Physical Chemistry , 1st ed.

that has units of action can be related to h, and Bohr did just that. He assumed

that the possible quantized values of the angular momentum were some mul-

tiples ofh:

Lmevr

2

n



h (9.33)

where his Planck’s constant and nis some integer (1, 2, 3,.. .) indicating that

the angular momentum is some integral multiple of Planck’s constant. The

value ofn0 is not allowed, because then the electron would have no mo-

mentum and wouldn’t be orbiting the nucleus. The 2in the denominator of

equation 9.33 accounts for the fact of a complete circle having 2radians, and

Bohr assumed that the orbits of the electron were circular.

Equation 9.33 can be rewritten as

v

2 

n

m

h

er



and we can substitute for velocity in equation 9.30, which is derived from

Bohr’s first assumption about forces. Performing this substitution and rear-

ranging the expression to solve for the radius r,we get

r



0

m

n^2

e

h

e^2

2

 (9.34)

where all variables are as defined above. It is easy to show that this expression

has units of length. Note that this equation implies that the radius of the or-

bit of an electron in the hydrogen atom will be a value determined by a col-

lection of constants: 0 ,h,,me,e, and the integer n. The only variable that

can change is n, but it is restricted by Bohr’s assumption 3 to be a positive in-

teger. Therefore the radius of the electron orbits in the hydrogen atom can only

have certain values, determined solely by n. The radius of the orbits of the elec-

tron is quantized. The integer denoted as nis termed a quantum number.A di-

agram of Bohr’s hydrogen atom having specific radii for the electron orbits is

shown in Figure 9.19.

Before leaving discussion of the radius, there are two other points to con-

sider. Note that the expression for rdepends on Planck’s constant h. If Planck

and others had not developed a quantum theory of light, the very concept of

hwould not exist, and Bohr would not have been able to rationalize his as-

sumptions. A quantum theory of light was a necessary precursor to a quantum

theory of matter—or at least, a theory of hydrogen. Second, the smallest value

ofrcorresponds to a value of 1 for the quantum number n. Substituting val-

ues for all of the other constants, whose values were known in Bohr’s time, one

finds that for n1:

r5.29 

10 ^11 m 0.529 Å

This distance ends up being an important yardstick for atomic distances and

is called the first Bohr radius. This meant, by the way, that the hydrogen atom

was about 1 Å in diameter. At that time, science (including theoretical work by

Einstein on Brownian motion) was just beginning to estimate the size of atoms.

This predicted radius fell exactly where it should be from experimental

considerations.

The total energy of a system is of paramount interest, and by using the ex-

pression for the quantized radius for the electron in a hydrogen atom, one can

264 CHAPTER 9 Pre-Quantum Mechanics