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3-3: The Heat Capacity of a Vacuum
We have discussed the heat capacities of matter or gas on the preceding page. Let us study the heat capacity of a vacuum here.
You might feel it strange that we will discuss the heat capacity of a vacuum which contains nothing. Does there exist a concept of temperature or energy in a vacuum where nothing exists? Yes there is. There is an energy named radiation.



For example, heat (or mainly infrared light) is emitted from a highly heated electric heater (stove). This warms up human bodies and others around. This is called heat radiation. (See the above figure).
Heat radiation is not a heat conduction through the air. Even if there is nothing in between, heat can be conducted even through a vacuum. Namely, radiation can exist even in a vacuum; i.e., there exists an energy in a form of light or electromagnetic wave in a vacuum.
Accordingly, since even a vacuum can contain energy, we can measure the heat capacity of a vacuum, if we can define the temperature of the vacuum. How can we consider the temperature of a vacuum?

[Cavity Radiation]
Suppose that we put an object in a heat bath (or heat reservoir) of the absolute temperature T. If the temperature of the object is lower than T, the object absorbs heat from the heat bath, but if it is higher than T, then the object emits heat into the bath. After a while, the object would be of the same temperature T and comes in a thermal equilibrium. Looking at this thermal equilibrium in detail, the object is always emitting and absorbing some amount of heat. The amounts of the emitting or absorbing heat are equal on average and they are keeping a balance.
Consider that we put a vacuum in the heat bath instead of an object. For example, make a hollow space (cavity) inside a block of iron and keep the temperature of this iron block to be T. In a thermal equilibrium, the outer part of the iron block would be a kind of heat bath and the temperature of the cavity (vacuum) can be thought to be T. In this situation, the cavity would be filled with radiation, i.e. lights or electromagnetic waves. This radiation is called the cavity radiation. (See the following figure.)



In order to know what frequency of light exists in the cavity radiation, and how strong the intensity of the radiation is, we may observe the inside of the cavity through a small hole on the wall. It should be small enough so as not to disturb the inside too much. Practically, the inside of a blast furnace in a steel industry may be considered to present a kind of state of cavity radiation. Therefore we can observe an example of cavity radiation through a small window on the wall of the blast furnace.

[Difficulty in the Heat Capacity of a Vacuum]
As studied in the part of Heat Capacities of Solids on the preceding page, a solid is a collection of harmonic oscillators. We obtained the heat capacity of this solid, considering that the constant amount of energy kT is partitioned to these harmonic oscillators. The result is well fit to experiment as far as the temperature of the solid is not so low.
According to Maxwellian electromagnetism, electromagnetic wave (light) in a vacuum is oscillation of electromagnetic field. It is understood to be equivalent to the vibration of continuous elastic body. If it is a one-dimensional vibration, we can imagine vibration of bowstring like a guitar or a harp, and if it is two-dimensional, we can associate it with a drum. In the case of electromagnetic wave (light), it is three-dimensional, so that we can associate it with the oscillation of the air, i.e. the sound, assuming that the air is a continuous substance.
These oscillations are formally equivalent to collections of vibrations of springs (harmonic oscillators). Therefore the way to obtain the heat capacities of these systems is exactly the same as that to calculate the heat capacity of a solid matter. Difference is in the number of the harmonic oscillators, i.e. the number of the degrees of freedom. In a case of 1 mole of solid, the number of the degrees of freedom is considered to be 3 times as large as the number of the molecules, i.e. 3 times of Avogadro's constant, 3NA. However, the electromagnetic field in a vacuum is a continuous elastic body. This is an essentially different point.
The normal oscillations of a string with length L
The waves with wavelengths, 2L, 2L/2, 2L/3, 2L/4, and so on, are possible.

Let us consider an oscillation of a string with the length L . Its normal modes of oscillation are shown in the above figure. Let the wavelength of the normal oscillation be and the frequency . Hence the allowed wavelength is

Let the sound speed in the string be c. Because

the corresponding frequency is Taking the frequency as one-dimensional coordinate axis, we plot dots at the positions of the values of the normal mode frequencies on this axis, then they distribute uniformly at equal interval in units of c/(2L). Hence, the number of the normal oscillations with frequencies between and is

Because the electromagnetic field in a vacuum (or cavity) is three-dimensional, we have to extend the above discussion to a three-dimensional space. This is somewhat tedious and we omit here, but the result is

where V is the volume of the cavity and c is the light speed.
Since the oscillation of the electromagnetic field in a vacuum is just the normal oscillation of a perfectly continuous body, there must be no upper limit in its frequency; namely any high frequency is possible. This means that the number of the corresponding harmonic oscillator is infinite in the cavity. In other words, the number of degrees of freedom in a vacuum is infinite. If the same amount of energy kT is partitioned to these infinite number of degrees of freedom, the total energy of a vacuum becomes infinite, and the heat capacity of a vacuum would be infinite. This implies that a vacuum can absorb an infinite amount of energy. But such a strange situation never happens in practice. Therefore there must be something wrong in the above discussion.
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