A system of physical chemistry - Index of

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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

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