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# A system of physical chemistry (Volume 3)

. (page 12 of 22)

to the most stable position is given by

W = 2Tr 2 e*m/tf

(compare equation (3), remembering that the most stable or permanent
condition means that T = i).

Hence for a system containing n electrons with nuclear charge N^
we can write

W = W F 2 .

Also, for the simplest atom in its permanent state we have seen already
(equation (3)) that

a = /& 2 /47T 2 ^ 2 .

Hence for a system containing n electrons, the charge on the nucleus
being N<?, we can write

a = a /F ; and w = o> F 2 .

1 The charge on a single electron being here regarded as the " unit " of (nega-
tive) electricity.

io8

A SYSTEM OF PHYSICAL CHEMISTRY

Using the numerical values already employed for the hydrogen atom we
see that

W = 2'Q x io

a o = o*55
o) = 6-2 x

~ n

erg.

io~ 8 cm.
io 18 revolutions per second,

and therefore, for the system containing n electrons and charge N* on
the nucleus

iV = 2-0 x io- 11 x F 2 . . (i)

= 0-55 x io- 8 /F .... (ii)

= 6-2 x io 15 F 2

(in)

In order to use these expressions in actual calculation it is necessary to
know F , i.e. (N - s n ). Bohr gives the following table of values for s n
for various values of n up to n 16. For n = i, s n is of course zero :

n.

Minimum Value

n.

s

Minimum Value

of N.l

of N.

I

O

9

3-328

13-14

2

0-25

0-25

10

3-863

18-13

3

0*577

0-58

ii

4-416

23-60

4

o-957

1-41

12

4-984

30-8

5

1*377

2*43

13

5-565

38-57

6

1-828

4-25

14

6-159

48-38

I

-2-305
2-805

6-35

15

16

6-764
7*379

71-65

The numbers of electrons represented by n refer to the number of
electrons in a single ring. This is in general a smaller number than
that represented by N, for in a neutral atom N is necessarily equal to the
total number of electrons present, some of which are present in the
innermost ring, others in outer rings. We see from the table that the
number of electrons which can rotate in a single ring round a nucleus
of charge Ne increases only very slowly with increase in N ; for N = 20,
the maximum value of n is io; for N = 40, n = 13; for N = 60,
n = 15. We see further that a single ring of n electrons cannot rotate
in a single ring round a nucleus of charge ne unless n is less than 8.
That is, if we consider an atom which has but one ring of electrons
(and therefore n = N), the maximum number of electrons in this ring is
7. If more electrons be present at least two rings must be formed.

Hydrogen.

We have already considered the values of W, a, and <> (i.e. W , a ot
and w ) which satisfy the condition of stability for the electrically neutral
atom of hydrogen. It is of interest, however, to consider the case of a

1 Of course N cannot be fractional. The nearest whole number greater than
the numbers quoted will give the smallest value for N.

RUTHERFORD-BOHR ATOM-MODEL 109

negatively charged hydrogen atom, that is, one which has picked up an
extra electron. In this case n = 2 and N = i, for N* = the nuclear
charge = e in the case of the hydrogen nucleus. On substituting n = 2
and the corresponding value of s n from the above table in the expres-
sions (i), (ii), and (iii), we find that this system which can be repre-
sented by the symbol 1, (2), where (2) refers to the total number of
electrons in the atmosphere and 1 refers to N, the number of unit
charges on the positive nucleus will give rise to a permanent state
conditioned by the following values of a, w, and W :

a = i'33 a o\ <* = 0-563 <D;; W = 1-13 W .

Since W is here greater than it is when the atom is neutral, it follows
that the hydrogen atom is actually capable of taking on a negative
charge, for in doing so, the term W is increased from W to 1-13 W ,
and one of the conditions for stability is, as we have seen, a maximum
value for W. The fact that the hydrogen atom can take on a negative
charge probably accounts for the fact that in the periodic table we
usually place hydrogen with the halogens partly because hydrogen can
replace chlorine in organic compounds.

Bohr has also calculated the value of W which would result if the
hydrogen atom acquired. two extra electrons, thus becoming (1, (3)).
It is found that W now becomes equal to 0-54 W . This value of
W is now less than W , its value for the neutral atom, and consequently
still less than for the atom carrying one extra electron. Hence such a
doubly charged hydrogen atom is unstable and will not exist. Experi-
ments on positive rays show that the hydrogen atom can acquire a single
negative charge, but it is doubtful if it can acquire a double negative
charge.

Helium.

In this case N = 2, and the neutral atom has therefore two elec-
trons. We have already considered Bohr's theory in relation to the
spectrum of helium. We have now to take up the process whereby first
one, then a second, and finally a third electron can be added to the
doubly charged helium nucleus. First, for the formation of the system,
(2, (i)), by bringing an electron from infinity up to the nucleus, since
n = i and therefore s n = o, and further since N = 2, the permanent
state is given by

a = 0-5 a ; w = 4<o ; W = 4W .

If we now bring the second electron up to the (2, (i)) system we find
for the permanent state since s n = 0-25 for n equal to 2

a = 0-571 a \ o> = 3'o6w ; W = 6'i3W .

Further, since

(W a>(2) - W 2)(I) ) = 2-i 3 W

no A SYSTEM OF PHYSICAL CHEMISTRY

it follows that each electron in a helium atom is more firmly held than
the single electron in the hydrogen atom. It follows also that the
ionisation potential of helium should be correspondingly high, as we
would expect from the chemical inertness of helium. Bohr regards the
two electrons of helium as being in the same ring.

If we now consider the process of adding an extra electron to the
neutral helium atom so as to produce the configuration (2, (3)), it
follows that the permanent state of the resulting system (since
.\$ = o 577 when n = 3) would be given by

a = 0*703^0; o) = 2'O2w ; W = 6*o7\V .

Since for this configuration W is less than it was for the neutral atom,
the theory indicates that a helium atom cannot acquire a negative
charge. This is in agreement with experiment, for Franck (Ber. Detitsch.
phys. Ges., 12, 613 (1910)) has shown that helium atoms have no
affinity for electrons.

Lithium.

In this case N = 3, the nuclear charge being 3*. If one electron
be brought up so as to form the system (3, (i)), i.e. an atom of lithium
carrying effectively two positive charges, we get for the permanent state
of the configuration (3, (i))

a = 0-33^0 ; u> = 9<o ; W = 9\V .

If a second electron be brought up and enters the ring containing the
first, the. stable state for the resulting system (3, (2)) is

a = 0-364^; G> = 7-560)0 ; W = i

Since W 3i ( X ) = 9W , and since (W 3t (2 > W 3i ( X )) = 6*i3\V , it follows
that the first two electrons of the lithium atom are bound much more
firmly than the electron of the hydrogen atom.

If we now consider the process of adding a third electron to the
lithium nucleus so as to form a neutral atom, chemical considerations,
i.e. considerations of valency, would suggest that the third electron
should lie on a ring outside the ring containing the first two electrons,
both rings being concentric. The single electron in the outer ring
would be the valency electron of the lithium atom. The arrangement
is represented by the symbol (3, (2) (i)), the large numeral indicating
the nuclear charge. Proceeding as before, Bohr calculates that the
stable state would correspond to the following :

inner ring : a^ = o'\$62a ; wj = 7'65<o

outer ring : a% = i'i&2a ; <o 2 o'7i6o> ; W = i6'02W .

Since (W 3i(2 ) (l) - W 3f (2 )) = o-&gW , we see that the outer electron of
the neutral atom is less firmly held than the electron of the hydrogen
atom.

For a negatively charged lithium atom, i.e. the configuration (3, (2)

RUTHERFORD-BOHR ATOM-MODEL in

(2)) we would have W = i6'i6W . This large value of W would
suggest that negatively charged lithium might under certain circumstances
manifest itself.

Bohr also considers beryllium, but enough has been said in connec-
tion with light atoms, in which the number of electrons is small, to
indicate the line of reasoning.

We have now to consider the question of series of rings present in
neutral atoms which contain a considerable number of electrons.

These rings are regarded as co-axial, lying in the same plane.
Bohr calculates that only in the case of systems containing a very large
number will there be any tendency for the rings to separate into different
planes. It is unnecessary to discuss this case. Considering systems
of rings in the same plane Bohr is led to important conclusions regard-
ing the maximum number of electrons which can rotate in a single ring.
We have dealt with this point earlier. It must be pointed out, however,
that the value for n simply gives us the maximum number of electrons
which could exist in a stable single ring ; it does not follow that this
number is reached in actual atoms. The formation of a second or of a
third ring is possible, even when the nuclear charge does not exceed Se,
i.e. when the total number of electrons does not exceed 8. The possi-
bility of this becomes clear when we take into account the question of
valency, for valency will be due essentially to the external ring. Bohr
further shows that in the stable system the number of electrons in any
ring diminishes as we proceed outwards from the nucleus. In dealing
with actual atoms Bohr makes use of the now generally accepted relation
(which will be discussed later), that the atomic number gives us directly
the number of electrons in the atom. The atomic number of an
element represents the position of the element in the whole series of
elements, arranged in order of increasing atomic weight. It has been
found that the atomic number, and consequently the number of
electrons present in an atom is approximately one half of the atomic
weight, i.e. helium 2, lithium 3, beryllium 4, oxygen 8, and so on.

It has already been stated that considerations of valency are a guide
to the number of electrons in the outermost ring. For monovalent
atoms the number is unity and so on for other atoms. A different con-
sideration also enters into the problem, viz. the possibility of confluence
of rings, especially when a considerable number of electrons are present.

Two rings may run together to form a single ring, and Bohr shows
that this will occur most easily when the two rings contain the same
number of electrons. Considering the binding of successive electrons
by a positive nucleus we conclude from this that, unless the charge on
the nucleus be very great, rings of electrons will join together if they
contain the same number of electrons and rotate in the same direction,
and that accordingly the number of electrons in inner rings will be the
numbers, 2, 4, 8, 16, etc. This assumption in regard to the number of
electrons in the rings is strongly supported by the fact that the chemical
properties of the elements of low atomic weight vary with a period of
eight. Further, it follows that the number of electrons on the outermost

112

A SYSTEM OF PHYSICAL CHEMISTRY

ring will always be' odd or even according as the total number of
electrons in the atom is odd or even. This suggests a relation with the
fact that the valency of an element of low atomic weight is always odd
or even according as the number of the element in the periodic series
is odd or even.

The fact that several considerations have been introduced into the
problem shows that the criterion of stability constancy of angular
momentum and a maximum value for W is not sufficient to determine
completely the constitution of an atomic system. In very simple cases
the constitution of the atom is in relatively little doubt. It is possible,
however, by making use of all the foregoing considerations to arrive at
a moderately satisfactory statement of the constitution of atoms which
contain up to twenty-four electrons, that is, up to an atomic weight
of forty-eight approximately. Bohr gives the following table for these
atomic systems. The large numeral indicates the value of N, the total
number of electrons in the neutral atom, whilst the smaller figures in
brackets give the composition of the rings starting from the innermost.

Thus the symbol 9 (4, 4, i) denotes a neutral atom which contains in
all nine electrons, arranged in three concentric rings, the innermost ring
containing four electrons, the second likewise four, and the outermost one
electron, thereby indicating that the normal valency of this atom is one.
The atomic weight of the element in question is approximately 2x9
= 1 8, i.e. fluorine. It will be observed that Bohr places hydrogen
along with the halogens, and not with the alkali metals. Bohr does not
give the name of the corresponding element to these configurations but
the attempt is here made to do so.

TABLE SHOWING STRUCTURE OF ATOMS.

Probable

Element.

Structure of
Atom.

Probable
Element.

Structure of
Atom.

Probable
Element.

Structure of
Atom.

Hydrogen

MI)

Fluorine .

9 (4, 4 J )

Chlorine .

17 (8, 4, 4, i)

Helium .

2(2)

Neon

10 (8, 2)

Argon

18 (8, 8, 2)

Lithium .

3 (2, i)

Sodium

II (8, 2, I)

Potassium

19 (8, 8, 2, i)

Beryllium

4 (2, 2)

Magnesium

12 (8, 2, 2)

Calcium .

20 (8, 8, 2, 2)

Boron

S (2, 3)

Aluminium

13 (8. 2, 3)

21 (8, 8, 2, 3)

Carbon .

6 (2, 4)

Silicon

14 (8, 2, 4)

22 (8, 8, 2, 4)

Nitrogen

7 (4, 3)

Phosphorus

15 (8 4, 3)

23 (8, 8, 4 , 3)

Oxygen .

8 (4, 2, 2)

Sulphur

16 (8, 4, 2, 2)

"""

24 (8, 8, 4, 2, 2)

Elements in the same horizontal line belong to the same group,
the vertical columns indicate the first three periods of the elements.

As regards elements of higher atomic weight, Bohr points out that
at the end of the third period of eight elements we meet with the iron
group, which takes a particular position in the system of elements since
it is the first time that elements of neighbouring atomic weights show
similar chemical properties. This circumstance indicates that the con-
figurations of the electrons in the elements of this group differ only in

RUTHERFORD-BOHR ATOM-MODEL 113

the arrangement of the inner electrons. The fact that the period in the
chemical properties of the elements after the iron group is no longer
eight but eighteen suggests that elements of higher atomic weight con-
tain a recurrent configuration of eighteen electrons in the innermost
rings. The deviation from 2, 4, 8, 16 may be due to a gradual inter-
change of electrons between the rings as already indicated when speak-
ing of the confluence of rings. Since a single ring of eighteen electrons
will not be stable, the electrons may be arranged in two parallel rings.
Such a configuration of the inner electrons will act upon the outer
electrons in very nearly the same way as a nucleus of (N 18) x e.
It is conceivable that with increase in N another configuration of the
same type will be found outside the first, such as is suggested by the
presence of a second period of eighteen elements.

Systems Containing Several Nuclei. Molecules.

According to Bohr a simple molecule such as the molecule of hydro-
gen consists of two positively charged nuclei, each carrying unit charge
with a ring of two electrons rotating between them, the rotation being
round the line joining the two nuclei. If the two nuclei are similar in
all respects the electron ring rotates at a position midway between the
nuclei, the plane of the ring being perpendicular to the line joining the
nuclei. If the nuclei be dissimilar the ring will rotate at some position
lying nearer the heavier and more complex nucleus.

The equilibrium of such a system is conditioned as before by the
constancy of the angular momentum of each electron, along with the
further condition that W the energy emitted in binding an electron to-
the system is a maximum.

Consider a system consisting of two positive nuclei of equal charges,
and a ring of electrons rotating round the line joining the nuclei. Let
the number of electrons be n, the charge on each be e, and the charge
on each nucleus be N*. It can be shown that the system will be in<
equilibrium if (i) the nuclei are equidistant from the ring and (2) if the
ratio between the diameter of the ring, 2a, and the distance apart of the
nuclei, 2^, be given by

It is assumed that the frequency of revolution to is a magnitude such
that for each electron the centrifugal force balances the radial force (due
to the attraction of the nuclei and the repulsion of the electrons).

g3

Denoting this radial force by -^F we get from the condition of the

universal constancy of the angular momentum of the electrons as already
shown

(9)

VOL. III. 8

ii4 A SYSTEM OF PHYSICAL CHEMISTRY

and VV = total energy necessary to remove all the charged particles to
infinite distances from one another = 2n7r * 2 . . . (10)
For the system in question we have

*-* - . (II)

where s n has its former significance. It will be observed that F is a
function of the radius of the ring.

As regards the stability of the ring of electrons with respect to dis-
placements perpendicular to the plane of the ring, Bohr shows that the
system will not be stable unless when N = i, n = 2 or 3. Throughout
the entire treatment of the problem it will be observed that Bohr regards
the question of stability with respect to such "perpendicular" displace-
ments as being capable of being dealt with by classical electrodynamics.
In the case of displacements of electrons in the plane of the ring classical
electrodynamics is incapable of indicating any position of stability, and
Bohr finds the condition by introducing the quantum theory in the form
of the constant angular momentum (/#/27r) possessed by each electron in
whatever orbit it may be rotating.

It is assumed that the motions of the nuclei with respect to one
another are so slow that the state of motion of the electrons at any
moment will not differ sensibly from that calculated on the assumption
that the nuclei are at rest. This assumption is permissible on account
of the great mass of the nuclei compared with that of the electrons,
which means that the vibrations resulting from a displacement of the
nuclei are very slow compared with those resulting from a displacement
of the electrons.

Let us now imagine that, by the help of extraneous forces acting on
the nuclei, we slowly alter the distance between them. During this dis-
placement the radius of the ring of electrons will alter in consequence
of the alteration in the radial force due to the attraction of the nuclei
for the electrons. During this alteration we suppose that the angular
momentum of the electrons remains constant. If the distance apart of
the nuclei increases, the radius of the electron ring will likewise increase.
It can be shown, however, that the radius of the ring will increase at a
slower rate than does the distance between the nuclei. On account of
this difference in the rate the attraction on one of the nuclei due to the
ring will be greater than the repulsion from the other nucleus. The
work done during the displacement by the extraneous forces acting upon
the nuclei will therefore be positive, and the system will be stable for
this displacement. The same result will also hold in the case in which
the distance between the nuclei diminishes.

For a system consisting of a ring of electrons and two nuclei of un-
equal charge the investigation becomes more complicated. During a
variation of the distance of the nuclei apart not only will the radius of
the electron ring vary, but also the ratio in which the plane of the ring

RUTHERFORD-BOHR ATOM-MODEL 115

divides the line connecting the nuclei. If we consider a neutral system,
containing two nuclei carrying large charges, in the stable configuration
the greater part of the electrons must be arranged round each nucleus,
approximately, as if the other nucleus were absent ; and only a few of
the outer electrons will be arranged differently, rotating in a ring round
the line connecting the nuclei. This ring which keeps the system
together represents the chemical bond.

Systems Containing few Electrons ; the Hydrogen Molecule.

As already stated, the neutral hydrogen molecule consists of two
similar nuclei each carrying unit charge e with two electrons rotating in
a ring between the nuclei. Denoting the radius of the ring by a, and
the distance apart of each nucleus from the plane of the ring by ^, we
get from equation (8), on putting N = i, and n = 2

From (n) we get further

F = 3 = 1-049.

From (9) and (io) we get denoting the values of a, w, and W for a
system consisting of a single electron and single nucleus (i.e. the
hydrogen atom) by a , o> , and W the following values for the hydro-
gen molecule :

a = o'95<z ; &> = I'lowo ', W = 2'2oW .

Since W is greater than 2\Y , it follows that two hydrogen atoms
combine to form a molecule with emission of energy. That is a
hydrogen molecule dissociates with absorption of energy. This is in
agreement with experiment. Putting W = 2'o x io~ n erg, and
NO = 6-2 x io 23 , where N is the number of molecules in one gram-
molecule, we get for the energy emitted during the formation of one
gram-molecule of hydrogen from two gram-atoms, the quantity
N (W - 2 W^) = 2-5 x io 12 erg, or 60,000 cals. This value is of the.
correct order of magnitude, the value observed by Langmuir being
approximately 80,000 cals.

In considering displacements of electrons perpendicular to the plane
of the ring, we have seen that the molecule will become unstable unless
when N = i, 3 or 3, Hence if we remove one of the two elec-
trons from the hydrogen molecule the whole system will become un-
stable and will break up into a single nucleus positively charged and
a neutral hydrogen atom. This process corresponds to the change
from a stationary state represented by the complete neutral molecule
and a second stationary state represented by the dissociated molecule.
On Bohr's theory when such a change takes place from one stationary
to another there is involved homogeneous radiation of frequency v, the

n6 A SYSTEM OF PHYSICAL CHEMISTRY

energy per molecule " decomposed " being hv. In this case hv repre-
sents energy absorbed given by the relation

h v = W - W = i-2oW .
Hence v = 37 x io 15 .

The value for the frequency in the farthest ultra violet calculated from
dispersional measurements in the case of hydrogen is 3*5 x io 15 , so
that the agreement is good. It may be pointed out that the " chemi-
cal" process involved in the above is H2->H + H+ + It is not
clear why this process should be identified with the dispersional
frequency.

The above calculation refers to the quantum involved in adding or
removing an electron. We have now to consider the frequency and
consequent magnitude of the quantum involved if one nucleus is removed
from the other. Bohr shows that this frequency is given by l

m

where m is the mass of an electron and M the mass of the nucleus.
Putting M/m = 1835 and o> = 6*2 x io 15 , we get

v r =1-91 x io u ,

or \r = i -5 7/*, t.e. the short wave-length region of the infra-red spec-
trum. This frequency is of the same order of magnitude as that cal-
culated for the nuclear or atomic vibration of hydrogen on the basis of
molecular heat (cf. Chap. IV.). On the other hand, no absorption of
radiation in hydrogen gas corresponding to this frequency is observed.
According to Bohr this is to be expected on account of the symmetri-
cal structure of the hydrogen molecule and the great ratio between the
frequencies corresponding to the displacements of the electrons and the
nuclei respectively.

A system of two nuclei, each of charge <?, and a ring of 3 electrons
rotating between them that is, a negatively charged hydrogen molecule
will be stable for displacements perpendicular to the ring, for the
system is stable if N i and n = 2 or 3. Bohr calculates that for
such a system

a = 1-14^0 ; w = 0770)0 ; W = 2'32W .

Since the W is greater in this case than it is in the case of the neutral
hydrogen molecule (W = 2'2oW ), the negatively charged molecule
should be capable of existence. Proof of the existence of negatively
charged hydrogen molecules has been obtained by Sir J. J. Thomson

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