If two marbles collide, they rebound with less total kinetic energy than they had before the collision. Each is warmed slightly by the collision, since the lost energy has spread as internal energy within the marbles. Some of the original macroscopic *ordered* energy has been converted to microscopic *disordered* energy. This is a simple example of the equipartition theorem at work. Energy concentrated initially in one mode of motion tends to distribute itself evenly over all of the available modes of motion.

The situation is usually very different for atomic collisions. A helium atom, like a marble, is a system composed of smaller constituents. Yet when two helium atoms collide at low energy, there is no energy dissipation. The total kinetic energy of the atoms after the collision is exactly the same as before, and the internal structure of the atoms is completely unaffected by the collision. This behavior is a consequence of *quantization *in the atom and is something that could not have been foreseen or explained in the framework of classical physics. If such collisions without dissipation could be arranged in the macroscopic world, it would make possible perpetual-motion machines. Indeed, on the atomic scale, perpetual motion is a reality!

The internal energy of a single atom is quantized; that is, it can take on only a certain definite set of values. For each atom there is a lowest energy state, called the ground state. At higher energies lie the “excited states” of the atom. Between the ground state and the first excited state is an energy gap that represents the smallest amount of energy the atom can absorb. If offered less than this minimum amount, the atom cannot accept it. Instead it remains unchanged in its ground state.

Picture, for instance, a collision between two helium atoms, one initially at rest, the other having an initial kinetic energy of 0.2 eV. The collision results in energy sharing. One atom gains energy, the other loses energy. Because the minimum energy needed to excite a helium atom is 20 eV, neither atom can be disturbed *internally* by the collision. The kinetic energy of a whole atom, on the other hand, is not restricted by such a quantum condition. The energies of motion of the atoms after the collision might be 0.12 eV and 0.08 eV, or 0.195 eV and 0.005 eV, or any other pair of values adding up to 0.2 eV. Energy sharing takes place only among the translational degrees of freedom of the whole atoms. The internal degrees of freedom associated with the motion of the electrons within the atom (and the protons and neutrons within the nucleus), because of their all-or-nothing requirement for energy gain, get none of the energy. These degrees of freedom can be called “frozen.” Since they do not ordinarily participate in energy sharing, they can be ignored. Indeed they must be ignored. To assign more than three degrees of freedom to a helium atom would lead to incorrect predictions (for example, of helium’s specific heat capacity).

This idea of frozen degrees of freedom was important at the birth of quantum theory. When physicists had tried to apply the equipartition theorem to the sharing of energy among a collection of molecules and electromagnetic waves trapped in a hollow box, they came up against the apparent infinite number of degrees of freedom of the electromagnetic radiation field. To explain why the waves did not take all the energy, leaving none for the matter, Max Planck postulated, in 1900, that energy could be transferred from matter to radiation, or from radiation to matter, only in multiples of a minimum amount—that energy transfer is *quantized.* In order to fit the already known facts on how the energy distributed itself in equilibrium, Planck had to assume that the minimum energy quantum increased in proportion to the frequency of the radiation, according to the simple formula,

*E = hf *,

in which *E *is the minimum energy quantum, *f *is the frequency of the radiation, and *h *is a constant of proportionality now known as Planck’s constant. By this revolutionary quantum postulate, Planck succeeded in “freezing out” the high-frequency radiation. Whenever less than the minimum quantum energy is available, no energy at all can go into radiation of a particular frequency. The infinitely many degrees of freedom associated with infinitely high frequencies became frozen degrees of freedom. The equipartition theorem was saved, the facts of electromagnetic energy sharing were explained, and the quantum theory was born.^{1}

The full description of Planck’s quantum theory is more complicated than is indicated by this discussion. Degrees of freedom, like butter, freeze only gradually. A particular mode of motion (or a particular electromagnetic wave) is never totally excluded from energy sharing, although its exclusion may be very nearly complete. Consider a sample of helium in which the *average *kinetic energy of each atom is 0.2 eV. In almost all collisions, the available energy will be less than the 20 eV required to produce internal excitation of an atom. Occasionally, however, a single atom will by chance acquire a kinetic energy greater than 20 eV. When it collides with a neighboring atom, one of them may emerge from the collision in an excited state of internal motion. The internal degrees of freedom are almost, but not completely, frozen out of the energy sharing. When the average kinetic energy of the atoms approaches the quantum energy of internal excitation, collisions producing internal excitation occur more often, and the internal degrees of freedom become gradually unfrozen.

At normal temperatures near 300 K, the average molecular kinetic energy of thermal motion is about 0.04 eV. Typical quantum energies of electronic excitation in atoms and molecules amount to several eV, about one hundredfold greater than average thermal energies. (Helium is not typical. Its quantum excitation energy of 20 eV is greater than that of any other atom.) This means that the internal electronic degrees of freedom are quite effectively frozen out. When heat is added to a substance at normal temperature, a negligible part of it goes into atomic excitation. At much higher temperatures the situation changes. When collision energies become large enough to excite atoms or molecules internally, more degrees of freedom come into play, and more of the system’s energy is in the form of atomic and molecular excitation, a smaller fraction residing in the translational kinetic energy that defines temperature. As a consequence, the specific heat capacity of most substances increases as temperature increases; the greater the number of “unfrozen” degrees of freedom, the more energy is required to produce a given temperature rise. Conversely, near absolute zero, where most degrees of freedom are frozen, specific heats are very small.

Vitally important factors governing the properties of a substance are its characteristic quantum energies relative to the characteristic thermal energy, ^{3}∕_{2}*kT*. The thermal behavior of a piece of matter at any particular temperature depends more than anything else on which of its degrees of freedom are open for energy sharing, and which are frozen. As an originally solid piece of matter is heated, it experiences successive marked changes—melting, boiling, dissociation, ionization—as successively higher quantum energies topple under the impact of increasing thermal energy. Ice melts at 273 K, when its thermal energy of molecular vibration becomes comparable to the energies binding one molecule to another. At the boiling point of water, 373 K (where the kinetic energy ^{3}∕_{2}*kT *is about one twentieth of an eV), the translational degrees of freedom become completely unfrozen. Melting and boiling are examples of “phase changes” occurring at well-defined temperatures. Transformations of form occurring at still higher temperature are gradual, requiring thousands of degrees or more to be completed. Water molecules dissociate (break apart into their constituent atoms) in the range 1,000 to 10,000 K. Ionization (the separation of electrons from their parent atoms) begins in this temperature range and is completed only above one million degrees. At such temperatures (which occur within stars), a gas of protons, electrons, and oxygen nuclei obviously bears little resemblance to ice or water or steam. This ionized gas is called a plasma. Because its particles are electrically charged, it has properties radically different from those of ordinary neutral gases. In the stellar plasma at a temperature of millions of degrees, the energy of thermal motion, ^{3}∕_{2}*kT*, is large compared with most atomic quantum energies, but is still small compared with the quantum excitation energy of the oxygen nucleus. Despite the intense heat, the internal degrees of freedom of the protons and neutrons in the nucleus remain frozen. Collisions leave the nucleus undisturbed in its ground state. Only if the temperature were to rise to many billions of degrees would the nuclear degrees of freedom thaw, allowing the individual nuclear particles to share energy. It is likely that such enormous temperatures are actually reached in the late stages of the evolution of a star.

1 In 1905, five years after Planck’s work, Einstein extended the quantum hypothesis by suggesting that not only is energy *transfer* to and from radiation quantized, but the energy *content* of radiation itself comes in quantum lumps (called corpuscles at the time, and later photons), each lump having energy *E = hf.*

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