Physical Chemistry Third Edition

642 14 Classical Mechanics and the Old Quantum Theory

T 25 2500 K

T 15 2000 K

0 1000 2000 3000

Infrared

region

T 35 3000 K

Rayleigh–Jeans

theory for

3000 K

Ultraviolet

region

Visible

region

λ/nm

Spectral radiant emittance

Figure 14.10 The Quantized Energies of an Oscillator as Postulated by Planck.The

horizontal line segments are plotted at the heights of the assumed energy values, 0,hν,

3 hν,4hν,5hν,6hν,7hν, etc.

walls, so that a conducting box can contain standing electromagnetic waves with nodes

at the walls. For a rectangular box, Rayleigh and Jeans counted the possible standing

waves of various wavelengths that could exist with nodes at the walls and computed

the average energy of each standing wave as a function of temperature using standard

methods of classical physics. Their result was

η(λ)dλ

2 πckBT

λ^4

dλ (14.4-2)

wherecis the speed of light,Tis the absolute temperature, andkBis Boltzmann’s

constant:

kB

R

NAv

 1. 38066 × 10 −^23 JK−^1

whereRis the ideal gas constant andNAvis Avogadro’s constant.
John William Strutt, 3rd Baron Rayleigh,
1842–1919, was the 1904 Nobel Prize
winner in physics, and Sir James Jeans,
1877–1946, was a British astronomer
and physicist.

Although the formula of Rayleigh and Jeans agrees with experiment for large values

of the wavelength, it predicts that the spectral radiant emittance becomes large without

bound in the limit of short wavelength, which contradicts experimental results. This

result is due to the fact that more and more standing waves can fit into a box if the

wavelength is made shorter and shorter. The failure of the Rayleigh–Jeans theory to

agree with experiment was called the “ultraviolet catastrophe.”