Heat Capacity One of the evidences for quantum theory comes from a study of the way the heat capacity of a substance varies with temperature. There is nothing difficult to comprehend about this subject, but it includes many details, and it requires at least a chapter to do it justice.

A good way to begin is to consider the concept of the "ideal gas". Think of a molecule as a point with no dimensions. It has zero size. However, unlike a mathematical point, it has an inertial mass somewhere between the mass of hydrogen and the mass of mercury. Although it has inertial mass, it has no gravitational mass. It has no attraction or repulsion for anything. If it collides, it rebounds. Having no parts, it has no internal motion. All of the energy of a molecule of ideal gas is translational kinetic energy. It has no potential energy whatever. At absolute zero temperature, the ideal gas has no pressure and no volume. Imagine an airtight chamber with an inside volume of 24.452 liters. When the molecule of ideal gas has a temperature of 1 K, it has an energy of 2.07039 x 10-16 erg. For every degree that the temperature rises, there is an additional 2.07039 x 10-16 erg of energy in the molecule. To find the kinetic energy of the ideal gas molecule, at 298 K, multiply 2.07039 x 10-16 erg by 298, and get 6.16976 x 10-14 erg. The molecules are not all at the same temperature at the same time. The average of the molecular temperatures is the temperature of the mole of molecules of ideal gas. The temperature of the mole of gas will not change at all, if energy is not transferred into or out of the chamber. It is sometimes more convenient to use other units. I convert ergs per molecule into ergs per mole by multiplying the numberof ergs per molecule by 6.022 x 1023, the number of molecules per mole. 2.07 x 10-16 x 6.022 x 1023 = 12.46 x 107 That is 12.46 x 107 erg per degree. There are 4.184 x 107 ergs per calorie. Convert ergs to calories by dividing by 4.184 x 107. 12.46 x 107 / 4.184 x 107 = 2.98 That is 2.98 calories per mole per degree. It takes an addition of 2.98 calories to make one mole of ideal gas raise its temperature one degree, when the ideal gas is in a chamber of constant volume. The quantity, 2.98 calories is called the heat capacity at constant volume of the ideal gas. Its symbol is Cv. The heat capacity is the same whether the temperatire goes up from 10 K to 11 K, or from 297 K to 298 K. There is another kind of chamber. A cylinder with a massless piston. The volume is adjusted by the position of the piston. The piston is always so situated as to have the pressure of the ideal gas inside the cylinder, equal the pressure of the air outside the cylinder. The ideal gas inside the cylinder does not have a constant volume. It has a constant pressure. At absolute zero, the mole of ideal gas takes up no space. The piston is in contact with the opposite wall of the cylinder. Adding energy to the molecules raises their temperature to 1 K, when the translational kinetic energy per molecule is 2.07 x 10-16 erg. At 1 K, the pressure of the ideal gas is one atmosphere, and the volume of one mole is .082 liter. For every rise of one degree, the ideal gas molecule gains 2.07 x 10-16 erg of translational kinetic energy, and the mole of ideal gas gains .082 liters of volume. The volume of one mole of ideal gas at atmospheric pressure at 298 K is equal to 298 x .o82 liters, or 24.4 liters. More precise figures are .082054 for .082, and 24.452 for 24.4.

When the ideal gas is in its constant volume chamber, its heat capacity per molecule is 2.07 x 10-16 erg per degree. When the ideal gas is in its constant pressure cylinder, its heat capacity is much greater. The way to measure the heat capacity is to observe how much energy from the environment must be transferred to the ideal gas to increase its translational kinetic energy per molecule by 2.07 x 10-16 erg. I start with a mole of ideal gas at 100 K in a constant pressure cylinder. The volume at 100 K is 8.2 liters. I add 3.45 x 10-16 erg per molecule, and the temperature rises to 101 K. However the translational energy is only 2.07 x -16 erg per molecule. The difference between 3.45 x 10-16 erg and 2.07 x 10-16 erg is 1.38 x 10-16 erg. The quantity 1.38 x 10-16 is called the Boltzmann constant, or the gas constant per molecule. The gas constant per mole is 1.9827 calories, .082 liter atmosheres. At a pressure of 1 atmosphere, the air pressure against any flat surface with a force of 1.013 x 106 dyne/cm2. Assume that the massless piston has an area on its inside surface of 100 cm2. The total force at a pressure of one atmosphere on a surface of 100 cm2 is 101.3 x 106 dyne or 1.013 x 108 dyne. In going from 100 K to 101 K, the ideal gas gains .082 liters of volume. A liter is 1000 cm3. .082 liters equal 82 cm3. The piston moves .82 cm, increasing the volume inside the cylinder by .82 x 100 cm3. The volume is the product of the length times the area of one surface. The product is 82 cm3.

The massless piston moves outward when the pressue inside the cylinder is infinitesimally greater than the pressure of the atmosphere. The ideal gas pushes the piston with a constant force of 1.013 x 108 dyne. With this information, I can calculate the work done, or the energy expended, by the ideal gas when it moves the piston .82 cm. I substitute in the equation for work: w = Fd w = (1.03 x108)(,82) w erg = 8.31 x 107 erg I find the work per molecule by the number of molecules in a mole: (8.31 x 107) / (6.022 x 1023) = 1.38 x 10-16 w erg = 1,38 x 10-16erg That is Boltzmann's constant. I have to add 1.38x10-16 erg per molecule to the translational energy of 2.07 x 10-16 erg per molecule at constant pressure, which is 3.45 x 10-16 erg per molecule. The symbol for the heat capacity at constant pressure is Cp.

Notice that the 1.38 x 10-16 erg does not reside in the molecule. The 1.38 x 10-16 erg is transferred to the atmosphere, which now holds it as potential energy. The space that the atmosphere is pushed out of, becomes potential space into which the atmosphere can push back. If, at a later time, the temperature inside the cylinder should fall one degree, the atmosphere will do work on the ideal gas, and restore the 1.38 x 10-16 erg to the molecule. It won't stay with the molecule, because, in order to lose one degree, the molecule must transfer 3.45 x 10-16 erg. A real gas, like helium, behaves like the ideal gas in many ways. One difference between helium and the ideal gas, is that the helium atom has volume. At absolute zero, helium does not have zero volume. Another difference between helium and the ideal gas, is that helium atoms have some attraction for each other. The attraction is negligible above 4.18 K, its boiling point at one atmosphere. Helium certainly does not behave like an ideal gas, when it is a liquid. As a gas, helium has the same Cv and Cp as the ideal gas. Helium is like the ideal gas, because it is a single atom, not a molecule. Hydrogen is different from the ideal gas, because it has two atoms per molecule. Hydrogen is a liquid when its temperature is below 33.24 K. The heat capacity at constant volume for hydrogen is 2.07 x 10-16 erg per molecule per degree, when the temperature is about 40 K. The Cv of hydrogen rises gradually to about 3.45 x 10-16 erg per molecule per degree at 298 K. The Cv stays at about 3.45 x 10-16 erg per molecule until about 800 K, when it starts to rise again. The increase in Cv as the temperature rises was taken as evidence for quantum theory. The reasoning was that there are energy levels for rotaional energy and vibrational energy. Although there is a minimum energy to satisfy the uncertainty principle, there is no heat capacity until the energy rises above the minimum to the first higher level. At low temperatures, there is not enough energy per collision to get to the first higher energy of rotation. It takes even more energy to reach the first higher level of vibration. Taken by the mole, when the temperature is 100 K, the individual molecules have temperatures that range from 0 K to possibly 1000 K. The number of molecules with energy too low for rotation is very large. When the molar temperature is 200 K, the number of molecules with temperatures high enough for rotation is much greater, and a good fraction of the molecules can support vibration.

The same evidence can be used to support neg-pos theory, in which there are no energy levels. Each collision between molecules is unique. I examine as many individual collisions as I can, to discover how the energy of the collision is divided between internal energy and translational energy. After all, that is the principle of heat capacity. The larger the ratio of internal energy to translational energy, the greater the heat capacity. One helium atom collides with one hyrogen molecule. (Fig. 1) Click on figure to enlarge. Click Back to return.

Fig. 1

1. The H2 is stationary. The He is moving with a temperature of 50 K. All three nuclei fall on one staight line. 2. The He is pushing its way into the region of repulsive force. It loses some speed, due to the force. The near H2 proton accelerates much faster than the He decelerates, because it has only 1/4 the mass of He. The H2 proton is being pushed into the repulsive field between the protons . This repulsive force pushes both protons equally. The force can only slow down the first proton a little against the acceleration that it receives from the He atom. At the same time, the second proton begins to accelerate. 3. The space between H2 protons stops decreasing, and the space between He and H2 stops decreasing. The proton on the right is moving as fast as the proton on the left. The He is still slowing down, because of the force between He and H2, although it begins to diminish, continues to decelerate the He atom as long as some force exists. 4. The proton on the right, with nothing pushing it back, makes better headway than the proton on the left. The space in the H2 molecule widens. The He atom has slowed down to a slower speed than the proton on the left. 5. The He atom can't keep up with the H2 molecule. The H2 protons find themselves out of their positions of zero force, because of the increased distance between the protons. The protons move toward each other, toward their points of zero force. They overshoot those points, and are forced to return. They overshoot again in the reverse direction. The protons continue to vibrate like that, until the molecule has another collision. The electrons can be neglected for now. I shall return to them later. The energy of the H2 molecule moving as a unit, is its translational energy. The H2 molecule started at rest, and was accelerated to more than the original speed of the He atom. The only internal energy it picked up was the weak vibration of the protons. At 50 K there is very little internal energy. I observe the same interaction at 200 K. The H2 molecule is at rest, and the He atom approaches much faster than in the first interaction. (Fig. 2)

Fig. 2

1. The He atom arrives at the region of repulsive force. 2. The He atom penetrates deeper into the repulsive region than it did at 50 K. The proton on the left starts to accelerate, and the He atom starts to decelerate. 3. The proton on the right has barely begun to accelerate, when the proton on the left penetrates deeply into the repulsive region between protons. 4. The proton on the right accelerates very strongly because of the compression of the repulsive region between protons, and because nothing bars its way. At the same time, the acceleration of the proton on the left is retarded by the rearward force from the same, compressed, repulsive region. This keeps the distance between He and H2 short. The proton on the right finds itself far to the right of its point of zero force. 5. The proton on the right snaps back, and takes up vibration. The proton on the left vibrates in step with the proton on the right, because each time the bond stretches, there is no retarding force on the left proton. It speeds up toward the right, only to be retarded again, when the repulsive force between the protons increases once more. 6. The erratic progress of the proton on the left, due to the vibration, thrusts the He atom and the proton deeply into the region of repulsive force repeatedly. This rearward force on the He atom, decelerates it. 7. The He atom, having decelerated throughout the interaction, can't keep up with the H2 molecule. The H2 molecule has gained some translational energy, but it has also gained a significant amount of vibrational energy. There is no doubt that this kind of collision accounts for a greater heat capacity at higher temperatures. I chose to describe a collision between a moving atom and a stationary molecule because of its simplicity, as compared with the case of two molecules in motion. The principle is the same. When the collision is head-on, the energy transfers from the molecule with more momentum to the molecule with less momentum. Otherwise the energy transfers from the overtaker to the overtaken. The hydrogen molecule passes energy back to the helium atom, when the occasion arises. In that event, the helium atom gains more temperature than the hydrogen molecule loses. I observe another collision. (Fig.3)

Fig. 3

1. The helium atom is moving toward the stationary hydrogen molecule. This time, the helium atom hits the hydrogen molecule broadside. The temperature of the helium atom is 50 K. 2. The He invades the region of repulsive force. All of the particles of the hydrogen molecule yield to the repulsive force. There is nothing blocking their progress. The H2 accelerates. 3. The He decelerates a bit. The H2 accelerates enough to get well ahead of the He. The electrons vibrate a little. The protons vibrate a little. The net effect is that most of the energy that was transferred in this interaction was translational. A collision does not contribute to the heat capacity. I repeat the experiment with more energy: (Fig. 4)

Fig. 4

1. The He moves at 100 K toward a stationary H2. 2. The He hits the H2 repulsive field so fast, that the two protons of H2 have no opportunity to respond. 3. The protons are forced apart by the He that wedges in between them. They also accelerate in the forward direction. 4. The He slows down a little. The H2 gets away from the He. The He molecule is badly distorted. Each proton returns to the point of zero force, but overshoots the point. This is followed by oscillation of the protons, which continues until the next collision. The H2 picked up a little translational energy this time, and much vibrational energy. I have more to say about electrons farther down the page. I repeat the experiment at higher temperatures, and get more internal energy each time the temperature goes up. At very high temperatures, a proton vibrates with so much amplitude, that it spends most of its time moving slowly at the more distant points of its swing. Potential energy increases with distance. Imagine a camera in a dark room with its shutter wide open. A light flashes at a steady high frequency. None of this light goes into the camera, because the walls of the room are perfectly black, and the flasher is behind the camera. An oscillating proton in front of the camera reflects the light. After one passage of the proton before the camera, the shutter closes. The developed picture looks like (Fig. 5)

Fig. 5

I count the dots. Each dot represents an equal time interval. There are twelve dots in the regions of high potential energy, and four dots in the region of high kinetic energy, That means that the time average for potential energy is three times the time average for kinetic energy, and it appears only at higher temperatures. In the next experiment, the collision would be broadside, but the He does not strike the H2 in the center. (Fig. 6)

Fig. 6

1. The He is moving at 50 K. The H2 is stationary. 2. The He invades the repulsive region of the lower proton, but not the upper proton. 3. The He is deflected downward, and the lower proton is forced into a curved path around the center of mass between the two protons of H2. Now the H2 has some rotational energy. I try the same experiment at higher temperature and get additional rotational energy. By itself, this does not change the heat capacity of H2. A constant Cv at all temperatures must accept more energy as the temperature rises. For each additional degree, a quantity equal to Cv is absorbed. Helium, which has a fixed Cv, at all temperatures, gains 2.07 x 10-16 erg per degree per molecule. If hyrogen were to accumulate internal energy per degree at the same rate that it accumulates translational energy per degree, it would have a constant Cv. At higher temperatures, additional rotational energy causes the protons to move apart somewhat, due to centrifugal force. The separation means additional potential energy. This potential energy is another way of storing internal energy, and it entails a higher heat capacity at constant volume at high temperatures. When a proton is pushed this way in a collision, it is not pushed along the circumference of the circle. It is pushed slightly outward, away from the center of the circle. (Fig. 7)

The proton is prevented from moving away from the atom by the attractive forces. However, it increases the radius of its path. To the extent that the proton has outward motion, it is being decelerated by the attractive forces. Gaining speed, it has increased centrifugwal force, giving the proton a larger radius. In that way, the proton alternates between inward and outward motion, while it is rotating around the molecule's center of mass. Relative to the center of mass, both protons are moving equally and symmetrically. (Fig. 8)

Since the radii are alternating between shorter and longer, the protons are in vibrational motion at the same time as they are in rotational motion. Their paths are elliptical. The center of mass is not a particle. It is a point. For protons, which have equal masses, the point is at the center of the line that connects the two protons. For particles of unequal masses, the center of mass is at the point where one would put the fulcrum of a see-saw, if the unequal masses were riding on it. The center of mass is closer to the partcle with more mass. Centrifugal force is not necessarily a force in essence. It acts just like a force. A body in motion continues in motion in a straight line, unless disturbed by an outside force. (Fig. 9)

1. The body is pushed in the direction of the arrow. 2. The push stops, and the proton is carried by its momentum in the direction of the dotted line. Meanwhile there is an attractive force toward the center. 3. The proton compromises, and takes an itermediate path. But the force toward the center continues. 4. The proton is forced to compromise continuously. This causes it to take a circular path. 5. Going in a circle, the proton is not getting any closer to the center. It is not losing speed, because nothing is pushing in the direction opposite to the direction of its motion. A body must move toward an attractive force, unless it is acted on by an equal and opposite force. The proton is not moving inward. One is compelled to conclude that there is a force equal and opposite to the attractive force. The force directed toward the center is centripetal force. The opposite force is a centrifugal force. Some people regard centrifugal force as a fictitious force. I consider force an abstraction at all times. It is interesting that alternative explanations are frequently proposed for centrifugal force, while common forces are accepted as real. In a head-on collision or an overtaking collision, all of the energy of the molecule is taken into account. It does not work that way in collisions on an angle. If molecule A and molecule B approach each other in a head-on collision, the rate at which they approach each other is the sum of their speeds. If molecule C is overtaking molecule D, they are approaching each other with a speed equal to the difference between the speed of C and the speed of D. If moecule E and molecule F approach each other, (Fig. 10)

Along paths that are other than 180o, their rate of approach is the rate of shortening of the straight line between E and F. The dotted lines show successive positions of the line, equally spaced. If I learn how fast the line shrinks, I find out how fast E and F approach each each other. The lines in this sereies are parallel. The information is directly applicable only if the line between centers of the molecules, when they meet, is parallel to the series of lines. (Fig. 11)

In this diagram the molecules are spherical. The molecules meet when the circles touch at one point. The line beween centers passes through the point where the circles touch. The calculation of the exchange of energy in the collision is performed as if the molecules were traveling along that line at a total speed equal to the rate of shortening of that line. As if that were not complicated enough, it is most unlikely that the meeting will take place along a parallel line. Most often, the molecules meet at some other angle. No matter what the angle, the molecules participate with only a fraction of their energy. One interesting collision has a slow molecule passing energy to a fast molecule. The fast-moving molecule is directly in the intersection of the paths, when the slow-moving molecule hits it along the line of the path of the slow molecule. (Fig. 12)

Let the speed of the slow molecule be represented by a short arrow S in the proper direction. Let the speed of the fast molecule be represented by a long arrow F in the proper direction.The length of each arrow is proportional to the speed. In the collision, the slow molecule is decelerated, and the fast molecule is accelerated. The slow molecule is much slower after the collision. (Fig. 13)

The velocity of the fast molecule is the combination of two arrows, or the dotted line, which is longer than the original arrow of the velocity of the fast molecule. Whenever a diagram of a collision shows the motion of each particle, atoms, nuclei, electrons, et al, an equally valid diagram can be drawn for the same particles going in the reverse direction. If the arrows are reversed, step 5 becomes step 1, step 4 becomes step 2, and so forth, with the former step 1 as the final step. This is not to say that particles can turn around and go back where they came from. What it really means is that the reverse process transfers energy in the same quantities, but in a reverse direction. If a helium atom loses 2 x 10-16 erg of translational energy in the forward process, and a hydrogen molecule gains 1 x 10-16 erg translational, .4 x 10-16 erg vibrational, and .6 x 10-16 erg rotational: the helium gas gains 2 x 10-16 erg of tanslational energy in the reverse process, and the hydrogen molecule loses 1 x 10-16 erg translational, .4 x 10-16 erg vibrational, and .6 x 10-16 erg rotational. In the reverse process, all of the forces are the same as they are in the forward process, but the forces grow in the reverse process at the point where they would be shrinking in the forward process. Of course, the exact reverse, in all of its details, never really happens, for the same reason that no two molecules are exactly alike. The positions and motions of the parts keep changing. Even the quantity of neg-pos is different in any two molecules of the same species.

I have not shown the contribution of electrons to the heat capacity, because the heat capacity of helium is the same as that of the ideal gas. According to quantum theory, the helium atom cannot absorb an energy of less than 19.75 ev of electronic vibrational energy. Not even one molecule per mole has that much energy when the molar temperature is as high as 1000 K. According to neg-pos theory, the electron can have any intermediate energy. The difficulty is in getting the energy delivered. The electron has very little mass compared with a nucleus. As soon as an interaction starts, the electron responds by moving to each new point of zero force without delay. In a molecular collision, the rate of change of position of the point of zero force for an electron, is very slow, compared with the speed with which an electron can respond. The force gets no chance to build up. It remains near zero. In the receding phase of the collision, when the molecules separate, the electron falls back to its new point of zero force promptly, It never picks up kinetic energy worth mentioning, and it never picks up potential, because it is always at a point of zero force. The helium electron has no trouble picking up intermediate energies from free electrons. The catch is that free electrons are seldom encountered. Should a helium electron take up oscillation, it promptly loses the oscillation in the next collision. The shifting of the point of zero force during a collision, changes the frequency of resonance of the oscillation of the electron. This is like pushing a playground swing with the wrong timing. It slows the vibration of the electron, and quickly brings it to a halt. Sometimes a collision between helium atoms takes place with the nuclei slightly off the line between them. (Fig. 14)

It is tantamount to a collision with a free electron. This does give the electron intermediate electronic vibrational energy. However it is rare and soon lost in an interaction. In a mole of helium atoms, a few excited electrons more or less, make no difference in the Cv. It works the same way for hydrogen molecules and for other molecules. At ordinary temperatures, the electrons cannot pick up energy in a collision. When the temperature of the mole of gas is high enough to make the mass glow, plenty of electrons are caused to oscillate by molecular collisions. Many electrons in molecules have resonant frequencies that are much lower than the resonant frequencies in the helium atom. At very high temperatures, molecules collide at such high speeds that the points of zeo force of the electrons move as fast as electrons in vibration. As the colliding molecules separate, the electron is left oscillating through its original point of zero force. The heat capacity of a gas increases at very high temperatures, because of the electronic vibration. A molecule with more than two atoms has more heat capacity, because it has more particles that can rotate and vibrate. The principles that apply to the hydrogen molecule, also apply to gases with three or four atoms per molecule.

In liquds the situation is very different. The substance is in the liquid state because the molecules attract each other. Each molecule has its rotation and vibration. In addition, there is potential energy that increases as the space between molecule increases. When a molecule vibrates with more energy, it makes a wider swing, and it bumps into other molecules. As molecules make room for each other,the entire liquid expands. There is more space between molecules. As the separation between molecules increases, the potential energy increases. H2O gas molecules have a heat capacity at constant pressure, Cp, of 9 calories per mole per degree at temperatures close to 700 K. At higher temperatures, the Cp of H2O may go as high as 12 cal/deg mol. H2O liquid has a heat capacity of 18 calories per mole per degree. It is fairly constant between 273 K and 373 K. The variations in heat capacity at different temperatures in that range are small enough to be negligible. The remarkable thing about the liquid is that it has so much more heat capacity than the solid or the gas. In ice, the H2O molecule must vibrate in its place in the crystal. It doesn't get much chance to spend time in the region of weaker attraction. In the liquid, the H2O molecule has enough kinetic energy to carry it away from its neighbors. The H2O molecules in the liquid continue to attract each other, so their potential energy is only part-time. As the molecules approach each other,their potential energy becomes kinetic energy, for each increase of 1 degree, their kinetic energy increases and their separation increases. With an increase in separation, the H2O molecule has more potential energy.

The separation of the positive part of one H2O molecule from the negative part of the nearest H2O molecule is what counts, not the separation between centers of mass. One of the reasons why the heat capacity of liquid H2O is higher than the heat capacity of ice, is that the molecules can rotate in the liquid. In ice, the molecules are so oriented that the hydrogen part of one molecule is close to the oxygen part of the other molecule, and the molecules can't rotate, because of the strong attraction between hydrogen and oxygen. The strongest evidence for quantum theory was the decrease in the heat capacity of solids as they lose temperature and approach absolute zero. It was reasoned that if the parts of a solid, the ions, atoms, or molecules, could absorb energy at any value, then the solid would have a heat capacity at 1 K nearly as large as its heat capacity at 300 K. As a matter of fact, the heat cpacity of a solid is near zero at 1 K, and about 4.14 x 10-16 erg per atom per degree at 300 K. Some solids, like copper, reach high values of Cv before other solids, like silicon, as the temperature rises. Among the slowest solids to reach high values with rise in temperature is the diamond. According to quantum theory, the nucleus of each atom in a solid has a minumum vibrational energy which it cannot transfer. Since that enegy can't be transferred, it has nothing to do with heat capacity, which is the quantity of energy that is transferred when the temperature changes one degree. Above the minimum energy level, there are successive energy levels.

The nucleus cannot vibrate at intermediate energy levels. The individual atom in a solid must gain Cv only by quantum jumps. At each temperature, no matter how low, as long as it is above 0 K, a few of the atoms have higher than minimum energy. If the total of all of the atoms in the solid is divided by the number of atoms, the quotient is the energy per atom of the solid. The minimum energy is not counted. Most of the atoms are counted as zeros. As the temperature rises, a larger fraction of the atoms have some energy in addition to the minimum. The temperature has to be somewhere in the range of 300 K before nearly every atom has an energy above the minimum. According to neg-pos theory, a nucleus can have zero energy at 0 K and every value of energy between zero and the energy that makes the atom leave the solid. Each nucleus in a solid is at rest at its point of zero force, at zero K. When the temperature rises one degree, in the environment of a solid, outside particles strike the surface atoms of the solid. The environmental paricles are moving very slowly at 1 K. The approaching particle, with its forces of attraction and repulsion, shifts the point of zero force of the nucleus of one of the atoms of the solid very gradually, because of the low speed. At the slow rate of motion of the point of zero force, the inertia of the nucleus is not sufficient to make the nucleus lag behind. The nucleus shifts almost as promptly as the point of zero force. Remember that a slight departure from the point of zero force is accompanied by a slight restoring force. With little force, there is little energy. Therefore there is hardly any transfer of energy to a solid at 1 K, and the heat capacity is near zero.

At a temperature of 50 K, the environmental particle has a much greater speed than at 1 K. The inertia of the nucleus of an atom in the solid is enough to make it lag behind the rate of shift of the point of zero force, the nucleus is farther separated from the point of zero force, At a larger separation, there is a larger force. With a larger force, there is a larger potential energy. Therefore the heat capacity of the solid increases with an increase of temperature. The heat capacity stops increasing in the range of 300 K, because a lag becomes less significant as the nucleus gets closer to the close range repulsive force. An infinitessimal difference in separation, deep in the region of short range repulsive force, makes a big difference in the magnitude of the force. I have discussed only the surface atoms of the solid so far. All of the nearest neighbors of the surface atom are affected by the shift of the point of zero force for the surface atom. After all, the point of zero force of the nucleus of the surface atom, when it is not involved in a collision, is determined by the forces from its nearest neighbors. Each force acts in two opposite directions. After a collision, the surface nucleus vibrates. The nucleus of the surface atom shifts relative to the nuclei of all its nearest neighbors. Thus there is a vibration of points of zero force, as well as a vibration of nuclei. This causes vibration to pass to other atoms.

Of course, the transfer of energy incurs the loss of some energy in the surface atom. The wider the energy is distributed, the less energy there is per atom. However, if the solid is exposed to an environment at constant temperature, it keeps receiving energy, until the entire solid has as much energy as it can hold at that temperature. The rate of growth of the heat capacity is different for different solids. As a rule, those solids with more massive nucei, have a faster rate of growth of heat capacity with increasing temperature. Carbon in the solid state as a diamond has a very slow rate of growth of heat capacity with temperature. The carbon atom has only 6 protons. The nucleus has very little inertia. It hardly lags behind the shift in the point of zero force. Therefore, the nucleus doesn't get into the region of strong repulsive force, and it doesn't acquire much energy. Finally, at a temperature close to 1000 K, the point of zero force shifts fast enough to get well ahead of the carbon nucleus. Then the heat capacity per atom of carbon in a diamond is about 4.14 x 10-16 erg per degree. It should be evident by now, that the manner in which the heat capacity of most substances changes when the temperature changes, does not reinforce quantum theory. Neither does it detract from quantum theory.