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