Assumption 1, regarding the relationship between forces, can be written as
FcentFcoul (9.26)where Fcentand Fcoulare the centripetal force and coulombic force, respectively.
Expressions for each of these quantities are known from classical mechanics,
and substituting them yields
m
rev^2
4 e
20 r^2(9.27)
where ris the radius of the circular orbit,eis the charge on the electron,me
is the mass of the electron,vis the velocity of the electron, and 0 is a physi-
cal constant called the permittivity of free space(and equals 8.854
10 ^12
C^2 /Jm). The total energy of a system is simply the sum of the kinetic energy
Kand the potential energy V:
EtotKV (9.28)The expressions for the kinetic energy of a moving electron,^12 mev^2 , and the po-
tential energy of two charged, attracting particles,e^2 /4 0 r, are also known,
giving
Etot1
2
mev^2
4 e
20 r(9.29)
If we rewrite Bohr’s equivalence of centripetal force and coulombic force,
equation 9.27, as
mev^2
4 e
20 r(9.30)
we can substitute for the kinetic energy term in equation 9.29 and combine the
two terms to get
Etot1
2
4 e
20 r(9.31)
Now Bohr’s assumption 3 can be applied. Classically, if an object of mass mis
traveling in a circular path with radius rabout a center, the magnitude* of the
angular momentum Lis
Lmvr (9.32)In the SI system of units, mass has units of kg, velocity has units of m/s, and
distance (the radius) has units of m. Angular momentum therefore has units
of kgm^2 /s. But also recognize that Js can be rewritten as
Js Nms
kg
s2m
ms kg
sm^2
That is, Planck’s constant has the same units as angular momentum! Or, re-
stated, angular momentum has units of action. As hinted earlier, any quantity
9.9 Bohr’s Theory of the Hydrogen Atom 263*Angular momentum is a vector whose formal definition includes the cross product of
the velocity vector,v, and the radius vector,r:
Lmr vEquation 9.32 relates the magnitude of the angular momentum only, and assumes that the
velocity vector is perpendicular to the radius vector.