648 14 Classical Mechanics and the Old Quantum TheoryThe centripetal force is provided by the electrostatic attraction of the positive nucleus
for the negative electron. Coulomb’s law for the force between a chargeQ 1 and another
chargeQ 2 separated by a vacuum isF 12
Q 1 Q 2
4 πε 0 r 12(14.4-12)
whereε 0 is thepermittivityof the vacuum, equal to 8. 8545 × 10 −^12 C^2 N−^1 m−^2 , and
wherer 12 is the distance between the charges. If the two charges are of opposite sign,
the force is negative, corresponding to attraction. We equate the centripetal force and
the electrostatic force:mev^2
re^2
4 πε 0 r^2(14.4-13)
Coulomb’s law is named for Charles
Augustin de Coulomb, 1736–1806, the
French physicist who discovered
this law.
The angular momentum of the electron in a circular orbit is a vector denoted byL
and is given by Eq. (E-19) of Appendix E. It is quantized, according to assumption 2,
taking on the possible magnitudesLrmeνnh
2 π(14.4-14)
where thequantum numbernis a positive integer. Equation (14.4-14) is solved for the
speedνand the result is substituted into Eq. (14.4-13). The resulting equation is solved
for the radius of the orbit to giverh^24 πε 0
4 π^2 me^2n^2 a 0 n^2 (14.4-15)wherea 0 is equal to 5. 29198 × 10 −^11 m (52.9198 pm or 0.529198 Å) and is called
theBohr radius. This value corresponds to an infinitely massive nucleus. The radius
of the orbit is quantized and is proportional to the square of the quantum numbern.
Figure 14.13 depicts the first few Bohr orbits.a4 a 9 a 16 a
25 a
36 aFigure 14.13 The Quantized Bohr
Orbits.
Exercise 14.15
a.Obtain Eq. (14.4-15) from Eqs. (14.4-13) and (14.4-14).
b.Using the accepted values of the physical constants, verify the value of the Bohr radius.The potential energy that corresponds to the Coulomb force of Eq. (14.4-11) isV
Q 1 Q 2
4 πε 0 r 12(14.4-16)
The potential energy for an electron of charge−eat a distancerfrom a nucleus of
chargeeisV−
e^2
4 πε 0 r