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# Construct Definitions

Thermal Equilibrium and Equipartition of Energy Among Modes

There are several important ideas here that all go together. By thermal equilibrium we mean that the random energy fluctuations associated with the motions of the atoms and molecules about their equilibrium positions in a solid or liquid or their random motions when in the gas phase, will over time, become uniformly distributed throughout the entire sample. That is, there will be about as much energy associated with the random energies of a small piece of the sample, but still containing 1015 or so particles, as in any other same size small piece. This is what we mean by thermal equilibrium on a particle basis. It is also similar to what we would say about the temperature. If we wait for a sufficiently long time, the temperature will become uniform throughout the sample. There would seem to be a direct connection between temperature and the disordered random motion associated with thermal energy. In fact there is a very definite connection.

In the Intro Particle Model of Matter we saw that each atom in a liquid or solid acted as if it vibrated like a spring-mass in each of three dimensions. An interesting question is how many ways does each of these particles “have energy?” We need to think about how many ways a spring-mass has energy. It has a KE and also a PE. We simply stated at the time that the average PE was the same as the average KE. We will simply take this as reasonable at this point. Because of the randomness or disordered-ness of the thermal motions of all the little mass springs in all three of the directions in space, it is plausible that on average, each spring would have the same average PE as would any other spring. And also the same KE as any other spring. In fact this is exactly what happens. It can be rigorously proven for all energies that depend on the square of a position or speed variable. Thus, in addition to working for spring-mass systems, it works for unbound atoms in the gas phase, which will have translational kinetic energies, since these energies depend on the square of their translational speed. It works for molecules that rotate, which will have rotational kinetic energies, since these energies depend on a square of a rotational speed.

So back to our question. How many ways does each spring have “to have” energy? The answer is two: one KE and one PE. How many ways does each particle in a solid or liquid have to have energy? Well, there are three springs and two ways per spring, so it must be six. Each particle in a solid or liquid has six ways to have energy. Now combine this with what we just argued regarding thermal equilibrium. On average, each “way to have energy” would have the same amount of energy when averaged over a sufficiently long period of time. There is a name, or label, for “way to have energy.” The name is “mode.” So we say that each particle has six modes in a solid or liquid. And on average when the sample is in thermal equilibrium, each mode has the same amount of energy (on average). This principle is referred to as the principle of equipartition of energy.

Freezing out of modes

Sometimes, however, the modes don’t “get excited” due to the quantization of energy levels. At low temperatures, the quantum splitting between energy levels, which you are familiar with from chemistry, keeps all but the ground state level from being populated, or having any energy. When this happens, we say that mode is “frozen out.” It is as if it didn’t exist. Frozen out modes cannot share thermal energy. So in the following statements, we usually put in the qualifier, “active modes,” meaning that only active modes share the thermal energy equally among themselves.

Heat Capacity at constant volume

Because we will want to compare values of heat capacity to our predicted values of thermal energy from the particle model of matter, we need to be careful that we are actually comparing the same things. We know that if a force acts through a distance, work will be done by one physical object on another. When we make a heat capacity measurement, we don’t want the sample doing work on the atmosphere or the container it is in. Therefore, we specify that the sample be kept at constant volume during the heat capacity measurement. This is designated with a subscript “v.” The important point here is that we have a way to directly measure the change in the thermal energy by measuring the heat capacity of a sample at constant volume, ensuring all the heat we put into the sample goes to changing its thermal energy and not doing some work by expanding the container or pushing against the air in the room.

# Meaning of the Model Relationships

1) In gases, the translational kinetic energy of each particle can be divided into three independent “pieces”, each one corresponding to one of the three independent spatial dimensions; each particle in a gas has at least these three independent modes.

Because there are no springs connecting the particles in a gas, there are only three modes per particle, if the particle itself has no internal modes.

2) In liquids and solids, the oscillations of each particle in its single-particle potential can be modeled as a mass held in place by three perpendicular springs. The potential and kinetic energies that are associated with those oscillations can each be divided into three independent modes, each one corresponding to one of the three independent spatial dimensions; each particle in a liquid or solid has at least these six independent modes.

3) In all phases (s, l, g) polyatomic molecules may have additional energies associated with rotations and internal vibrations of the molecule. These might contribute additional modes, depending on whether or not they are frozen out at the temperature in question. At room temperature the vibrational modes of most diatomic molecules (but not translational or rotational modes) are frozen out.

A diatomic molecule, for example, might vibrate or it might rotate. These could contribute additional modes.

4) The thermal energy of a substance is the total energy in all the active modes of all the particles comprising the substance.

5) In thermal equilibrium all active modes have, on average, the same amount of energy. This principle is referred to as “equipartition of energy.”

6) The amount of energy, on average, in an active mode is directly proportional to the temperature. The proportionality constant, for historical reasons, is written kB/2, where kB is the Boltzmann constant. (kB = 1.38 ´ 10-23 J/K)

Now we get the connection with temperature and modes. Temperature is actually a measure of the average energy in an active mode when the sample is in thermal equilibrium.

7) The total thermal energy of an object in thermal equilibrium is equal to the product of [the total number of active modes] and [the average energy per mode].

This last relationship gives us a precise notion of what thermal energy really is.

8) When the change in thermal energy is due solely to the addition or removal of energy as heat, the constant volume heat capacity, CV, is given by the rate of change of thermal energy with respect to temperature.

CV is the macroscopic variable that corresponds to how many modes there are on a particle basis.

## Algebraic Representations

The three essential relationships (6., 7, and 8) summarize the three really big ideas here.

Relationship (6) $$E_{thermal} / \text {mode} = \frac{1}{2}k_{b}T$$

This is the Big One!

Relationship (7) $$E_{thermal-tot} = \text{(total number of active modes)} \times \frac{1}{2}k_{b} T$$

Relationship (8) $$C_{v} = \frac{dE}{dT}$$