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D. Einstein's solution illustrated
The problem with 19th-century atomic theory
was the assumption that atoms could vibrate with just any old energy. If
atoms could only vibrate with some energies but not others, then the fraction
of atoms having energies in between would be 0%. That by itself is
enough to affect the average energy per atom. In fact, the average
energy per atom is guaranteed to be less if (a) the possible energies
are the same for every atom, (b) the differences between each possible
energy and the next are all the same size, and (c) the minimum possible
energy is the same as for a "19th-century" atom.
Under these conditions, the average energy of an atom
would not be directly proportional to the absolute temperature,
as earlier theory implied, though if the temperature were high enough,
the average energy would almost be proportional to it. The energy
per atom that would raise the temperature of a warm solid by one degree
would raise a cold solid's temperature by more than one degree-much
more, if the starting temperature were very low. The exact way in
which this energy would depend on starting temperature is illustrated in
the figure below.
Figure 1. Vertical axis indicates the amount of
energy per atom one would have to add to a solid to raise the solid's temperature
by one unit. The curve shows what this amount would be according
to Einstein's assumptions, while the horizontal line at the top of the
graph shows the amount expected from 19th-century assumptions. The
horizontal axis represents temperature, in units of one kelvin for every
20.8 gigahertz of atomic vibrational frequency (every 20.8 billion vibrations
per second). For example, if all the atoms in a solid had a vibrational
frequency of 600 gigahertz, one unit on the horizontal axis would represent
a temperature of (600/20.8) kelvins ≈ 28.8 kelvins.
The horizontal line across the top of Figure 1 represents
how much energy it would take to raise a solid's temperature one unit according
to 19th-century atomic theory-the same amount, no matter what
the starting temperature was. The curve illustrates the effect of
atoms only being able to vibrate at certain energies.
If the possible energies are the same for all atoms
in the solid, their vibration frequencies must also be the same. In
the figure, the size of the temperature unit is proportional to this frequency-one
kelvin for every 20.8 billion vibrations per second (every 20.8 gigahertz
of frequency). This would be a large number of vibrations per second
for a large object, but for atom-sized entities it's actually very few. Typical
solids interact most strongly with infrared light waves, which vibrate trillions of
times per second, and it seemed reasonable from Maxwell's laws to expect
that the atoms of those solids would vibrate at the same frequencies. For
a solid whose atoms vibrate 5 trillion times per second (5,000 gigahertz),
a temperature of one unit in Figure 1 would be (5,000/20.8) kelvins, or
about 240 kelvins (-32.8°C).
According to the figure, a solid whose temperature is
one unit requires almost as much heat as 19th-century theory
suggested to raise its temperature one more unit. So if the unit
is 240 kelvins, 19th-century theory almost works even at rather
cold temperatures. But suppose we had a solid whose atoms vibrated
several times faster, say at 35,000 gigahertz. Figure 1's one-unit
temperature would then be about 1680 kelvins (1410°C), which would put
room temperature at about 0.17 unit. The graph for that temperature
shows that, for this solid, the heat input per unit temperature increase
is much less than the 19th-century prediction. In this
case the limits on the possible energies of atoms would make a considerable
difference.
When Einstein presented these findings in 1906, he was
simply trying to demonstrate that Planck's discovery about light had implications
for the way matter absorbed heat, which at least roughly agreed with actual
observations. But he also pointed out that this elementary analysis
had not taken all relevant thermodynamic considerations into account. For
one thing, most solids expand when heated, meaning that their atoms have
more room between them; having a different amount of space to vibrate in
could affect the atoms' vibrational frequencies. It's also conceivable
that atoms might vibrate at different frequencies when absorbing heat than
when interacting with light, contrary to Einstein's working assumption. Furthermore,
the atoms' vibrational frequency itself might even change with temperature.
Still, when Einstein compared his analysis with actual
data about real solids, he saw that the general trends matched. He
found that, in general, solids that gain little heat per degree temperature
increase also have low atomic masses, which should be the case for atoms
with high atomic vibrational frequencies-the less massive something is,
the faster it can vibrate under the same conditions. Furthermore,
the expected vibrational frequencies of some transparent solids were at
least in the same ballpark as the frequencies of light with which the solids
interact most strongly. Finally, Einstein compared his analysis with
observations of heat absorption by diamond. Diamond (carbon) has
atoms of low mass, and absorbs far less heat per degree temperature increase
than many other materials, even at room temperature. This is in clear
disagreement with earlier theory, but does follow the general trend of
Figure 1, as we can see in Figure 2 below.
Figure 2. The dots represent actual experimental
observations for diamond, under the assumption that at a temperature of
331.3 kelvins, diamond absorbs heat at exactly the rate expected from the
theory. While the other dots don't all lie on the curve, they do
match its general trend. (Based on data from Einstein's paper "Die
Plancksche Theorie der Strahlung und die Theorie der spezifischen Wärme", Annalen
der Physik 22 (1907), pp. 180-190. According to that data, one
temperature unit in this figure should correspond to 1325 kelvins, or 1052°C.) (.....continued)
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