Page 252 - Essentials of physical chemistry

214 Essentials of Physical Chemistry

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fibers in a vacuum of 10 mmHg. Chemistry students should be pleased that the HUGA equations
we learned earlier are also important in astronomy. Suffice it to say there is such pressure as related
to the density of the radiation r(T ), which is a function of the absolute temperature. Then we have
U
so that U ¼ r(T)V so the energy is the energy density per unit volume times the volume
r(T) ¼
V

qU r
and ¼ r. Then, when we express the pressure in just one of three dimensions, P ¼ ,we
qV 3
T

qP 1 dr qU qP
find ¼ . Now the energy expression becomes ¼ T P so that
qT 3 dT qV qT
V T V

T dr(T) r(T) 4r(T) dr(T)
. Rearranging we find and further
) r(T) ¼ ¼
3 dT 3 T dT
ð ð
4dT dr(T)
) 4 ln T þ C ¼ ln r(T). This can be put in final form using experimental meas-
¼
T r(T)
16 3 4
urements of the constant to find that a ¼ anti ln (C) ffi 7:5657 10 (J=m K ) and that
4
r(T) ¼ aT :
This expression is used by astronomers to estimate the temperature of stars, and later in this chapter
we will ask if the Planck theory agrees with this macroscopic relationship.
We have shown the Stefan–Boltzmann Law to indicate a boundary between macroscopic
thermodynamics and a revolutionary new way of thinking about energy in terms of quantization.
BLACKBODY RADIATION: RELATING HEAT AND LIGHT—PART II
We are now going to show the details of one of the most important advances in science in the early
twentieth century and we are going to give all the details so you appreciate there is no philosophical
‘‘wiggle-room’’ around the inescapable conclusion of energy quantization. The derivation is in two
parts. The first part has to do with counting how many waves can fit into a cubical volume and is
tricky, but correct. The wave-counting part of the problem is just a necessary exercise to compare
the final equation to experimental data. The second part of the derivation is the revolutionary
concept but the mathematics is much simpler and only involves summing a geometric series. While
the ‘‘mode counting’’ part of the derivation is necessary, a student should ponder deeply over the
summation of a discrete series to get the average energy per mode! The second step is easily learned
and it really is an important part of science education!
One of the unsolved mysteries of the late 1800s was an understanding of the blackbody radiation
spectrum. Ideally, a black body is one which is in thermal equilibrium between absorbing and
emitting radiation, as such it would appear ‘‘black.’’ A real blackbody can be approximated by a box
with a pinhole since light going in the hole would be absorbed but what light comes out would be
characteristic of the temperature of the interior and the spectrum does not depend on the material
used for the box. We offer a very crude schematic in Figure 10.1 to convey the sense of the problem.
Common experience teaches us that hot objects can give off a color that changes from red to orange
to yellow to white as the temperature increases. In principle, careful measurements can be made of
the spectrum of an object using a dispersing element such as a prism or a grating and recording the
dispersed light on film or more modern electronic devices (Figure 10.2). Due to the success of the
Maxwell equations and the experiments of Hertz, there was an air of confidence among physicists
that all the equations were available to explain the blackbody spectrum. If you read some of the
histories of this period in Science on the Internet, you will better appreciate the mental groping
regarding the nature of light. The Rayleigh–Jeans treatment of this problem was carried out from
1900 to 1905 by British scientists and perhaps overlooked the 1901 work by Max Planck (1858–
1947), a German scientist who later received the Nobel Prize for his work in 1918 (Figure 10.3).