second from a black-body or perfect radiator at a temperature T,
and is denoted by oT 4 , where a is the Stefan-Boltzmann constant
of full radiation. By the geometrical conditions of the problem, the
quantity aT 4 is A/4 times the latent heat per unit volume, or
A/3 times the energy-density in an isothermal enclosure at T, where
A is the velocity of light. The qualitative verification of the fourth-
power law requires only a receiver capable of giving correct relative
values of the radiation received, and is now generally accepted as
satisfactory; but the absolute measurement of the value of the con-
stant a is a much more difficult problem, which has frequently
been attacked in recent years without obtaining so high a degree
of concordance as is desirable in so fundamental a research. The
value 5-32 Xio 5 ergs per sq. cm. per second, found by F. Kurl-
baum in 1898 (see 13.155), was accepted for several years, though
it rested on a somewhat doubtful value of the absorption coefficient
of the bolometer. Moreover, the assumption that the radiant
energy measured was equivalent to the electric energy required to
produce the same rise of temperature in the bolometer, was rendered
somewhat uncertain by conduction effects at the ends of the strips.
A similar bolometer, with the end-effects compensated, as employed
in the solar eclipse of 1905, gave the somewhat higher value 5-60 X
jo 5 . Kurlbaum (1912) gave the corrected result 5-45 Xio 5 .
F. Paschen and W. Gerlach, by a modification of Angstrom's method
(Ann. Phys., 38, p. 41, 1912), found the value 5-8oXio 5 , which was
confirmed by G. A. Shakespear (Proc. R. S., A, 86, p. 180, 1912), and
by H. B.Keene(Proc.^.5.,A,88, p. 49, 1913), who found 5- and s-SgXio" 6 , respectively. W. Coblentz (U.S.A. Bur. 12, P- 553. 1916), by a method similar to that of Paschen and Gerlach, found the value 5-72 Xio 5 , which is a fair mean of the previous results. One of the most promising methods is that of the radio- balance (Proc. Phys. Soc., 23, pp. 1-34, 1910), in which radiation received through a measured aperture is completely absorbed in a small copper cup, and is compensated by the Peltier cooling-effect due to a current through a thermojunction. Unfortunately, these experiments were interrupted by the war, and the final reductions have not yet been completed. There seems to be little doubt that Kurlbaum's original value was too low, but there are many pit- falls in such difficult experiments, and most of the methods adopted are liable to some objections.
It is generally admitted that the distribution of energy in the spectrum may be represented within the limits of experimental error by Planck's formula (see 13.156), namely,
EdX = C'X- . . . (13).
If this formula is integrated from o to , and equated to aT 4 , assuming that it represents the distribution of energy in the spec- trum as observed experimentally, we find for the constant C , in terms of c' and a, C' = i$a(c ITC}*. The value of the distribu- tion constant c 1 is most readily deduced from the wave-length \ m corresponding to the maximum ordinate of the energy curve at T, since by Wien's law the product X^T is the same for all tempera- tures. According to Planck's formula the maximum occurs at the point X m T=c'/4'965i. Planck took X m T = o-294, and cr = 5'3oX IO 5 , giving C' =3-735 Xlo 6 , and ' = 1-460. But if X m T = 0-289, and = 5-72Xlo 5 , then C' = 3'7o8Xlo 5 ande' = 1-435, according to the latest values of X m T and a. A comparatively small error in c', which is raised to the fourth power, suffices to neutralize the error jn a. The weak point of the method is that the position of the max- imum of an experimental curve cannot be fixed with any certainty when the curve (as in this case) is far from symmetrical on either side of the maximum.
It is too commonly assumed that Planck's radiation formula, in spite of the weighty objections that have been repeatedly urged against it, is so firmly founded in theory and experiment, that no other formula is worth considering in comparison with it. It is also frequently asserted that no formula based on the " classical " mechanics can possibly satisfy the required conditions. The argu- ment is somewhat as follows. The number of possible vibrations per unit volume of a continuous medium possessing the properties of the aether, between the limits X and X+ of wave-length, should be represented by 8ir\ 4 d\, according to Lord Rayleigh's method of calculation (Phil. Mag., 49, p. 539, 1900), if the length of path between each reflection is restricted to an integral multiple of half a wave-length. If the different frequencies are regarded as separate inconvertible entities, like the molecules of different gases, between which the energy must be equally divided, the whole of the energy would accumulate in the infinitely short waves, which is absurd and contradicts experiment. It would be more natural, however, from a physical standpoint to regard Lord Rayleigh's formula
(8irRT/N)e-"/ATx-yx . . . (14)
as corresponding to the partition of energy among a number of similar molecules, according to Maxwell's law, which is universally admitted in the kinetic theory of gases, as resulting from the steady state produced by collisions. The steady distribution of energy of radiation in equilibrium with matter arises in a similar manner from the Doppler effect, by which the energy of a group of waves is changed in the same proportion as the frequency at each encounter with a moving obstacle. The frequency, or the reciprocal of the wave-length, corresponds to the energy, and occurs in much the
same way in Rayleigh's formula, as the square of the velocity, or the kinetic energy, in Maxwell's law. On this view, Lord Rayleigh's formula evidently represents the distribution of pressure-energy between the different wave-lengths about a mean value RT/N, which, according to the law of equipartition, should be the same as the pressure-energy of a single gas-molecule at the same temperature.
If we take Rayleigh's formula as representing the pressure dis- tribution in full radiation, the expression for the latent heat of absorption L as measured experimentally (corresponding to (4) above, but expressed in terms of the wave-length X in the normal spectrum) may be written
, . . . (15).
Integrating from o to oo we find C" = 3 /8- The maximum of this curve occurs at the point where c "/XT = 2 +2 V2i whence c" ' = 4-8284X. The absolute value of the maximum ordinate comes out o-65754(oT t /\m). The value of the same ordinate, calculated in the same way for Planck's formula (13), but with c' = 4-<)6$i\ m T comes out o-65755(erT 4 /X). It is a curious and significant fact that the maxima should be so exactly the same when the same values of the experimental data are assumed for both curves. The total areas of the two curves are the same, and they agree so closely throughout their whole extent that it would be practically impos- sible to distinguish between them with certainty by experiments on the distribution of heat in the spectrum. The greatest differ- ence amounts to about I % of the maximum ordinate, and occurs near the point X = X m /2 on the short wave-length side, where the curve is very steep. This difference becomes quite appreciable in the specific heats, when the curves are differentiated, and seems to lead to better agreement with experiment than Planck's formula as explained above.
The most serious difficulty from an experimental standpoint in applying Planck's formula, is that the latent heat of emission per unit volume is always tacitly assumed (following Planck) to be the same as the energy-density, without taking any account of the pressure, whereas the existence of the radiation pressure is uni- versally admitted as the basis of the deduction of the fourth-power law. The work done by the pressure, if it exists, cannot consistently be neglected in experimental measurements of radiation in steady flow. This is one of the most fundamental points in practical thermo- dynamics, but had not up to 1921 received sufficient attention from the mathematicians who had worked so elaborately on the theory.
VAPORIZATION
A good deal of attention has been devoted in recent years to the study of the properties of vapours employed in heat engines and refrigerating machines. The importance of the thermo- dynamical aspect of the problem has been widely recognized by engineers as the only sure guide to improvements in efficiency, and it has been realized that equations employed to represent the properties of the working fluid must be exactly consistent with the laws of thermodynamics, if it is desired to avoid dis- crepancies in the results of calculations by different methods. The principal properties of vapours were discussed from this point of view in the earlier article (see 27.897). The theory there given still holds good, but it will be of interest to discuss some of the evidence which has since accumulated on the experi- mental side. The case of steam, for which the experimental data are more accurate than for any other substance, will be taken, as being far the most important to engineers, and as illustrating the properties of vapours at moderate pressures. At high pressures, on the other hand, in the neighbourhood of the critical point, the data for steam are almost entire deficient, owing to the difficulty of the experiments, and the impractica- bility of using steam as a working fluid under these conditions. In the critical region the properties of carbonic acid have been most widely studied on account of its use for refrigeration.
Properties of Steam. The equations for steam, first proposed by Callendar in the loth ed. of the E.B. (1902), were founded on experi- mental measurements, (i) of the specific heats, s and S, of water and steam by the continuous electric method, (2) of the Joule- Thomson cooling-effect C with a differential throttling calorimeter, and (3) on the adiabatic index 7 for dry steam with a very sensitive platinum thermometer. These experiments, when taken in con- junction with the laws of thermodynamics, sufficed to determine all the required properties with a fair degree of accuracy at moderate pressures.
The experiments on the specific heat of water extended from o" to iooC., and, when taken in conjunction with those of Regnault at higher temperatures, showed that the total heat h under satura- tion pressure could be represented, with sufficient accuracy for the purpose, by the thermodynamic formula
h = st+avT(dpldT),=st+vL/(V,-v), . . . (16)
