orbit is giving off radiation continuously, and in so doing its orbit
gradually becomes smaller so that the electron would finally drop into
the nucleus. Furthermore, with the alteration in orbit the wave-length
of the light emitted would gradually change, a conclusion which is op-
posed to direct observation, for spectra are clearly characteristic and
unchangeable properties of an atom. Bohr assumes that no radiation
at all is given out when the electron is rotating in an orbit.
VOL. III. 7



98 A SYSTEM OF PHYSICAL CHEMISTRY

Let us now consider the hydrogen atom one in which there is a
single nucleus carrying unit positive charge with one valency electron
rotating round it in a state of equilibrium, the electron traversing
an elliptical orbit with a frequency of revolution o>, the major axis of
the orbit being 20. The amount of energy W which must be given
to the system (i.e. which the system must absorb) in order to remove
the electron to an infinitely great distance (i.e. to dissociate the atom
into a positively charged nucleus and a free electron), is connected
with w and 2a by the two following relations :

W 3 / 2 eE

a = -



where e is the charge on an electron, E the charge on the nucleus, and
m the mass of an electron. Further it can be shown that the kinetic
energy of the electron taken for a complete revolution when it is rotating
in one of it orbits is equal to W, the work required to eject the electron
entirely from the system. Note that removal of the electron necessitates
absorption of radiant energy.

Now let us consider the reverse process, namely, the act of binding
a free electron to the nucleus. At the beginning the electron may be
regarded as possessing no sensible velocity with respect to the nucleus,
i.e. its frequency of revolution is zero relatively to the nucleus. The
electron, after interaction has taken place, settles down to a station-
ary orbit the word stationary refers to the orbit, the electron itself
is in rapid rotation round the orbit. Bohr now takes the orbit as circular
for reasons given later in connection with the physical significance of h.
The initial free state of the electron represents the extreme limit of a
series of stationary states through which the electron is capable of
passing. The other limit is given by the smallest value of za or the
greatest value of o> which necessitates the maximum value of W.

Let us assume that during the act of binding of the free electron to
the nucleus, a homogeneous radiation of frequency v is emitted. This
frequency is, according to Bohr, just half the frequency of rotation o>
of the electron in its final orbit. On the basis of Planck's theory we
would expect that the total amount of radiant energy thus emitted
would be rhv where T is a whole number. The assumption that the
frequency v is just w/2 suggests itself, since the frequency of revolution
at the beginning of the binding process is zero and at the end of the
process is w, the mean or average of the two being <*>/2. Bohr gives
a more rigorous treatment of this point in the first paper referred to.
Since rhv is the amount of energy emitted whilst the electron is
approaching the nucleus from an infinite distance, and W is the amount
of energy which has to be absorbed to make it reverse the operation,
it follows that

W = rhv = T/&- . . . . (2)



RUTHERFORD-BOHR ATOM-MODEL 99

Combining thus with the equation (i) we obtain

If in these expressions we give T different values, we get a series of
values for W, o>, and a, corresponding to a series of configurations of
the system, such configurations being stationary states in which there
is no radiation of energy, and in which the electron will remain as long
as the system is not disturbed from outside. We see that the value of
W is greatest if T has its smallest value ', viz. unity. This will therefore
correspond to the most stable state of the system : i.e. it will correspond
to the binding of the electron for the removal of which the greatest
amount of energy is required. Substituting in the above expressions
T = i, and E = e = the charge on a single electron, and introducing
the experimental values, viz. e 47 x io- 10 , elm = 5*31 x io 17 ,
h = 6*5 x io 27 we obtain : -

2a = i'i x io- 8 cm. ; o> = 6*2 x io 15 per sec. ; W/e = 13 volt.
We see that these values are of the accepted order of magnitude for the
linear dimensions of the atoms, the optical frequencies, and the ionisa-
tion potentials respectively.

It is to be clearly borne in mind that the results so far obtained
rest on two main hypotheses :

1. That the dynamical equilibrium of the systems in the stationary
states can be discussed by help of the ordinary electro-dynamics, whilst
the passage of the systems between different stationary states cannot be
treated on that basis.

2. That the latter process is followed by the emission of homogeneous
radiation, for which the relation between the frequency and the amount
of energy emitted is the one given by Planck's theory.

The next problem which Bohr takes up is to show that his theory
is capable of accounting for the line spectra of hydrogen.

If we consider the act of binding a free electron to a hydrogen
nucleus carrying a positive charge equivalent to the charge on a single
electron so that the electron finally rotates in one of the stationary
states, the energy radiated out in the formation of this stationary state
is given by equation (3), viz.

W T =

The amount of energy emitted by the passing of the electron from a
state where T T : to a state where T = r 2 is consequently



The state corresponding to r 2 is one in which the electron is closer to
the nucleus than in the state corresponding to r x ; in short, T S is less
than T 1} and the linear dimension a of the atom is smaller when T = r 2
than when T = n.

7*



ico A SYSTEM OF PHYSICAL CHEMISTRY

If we now suppose that the radiation in question is homogeneous,
and that the amount of energy emitted is equal to hv, we get

W T2 - W TX = hv
and hence



27T me



-X? - *)



This expression gives the frequency of the homogeneous radiation
emitted by the gas when the atomic system changes from the stationary
state denned by r x to the stationary state defined by r 2 . If these two
states were the only possible ones the gas could only give rise to a single
frequency v. The quantities r 2 and T I are whole numbers, and a single
atom is capable of existing in a corresponding number of stationary
states, so that in general a number of lines will be emitted as is borne
out by experiment. The above expression is capable of accounting for
series of lines emitted by incandescent hydrogen. If we put T Z = 2
and allow TJ to vary, i.e. TJ can take on the values 3, 4, 5, etc., we get
the well-known Balmer series of lines in the hydrogen spectrum. It is
noteworthy that to account for the Balmer hydrogen line spectra we
have to assume that the electron from an outer orbit passes to the
second orbit (T S = 2) and not to the first or innermost orbit, which re-
presents maximum stability. If we put r 2 = 3 and allow TJ to vary
(TI = 4, 5, 6, etc.), we get the infra-red series of lines observed by
Paschen (Ann. Physik, 27, 565 (1908)). If we put r 2 = i and allow
TJ to vary we get a series in the extreme ultra-violet, the furthest Lyman
region. In the case of elements heavier than hydrogen, e.g. platinum,
the kind of radiation to be expected on putting r 2 = i corresponds to
X-rays. If we put r 2 = 4, 5, etc., we get series of lines in the extreme
infra-red not yet observed. It will be observed that Bohr's theory
accounts excellently for those series of lines which have been observed
in the case of hydrogen and is even capable of predicting other series
in regions not yet examined. Further, the agreement is quantitative
as well as qualitative. Putting e 4*78 x io~ 10 electrostatic units;
e\m 5*31 x io 17 ; and h = 6^55 x io~ 27 , we get

= 3-26 x io 15 ,

whilst the value obtained experimentally for the factor (the so-called
Rydberg constant) outside the bracket in formula (4) is 3*290 x io 15 .
The agreement between the observed and calculated value is within the
error due to experimental determination of the various quantities in-
volved.

It may be pointed out that it has not been possible to observe more
than twelve lines of the Balmer series in experiments with vacuum tubes
whilst thirty-three lines are observed in the spectra of certain celestial
bodies. This may likewise be anticipated on the basis of Bohr's
theory. According to equation (3) the diameter of the orbit of the



RUTHERFORD-BOHR ATOM-MODEL 101

electron in the different stationary states is proportional to T 2 . For
T = 12, the diameter is equal to 1-6 x io~ 6 cm., or equal to the mean
distance apart of the molecules of a gas at a pressure of about 7 mm. of
mercury; for T = 33 the diameter is equal to 1*2 x io~ 5 cm., corre-
sponding to the mean distance apart of the molecules at a pressure of
about 0-02 mm. of mercury. According to the theory, the necessary
condition for the appearance of a great number of lines is therefore a
very small density of the gas, and if this were realised in a vacuum tube
the whole mass of gas present would be insufficient to give rise to a
measurable intensity of emission. In solar conditions we have large
masses of incandescent gas at extreme rarefaction, and hence lines are
observed under these conditions which cannot be observed in a vacuum
tube,

Bohr next considers the spectrum of helium.

The neutral atom of helium consists, according to Rutherford, of a
single positive nucleus of charge 2e, with two electrons rotating in the
same orbit around it. Now, considering the binding of a single electron
by a helium nucleus, we get, on putting E = 2<?, in expression (3)




If we put T 2 = i or T 2 = 2 in this formula we get series of lines in the
extreme ultra-violet. If we put r 2 = 3 and let T I vary we get a series
of lines which' includes two of the series observed by Fowler and
ascribed by him to hydrogen. If we put r 2 = 4 we get the series observed
by Pickering in the spectrum of Puppis. Every second one of the
lines of this series is identical with a line in the Balmer series of the
hydrogen spectrum. It is not surprising therefore that these lines from
Puppis were ascribed to hydrogen. In Fowler's experiments hydrogen
was mixed with helium in the vacuum tube. With helium alone in a
vacuum tube the lines given by putting r 2 = 3 are not shown, as the
ionisation of helium is so small. The presence of hydrogen appears to
increase the ionisation of helium, due very probably to the fact that a
positively charged hydrogen nucleus has a considerable affinity for
electrons and consequently causes the helium atom to lose both of its
electrons more easily than it would do in the absence of ionised hydrogen.
The lines predicted by Bohr's theory refer to the binding of an electron
by the helium nucleus which carries a positive double charge. Further,
hydrogen atoms are known to be capable of acquiring a negative
charge, so that their " catalytic " effect on the helium may be partly due
to the affinity of neutral hydrogen atoms for electrons. Bohr explains
in this manner a certain set of the lines obtained from a vacuum tube
containing hydrogen and helium as due to the presence of the doubly
charged helium nuclei combining with electrons. Finally, if we put
T 2 = 5> 6, etc., in the above formula we get series, the strong lines of
which are to be expected in the infra-red region.



102 A SYSTEM Of PHYSICAL CHEMISTRY

We are now in a position to give something like a physical interpre-
tation of Planck's constant h. Consider equation (2). If we denote
the angular momentum of an electron rotating round a nucleus by M, it

T
follows that for a circular orbit TrM = where w is the frequency of

O)

revolution and T the kinetic energy of the electron. For a circular
orbit we further have T = W, as we have already seen. Hence

M = rM
where M = h^w = 1*04 x io~ 27 .

If we assume therefore that the orbit of the electron in the stationary
state is circular we can say : the angular momentum of the electron
round the nucleus in any one of the stationary states of the system is
equal to an entire multiple of a universal value M or ^/2ir, independent
of the charge on the nucleus. The possible^importance of the angular
momentum in relation to Planck's theory ha's also been emphasised by
Nicholson.

We can only observe a great number of different stationary states by
investigating emission and absorption of radiation. In most of the
other physical phenomena we deal with atoms in a single state, the state
which corresponds to low temperatures. From these considerations we
are led to the assumption that the permanent state is the one among
the stationary states during the formation of which (from free electron
and nucleus) the greatest amount of energy is emitted. According to
equation (3) this state corresponds to T = i.

As regards the process of absorption of radiation, the existence of
Kirchhoff's law (viz. that a body absorbs radiation of the same wave-
length as it emits), naturally suggests that the same mechanism is to be
attributed to absorption as to emission, absorption taking place when
an electron passes from an inner orbit to an outer.

Bohr's theory is likewise in agreement with Einstein's expression
for the photo-electric effect, viz. T = hv W, where T is the kinetic
energy of the electron ejected under the influence of light of frequency
v, and W is the total amount of energy which would be emitted during
the act of binding a free electron to the nucleus of the atom (cf.
following chapter).

We have now to consider in more detail what has been called the
permanent state of an atomic system. We have seen that in the case of
a system consisting of a nucleus and a single electron, the permanent
state is defined by the condition that the angular momentum of the
electron is h\2ir. The atom of hydrogen corresponds to this system,
but the atom of hydrogen is difficult to deal with because of the very
slight dissociation of gaseous molecular hydrogen. In order to get a
closer comparison with experiment it is necessary to consider more
complex systems. Let us consider a ring of n electrons rotating round
a nucleus of charge E, the electrons being arranged at equal angular
intervals round the circumference of a circle of radius a.



RUTHERFORD-BOHR ATOM-MODEL



103



The total potential energy of the system consisting of the electrons
and the nucleus is



where



s n = -



1 The above expressions require probably a little explanation. Let us consider a
positively charged nucleus, charge E surrounded by n electrons, each of charge e,
the electrons being arranged at equal distances on the circumference of a circle of
radius a. We shall number the electrons i, 2, 3, /

etc., up to n. The problem is to find the expression -^\**'

for the potential energy of this entire system, the
potential energy being measured by the work
which it is necessary to do upon the system in
order to remove all the electrons to an infinite dis-
tance from the nucleus and from each other. First
of all consider the nucleus with respect to electron
i. There is a force of attraction between the elec-
tron and the nucleus amounting to ^E/a 2 . The
potential of the electron with respect to the nucleus
being - e-^E/a, the minus sign being introduced
because, in pulling the electron away, we oppose
the natural direction in which the electron tends to
move (viz. towards the nucleus). For electron 2
we have a similar term, so that for the n electrons
we have the potential energy of the electrons with
respect to the nucleus given us by the expression
- neE/a. We have now to consider the effect of the
electrons upon each other. The force is now one
o repulsion. It therefore aids the process of pulling
apart. The expression for the potential energy of
each electron with respect to every other appears
therefore with a positive sign. Consider electron




electron 2. The force of repulsion is



where



first of all with respect to
2 is the chord connecting



the two electrons. The potential of electron i with respect to 2 is therefore ^ 2 /
The potential of electron i with respect to electron 3 is given by e^\r^ 3 and so on
up to the nth electron. The total potential energy due to the (n - i) electrons
acting upon electron i is thus given by a sum of (n - i) terms :

* 2 /'l, 2 + * 2 /'l> 3 + ''/'I. 4 + + ' 2 /'l>

Now from the figure it is seen that

r lt 2 = 2a sin ir\n where 2ir/w is the angle between successive electrons.
r lt 8 = 2a sin 2w/

r j, 4 ~ aa sin 3*> ,,

fj, n = 2a sin (n- i) ir\n ,, ,, ,,

Hence the sum of the potential energy terms of electron i with respect to all the
others is



[cosec IT In + cosec 2ir/n + cosec



. . . + cosec (n -



For electron 2 we can write down an analogous set of terms and so on up to the
nth electron. The sum of all such terms is therefore n times the series given. That
is, for all electrons mutually acting upon one another we have



- [cosec IT/M + cosec zirjn + cosec



. . . + cosec (n - i)v/n].



104 A SYSTEM OF PHYSICAL CHEMISTRY

For the radial force exerted on an electron by the nucleus and the other
electrons we get



Denoting the kinetic energy of an electron by T and neglecting the
electro-magnetic forces due to the motion of the electrons, we get on
equating the centrifugal force on an electron to the radial force



or T = (E - es n ).

From this we get for the frequency of rotation

e(E - ^

The total amount of energy W which must be given to the system
in order to remove the electrons to infinite distances apart from the
nucleus and from each other is

W = - P - T = (E - es n ) - (E - es H ) = (E - es n ) = T . (6)



But it is evident at once that in adding up all the terms which go to make up the
above expression we have counted each potential term twice over. Thus, the
potential between i and 3 appears when we are considering electron i by the term
* 2 / r if 3- Similarly when we consider electron 3 with respect to the others we have
amongst other terms the term * 2 / y 3 i> which is simply the potential a / r i 3 over a g a i n *
Hence to get the true potential of all the electrons with respect to one another we
have to divide the above expression by two. That is we get

[cosec irfn + cosec 2ir/ + cosec 3?r/w + . . . + cosec (n - i )*/].

Now the total potential P of the system is made up of terms due to the nucleus
acting upon the electrons and the electrons acting upon on 2 another.
That is

P = : H [cosec ir\n + cosec 2ir/ -f . . . + cosec (n - i)/],

s = n - i
If we now write the bracket in the form J cosec sir In



and further write the symbol s n for 2, cosec sir/n

s = i
it follows that we can write

nEe ne*
P = + sn

a a

\

or P = (E - )

a

which is the expression given by Bohr.



RUTHERFORD-BOHR ATOM-MODEL 105

That is, the energy W is equal to the kinetic energy taken for a whole
revolution of all the electrons when rotating in the stationary orbit.
This is the same relation as holds for the case of a single electron
rotating in an orbit. Notice also that the total potential energy P of
the system is just double the kinetic energy of the electrons.

We see that the only difference in the above formula (6) and that
holding for the motion of a single electron in a circular orbit round a
nucleus is the exchange of E for (E - es n ). We are therefore led to
suppose that the kinetic energy per electron rotating in a stationary

state is again rA-. The permanent configuration, that is, the configura-
tion in the formation of which the maximum amount of energy is
emitted, is the one for which r = i. The angular momentum of each
single electron in this state is again hJ2ir. Also, we can express angular
momentum in all cases as mva where m is the mass of the rotating
particle travelling with a velocity v round a circle of radius a. That is

i h . .

v = (7)

a 2irm

There may be many stationary states which a single ring of electrons may
assume. This seems necessary to account for the line spectra of sub-
stances containing more than one electron per atom. Further there
may be stationary configurations of a system of n electrons and a
nucleus of charge E in which all the electrons are not arranged in a
single ring.

As regards the stability of a ring of electrons, two problems arise.
First as regards the stability for displacement of the electrons in the
plane of the ring, and secondly, as regards displacements perpendicular to
this plane. Nicholson has shown that the question of stability is very
different in the two cases. While the ring for the latter displacements
(displacements perpendicular to the ring) is in general stable, the ring
is in no case stable for displacements in the plane of the ring. This
conclusion is based upon classical electro-dynamics, and Bohr avoids
the difficulty by making use of the quantum idea. In fact he shows
that the stability of a ring of electrons rotating round a nucleus is
secured through the condition of the universal constancy of the angular
momentum hJ2ir of each electron together with the further condition
that the configuration of the electrons is that in the formation of which
from free electrons and nucleus the amount of energy, W, emitted
is a maximum ; in other words, the stable configuration possesses the
least energy. That this assumption leads to the condition for stability
may be shown as follows.

Consider a ring of electrons rotating round a nucleus and let us
assume that the system is in dynamical equilibrium, the radius of the
ring being a > the velocity of the electrons # , their total kinetic energy
T , and the total potential energy of the entire system P . We have
already seen from equation (6) that P =* - 2T . Next' consider a



106 A SYSTEM OF PHYSICAL CHEMISTRY



configuration of the system in which the electrons, under the influence
of extraneous forces, rotate with the same angular momentum round
the nucleus in a ring of radius a where a = aa Q , a being greater than

unity. In this case we have P = - P (as follows from the definition of

a

potential energy given by equation (5)) ; also on account of the con-
stancy of the angular momentum it follows that v - VQ (compare equa-

a

tion (7)), and T = ^T , for T involves the square of v and therefore
the square of a. It follows then that

P + T - P + I 2 T - (P + TO) + T

We see that the total energy (P + T) of the new configuration is
greater than the (P + T ) of the original configuration. The system
is therefore stable for the displacement considered, that is it will tend
to revert to the first configuration, for the first configuration corresponds
to a maximum value of W as defined previously.

The condition of stability which we have just been considering refers
to possible displacements of the electrons in the plane of the ring. It
is necessary to consider the problem of stability in more detail howevef
before we take up the problem of actual atoms containing several
electrons.

Configuration and Stability in Systems Possessing one Nucleus.

Let us consider an electron of charge e and mass m moving in a
circular orbit of radius a with a velocity v, which is small compared
with the speed of light. Let us denote the radial force acting on the

e 2

electrons by -^ F ; F will be dependent in general upon a. The con-
dition of dynamical equilibrium gives



Introducing the condition of universal constancy of the angular
momentum of the electron we have

mva ti/27r.
From these relations we get

v =
and for the frequency of revolution



RUTHERFORD-BOHR ATOM-MODEL 107

[f F is known, the dimensions and frequency of the corresponding
orbit are determined by these equations.

We have seen already (equation (50)) that the radical force F is
given by

F = - 1(E - )

where F is the force acting upon a single electron in a system contain-
ing n electrons, E being the charge on the nucleus. Now E = N<?,
where N denotes the number of unit l charges carried by the nucleus.
It follows therefore that



i



But we have just seen that

*-**-

Hence F = (N - j n ).

Further, we have seen that the energy term, W, is given by

W = - P/2 = (E - ) = ^(N* - .)- ^(N - S n )

ne*

= Fo.

2d

W refers of course to all the electrons in the system. Further

a =
Hence W =

If we now consider the simplest type of atom, viz. the hydrogen atom,
the energy emitted, W , in bringing the single electron from infinity up