BioPHYSICAL chemistry

material forming the blackbody as that material vibrates due to its thermal
energy. Classically, it was assumed that the vibrators could have any fre-
quency and any value of energy. Instead of using this assumption, Planck
proposed that the energy was proportional to the frequency, according to:

E=nhν where n=0,1,2,... and h=6.626 × 10 −^34 Js (9.10)

where his now known as Planck’s constant. Unlike classical mechanics,
which allows objects to have any energy, this equation predicts that light
can have energy only at certain discrete values, or, in other words, it is
quantized according to the frequency.
Using statistical arguments (eqn 8.17) but with the different dependence
for the energy, Planck derived a new dependence for the energy density
that agreed with the experimental data (Figure 9.2):

(9.11)

At long wavelengths, this equation agrees with the classical prediction
(eqn 9.8). At long wavelengths the exponential term is very small and
the exponential can be written approximately as:

ex− 1 ≈x when x<< 1 (9.12)

Using this approximation, the energy density can be written as:

(9.13)

At long wavelengths, the new equation gives the classical prediction but
at small wavelengths only the quantum model correctly predicts that
the energy density will decrease with decreasing wavelength. Why does
this new assumption predict that the distribution will decrease at small
wavelengths? Blackbody radiation is associated with a temperature and
arises from motion of the walls of the blackbody. According to classical
mechanics, the atoms can vibrate at all wavelengths and so energy should
be emitted at all wavelengths. According to quantum mechanics the wave-
length of the vibration is coupled to the energy of the system. At a given
temperature there is only a limited amount of thermal energy available
to vibrate the atoms and not enough energy for high-energy vibrations.
The effect of the quantum theory is to remove the high-energy vibrations
from consideration and hence to decrease the predicted amount of energy
emitted at large frequencies or correspondingly small wavelengths. For
his theoretical work, Planck received the Nobel Prize in Physics in 1918.

ρ
π
λλ

π

λ

≈

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =

−

88

5

1

4

hc hc

kT

kT

ρ
π
λ /λ

=

−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

81

(^51)
hc
ehc kT

CHAPTER 9 QUANTUM THEORY 179