of mole-
cules in one gram-molecule, the above expression becomes
! = N(i - *-Vi/ RT ),
where R is the gas constant per gram-molecule.
It follows from this that the number of molecules which possess
potential energy between W l and infinity is (i - J which is equal to
N<r w i/* T or N<r N o w i/ RT .
Distribution of Molecular Velocities and Temperature. On the kin-
etic theory it is to be expected that the temperature of a gas should
be expressible in purely mechanical terms. We are already familiar
with the concept that temperature is measured by the kinetic energy of
the molecules. In view of the distribution of velocities and therefore
of kinetic energy, among molecules, as expressed in Maxwell's law, it
is evident that the kinetic energy of a given individual molecule may be
very different from that possessed by another molecule of the same
system. Further, the kinetic energy of one and the same molecule
varies from moment to moment as a result of collisions. The tempera-
ture of the system measured in the ordinary way, by means of a ther-
mometer is a perfectly definite quantity for the gas system as a whole
in the steady state. The temperature, in fact, is determined by the
average kinetic energy. It is therefore meaningless to speak of the
temperature of a single molecule in a gas. Temperature is essentially
a statistical effect due to the presence of a large number of molecules
each contributing its own share to the total effect. Two independent
systems are at the same temperature when the average kinetic energy
of each is the same. This is true whether the systems be gaseous,
liquid, or solid, homogeneous or heterogeneous.
It will be appreciated at the same time that pressure is likewise a
statistical effect. A single gas molecule cannot be conceived of as
exerting observable pressure, though each molecule exerts a certain
force against the walls of the containing vessel, the total effects of which,
when numerous molecules take part, is manifested as a uniform gas
pressure.
ENTROPY AND THERMODYNAMIC PROBABILITY.
It is proposed to indicate how the second law of thermodynamics
can be deduced on the basis of statistical mechanics. This was first
ENTROPY AND THERMODYNAMIC PROBABILITY n
demonstrated by Boltzmann. Hitherto we have regarded the second
law as a law of experience, its validity depending upon the fact that no
contradiction to it has been met with in nature. It is important to see
that this law possesses at the same time a mechanical basis. The
demonstration consists in showing the connection between the entropy
of a system the concept of entropy involving necessarily the concept
of the second law and a statistical quantity known as the thermo-
dynamic probability of the system.
It is necessary to recall first of all what is meant by thermodynamical
equilibrium, as stated in terms of entropy, that is as stated in terms of
the second law. Planck's definition of such equilibrium is as follows
(cf. Planck, Theory of Heat Radiation, English ed., p. 22) : "A system
of bodies of arbitrary nature, shape, and position, which is at rest and
is surrounded by a rigid cover impermeable to heat, will, no matter what
its initial state may be, pass in the course of time into a permanent
state in which the temperature of all bodies in the system is the same.
This is the state of thermodynamic equilibrium, in which the entropy
of the system has the maximum value, compatible with the total energy
of the system as fixed by the initial conditions. This state being reached,
no further increase in entropy is possible."
We know that heat, from the kinetic molecular point of view, is
represented by the kinetic energy of the molecules of a system, the
molecules moving about in a completely chaotic manner as a result of
collisions. Owing to collisions any ordered arrangement which the mole-
cules might be conceived of as possessing initially would be quickly
annulled, and completely disordered distribution, both as to position
and to molecular velocities, would ensue. This represents the direction
of change in any spontaneous or naturally occurring process. That is,
from the molecular standpoint a system always changes from an ordered
to a chaotic state, and the change will go on until the molecular motion
has become as disordered as possible. When this stage is reached,
there is no longer any reason for further change. When equilibrium is
reached the system has at the same time reached a maximum disorder
or " mixed-up-ness ". This involves the idea that a system in equili-
brium possesses a maximum value of the probability of the state, the
probability here referred to dealing with possible modes of molecular
arrangement and velocity. We may call this the thermodynamic prob-
ability.
According to Boltzmann the thermodynamic probability of an ideal
monatomic gas is a number which denotes by how many times or by
how much the actual state of a gas system is more probable than a
state of the same gas system (i.e. possessing the same total energy and
volume) in which the molecules are equally spaced and all possess the
same velocity. This "standard" state represents perfect order or
arrangement of the molecules. It is of course never realised in practice
owing to the disorder brought about as a result of collisions. The
standard state represents the stage farthest away from the equilibrium
state finally attained by the gas, in which final state the system is
12
A SYSTEM OF PHYSICAL CHEMISTRY
completely chaotic. The probability of disorder is very much greater
than the probability of complete order.
The thermodynamic probability of this "standard" state (that is
the probability of complete order) is taken to be unity. It follows there-
fore that the thermodynamic probability of a real equilibrium state is
an integral number usually much greater than unity. Thermodynamic
probability differs therefore from mathematical probability in that the
latter is always a fractional quantity, i.e. it denotes the ratio of the
number of favourable cases to the total number of possible cases.
Thermodynamic probability is proportional to but is not identical with
mathematical probability.
It has just been stated that a system in equilibrium possesses a dis-
tribution or arrangement which is characterised by a maximum value for
the thermodynamical probability of the state. It is necessary before
going further to give a somewhat more concrete idea of what we mean
by states or arrangements and the probability of arrangements. Let us
leave the problem of molecules and turn to a very simple kind of
system which can undergo various arrangements.
Let us suppose that we have two squares or areas denoted by the
symbols I and II, and further let us suppose that we have five letters,
a, b, c> d, e, and we wish to distribute or arrange these letters between
the squares in every possible way. It is evident that all possible ways
are included in the following :
I.
II.
Nature of Arrangement.
First arrangement :
ist way of arranging the letters is
abcde
All letters in I none in II.
Second arrangement :
2nd way of arranging the letters is
3rd ,, ,, ,,
abed
abce
e
d
Four letters in square I ; one
letter in II. There are five
4th .
abde
c
ways of producing this ar-
5th
acde
b
rangement or distribution.
6th
bcde
a
Third arrangement :
7th way of arranging the letters is
8th
9th
abc
abd
acd
de
ce
be
Three letters in square I ; two
letters in square II. There
are ten ways of producing
loth ,
bed
ae
this arrangement or dis-
nth ,
abe
cd
tribution.
i 2th ,
ace
bd
I3th
bee
ad
I4th ,
ade
be
I5th ,
bde
ac
i6th
cde
ab
We now begin with an arrangement similar to the first, but place
all the five letters in square II and none in square I. Similarly we
ENTROPY AND THERMODYNAMIC PROBABILITY 13
can interchange squares I and II in the other two arrangements, so
that in all, there are six arrangements possible in this system, and in
all, thirty-two different ways in which these six arrangements or dis-
tributions may be carried out. Each of these ways is analogous to
a " complexion " on Boltzmann's nomenclature. Every way or com-
plexion is to be regarded as equally probable. It is very necessary to
distinguish between arrangement or distribution and complexion. Thus
in the first case, the arrangement is five letters in square I, none in
square II. There is only one way or complexion of doing this. In
the second arrangement, which consists of four letters in square I and
one letter in square II, there are five ways or complexions of doing
this. It is simply a question of combinations. Thus there are five
letters to be divided in such a way that four are in one square, one in
another square. The number of possible ways of doing this particular
distribution is -= = 5. In the third arrangement or distribution
we have to distribute the letters so that there are always three in
square I and two in square II. There are ten ways or complexions
corresponding to this single distribution. This number is given by
-i= =10. Note that the problem is not which letters are in the
2
squares, but how many different ways can they bed ivided to cor-
respond with any particular arrangement, such as three letters in one
square, two in the other.
If instead of five letters we had N letters and divided them between
two squares, in such a way that n letters are in square I and N- letters
in square II, the possible ways or complexions possible to this particular
distribution are
|N
|N - n n
If, instead of two squares, we had m squares, the total number of
complexions in a particular distribution would be given by
IN
where n-^ + n^ + n% + . . . n m = N.
To return to the simple case of five letters arid two spaces. We
have seen that there are six possible arrangement or distributions, viz.
(5) (o); (4) (I); (3) (2); (o) (5); (i) (4); (a) (3). .Each of these
arrangements has its own number of ways or complexions. Thus for
arrangement (5) (o), the number of complexions is one. For arrange-
ment (4) (i), the number of complexions is five ; for arrangement
(3) (2), the number of complexions is ten. Similar numbers of
complexions are found in the remaining arrangements or distributions.
A SYSTEM OF PHYSICAL CHEMISTRY
It is evident that the arrangement (3) (2) has the same number of
complexions as the arrangement (2) (3), and that either of these two
arrangements or distributions possesses the maximum number of com-
plexions. But the number of complexions is identical with the thermo-
dynamic probability of the particular distribution under consideration.
Either of the systems (3) (2) or (2) (3) possesses the maximum prob-
ability. Either distribution represents therefore the equilibrium dis-
tribution. The distribution (3) (2) or (2) (3) is the most likely
distribution of five letters scattered at random between two squares.
The equilibrium distribution in the above system is ten times more
likely than the distribution (5) (o) or the distribution (o) (5), for the
latter only possesses one complexion whilst the equilibrium distribution
possesses ten.
The choice of five letters has somewhat obscured an important
point in that we are led to two distributions of equal maximum
probability. To find a closer analogue to the actual case of molecular
systems containing an enormous number of molecules it is better to
choose as the simplest model the distributions of six letters between
two squares. With this system we have the following distributions
and corresponding complexion-number for each distribution :
Complexion-
Number.
ist distribution or arrangement
6 letters in I
o letters in II
I
2nd
5
i
6
3rd
4
2
15
4th
3
3
20
5 th
2
4
15
6th
I
5
6
7 th
O
6
i
Total number of complexions = 64
The single arrangement or distribution which contains the maximum
number of complexions is that represented by three letters in each
square. This distribution has therefore the maximum probability, i.e.
it is the one most likely to occur if six letters be scattered at random
over two squares. This distribution is consequently the " equilibrium "
distribution of the constituents of this system which consists of six
letters.
Now let us apply this idea to a system of molecules. In this case
also that distribution or arrangement which has the greatest number
of complexions or ways is the most probable arrangement, its degree
of probability, or simply its probability, being measured numerically
by the number of complexions possible to it. The most probable
distribution is the equilibrium distribution or state or condition of the
system. A system changes in the sense that it tends to approach and
finally reach an equilibrium state. In this state the number of possible
ENTROPY AND THERMODYNAMIC PROBABILITY 15
complexions is a maximum, i.e. greater than the number of complexions
in any other distribution of the constituents of the system. A maximum
number of complexions is identical with the idea of maximum disorder
or maximum molecular chaos.
In all spontaneous processes the thermodynamical probability tends