bei48482_FM

Atomic Structure 125

holding the electron in an orbit rfrom the nucleus is provided by the electric force

Fe

between them. The condition for a dynamically stable orbit is

FcFe

 (4.3)

The electron velocity is therefore related to its orbit radius rby the formula

 (4.4)

The total energy Eof the electron in a hydrogen atom is the sum of its kinetic and

potential energies, which are

KE m^2 PE

(The minus sign follows from the choice of PE 0 atr , that is, when the

electron and proton are infinitely far apart.) Hence

EKEPE

Substituting for from Eq. (4.4) gives

E

E (4.5)

The total energy of the electron is negative. This holds for every atomic electron and

reflects the fact that it is bound to the nucleus. If Ewere greater than zero, an electron

would not follow a closed orbit around the nucleus.

Actually, of course, the energy Eis not a property of the electron alone but is a prop-

erty of the system of electron nucleus. The effect of the sharing of Ebetween the

electron and the nucleus is considered in Sec. 4.7.

Example 4.1

Experiments indicate that 13.6 eV is required to separate a hydrogen atom into a proton and an

electron; that is, its total energy is E13.6 eV. Find the orbital radius and velocity of the

electron in a hydrogen atom.

e^2



8  0 r

Total energy of

hydrogen atom

e^2



4  0 r

e^2



8  0 r

e^2



4  0 r

m^2



2

e^2



4  0 r

1



2

e



 4  0 mr

Electron velocity

e^2



r^2

1



4  0

m^2



r

e^2



r^2

1



4  0

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