A system of physical chemistry - Index of

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 w, the major axis **of**

the orbit being 2a. 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 :—

J2 W'U , eE

w = -^^^ .

-F^-y= and 2a = . . . ^ (i) ^ ^

TT eE Jm W

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** /i.

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** 2a or the

greatest value **of** w 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 w

**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 tAi' 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 is process zero and at the end **of** the

process is w, the mean or average **of** the two being (0/2. Bohr gives

a more rigorous treatment **of** this point in the first paper referred to.

Since tAv 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 = t/iv = t/i- . . . . (2) ^ '

2