where R denotes the universal gas constant. N denotes the number of “real molecules” in a gram equivalent, and T the absolute temperature. The energy E is equal to two-thirds the kinetic energy of a free monatomic gas particle because of the equality the time average values of the kinetic and potential energies of the oscillator. If through any cause—in our case through radiation processes—it should occur that the energy of an oscillator takes on a time-average value greater or less than E , then the collisions with the free electrons and molecules would lead to a gain or loss of energy by the gas, different on the average from zero. Therefore, in the case we are considering, dynamic equilibrium is possible only when each oscillator has the average energy Ē .
We shall now proceed to present a similar argument regarding the interaction between the oscillators and the radiation present in the cavity. Herr Planck has derivedq the condition for the dynamics equilibrium in this case under the supposition that the radiation can be considered a completely random process.r He found
where (Eν ) is the average energy (per degree of freedom) of an oscillator with eigenfrequency ν, L the velocity of light, ν the frequency, and ρνdν the energy per unit volume of that portion of the radiation with frequency between ν and ν + dν.
If the radiation energy of frequency ν is not continually increasing or decreasing, the following relations must obtain:
These relations, found to be the conditions of dynamic equilibrium, not only fail to coincide with experiment, but also state that in our model there can be not talk of a definite energy distribution between ether and matter. The wider the range of wave numbers of the oscillators, the greater will be the radiation energy of the space, and in the limit we obtain
2. CONCERNING PLANCK’S DETERMINATION OF THE FUNDAMENTAL CONSTANTS
We wish to show in the following that Herr Planck’s determination of the fundamental constants is, to a certain extent, independent of his theory of blackbody radiation.
Planck’s formula,s which has proved adequate up to this point, gives for ρν
For large values of T/ν; i.e. for large wavelengths and radiation densities, this equation takes the form
It is evident that this equation is identical with the one obtained in Sec. 1 from the Maxwellian and electron theories. By equating the coefficients of both formulas one obtains
or
i.e., an atom of hydrogen weighs 1/N grams = 1.62 × 10−24 g. This is exactly the value found by Herr Planck, which in turn agrees with values found by other methods.
We therefore arrive at the conclusion: the greater the energy density and the wavelength of a radiation, the more useful do the theoretical principles we have employed turn out to be: for small wavelengths and small radiation densities, however, these principles fail us completely.
In the following we shall consider the experimental facts concerning blackbody radiation without invoking a model for the emission and propagation of the radiation itself.
3. CONCERNING THE ENTROPY OF RADIATION
The following treatment is to be found in a famous work by Herr W. Wien and is introduced here only for the sake of completeness.
Suppose we have radiation occupying a volume v. We assume that the observable properties of the radiation are completely determined when the radiation density ρ(ν) is given for all frequencies.t Since radiation of different frequencies are to be considered independent of each other when there is no transfer of heat or work, the entropy of the radiation can be represented by
where ϕ is a function of the variables ρ and ν.
ϕ can be reduced to a function of a single variable through formulation of the condition that the entropy of the radiation is unaltered during adiabatic compression between reflecting walls. We shall not enter into this problem, however, but shall directly investigate the derivation of the function ϕ from the blackbody radiation law.
In the case of blackbody radiation, ρ is such a function of ν that the entropy is maximum for a fixed value of energy; i.e.,
providing
From this it follows that for every choice of δρ as a function of ν
where λ is independent of ν. In the case of blackbody radiation, therefore, ∂ϕ/∂ρ is independent of ν.
The following equation applies when the temperature of a unit volume of blackbody radiation increases by dT
or, since ∂ϕ/∂ρ is independent of ν.
Since dE is equal to the heat added and since the process is reversible, the following statement also appliesdS = (1/T) d E.
By comparison one obtains∂ϕ/∂ρ = 1/ T.
This is the law of blackbody radiation. Therefore one can derive the law of blackbody radiation from the function ϕ, and, inversely, one can derive the function ϕ by integration, keeping in mind the fact that ϕ vanishes when ρ = 0.
4. ASYMPTOTIC FROM FOR THE ENTROPY OF MONOCHROMATIC RADIATION AT LOW RADIATION DENSITY
From existing observations of the blackbody radiation, it is clear that the law originally postulated by Herr W. Wien,ρ = αν3e −βν/T ,
is not exactly valid. It is, however, well confirmed experimentally for large values of ν / T. We shall base our analysis on this formula, keeping in mind that our results are only valid within certain limits.
This formula gives immediately(1/T) = − (1/βν) ln (ρ/αν3)
and then, by using the relation obtained in the preceeding section,
Suppose that we have radiation of energy E, with frequency between ν and ν + dν , enclosed in volume ν. The entropy of this radiation is:
If we confine ourselves to investigating the dependence of the entropy on the volume occupied by the radiation, and if we denote by S0 the entropy of the radiation at volume v0, we obtainS −S0 = (E/βν) in (v/v0) .
This equation shows that the entropy of a monochromatic radiation of sufficiently low density varies with the volume in the same manner as the entropy of an ideal gas or a dilute solution. In the following, this equation will be interpreted in accordance with the principle introduced into physics by Herr Boltzmann, namely that the entropy of a system is a function of the probability its state.
5. MOLECULAR—THEORETIC INVESTIGATION OF THE DEPENDENCE OF THE ENTROPY OF GASES AND DILUTE SOLUTIONS ON THE VOLUME
In the calculation of entropy by molecular–theoretic methods we frequently use the word “probability” in a sense differing from that employed in the calculus of probabilities. In particular “gases of equal probability” have frequently been hypothetically established when one theoretical models being utilized are definite enough to permit a deduction rather than a conjecture. I will show in a separate paper that the so-called “statistical probability” is fully adequate for the treatment of thermal phenomena, and I hope that by doing so I will eliminate a logical difficulty that obstructs the application of Boltzmann’s Principle. Here, however, only a general formulation and application to very special cases will be given.
If it is reasonable to speak of the probability of the state of a system, and futhermore if every entropy increase can be understood as a transition to a state of higher probability, then the entropy S1 of a system is a function of W1, the probability of its instantaneous state. If we have two noninteracting systems S1 and S2, we can write
If one considers these two systems as a single system of entropy S and probability W, it follows thatS = S1 + S2 = ϕ(W )
andW = W1 · W2.
The last equation says that the states of the two systems are independent of each other.
From these equation it follows thatϕ (W1 · W2) = ϕ1 (W1) + ϕ2 (W2)
and finally
The quantity C is therefore a universal constant; the kinetic theory of gases shows its value to be R/N, where the constants R and N have been defined above. If S0 denotes the entropy of a system in some initial state and W denotes the relative probability of a state of entropy S, we obtain in generalS − S0 = ( R/N) ln W.
First we treat the following special case. We consider a number
(n) of movable points (e.g., molecules) confined in a volume v0. Besides these points, there can be in the space any number of other movable points of any kind. We shall not assume anything concerning the law in accordance with which the points move in this space except that with regard to this motion, no part of the space (and no direction within it) can be distinguished from any other. Further, we take the number of these movable points to be so small that we can disregard interactions between them.
This system, which, for example, can be an ideal gas or a dilute solution, possesses an entropy S0. Let us imagine transferring all n movable points into a volume v (part of the volume v0) without anything else being changed in the system. This state obviously possesses a different entropy (S), and now wish to evaluate the entropy difference with the help of the Boltzmann Principle.
We inquire: How large is the probability of the latter state relative to the original one? Or: How large is the probability that at a randomly chosen instant of time all n movable points in the given volume v0 will be found by chance in the volume v?
For this probability, which is a “statistical probability”, one obviously obtains:W = (v/v0)n ;
By applying the Boltzmann Principle, one then obtainsS −S0 = R (n/N) ln (v/v0) .
It is noteworthy that in the derivation of this equation, from which one can easily obtain the law of Boyle and Gay–Lussac as well as the analogous law of osmotic pressure thermodynamically,u no assumption had to be made as to a law of motion of the molecules.
6. INTERPRETATION OF THE EXPRESSION FOR THE VOLUME DEPENDENCE OF THE ENTROPY OF MONOCHROMATIC RADIATION IN ACCORDANCE WITH BOLTZMANN’S PRINCIPLE
In Sec. 4, we found the following expression for the dependence of the entropy of monochromatic radiation on the volumeS −S0 = (E/βν) in (v/v0).
If one writes this in the fromS − S0 = (R/N) ln [(v/v0)(N/R)(E/βν) ].
and if one compares this with the general formula for the Boltzmann principleS − S0 = (R/N) ln W,
one arrives at the following conclusion:
If monochromatic radiation of frequency ν and energy E is enclosed by reflecting walls in a volume v0, the probability that the total radiation energy will be found in a volume v (part of the volume v0) at any randomly chosen instant isW = (v/v0)(N R)(E βν) .
From this we further conclude that: Monochromatic radiation of low density (within the range of validity of Wien’s radiation formula) behaves thermodynamically as though it consisted of a number of independent energy quanta of magnitude Rβ ν/N.
We still wish to compare the average magnitude of the energy quanta of the blackbody radiation with the average translational kinetic energy of a molecule at the same temperature. The latter is 3/2(R/N)T, while, according to the Wien formula, one obtains for the average magnitude of an energy quantum
If the entropy of monochromatic radiation depends on volume as though the radiation were a discontinuous medium consisting of energy quanta of magnitude Rβν/N, the next obvious step is to investigate whether the laws of emission and transformation of light are also of such a nature that they can be interpreted or explained by considering light to consist of such energy quanta. We shall examine this question in the following.
7. CONCERNING STOKES’S RULE
According to the result just obtained, let us assume that, when monochromatic light is transformed through photoluminescence into light of a different frequency, both the incident and emitted light consist of energy quanta of magnitude Rβν/N, where ν denotes the relevant frequency. The transformation process is to be interpreted in the following manner. Each incident energy quantum of frequency ν1 is absorbed and generates by itself–at least at sufficiently low densities of incident energy quanta—a light quantum of frequency ν2; it is possible that the absorption of the incident light quanta can give rise to the simultaneous emission of light quanta of frequencies ν3, ν4 etc., as well as to energy of other kinds, e.g., heat. It does not matter what intermediate processes give rise to this final result. If the fluorescent substance is not a perpetual source of energy, the principle of conservation of energy requires that the energy of an emitted energy quantum cannot be greater than that of the incident light quantum; it follows thatR βν2/N ≤ R βν1/N
orν2 ≤ ν1.
This is the well–known Stokes’s Rule.
It should be strongly emphasized that according to our conception the quantity of light emitted under conditions of low illumination (other conditions remaining constant) must be proportional to the strength of the incident light, since each incident energy quantum will cause an elementary process of the postulated kind, independently of the action of other incident energy quanta. In particular, there will be no lower limit for the intensity of incident light necessary to excite the fluorescent effect.
According to the conception set forth above, deviations from Stokes’s Rule are conceivable in the following cases:1. when the number of simultaneously interacting energy quanta per unit volume is so large that an energy quantum of emitted light can receive its energy from several incident energy quanta;
2. when the incident (or emitted) light is not of such a composition that it corresponds to blackbody radiation within the range of validity of Wien’s Law, that is to say, for example, when the incident light is produced by a body of such high temperature that for the wavelengths under consideration Wien’s Law is no longer valid.
The last-mentioned possibility commands especial interest. According to the conception we have outlined, the possibility is not excluded that a “non-Wien radiation” of very low density can exhibit an energy behavior different from that of a blackbody radiation within the range of validity of Wien’s Law.
8. CONCERNING THE EMISSION OF CATHODE RAYS THROUGH ILLUMINATION OF SOLID BODIES
The usual conception that the energy of light is continuously distributed over the space through which it propagates, encounters very serious difficulties when one attempts to explain the photoelectric phenomena, as has been pointed out in Herr Lenard’s pioneering paper.v
According to the concept that the incident light consists of energy quanta of magnitude Rβν/N, however, one can conceive of the ejection of electrons by light in the following way. Energy quanta penetrate into the surface layer of the body, and their energy is transformed, at least in part, into kinetic energy of electrons. The simplest way to imagine this is that a light quantum delivers its entire energy to a single electron: we shall assume that this is what happens. The possibility should not be excluded, however, that electrons might receive their energy only in part from the light quantum.
An electron to which kinetic energy has been imparted in the interior of the body will have lost some of this energy by the time it reaches the surface. Furthermore, we shall assume that in leaving the body each electron must perform an amount of work P characteristic of the substance. The ejected electrons leaving the body with the largest normal velocity will be those that were directly at the surface. The kinetic energy of such electrons is given byRβν/N − P.
In the body is charged to a positive potential Π and is surrounded by conductors at zero potential, and if Π is just large enough to prevent loss of electricity by the body, if follows that:Π∈ = Rβν/N − P
where ∈ denotes the electronic charge, orΠE = Rβν − Pʹ
where E is the charge of a gram equivalent of a monovalent ion and Pʹ is the potential of this quantity of negative electricity relative to the body.w
If one takes E = 9.6 × 103, then Π · 10−8 is the potential in volts which the body assumes when irradiated in a vacuum.
In order to see whether the derived relation yields an order of magnitude consistent with experience, we take Pʹ = 0, ν = 1.03 × 1015 (corresponding to the limit of the solar spectrum toward the ultraviolet) and β = 4.866 × 10−11. We obtain Π . 107 = 4.3 volts, a result agreeing in order magnitude with those of Herr Lenard.x
If the derived formula is correct, then Π, when represented in Cartesian coordinates as a fun
ction of the frequency of the incident light, must be a straight line whose slope is independent of the nature of the emitting substance.
As far as I can see, there is no contradiction between these conceptions and the properties of the photoelectric observed by Herr Lenard. If each energy quantum of the incident light, independently of everything else, delivers its energy of electrons, then the velocity distribution of the ejected electrons will be independent of the intensity of the incident light; on the other hand the number of electrons leaving the body will, if other conditions are kept constant, be proportional to the intensity of the incident light.y
Remarks similar to those made concerning hypothetical deviations from Stokes’s Rule can be made with regard to hypothetical boundaries of validity of the law set forth above.
In the foregoing it has been assumed that the energy of at least some of the quanta of the incident light is delivered completely to individual electrons. If one does not make this obvious assumption, one obtains, in place of the last equation:ΠE + Pʹ ≤ Rβν.
For fluorescence induced by cathode rays, which is the inverse process to the one discussed above, one obtains by analogous considerations:ΠE + Pʹ ≥ Rβν.
The Dreams That Stuff is Made of Page 3
