BioPHYSICAL chemistry

A blackbody can be constructed, allowing the radiation output to

be measured experimentally. The amount of light emitted at any given

temperature is characterized by the energy density, ρ, which is a measure

of how much energy is emitted within a specific wavelength region λto

λ+δλ, where δλis a small number. The energy density of a blackbody

has a characteristic dependence upon both wavelength and temperature

(Figure 9.2). At small values of wavelength, the amount of energy emitted

is low. As the wavelength is increased, the amount of energy emitted

increases until a peak value is reached. Increasing the wavelength further

results in smaller amounts of energy being emitted.

The dependence of ρon the wavelength was determined for many

materials and was found to always follow the same general dependence.

The dependence of the energy distribution can be modeled using statistical

arguments for classical thermodynamics. Assuming that the radiated energy

follows the classical dependence on the amplitude and is independent of

the wavelength (eqn 9.8), the long-wavelength part of the dependence

is predicted but fails for short wavelengths. Instead of predicting that

the density will drop to zero, the theory predicts the so-called ultraviolet

catastrophe: that the energy density will become infinite:

(9.9)

In 1900, Max Planck realized that he could derive the observed distribution

if he made an unusual change to classical theory. Light is emitted from the

ρ

π

λ

=

8

4

kT

178 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Container at a

temperature T

(a)

Pinhole

Detected radiation

Classical

prediction

(b)

Energy density,

ρ

Wavelength, λ

Experimental

data

Figure 9.2Blackbody radiation. (a) In the sphere light undergoes a series of reflections causing
the light radiated through the pinhole to be uniform, with a spectrum characteristic of the
temperature of the sphere. (b) The wavelength dependence for the energy density at both a high
temperature and a low temperature. Also shown is the classical prediction for the dependence
(dashed line).