A system of physical chemistry - Index of

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A system of physical chemistry - Index of

114

A SYSTEM OF PHYSICAL CHEMISTRY

and W = total energy necessary to remove all the charged particles to

infinite distances from one another = ~.—— , . . (lo)

For the system in question we have—

where 5,, has its former significance. It will be observed that Yq is a

function of the radius of the ring.

As regards the stability of the ring of electrons with respect to displacements

perpendicular to the plane of the ring, Bohr shows that the

system will not be stable unless when N = i, « = 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" displacements

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 stabihty, 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 displacement

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 unequal

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

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