Stefan-Boltzmann Law of Radiation

61. For the following we imagine a perfectly evacuated hollow cylinder with an absolutely tight-fitting piston free to move in a vertical direction with no friction. A part of the walls of the cylinder, say the rigid bottom, should consist of a black body, whose temperature $T$ may be regulated arbitrarily from the outside. The rest of the walls including the inner surface of the piston may be assumed as totally reflecting. Then, if the piston remains stationary and the temperature, $T$, constant, the radiation in the vacuum will, after a certain time, assume the character of black radiation (Sec. 50) uniform in all directions. The specific intensity, $K$, and the volume density, $u$, depend only on the temperature, $T$, and are independent of the volume, $V$, of the vacuum and hence of the position of the piston.

If now the piston is moved downward, the radiation is compressed into a smaller space; if it is moved upward the radiation expands into a larger space. At the same time the temperature of the black body forming the bottom may be arbitrarily changed by adding or removing heat from the outside. This always causes certain disturbances of the stationary state. If, however, the arbitrary changes in $V$ and $T$ are made sufficiently slowly, the departure from the conditions of a stationary state may always be kept just as small as we please. Hence the state of radiation in the vacuum may, without appreciable error, be regarded as a state of thermodynamic equilibrium, just as is done in the thermodynamics of ordinary matter in the case of so-called infinitely slow processes, where, at any instant, the divergence from the state of equilibrium may be neglected, compared with the changes which the total system considered undergoes as a result of the entire process.

If, e.g., we keep the temperature $T$ of the black body forming the bottom constant, as can be done by a suitable connection between it and a heat reservoir of large capacity, then, on raising the piston, the black body will emit more than it absorbs, until the newly made space is filled with the same density of radiation as was the original one. Vice versa, on lowering the piston the black body will absorb the superfluous radiation until the original radiation corresponding to the temperature $T$ is again established. Similarly, on raising the temperature $T$ of the black body, as can be done by heat conduction from a heat reservoir which is slightly warmer, the density of radiation in the vacuum will be correspondingly increased by a larger emission, etc. To accelerate the establishment of radiation equilibrium the reflecting mantle of the hollow cylinder may be assumed white (Sec. 10), since by diffuse reflection the predominant directions of radiation that may, perhaps, be produced by the direction of the motion of the piston, are more quickly neutralized. The reflecting surface of the piston, however, should be chosen for the present as a perfect metallic reflector, to make sure that the radiation pressure (66) on the piston is Maxwell’s. Then, in order to produce mechanical equilibrium, the piston must be loaded by a weight equal to the product of the radiation pressure $p$ and the cross-section of the piston. An exceedingly small difference of the loading weight will then produce a correspondingly slow motion of the piston in one or the other direction.

Since the effects produced from the outside on the system in question, the cavity through which the radiation travels, during the processes we are considering, are partly of a mechanical nature (displacement of the loaded piston), partly of a thermal nature (heat conduction away from and toward the reservoir), they show a certain similarity to the processes usually considered in thermodynamics, with the difference that the system here considered is not a material system, e.g., a gas, but a purely energetic one. If, however, the principles of thermodynamics hold quite generally in nature, as indeed we shall assume, then they must also hold for the system under consideration. That is to say, in the case of any change occurring in nature the energy of all systems taking part in the change must remain constant (first principle), and, moreover, the entropy of all systems taking part in the change must increase, or in the limiting case of reversible processes must remain constant (second principle).

62. Let us first establish the equation of the first principle for an infinitesimal change of the system in question. That the cavity enclosing the radiation has a certain energy we have already (Sec. 22) deduced from the fact that the energy radiation is propagated with a finite velocity. We shall denote the energy by $U$. Then we have

where $u$ the volume density of radiation depends only on the temperature $T$ of the black body at the bottom.

The work done by the system, when the volume $V$ of the cavity increases by $dV$ against the external forces of pressure (weight of the loaded piston), is $pdV$, where $p$ represents Maxwell’s radiation pressure (66). This amount of mechanical energy is therefore gained by the surroundings of the system, since the weight is raised. The error made by using the radiation pressure on a stationary surface, whereas the reflecting surface moves during the volume change, is evidently negligible, since the motion may be thought of as taking place with an arbitrarily small velocity.

If, moreover, $Q$ denotes the infinitesimal quantity of heat in mechanical units, which, owing to increased emission, passes from the black body at the bottom to the cavity containing the radiation, the bottom or the heat reservoir connected to it loses this heat $Q$, and its internal energy is decreased by that amount. Hence, according to the first principle of thermodynamics, since the sum of the energy of radiation and the energy of the material bodies remains constant, we have

According to the second principle of thermodynamics the cavity containing the radiation also has a definite entropy. For when the heat $Q$ passes from the heat reservoir into the cavity, the entropy of the reservoir decreases, the change being

Therefore, since no changes occur in the other bodies—inasmuch as the rigid absolutely reflecting piston with the weight on it does not change its internal condition with the motion—there must somewhere in nature occur a compensation of entropy having at least the value $\frac{Q}{T}$, by which the above diminution is compensated, and this can be nowhere except in the entropy of the cavity containing the radiation. Let the entropy of the latter be denoted by $S$.

Now, since the processes described consist entirely of states of equilibrium, they are perfectly reversible and hence there is no increase in entropy. Then we have

or from (71)

In this equation the quantities $U$, $p$, $V$, $S$ represent certain properties of the heat radiation, which are completely defined by the instantaneous state of the radiation. Therefore the quantity $T$ is also a certain property of the state of the radiation, i.e., the black radiation in the cavity has a certain temperature $T$ and this temperature is that of a body which is in heat equilibrium with the radiation.

63. We shall now deduce from the last equation a consequence which is based on the fact that the state of the system considered, and therefore also its entropy, is determined by the values of two independent variables. As the first variable we shall take $V$, as the second either $T$, $u$, or $p$ may be chosen. Of these three quantities any two are determined by the third together with $V$. We shall take the volume $V$ and the temperature $T$ as independent variables. Then by substituting from (66) and (70) in (73) we have

From this we obtain

On partial differentiation of these equations, the first with respect to $V$, the second with respect to $T$, we find

or

and on integration

and from (21) for the specific intensity of black radiation

Moreover for the pressure of black radiation

and for the total radiant energy

This law, which states that the volume density and the specific intensity of black radiation are proportional to the fourth power of the absolute temperature, was first established by J. Stefan17 on a basis of rather rough measurements. It was later deduced by L. Boltzmann18 on a thermodynamic basis from Maxwell’s radiation pressure and has been more recently confirmed by O. Lummer and E. Pringsheim19 by exact measurements between $100^{\circ }$ and $1300^{\circ }C.$, the temperature being defined by the gas thermometer. In ranges of temperature and for requirements of precision for which the readings of the different gas thermometers no longer agree sufficiently or cannot be obtained at all, the Stefan-Boltzmann law of radiation can be used for an absolute definition of temperature independent of all substances.

64. The numerical value of the constant $a$ is obtained from measurements made by F. Kurlbaum.20 According to them, if we denote by $S_t$ the total energy radiated in one second into air by a square centimeter of a black body at a temperature of $t\circ C.$, the following equation holds

Now, since the radiation in air is approximately identical with the radiation into a vacuum, we may according to (7) and (76) put

and from this

therefore

Recently Kurlbaum has increased the value measured by him by $2.5$ per cent.,21 on account of the bolometer used being not perfectly black, whence it follows that $a=7.24\cdot 10^{-15}$.

Meanwhile the radiation constant has been made the object of as accurate measurements as possible in various places. Thus it was measured by Féry, Bauer and Moulin, Valentiner, Féry and Drecq, Shakespear, Gerlach, with in some cases very divergent results, so that a mean value may hardly be formed.

For later computations we shall use the most recent determination made in the physical laboratory of the University of Berlin22

From this $a$ is found to be

which agrees rather closely with Kurlbaum’s corrected value.

65. The magnitude of the entropy $S$ of black radiation found by integration of the differential equation (73) is

In this equation the additive constant is determined by a choice that readily suggests itself, so that at the zero of the absolute scale of temperature, that is to say, when $u$ vanishes, $S$ shall become zero. From this the entropy of unit volume or the volume density of the entropy of black radiation is obtained,

66. We shall now remove a restricting assumption made in order to enable us to apply the value of Maxwell’s radiation pressure, calculated in the preceding chapter. Up to now we have assumed the cylinder to be fixed and only the piston to be free to move. We shall now think of the whole of the vessel, consisting of the cylinder, the black bottom, and the piston, the latter attached to the walls in a definite height above the bottom, as being free to move in space. Then, according to the principle of action and reaction, the vessel as a whole must remain constantly at rest, since no external force acts on it. This is the conclusion to which we must necessarily come, even without, in this case, admitting a priori the validity of the principle of action and reaction. For if the vessel should begin to move, the kinetic energy of this motion could originate only at the expense of the heat of the body forming the bottom or the energy of radiation, as there exists in the system enclosed in a rigid cover no other available energy; and together with the decrease of energy the entropy of the body or the radiation would also decrease, an event which would contradict the second principle, since no other changes of entropy occur in nature. Hence the vessel as a whole is in a state of mechanical equilibrium. An immediate consequence of this is that the pressure of the radiation on the black bottom is just as large as the oppositely directed pressure of the radiation on the reflecting piston. Hence the pressure of black radiation is the same on a black as on a reflecting body of the same temperature and the same may be readily proven for any completely reflecting surface whatsoever, which we may assume to be at the bottom of the cylinder without in the least disturbing the stationary state of radiation. Hence we may also in all the foregoing considerations replace the reflecting metal by any completely reflecting or black body whatsoever, at the same temperature as the body forming the bottom, and it may be stated as a quite general law that the radiation pressure depends only on the properties of the radiation passing to and fro, not on the properties of the enclosing substance.

67. If, on raising the piston, the temperature of the black body forming the bottom is kept constant by a corresponding addition of heat from the heat reservoir, the process takes place isothermally. Then, along with the temperature $T$ of the black body, the energy density $u$, the radiation pressure $p$, and the density of the entropy $s$ also remain constant; hence the total energy of radiation increases from $U=uV$ to $U\prime =uV\prime$, the entropy from $S=sV$ to $S\prime =sV\prime$ and the heat supplied from the heat reservoir is obtained by integrating (72) at constant $T$,

or, according to (81) and(75),

Thus it is seen that the heat furnished from the outside exceeds the increase in energy of radiation $(U\prime -U)$ by $\frac{1}{3}(U\prime -U)$. This excess in the added heat is necessary to do the external work accompanying the increase in the volume of radiation.

68. Let us also consider a reversible adiabatic process. For this it is necessary not merely that the piston and the mantle but also that the bottom of the cylinder be assumed as completely reflecting, e.g., as white. Then the heat furnished on compression or expansion of the volume of radiation is $Q=0$ and the energy of radiation changes only by the value $pdV$ of the external work. To insure, however, that in a finite adiabatic process the radiation shall be perfectly stable at every instant, i.e., shall have the character of black radiation, we may assume that inside the evacuated cavity there is a carbon particle of minute size. This particle, which may be assumed to possess an absorbing power differing from zero for all kinds of rays, serves merely to produce stable equilibrium of the radiation in the cavity (Sec. 51 et seq.) and thereby to insure the reversibility of the process, while its heat contents may be taken as so small compared with the energy of radiation, $U$, that the addition of heat required for an appreciable temperature change of the particle is perfectly negligible. Then at every instant in the process there exists absolutely stable equilibrium of radiation and the radiation has the temperature of the particle in the cavity. The volume, energy, and entropy of the particle may be entirely neglected.

On a reversible adiabatic change, according to (72), the entropy $S$ of the system remains constant. Hence from (80) we have as a condition for such a process

or, according to (77),

i.e., it increases in proportion to the absolute temperature, although the volume becomes smaller.

69. Let us finally, as a further example, consider a simple case of an irreversible process. Let the cavity of volume $V$, which is everywhere enclosed by absolutely reflecting walls, be uniformly filled with black radiation. Now let us make a small hole through any part of the walls, e.g., by opening a stopcock, so that the radiation may escape into another completely evacuated space, which may also be surrounded by rigid, absolutely reflecting walls. The radiation will at first be of a very irregular character; after some time, however, it will assume a stationary condition and will fill both communicating spaces uniformly, its total volume being, say, $V\prime$. The presence of a carbon particle will cause all conditions of black radiation to be satisfied in the new state. Then, since there is neither external work nor addition of heat from the outside, the energy of the new state is, according to the first principle, equal to that of the original one, or $U\prime =U$ and hence from (78)

which defines completely the new state of equilibrium. Since $V\prime >V$ the temperature of the radiation has been lowered by the process.

According to the second principle of thermodynamics the entropy of the system must have increased, since no external changes have occurred; in fact we have from (80)

70. If the process of irreversible adiabatic expansion of the radiation from the volume $V$ to the volume $V\prime$ takes place as just described with the single difference that there is no carbon particle present in the vacuum, after the stationary state of radiation is established, as will be the case after a certain time on account of the diffuse reflection from the walls of the cavity, the radiation in the new volume $V\prime$ will not any longer have the character of black radiation, and hence no definite temperature. Nevertheless the radiation, like every system in a definite physical state, has a definite entropy, which, according to the second principle, is larger than the original $S$, but not as large as the $S\prime$ given in (82). The calculation cannot be performed without the use of laws to be taken up later (see Sec. 103). If a carbon particle is afterward introduced into the vacuum, absolutely stable equilibrium is established by a second irreversible process, and, the total energy as well as the total volume remaining constant, the radiation assumes the normal energy distribution of black radiation and the entropy increases to the maximum value $S\prime$ given by (82).