Physical Chemistry , 1st ed.

then used statistics to derive an expression for the energy density distribution

of blackbody radiation. The modern form of the equation that Planck pro-

posed is

d

8 

5

hc



ehc/^ k

1

T 1 d^ (9.22)

where is the wavelength of the light,cis the speed of light,kis Boltzmann’s

constant, and Tis the absolute temperature. The variable hrepresents a con-

stant, which has units of Js (joules times seconds) and is known as Planck’s

constant. Its value is about 6.626 

10 ^34 Js. Equation 9.22 is referred to as

Planck’s radiation distribution law,and it is the central part of Planck’s quan-

tum theoryof blackbody radiation.

An alternate form of Planck’s equation is given not in terms of the energy

density but in terms of the infinitesimal powerper unit area, or the power flux

(also known as emittance,which is related to the intensity). Recall that power

is defined as energy per unit time. In terms of the infinitesimal power per unit

area demitted over some wavelength interval d , Planck’s law is written as

follows (we omit the derivation):

d

2 

h

5

c^2



ehc/^ k

1

T 1 d^ (9.23)

Plots of equation 9.23 are shown in Figure 9.15. Note that they are the same

as the plots of blackbody radiation, but understand that Planck’s equation pre-

dicts the intensity of blackbody radiation at allwavelengths and alltempera-

tures. Thus, by predicting the intensities of blackbody radiation, Planck’s quan-

tum theory correctly models a phenomenon that classical science could not.

Planck’s equation can also be integrated from 0 to in a straightfor-

ward manner to obtain



1

2

5



c

5

2

k

h

4

 (^3) T
(^4) (9.24)
258 CHAPTER 9 Pre-Quantum Mechanics
Figure 9.15 A plot of the intensity of radiation versus wavelength at different temperatures
for a blackbody, assuming Planck’s radiation law is correct. Predictions based on Planck’s law
agree with experimental measurements, suggesting a correct theoretical basis—no matter what
its implications are.
3000 K
2000 K
4000 K
5  105
0
0
Wavelength (m)
Intensity (arbitrary units)
4  105
3  105
2  105
1  105
100 200 300 400 500 600