The previous argument then gives E(λ,T)δλ=c1λ−5e−c/λTδλ, a type of formula which was originally suggested by Wien on the basis of the analogy that it assigns the same distribution for the radiant energy, among the various frequencies of vibration, as for the energy of the molecules in a gas among their various velocities of translation. But the experimental inadequacy of this formula afterwards suggested a new procedure, as infra.
Processes may be theoretically assigned for the direct continuous transformation of radiant into mechanical energy. Thus we can imagine a radiating body at the centre of a wheel, carrying oblique vanes along its circumference, which reflect the radiation on to a ring of parallel fixed vanes, which finally reverse its path and return it to the centre. The pressure of the radiation will dr1ve the wheel, and in case its motion is not resisted, a very great velocity may be theoretically obtained. The thermodynamic compensation in such cases lies in the reduction of the effective temperature of the portion of the radiation not thus used up. We might even do away with the radiating body at the centre of the wheel, and consider a beam of definite radiation reflected backwards and forwards across a diameter. It is easy to see that its path will remain diametral; the work done by it in driving the wheel will be concomitant with increase of the wave-length, and therefore with expansion of the length occupied by the beam. The thermodynamic features are thus analogous to those of the more familiar case of an envelope filled with gas, which can change its thermal energy into mechanical energy by expansion of the envelope against mechanical resistances. In the case of the expanding gas pv = 23E0, where E0 is the total translator energy of the molecules, while in adiabatic expansion p=kv−γ. Thus the work gained in unlimited expansion, ∫pdv, is 23E0/(γ−1). The final temperature being absolute zero, this should by Carnot’s principle be equal to the total initial energy of the gas that is in Connexion with temperature, constitutive energy of the molecules being excluded; when ~y-1 is less than § there is thus internal thermal energy in the molecules in addition to the translator energy. In the case of the beam of radiation, of length l, between n and n+δn reflexions, where δn is an integer, its total energy E is by § 2 reduced according to the law δEE=−4cvδn(c+v)2. Also ¥=%L; thus %E-= -5% When 11 is small compared with c, this gives E=»<l“2; and p is then 2E/l, so that fpdl=E, the temperature of the beam being ultimately reduced to absolute, zero by the unlimited expansion. This is in accord with Carnot’s principle, in that the whole energy of the beam travelling in a vacuum is mechanically available when reduction to absolute zero of temperature is in our power.
12. Experimental Knowledge.—Under the stimulus of Wien’s investigation and of improvements in the construction of linear thermophiles and bolometers for the refined measurement of the distribution of energy along a spectrum, the general character of the curve connecting energy and wave-length in the complete radiation at a given temperature has been experimentally ascertained over a wide range. At each temperature there is a wave-length hm of maximum radiation, which is displaced towards the ultra-violet as the temperature rises, and Wien's law of homology (§ 6) shows that λmT should be constant. This deduction, and the law of homology itself, as also the law of Stefan and Boltzmann that the total radiation varies as T4, have been closely verified by the experiments of Rubens and Kurlbaum, Lummer and Pringsheim, Paschen and others. They established a steady held of radiation inside a material enclosure by raising the Walls to a definite temperature, and measured the radiant intensity emitted from it through an opening or slit in the walls, by means of a bolometer or thermophile, this being the radiation of the so-called perfectly black body. The principle here involved formed one of the foundations of Balfour Stewart's early treatment of the theory, and had already been employed by him and Stokes (1860) in experiments on the polarized emission from tourmaline: cf. Stokes, Math. and Phys. Papers, iv. 136. It has been remarked by Planck and by Thiesen that the coefficient of T4 in Stefan's law, and the value of λmT, are two absolute physical constants independent of any particular kind of matter, which in conjunction with the constant of gravitation would determine an entirely absolute system of physical units. The form of the function φ(Tλ) adopted by Wien and in Planck’s earlier discussions, namely, c1e−c/Tλ, was found to agree fairly with experiment over the range from 100° C., to 1300° C., when c1=1·24×10−5, and c= 1·4435 in c.g.s. measure, but not so well when the range is farther extended: it appeared that a larger value of c was needed to represent the radiation for high values of Th, that is, for high temperature or for very long wave-lengths. Thiesen proposed the somewhat more general form c, (T})'°e“'/TA, and suggested that the value k=12 agrees better with the experimental numbers than Wien's value k=0; Lord Rayleigh was led (Phil. Mag., June 1000) towards this form with k equal to' unity from entirely different theoretical considerations, on the assumption of the Maxwell-Boltzmann distribution of the energy of a system, consisting of an isolated block of aether, among its free periods of vibration, infinite in number; in some cases this form appeared to give as good results as Wien's own.
Acting on a suggestion advanced by Lord Rayleigh, Rubens and Kurlbaum soon afterwards widely extended the test of the formulae by means of the so-called Reststrahlen. A substance such as an aniline dye, which exhibits selective absorption of any group of rays, also powerfully reflects those rays; and Rubens has been able thus to isolate in considerable purity the rays belonging to absorption bands very far down in the invisible ultra-red, having wave-length of order 10−3 cm., which are intensely absorbed by substances such as sylvine, by means of five or six successive reflexions of the beam of radiation. By experiments ranging between temperatures −200° C. and + 1500° C. of the source of radiation, it has been found that the intensity of this definite radiation tends to vary simply as T, with close approximation, thus increasing indefinitely with the temperature, whereas Wien’s formula would make it tend to a definite limit. The only existing formula (except the one suggested by Lord Rayleigh) that proved to be in accord with this result was a new one advanced shortly before and supported on theoretical grounds by Planck, namely, Ei6l= Cλ−5/(e'/"T- 1), which for small values of XT agrees with Wien's original form, known to be there satisfactory, While for larger values it tends towards C /c .}“4T; the new formula is, in fact, the simplest and most likely form that satisfies these two conditions. The point of Lord Rayleigh's argument was that, at any rate at low frequencies, the law of distribution would suggest an equable partition of the energy between temperature heat and radiant vibrations, and that therefore the energy of the latter should ultimately vary as T; and this prediction, which has thus been verified, may be grafted on to any formula that is in other respects appropriate.
Recognizing that his previous hypothesis, restricting the nature of the entropy in addition to its property of continually increasing, had thus to be abandoned, Planck had in fact made a fresh start on the basis of a train of ideas which was introduced by Boltzmann in 1877, in order to obtain a precise 'physical conception of entropy. According to the latter, for an indefinitely numerous system of molecules, with known properties and in given circumstances, there is a definite probability of the occurrence of each statistical distribution of velocities, or' say each “complexion” of the system, that is formally possible when all velocities consistent with given total energy are considered to be equally likely as regards each molecule; the distribution of greatest possible probability is the state of thermal equilibrium of the system, and the probability of any other state is a function of the entropy of that state. This conception can be developed only in very simple cases; the application to an ideal monatomic gas-system led Boltzmann to take the entropy proportional to the logarithm of the probability. This logarithmic law is in fact demanded in advance by the principle that the entropy of a system should be the sum of the entropies of its parts. By means of a priori considerations of this nature, referring to the distribution of internal vibratory energy among a system of linear electric vibrators of given period, and its equilibrium of exchanges with the surrounding radiant energy, Planck has been guided to an expression for the law of dependence of the entropy of that system on the temperature, which corresponds to the form of the law of radiation above stated. The result gains support from the fact that the expressions for the coefficients to which he is led give determinations of the
