The Theory of Heat Radiation (1914) — part 8 of 12

(171) Hence, by taking account of (165), we obtain

W=(JT-}m (~Y' (-Y' (172)

124. Exactly the same method as in the case of the space dis tribution just considered may be used for the definition of a macroscopic state and of the thermodynamic probability in the general case, where not only the coordinates but also the veloci ties, the electric moments, etc., of the molecules are to be dealt with. Every thermodynamic state of a system of N molecules is, in the macroscopic sense, denned by the statement of the number of molecules, Ni, N2, N3, , which are con tained in the region elements 1, 2, 3, of the " state

space." This state space, however, is not the ordinary three- dimensional space, but an ideal space of as many dimensions as there are variables for every molecule. In o-ther respects the definition and the calculation of the thermodynamic probability W are exactly the same as above and the entropy of the state is accordingly found from (164), taking (166) also into account, to be

S=-kN2w1\ogwi,. (173)

where the sum S is to be taken over all region elements. It is obvious from this expression that the entropy is in every case a positive quantity.

125. By the preceding developments the calculation of the

1 Abridged in the sense that factors which in the logarithmic expression (173) would give rise to small additive terms have been omitted at the outset. A brief derivation of equation (173) may be found on p. 218 (Tr.).

2 See for example E. Czuber, Wahrscheinlichkeitsrechnung (Leipzig, B. G. Teubner) p. 22, 1903; H. Poincart, Calcul des Probabilit.es (Paris, Gauthier-Vitlars), p. 85, 1912.

FUNDAMENTAL DEFINITIONS AND LAWS 125

entropy of a system of N molecules in a given thermodynamic state is, in general, reduced to the single problem of finding the magnitude G of the region elements in the state space. That such a definite finite quantity really exists is a characteristic feature of the theory we are developing, as contrasted with that due to Boltzmann, and forms the content of the so-called hypo thesis of quanta. As is readily seen, this is an immediate conse quence of the proposition of Sec. 120 that the entropy S has an absolute, not merely a relative, value; for this, according to (164), necessitates also an absolute value for the magnitude of the ther modynamic probability W, which, in turn, according to Sec. 123, is dependent on the number of complexions, and hence also on the number and size of the region elements which are used. Since all different complexions contribute uniformly to the value of the probability W, the region elements of the- state space represent also regions of equal probability. If this were not so, the complexions would not be all equally probable.

However, not only the magnitude, but also the shape and posi tion of the region elements must be perfectly definite. For since, in general, the distribution density w is apt to vary appreciably from one region element to another, a change in the shape of a region element, the magnitude remaining unchanged, would, in general, lead to a change in the value of w and hence to a change in S. We shall see that only in special cases, namely, when the distribution densities w are very small, may the absolute magni tude of the region elements become physically unimportant, inas much as it enters into the entropy only through an additive con stant. This happens, e.g., at high temperatures, large volumes, slow vibrations (state of an ideal gas, Sec. 132, Rayleigh's radia tion law, Sec. 195). Hence it is permissible for such limiting cases to assume, without appreciable error, that G is infinitely small in the macroscopic sense, as has hitherto been the practice in statistical mechanics. As soon, however, as the distribution densities w assume appreciable values, the classical statistical mechanics fail.

126. If now the problem be to determine the magnitude (7 of the region elements of equal probability, the laws of the class ical statistical mechanics afford a certain hint, since in certain limiting cases they lead to correct results.

126 ENTROPY AND PROBABILITY

Let #i, 02, 03, ..... be the " generalized coordinates," ^i, fay ^3, ..... the corresponding " impulse coordinates" or "moments," which determine the microscopic state of a cer tain molecule; then the state space contains as many dimensions as there are coordinates 0 and moments \J/ for every molecule. Now the region element of probability, according to classical statistical mechanics, is identical with the infinitely small element of the state space (in the macroscopic sense) 1

rf0id02^03 ..... difridfadfa ..... (174)

According to the hypothesis of quanta, on the other hand, every region element of probability has a definite finite magnitude

G= Id0id02d03 . . dtidfad^s ..... (175)

-J

whose value is the same for all different region elements and, more over, depends on the nature of the system of molecules considered. The shape and position of the separate region elements are deter mined by the limits of the integral and must be determined anew in every separate case.

1 Compare, for example, L. Boltzmann, Gastheorie, 2, p. 62 et seg., 1898, or J. W. Gibbs, Elementary principles in statistical mechanics, Chapter I, 1902.

CHAPTER II IDEAL MONATOMIC GASES

127. In the preceding chapter it was proven that the introduc tion of probability considerations into the mechanical and electrodynamical theory of heat is justifiable and necessary, and from the general connection between entropy S and probability W, as expressed in equation (164), a method was derived for cal culating the entropy of a physical system in a given state. Before we apply this method to the determination of the entropy of radiant heat we shall in this chapter make use of it for calculating the entropy of an ideal monatomic gas in an arbitrarily given state. The essential parts of this calculation are already con tained in the investigations of L. Boltzmann1 on the mechanical theory of heat; it will, however, be advisable to discuss this simple case in full, firstly to enable us to compare more readily the method of calculation and physical significance of mechanical entropy with that of radiation entropy, and secondly, what is more important, to set forth clearly the differences as compared with Boltzmanris treatment, that is, to discuss the meaning of the universal constant k and of the finite'region elements G. For this purpose the treatment of a special case is sufficient.

128. Let us then take N similar monatomic gas molecules in an arbitrarily given thermodynamic state and try to find the corresponding entropy. The state space is six-dimensional, with the three coordinates x, y, z, and the three corresponding moments w£, my, m£, of a molecule, where we denote the mass by m and velocity components by £, r?, f . Hence these quantities are to be substituted for the </> and ^ in Sec. 126. We thus obtain for the size of a region element G the sextuple integral

, (176)

where, for brevity

dx dy dz d£ d-n d£ = d<r (177)

1 L. Boltzmann, Sitzungaber. d. Akad. d. Wissensch. zu Wien (II) 76, p. 373, 1877. Com pare also Gastheorie, 1, p. 38, 1896.

127

128 ENTROPY AND PROBABILITY

If the region elements are known, then, since the macroscopic state of the system of molecules was assumed as known, the numbers NI, Nz, Ns, ..... of the molecules which lie in the separate region elements are also known, and hence the dis tribution densities Wi, Wz, ws, ..... (166) are given and the entropy of the state follows at once from (173).

129. The theoretical determination of G is a problem as difficult as it is important. Hence we shall at this point restrict ourselves from the very outset to the special case in which the distribution density varies but slightly from one region element to the next — the characteristic feature of the state of an ideal gas. Then the summation over all region elements may be replaced by the inte gral over the whole state space. Thus we have from (176) and (167)

^• -- ^ * ^ *

l, (178)

in which w is no longer thought of as a discontinuous function of the ordinal number, i, of the region element,, where i = l, 2, 3, ..... HJ but as a continuous function of the variables, x> U> z) £> *?> T> of the state space. Since the whole state region contains very many region elements, it follows, according to (167) and from the fact that the distribution density w changes slowly, that w has everywhere a small value.

Similarly we find for the entropy of the gas from (173) :

i^^ -^ i\y> 3 /"•

S=-kN^wl logtt>i=-fctf— J w logw da. (179)

Of course the whole energy E of the gas is also determined by the distribution densities w. If w is sufficiently small in every region element, the molecules contained in any one region element are, on the average, so far apart that their energy depends only on the velocities. Hence:

E =

(180)

where £177 if i denotes any velocity lying within the region element 1 and EQ denotes the internal energy of the stationary molecules,

IDEAL MONATOMIC GASES 129

which is assumed constant. In place of the latter expression we may write, again according to (176),

(181)

130. Let us consider the state of thermodynamic equilibrium. According to the second principle of thermodynamics this state is distinguished from all others by the fact that, for a given volume V and a given energy E of the gas, the entropy S is a maximum. Let us then regard the volume

= fj*j'

dxdydz (182)

and the energy E of the gas as given. The condition for equi librium is 5>S = 0, or, according to (179),

and this holds for any variations of the distribution densities whatever, provided that, according to (167) and (180), they satisfy the conditions

This gives us as the necessary and sufficient condition for thermo dynamic equilibrium for every separate distribution density w:

log w+/3(£2+T72+£2)+ const. =0 or

w = ae-W+^W) (183)

where a and /3 are constants. Hence in the state of equilibrium the distribution of the molecules in space is independent of x, y, z, that is, macroscopically uniform, and the distribution of velocities is the well-known one of Maxwell.

131. The values of the constants a and |8 may be found from those of V and E. For, on substituting the value of w just found in (178) and taking account of (177) and (182), we get

C1

— • • - - - - =ay

130 ENTROPY AND PROBABILITY

and on substituting w in (181) we get

-0(*2+»2+f2) nz+^e d£ dr, *,

NV r r / ~J f J

or

3am'NV 1

Solving for a and 0 we have

!

From this finally we find, as an expression for the entropy S of the gas in the state of equilibrium with given values of N, V, and E,

132. This determination of the entropy of an ideal monatomic gas is based solely on the general connection between entropy and probability as expressed in equation (164); in particular, we have at no stage of our calculation made use of any special law of the theory of gases. It is, therefore, of importance to see how the entire thermodynamic behavior of a monatomic gas, especially the equation of state and the values of the specific heats, may be deduced from the expression found for the entropy directly by means of the principles of thermodynamics. From the general thermodynamic equation defining the entropy, namely,

(187)

the partial differential coefficients of S with respect to E and V are found to be

2 E-E0 T and

IDEAL MON ATOMIC GASES 131

Hence, by using (186), we get for our gas

'MN 3 M l (188)

= — = — (189)

The second of these equations

P = /by7- (190)

contains the laws of Boyle, Gay Lussac, and Avogadro, the last named because the pressure depends only on the number Nj not on the nature of the molecules. If we write it in the customary form:

P = ~^-' (191)

where n denotes the number of gram molecules or mols of the gas, referred to 02 = 320, and R represents the absolute gas constant

AWtfV

(192)

degree we obtain by comparison

If we now call the ratio of the number of mols to the number of molecules co, or, what is the same thing, the ratio of the mass of a

Tl

molecule to that of a mol, co = — , we shall have

N

k = uR. (194)

From this the universal constant k may be calculated, when co is given, and vice versa. According to (190) this constant k is nothing but the absolute gas constant, if it 'is referred to mole cules instead of mols. From equation (188)

E-EQ=*kNT. (195)

132 ENTROPY AND PROBABILITY

Now, since the energy of an ideal gas is also given by

E = AncvT+E0 (196)

where cv is the heat capacity of a mol at constant volume in calories and A is the mechanical equivalent of heat:

it follows that

A =419X105~ (197)

cal

_ _ Cv~2An

and further, by taking account of (193) 3# 3831X105

as an expression for the heat capacity per mol of any monatomic gas at constant volume in calories.1

For the heat capacity per mol at constant pressure, cp, we have as a consequence of the first principle of thermodynamics :

and hence by (198)

R

/»

Tt 1/V

§?• -4 2 A cv 3

as is known to be the case for monatomic gases. It follows from (195) that the kinetic energy L of the gas molecules is equal to

(200) 2

133. The preceding relations, obtained simply by identifying the mechanical expression of the entropy (186) with its thermo- dynamic expression (187), show the usefulness of the theory developed. In them an additive constant in the expression for the entropy is immaterial and hence the size G of the region ele ment of probability does not matter. The hypothesis of quanta, however, goes further, since it fixes the absolute value of the entropy and thus leads to the same conclusion as the heat theorem

1 Compare F. Richarz, Wiedemann's Annal., 67, p. 705, 1899.

IDEAL MONATOMIC GASES 133

of Nernst. According to this theorem the " characteristic func tion" of an ideal gas1 is in our notation

where a denotes Nernst's chemical constant, and b the energy constant.

On the other hand, the preceding formulae (186), (188), and (189) give for the same function $ the following expression:

-k log T-k log p+af } —^

where for brevity a' is put for:

»1

\kN •

a' = /clog — (27rw/b) ( e(jr

From a comparison of the two expressions for <£ it is seen, by taking account of (199) and (193), that they agree completely, provided

5

N> PI /o ^

a = — a' = R log — (2irm)

n

This expresses the relation between the chemical constant a of the gas and the region element G of the probability.2

It is seen that G is proportional to the total number, N} of the molecules. Hence, if we put G = Ng,we see that g, the molecular region element, depends only on the chemical nature of the gas.

Obviously the quantity g must be closely connected with the law, so far unknown, according to which the molecules act micro scopically on one another. Whether the value of g varies with the nature of the molecules or whether it is the same for all kinds of molecules, may be left undecided for the present.

1 E.g., M. Planck, Vorlesungen tiber Thermodynamik, Leipzig, Veit und Comp., 1911, Sec. 287, equation 267.

2 Compare also O. Sackur, Annal. d. Physik, 36, p. 958, 1911, Nernst-Featschrift, p. 405, 1912, and H. Tetrode, Annal. d. Physik, 38, p. 434, 1912.

134 ENTROPY AND PROBABILITY

If g were known, Nernst's chemical constant, a, of the gas could be calculated from (201) and the theory could thus be tested. For the present the reverse only is feasible, namely, to calculate g from a. For it is known that a may be measured directly by the tension of the saturated vapor, which at suffi ciently low temperatures satisfies the simple equation1

• (202)

(where r0 is the heat of vaporization of a mol at 0° in calories). When a has been found by measurement, the size g of the mo lecular region element is found from (201) to be

"E"1 (203)

Let us consider the dimensions of g.

According to (176) g is of the dimensions [erg3sec3]. The same follows from the present equation, when we consider that the dimension of the chemical constant a is not, as might at first be

P thought, that of R, but, according to (202), that of R log —5

T*

134. To this we may at once add another quantitative rela tion. All the preceding calculations rest on the assumption that the distribution density w and hence also the constant a in (183) are small (Sec. 129). Hence, if we take the value of a from (184) and take account of (188), (189) and (201), it follows that

— ° — i

—6e R must be small. T>

When this relation is not satisfied, the gas cannot be in the ideal state. For the saturated vapor it follows then from (202) that

_Ar0

e RT is small. In order, then, that a saturated vapor may be assumed to be in the state of an ideal gas, the temperature T

A r

must certainly be less than - r0 or --. Such a restriction is un-

R 2

known to the classical thermodynamics.

i M. Planck, 1. c., Sec. 288, equation 271.

CHAPTER III IDEAL LINEAR OSCILLATORS

135. The main problem of the theory of heat radiation is to determine the energy distribution in the normal spectrum of black radiation, or, what amounts to the same thing, to find the function which has been left undetermined in the general expres sion of Wien's displacement law (119), the function which con nects the entropy of a certain radiation with its energy. The purpose of this chapter is to develop some preliminary theorems leading to this solution. Now since, as we have seen in Sec. 48, the normal energy distribution in a diathermanous medium can not be established unless the medium exchanges radiation with an emitting and absorbing substance, it will be necessary for the treatment of this problem to consider more closely the processes which cause the creation and the destruction of heat rays, that is, the processes of emission and absorption. In view of the complex ity of these processes and the difficulty of acquiring knowledge of any definite details regarding them, it would indeed be quite hopeless to expect to gain any certain results in this way, if it were not possible to use as a reliable guide in this obscure region the law of Kirchhoff derived in Sec. 51. This law states that a vacuum completely enclosed by reflecting walls, in which any emitting and absorbing bodies are scattered in any arrangement whatever, assumes in the course of time the stationary state of black radiation, which is completely determined by one parame ter only, namely, the temperature, and in particular does not depend on the number, the nature, and the arrangement of the material bodies present. Hence, for the investigation of the properties of the state of black radiation the nature of the bodies which are assumed to be in the vacuum is perfectly immaterial. In fact, it does not even matter whether such bodies really exist somewhere in nature, provided their existence and their proper ties are consistent with the laws of thermodynamics and electro-

135

136 ENTROPY AND PROBABILITY

dynamics. If, for any special arbitrary assumption regarding the nature and arrangement of emitting and absorbing systems, we can find a state of radiation in the surrounding vacuum which is distinguished by absolute stability, this state can be no other than that of black radiation.

Since, according to this law, we are free to choose any system whatever, we now select from all possible emitting and absorbing systems the simplest conceivable one, namely, one consisting of a large number N of similar stationary oscillators, each consist ing of two poles, charged with equal quantities of electricity of opposite sign, which may move relatively to each other on a fixed straight line, the axis of the oscillator.

It is true that it would be more general and in closer accord with the conditions in nature to assume the vibrations to be those of an oscillator consisting of two poles, each of which has three degrees of freedom of motion instead of one, i.e., to assume the vibrations as taking place in space instead of in a straight line only. Never theless we may, according to the fundamental principle stated above, restrict ourselves from the beginning to the treatment of one single component, without fear of any essential loss of generality of the conclusions we have in view.

It might, however, be questioned as a matter of principle, whether it is really permissible to think of the centers of mass of the oscillators as stationary, since, according to the kinetic theory of gases, all material particles which are contained in substances of finite temperature and free to move possess a cer tain finite mean kinetic energy of translatory motion. This objection, however, may also be removed by the consideration that the velocity is not fixed by the kinetic energy alone. We need only think of an oscillator as being loaded, say at its positive pole, with a comparatively large inert mass, which is perfectly neutral electrodynamically, in order to decrease its velocity for a given kinetic energy below any preassigned value whatever. Of course this consideration remains valid also, if, as is now frequently done, all inertia is reduced to electrodynamic action. For this action is at any rate of a kind quite different from the one to be considered in the following, and hence cannot influence it.

Let the state of such an oscillator be completely determined by its moment f(t), that is, by the product of the electric charge

IDEAL LINEAR OSCILLATORS 137

of the pole situated on the positive side of the axis and the pole distance, and by the derivative of / with respect to the time or

(204)

Let the energy of the oscillator be of the following simple form:

U = ±Kf*+};Lf*} (205)

where K and L denote positive constants, which depend on the nature of the oscillator in some way that need not be discussed at this point.

If during its vibration an oscillator neither absorbed nor emitted any energy, its energy of vibration, U, would remain constant, and we would have:

d U = Kfdf+Lfdf = 0, (205 a)

or, on account of (204),

Kf(t)+Lf(t)=0. (206)

The general solution of this differential equation is found to be a purely periodical vibration:

/=Ccos (2irrt-0) (207)

where C and 0 denote the integration constants and v the number of vibrations per unit time:

-iVf (208)

136. If now the assumed system of oscillators is in a space traversed by heat rays, the energy of vibration, U, of an oscillator will not in general remain constant, but will be always changing by absorption and emission of energy. Without, for the present, considering in detail the laws to which these processes are subject, let us consider any one arbitrarily given thermodynamic state of the oscillators and calculate its entropy, irrespective of the surrounding field of radiation. In doing this we proceed entirely according to the principle advanced in the two preceding chapters, allowing, however, at every stage for the conditions caused by the peculiarities of the case in question.

The first question is: What determines the thermodynamic state of the system considered? For this purpose, according to

138 ENTROPY AND PROBABILITY

Sec. 124, the numbers Ni, Nz, N3, of the oscillators,

which lie in the region elements 1, 2, 3, of the " state

space" must be given. The state space of an oscillator contains those coordinates which determine the microscopic state of an oscillator. In the case in question these are only two in number, namely, the moment/ and the rate at which it varies,/, or instead of the latter the quantity

t=Lf, (209)

which is of the dimensions of an impulse. The region element of the state plane is, according to the hypothesis of quanta (Sec. 126), the double integral

$ = h. (210)

The quantity h is the same for all region elements. A priori, it might, however, depend also on the nature of the system con sidered, for example, on the frequency of the oscillators. The following simple consideration, however, leads to the assumption that h is a universal constant. We know from the generalized displacement law of Wien (equation 119) that in the universal function, which gives the entropy radiation as dependent on the energy radiation, there must appear a universal constant of the

C3U

dimension — and this is of the dimension of a quantity of action1 v*

(erg sec.). Now, according to (210), the quantity h has precisely this dimension, on which account we may denote it as "element of action" or "quantity element of action." Hence, unless a second constant also enters, h cannot depend on any other phys ical quantities.

137. The principal difference, compared with the calculations for an ideal gas in the preceding chapter, lies in the fact that we

do not now assume the distribution densities Wi, w2, wz

of the oscillators among the separate region elements to vary but little from region to region as was assumed in Sec. 129. Accord ingly the w's are not small, but finite proper fractions, and the summation over the region elements cannot be written as an integration.

1 The quantity from which the principle of least action takes its name. (Tr.)

IDEAL LINEAR OSCILLATORS 139

In the first place, as regards the shape of the region elements, the fact that in the case of undisturbed vibrations of an oscillator the phase is always changing, whereas the amplitude remains constant, leads to the conclusion that, for the macroscopic state of the oscillators, the amplitudes only, not the phases, must be considered, or in other words the region elements in the f\f/ plane are bounded by the curves C = const., that is, by ellipses, since from (207) and (209)

The semi-axes of such an ellipse are :

a = C Siiidb = 27rpLC. (212)

Accordingly the region elements 1, 2, 3, ..... n ..... are the concentric, similar, and similarly situated elliptic rings, which are determined by the increasing values of C :

0, Ci, C2, C3, ..... Cn-u Cn ..... (213)

The nth region element is that which is bounded by the ellipses C = (?„_! and C = Cn. The first region element is the full ellipse Ci. All these rings have the same area h, which is found by subtracting the area of the full ellipse Cn-i from that of the full ellipse Cn; hence

h = (anbn-an-1bn_1)7r or, according to (212),

/*=(Cn2-Cn_l2) 27T2^L,

where n = l, 2, 3, .....

From the additional fact that C0 = 0, it follows that :

(214)

Thus the semi-axes of the bounding ellipses are in the ratio of the square roots of the integral numbers.

138. The thermodynamic state of the system of oscillators is fixed by the fact that the values of the distribution densities wi, wz, ws, ..... of the oscillators among the separate region elements are given. Within a region element the distri bution of the oscillators is according to the law of elemental chaos (Sec. 122), i.e., it is approximately uniform.

140 ENTROPY AND PROBABILITY

These data suffice for calculating the entropy S as well as the energy E of the system in the given state, the former quantity directly from (173), the latter by the aid of (205). It must be kept in mind in the calculation that, since the energy varies appreciably within a region element, the energy En of all those oscillators which lie in the nth region element is to be found by an integration. Then the whole energy E of the system is:

E = E!+E2+ ..... En+ ..... (215)

En may be calculated with the help of the law that within every region element the oscillators are uniformly distributed. If the nth region element contains, all told, Nn oscillators, there are per

Nn Nn

unit area — — oscillators and hence — - df-d\t/ per element of area. h h

Hence we have:

In performing the integration, instead of / and ^ we take C and <£, as new variables, and since according to (211),

/= C cos </> ^ = IwLC sin 0 (216)

we get:

En = 2irpL N— f f U C dC d4> h J J

to be integrated with respect to </> from 0 to 2?r and with respect to C from €„-,- to Cn. If we substitute from (205), (209) and (216)

C7 = iKC2, (217)

we obtain by integration

and from (214) and (208):

that is, the mean energy of an oscillator in the nth region element is (n — ?)hv. This is exactly the arithmetic mean of the energies (n—)hv and rihv which correspond to the two ellipses C = Cn-i and C = Cn bounding the region, as may be seen from (217), if the values of Cn-\ and Cn are therein substituted from (214).

IDEAL LINEAR OSCILLATORS 141

The total energy E is, according to (215),

-l)wn. (219)

139. Let us now consider the state of thermo dynamic equi librium of the oscillators. According to the second principle of thermodynamics, the entropy S is in that case a maximum for a given energy E. Hence we assume E in (219) as given. Then from (179) we have for the state of equilibrium:

i where according to (167) and (219)

= 0 and S(n — %)dwn = Q i i

From these relations we find:

log wn+pn-- const. =0 or

wn = ay\ (220)

The values of the constants a and 7 follow from equations (167) and (219) :

2Nhv _2E-Nhp

2E-Nhv y2E+Nhv

Since wn is essentially positive it follows that equilibrium is not possible in the system of oscillators considered unless the total

energy E has a greater value than -— , that is unless the mean

2i

hv energy of the oscillators is at least — • This, according to

(218), is the mean energy of the oscillators lying in the first region element. In fact, in this extreme case all N oscillators lie in the first region element, the region of smallest energy; within this element they are arranged uniformly.

The entropy S of the system, which is in thermodynamic equilibrium, is found by combining (173) with (220) and (221)

142 ENTROPY AND PROBABILITY

140. The connection between energy and entropy just obtained allows furthermore a certain conclusion as regards the tempera ture. For from the equation of the second principle of thermo-

ITjl

dynamics, dS = — and from differentiation of (222) with respect to E it follows that

_hv

hv l+e kT

»-- 2

-ekT

Hence, for the zero point of the absolute temperature E becomes,

hv

not 0, but N—' This is the extreme case discussed in the pre ceding paragraph, which just allows thermodynamic equilibrium to exist. That the oscillators are said to perform vibrations even at the temperature zero, the mean energy of which is as large as

hv

— and hence may become quite large for rapid vibrations, may

at first sight seem strange. It seems to me, however, that certain facts point to the existence, inside the atoms, of vibrations independent of the temperature and supplied with appreciable energy, which need only a small suitable excitation to become evident externally. For example, the velocity, sometimes very large, of secondary cathode rays produced by Roentgen rays, and that of electrons liberated by photoelectric effect are inde pendent of the temperature of the metal and of the intensity of the exciting radiation. Moreover the radioactive energies are also independent of the temperature. It is also well known that the close connection between the inertia of matter and its energy as postulated by the relativity principle leads to the assumption of very appreciable quantities of intra-atomic energy even at the zero of absolute temperature.

For the extreme case, T = °° , we find from (223) that

E^NkT, (224)

i.e., the energy is proportional to the temperature and indepen dent of the size of the quantum of action, h, and of the nature of the oscillators. It is of interest to compare this value of the energy of vibration E of the system of oscillators, which holds at high temperatures, with the kinetic energy L of the molecular

IDEAL LINEAR OSCILLATORS

143

motion of an ideal monatomic gas at the same temperature as calculated in (200). From the comparison it follows that

E = \L (225)

This simple relation is caused by the fact that for high tem peratures the contents of the hypothesis of quanta coincide with those of the classical statistical mechanics. Then the absolute magnitude of the region element, G or h respectively, becomes physically unimportant (compare Sec. 125) and we have the simple law of equipartition of the energy among all variables in question (see below Sec. 169). The factor f in equation (225) is due to the fact that the kinetic energy of a moving molecule depends on three variables (£, 77, f ,) and the energy of a vibrating oscillator on only two (/, i/O-

The heat capacity of the system of oscillators in question is, from (223),

dE_ dT

= Nk

hv V

kT

(226)

It vanishes for T = 0 and becomes equal to Nk for T = °o . A. Einstein1 has made an important application of this equation to the heat capacity of solid bodies, but a closer discussion of this would be beyond the scope of the investigations to be made in this book.

For the constants a and 7 in the expression (220) for the dis tribution density w we find from (221) :

= kT -

a = e

kT

(227)

and finally for the entropy S of our system as a function of tem perature :

hv

jkT-

'kT -I

_

'kT

(228)

i A. Einstein, Ann. d. Phys. 22, p. 180, 1907. Compare also M. Born uiid Th. von Kdrman, Phys. Zeitschr. 13, p. 297, 1912.

CHAPTER IV

DIRECT CALCULATION OF THE ENTROPY IN THE CASE OF THERMODYNAMIC EQUILIBRIUM

141. In the calculation of the entropy of an ideal gas and of a system of resonators, as carried out in the preceding chapters, we proceeded in both cases, by first determining the entropy for an arbitrarily given state, then introducing the special condition of thermodynamic equilibrium, i.e., of the maximum of entropy, and then deducing for this special case an expression for the entropy.

If the problem is only the determination of the entropy in the case of thermodynamic equilibrium, this method is a roundabout one, inasmuch as it requires a number of calculations, namely, the determination of the separate distribution densities Wi, w2,

Ws, which do not enter separately into the final