By the early 20^th century, the way in which temperatures of solid objects changed as they absorbed heat was considered strong evidence that matter was not made of atoms.  Einstein used some recent discoveries about light to turn this assessment around.

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A.  A puzzle, and a surprising solution

Take equal masses of lead and aluminum.  Heat them until their temperatures are both 10 degrees higher.  Will it take the same amount of heat for each? 

Back in the 18^th century, the chemist Joseph Black discovered that different materials required different amounts of heat to raise their temperatures by equal amounts.  The amount by which the temperature of a material changes as it absorbs or gives off heat can even be used to help identify the material.  Among solid materials near room temperature, aluminum and lead differ almost as much as any two chemical elements in this respect:  to raise the temperature of aluminum 10 degrees takes more than five times the amount of heat that the same mass of lead requires for the same increase.

Why would different materials react so differently to heat?  The idea that heat is just the energy of the random motion of atoms offers a clue.  The atoms of any solid object move constantly, each atom continually bouncing off its neighbors.  At any given temperature, the atoms will have a certain average energy.  If we heat the solid to a higher temperature, we increase the agitation of its atoms to a higher average energy.

Whatever the combined energy is of all this agitation, the average energy of each atom is just this combined energy divided by the number of atoms.  If the solid is a material like lead, each atom will be massive, so it will take relatively few atoms to make one gram; with fewer atoms, each atom will have a large share of energy.  Materials made of less massive atoms will require more atoms to make an equal mass of solid, so each atom will have less energy.

So far, nothing about this seems especially complicated, but eventually a puzzling fact became known.  Given the ideas about atoms that were common in the 19^th century, one could deduce that the average energy per atom should be directly proportional to the solid's temperature-double the absolute temperature, and each atom should have twice the average energy.  Put another way, raising the temperature of a solid by one degree would require the same amount of energy per atom, no matter what the solid's original temperature was.  This conclusion agreed with observations of real solids made by Pierre-Louis Dulong and Alexis-Thérèse Petit, which they published in 1819.

But, in time, a puzzling fact emerged as other observations failed to agree with this theory.  Dulong and Petit had experimented with solids at ordinary temperatures.  At extremely low temperatures we find something quite different.  The lower the starting temperature, the less heat you need to raise the temperature of a solid by one degree.  If you start with extremely low temperatures close to absolute zero, you can raise the temperature of solids by several degrees with barely any heat input at all.

Even at ordinary temperatures, the theory was wrong about some materials.  If the theory were only slightly off, it might be basically sound; some small unaccounted-for disturbance might be affecting some types of atoms, or any atoms in some situations.  But since the atomic theory was way off for several materials, many physicists suspected there might be something fundamentally wrong with the very assumption that atoms even existed.

But by the early 20^th century, Albert Einstein had realized that one thermodynamic phenomenon pointed directly to the existence of atom-like units of matter, and could even allow a precise determination of molecular masses.  Einstein had found that, if liquids were made of molecules, small particles suspended in the liquid would share in the liquid molecules' random motion.  Einstein also proved an exact relation between how far the suspended particles would move on the average, how long they have been moving, and the size of the unit of molecular mass.  Within a few years, Jean Perrin and his students verified this relation in experiments and determined the molecular-mass unit.

While these experiments appeared to confirm that heat energy is the random motion of atoms, they didn't directly address the theory's apparent discrepancy with the behavior of solid objects.  Einstein found a resolution to this problem while thinking through the implications of a recent discovery by Max Planck.  Einstein found that atoms might be real after all, but if so, they behaved very differently from what had previously been suspected.

In the next section, we'll briefly describe what Einstein found, along with some of the questions this finding raised.  To answer those questions, we'll look first at how temperature is related to energy, and then, with the help of some graphs illustrating Einstein's findings use that relationship to answer the questions.  Finally, we'll see how these findings exemplify certain features of scientific progress.     (.....continued)

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