Page 258 - Essentials of physical chemistry

220 Essentials of Physical Chemistry

grade of students in a class to make sure we divide the thing we are averaging by the number of
students in the class, ‘‘normalizing’’ the weighted-average. It helps to simplify the summation if we

e
define x e k B T and note that Planck defined the discrete energy as E ¼ e,2e,3e,4e . . . You may
4
2
3
recall a geometric series from calculus? Let f(x) ¼ 1 þ x þ x þ x þ x þ for x < 1.
2 3 4
f (x) ¼ 1 þ x þ x þ x þ x þ
2
3
5
4
[xf (x) ¼ x þ x þ x þ x þ x ]
_______________________________
(1 x)f (x) ¼ 1
1
. Further, we can take the
(1 x)
As long as x < 1 we can do this trick to find the sum of f (x) ¼

df (x) d 1 ( 1) 2 3 4
derivative of f(x)as ¼ ¼ ¼ 1 þ 2x þ 3x þ 4x þ 5x þ , but recall
dx dx 1 x (1 x) 2
P n 2 3 4 2 3
1 (ne)x ¼ ex þ 2ex þ 3ex þ 4ex þ ¼ (ex)(1 þ 2x þ 3x þ 4x þ ) and this means
n¼0 e
2
[ex=(1 x) ] (ex) ee k B T e
in a closed form! Next, Planck
¼ ¼
[1=(1 x)] (1 x) 1 e k B T e þ k B T 1
that e ¼ ¼ e e
made a very famous assumption based on Maxwell’s 1865 hypothesis that light is an electromag-
netic wave and Hertz’ 1886 experiments that radio waves can be transmitted and diffracted. Planck
proposed that the energy of light is proportional to frequency, e ¼ hn. Note that at this point Planck
did not know h and in fact his idea was so revolutionary (energy chunks indeed!) that one of the
criticisms of his calculation was that he had to adjust the value of h by choosing a value that fit the
data. In fact if you check the dates, Jeans published additional work in 1905 using the Rayleigh–
Jeans formulation after Planck’s 1901 work! So now we are ready for the amazing result of Planck’s
assumption of quantized energy.
3
8pn 2 hn 8phn gn
Planck ) r(n)dn ¼ 3 hn dn ¼ þhn
c e B T 1 c e B T 1
3
k
k
Skeptics criticized h as an adjustable parameter, but when Planck chose h ¼ 6.626 10 34 J she
was able to fit the experimental to the experimental data for essentially an exact fit! One of the main
critics was Wilhelm Ostwald (1853–1932), a German physical chemist, who did not accept the
atomistic theory and believed energy is continuous. While Planck also was skeptical about the
existence of atoms, he had to adjust his thinking when his equation produced an exact fitto
experiment based on quantization. In 1909, Ostwald was awarded the Nobel Prize for his work
with catalysis. From this brief discussion, you can see that even at this late date Boltzmann’s 1866
KMTG prediction of tiny gas atoms was not widely accepted. The term ‘‘ultraviolet catastrophe’’
was only used later by Paul Ehrenfest in 1911 and Planck was motivated mostly by the shift in
wavelength peak with temperature due to his background in thermodynamics.
Note the long wavelength agreement of the Rayleigh–Jeans function with experiment, does
Planck’s formula satisfy this condition?
2 3

hn h
lim ¼ lim 5 ¼ k B T where we have used L’Hopital’s rule for the limit. That
4
n!0 hn n!0 h þhn
k
e B T 1 e B T
k
k B T
is amazing for the low frequency limit but what about the high-frequency limit? Recall from Chapter 0
n
x
x
4
3
2
that e ¼ 1 þ x þ x þ x þ x þ and that e will dominate any integer power of x because
x
e contains in its series every power of x and one higher power for any given single power of x!