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)