A system of physical chemistry - Index of

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A system of physical chemistry - Index of

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^. We are therefore led to

suppose that the kinetic energy per electron rotating in a stationary

state is again rh-. 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 t = i. The angular momentum of each

single electron in this state is again ^/27r. 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—

\ h .

f

V = (7)

a 2iTm

There may be many stationary states which a single ring of electrons may

assume. This seems necessary to account for the line spectra of substances

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 elecirons 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

and Bohr avoids

conclusion is based upon classical electro- dynamics,

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 h\2v 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 (Zq, the velocity of the electrons 5^0, their total kinetic energy

To, and the total potential energy of the entire system Pq. We have

already seen from equation (6) that Pq = - 2T0. Next consider a

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