Physical Chemistry Third Edition

14.4 The Old Quantum Theory 643

In 1900 Max Planck devised a new theory of black-body radiation that agreed with

experimental results. This theory is the first part of what we call the “old quantum

theory.” The following statements are a simplified version of the assumptions that led

to his result:^3

Max Planck, 1858–1947, received the
Nobel Prize in physics in 1918 for this
theory, although at first most physicists
were reluctant to believe that it was
correct.

1. The walls of the box contain oscillating electric charges. Each oscillator has a char-
   acteristic fixed frequency of oscillation, but many oscillators are present and many
   frequencies are represented.


2. The standing waves in the box are equilibrated with the oscillators so that the average
   energy of standing waves of a given frequency equals the average energy of the wall
   oscillators of the same frequency.


3. The energy of an oscillator isquantized. That is, it is capable of assuming only one
   of the values from the set:

E0,hν,2hν,3hν,4hν,...,nhν,... (14.4-3)

whereνis the frequency of the oscillator and wherehis a new constant, now known

asPlanck’s constant. The quantityn, which can take on any non-negative integral

value, is called aquantum number. Figure 14.11 schematically shows this energy

quantization. Quantization has been compared to standing on a ladder. A person can

stand on any rung of a ladder, but nowhere between the rungs. The energy can take

on any of the values in Eq. (14.4-3), but cannot take on any value between these

values.

6

4

2

0

Energy / (

hν

)

Figure 14.11 The Quantized Ener-

gies of an Oscillator as Postulated

by Planck.The horizontal line segments

are plotted at the heights of the assumed

energy values, 0,hν,3hν,4hν,5hν,6hν,

7 hν, etc.

4. The probability of a state of energyEis given by theBoltzmann probability distri-
   bution:

(probability of a state of energyE)∝e−E/kBT (14.4-4)

where the symbol∝stands for “is proportional to,” whereTis the absolute temper-

ature, and wherekBis Boltzmann’s constant. We discuss the Boltzmann probability

distribution in Parts 2 and 4 of this textbook, but simply state it as an assumption of

Planck at this point.

Planck’s theory removed the ultraviolet catastrophe because short wavelengths cor-

respond to large frequencies and large energy spacings. The excited states therefore

have very small populations. The result of Planck’s theory is that

η(λ)dλ

2 πhc^2

λ^5 (ehc/λkBT−1)

dλ (14.4-5)

Planck was able to achieve agreement with experimental data by choosing a value of

happroximately equal to the presently accepted value, 6.62608× 10 −^34 J s. Planck’s

formula agrees with the Stefan–Boltzmann law and withWien’s law, which states that

the wavelength at the maximum in the spectral radiant emittance curve is inversely

proportional to the absolute temperature.

(^3) M. Jammer,The Conceptual Development of Quantum Mechanics, McGraw-Hill, New York, 1966,
p. 10ff.