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E. The theories and the facts
The figures illustrating Einstein's result give us a
clue as to why 19th-century theory worked as well as it did,
even though it took no account of the yet-to-be-discovered energy quanta. For
all but the lower temperatures, the quantum hypothesis leads to practically
the same results as the classical theory. And how low the temperature
of a solid has to be to get very different results depends on how small
the energy quanta are that the solid can absorb. The smaller the
energy quanta (and the smaller the atoms' vibrational frequency), the more
quanta it would take to raise the solid from a low to a high temperature-and
the more closely the flow of energy quanta would resemble the continuous
energy stream expected from 19th-century thermodynamic theory,
instead of a "grainy" energy stream made of quanta.
We noted earlier that atoms' apparent inability to vibrate
with arbitrary energies is odd, given the apparent ability of larger objects
to vibrate at any energy whatsoever. If matter is made of such atoms,
why don't we see larger objects also limited to certain vibrational motions? It
turns out that we do, but it's not obvious. Again, the reason is
that the energy quanta for ordinary vibrational frequencies are very small. A
large weight on a spring will have almost the same motion if its energy
increases or decreases by one quantum. If the motion becomes very
different, a large number of the small energy quanta have to be gained
or lost, and that would look to our unaided senses like a continuous energy
change instead of a "grainy" one. So here, too, 19th-century
mechanics' lack of a quantum hypothesis doesn't make it noticeably inaccurate.
It turned out that 19th-century theory was
accurate in another respect as well. We noted earlier that certain
features of atoms, which explain why they cannot absorb just any
amount of energy, had yet to be discovered in 1906. Once we take
account of these features, we find that the average energy per atom should
be half a quantum higher than it would if Einstein's working assumptions
were correct. So, instead of real atoms having less average
energy than the atoms of classical theory-barely less at small temperatures,
approaching a half-quantum less at high temperatures-real atoms should
have more average energy-a half-quantum more at absolute zero temperature,
but very little more at high temperatures. The higher the temperature,
the more closely a real atom's average energy should equal that of a "classical" atom.
One other thing Einstein didn't take full account of
is the way each atom's motion affects that of its neighbors. When
one atom moves, it pushes against the atoms it approaches, and pulls at
the atoms from which it moves away. This tends to get those other
atoms moving in the same direction, or at least slow their motion if they
were already moving in the opposite direction. At the same time,
the neighboring atoms resist the motion of the first atom, tending to slow
it down, stop it, and reverse its original motion. Thus when some
atoms move back and forth, their neighbors begin to follow that motion,
lagging behind it a little. And these atoms in their turn start the
same process in their neighbors, spreading vibrations through the solid
in waves-sound waves.
If
an atom barely moved when its neighbors vibrated, the solid's energy would
increase with temperature in much the same way as for Einstein's hypothetical
solid. But in sound waves, each atom more or less keeps pace with its
neighbors. The more closely an atom follows its neighbors' motion,
the less it resists their motion, and the more time it will take all the
atoms to complete a vibration. In other words, atoms' moving together
lowers their vibrational frequency. The longer the sound waves are,
the more closely neighboring atoms pace each other and thus the more their
frequency lowers. So in a solid, the atoms have many possible vibrational
frequencies, not just one. Just this additional fact implies a slightly
different (and more accurate) relation between a solid's temperature and
the average energy of its atoms. Still, this more accurate relation
shows the same general trend as Figures 1 and 2-a high, nearly steady increase
of average energy with temperature at high temperatures, a small but variable
increase at low temperatures. (.....continued)
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