On the Equation of State of an Ideal Monatomic Gas According to the Quantum-Theory, In: KNAW, Proceedings, 16 I, 1913, Amsterdam, 1913, Pp

Huygens Institute - Royal Netherlands Academy of Arts and Sciences (KNAW)

Citation:

W.H. Keesom, On the equation of state of an ideal monatomic gas according to the quantum-theory, in: KNAW, Proceedings, 16 I, 1913, Amsterdam, 1913, pp. 227-236

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- 1 - 227 high. We are uncertain as to the cause of this differencc: most probably it is due to an nncertainty in the temperature with the absolute manometer. It is of special interest to compare these observations with NERNST'S formula. Tbe fourth column of Table II contains the pressUl'es aceord ing to this formula, calc111ated witb the eonstants whieb FALCK 1) ha~ determined with the data at bis disposal. FALOK found the following expression 6000 1 0,009983 log P • - --. - + 1. 7 5 log T - T + 3,1700 4,571 T . 4,571 where p is the pressure in atmospheres. The correspondence will be se en to be satisfactory considering the degree of accuracy of the observations. 1t does riot look as if the constants could be materially improved.

Physics. - "On the equation of state oj an ideal monatomic gflS accoJ'ding to tlw quantum-the01'Y." By Dr. W. H. KEEsmr. Supple ment N°. 30a to the Oommunications ft'om the PhysicaL Labora tory at Leiden. Oommunicated by Prof. H. KAl\IERLINGH ONNES.

(Communicated in the meeting of May' 31, 1913).

~ 1. Int1'oduction. Swnmal'y. DEBIJE2) has shown that agreement with tbe obser\'ations concerning the specific heat of solid sub::,tances ean be obtained by modifying the Itheory of EINSTEIN 3) in this sense, that the formula 1) which PLANCK has given for the mean energy at the tempel'ature T of a linear electrical oscillator is applied to the different principal modes of vibration of asolid. It seems natm'al to apply the same princi]!le to other materiaJ systems which can behave as an oscillator and hence to investigate the correctness of the consequences which follow ti'om the hypo thesis

1) F ALeK: Physik. Zeits. 1908, p. 433.\ 2) P. DEBlJE. Ann. d. Phys. (4) 39 (1912), p. 789. S) For iJie liLerature see: H. KAMERLINGH ONNES and W. H. KEESOM. Math. Enz. V 10, Leiden Comm. Suppl. N0. 23, § 74c. 4) DEBIJE makes use of the original formula of PLANCK, Wärmestrahlung, lstc Aufl., p. 157. The more 'recent formula, Wàrmestrahlung, 2te Aufl., p. 140, which diffel's from the first by th~ introduction of a zero point energy, leads to the same results as regards the specific heat as long as the frequencies do not dep end upon the temperalure. For processes in which 41e frequencies change, cf. P. DEBIJE, Progl'amme for the WOLFSKEHL lectul'e, Physik. Z S. 14 (1913), p. 259, 1t can gi 1'0 tlivcrging results fOL' solids Loo.

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228

that PJJANOK'S principle of fini te elements of energy holds for each principal mode of vlbration of au oscillater whose motions ean be descl'ibed by linear _differential equations, wbatever its constilution may otberwise be. ~ In tbis paper tbose consequences wiU be deduced tor an ideal monatomic gas. Similar applications of tbe quantum-tbeory to an ideal monatomic gas were already made by TETRODE 1) and by LENZ ~). SACKUR 3) aJso makes use of the quantum-theol'y for tbe deduction of the equation of state, but in a different way. The deductions of TETRoDE and of LENZ are based on 'the sup position that each principal mode of vibration of a gas enclosed in a given "essel exchanges its energy by whole energy elements at once. If tbis supposition is uccepted it can be made probable by contemplating the exchange of energy between tbe gas and tbe radiation which is in equilibrium with it, that these energy elements have a magnitude 11zv, v being tbe frequency of the (longitudinal) waves in the gas, h the well known PMNCK'S constant. The mean "temperatllre enel'gy" per mode of vibration is then found to be 1 'tV k bemg the weIl known constant of BOI,TZMANN'S entropy 2-l-I.- ekT - 1 principle. As Prof. KAl\IERUNGH ONNES and I communicated to tbe WOLFSKEHL congress last month, tl118 tl'eatment gives re su lts which do not conflict with observations on the equation of state of hèlillm onJy 4) on the condition that, in writmg down the mean ene.cgy pel' mode of vibratlOn for an ideal gas, a zero point energy to an amount of 1 - hv is added to the above-mentioned expression for tbe temperature 4 enel'gy 5). Tbe same zero point energy was recently assumed fol'

1) H. TETRODE. Physik. ZS. 14 (1913), p. 212.

ll) Cf. A. SOMMERFELD,. Programme for the W OLFSKEHL Iecture, Physik. ZS. 14 (1913), p. 262. 3) O. SAOKUR. Jahresber. der Schles. Gesellschaft fiIr vaterl. Cultur, Febr. 1913. 4) VIZ. on the supposition that the determination of the frequencies of a gas in a wa)' corresponding to that which DEBIJE follows for asolid may be considered as approximalely correct.

5) Hence the fundamental assumpllOlls of this paper diverge from these of TETRODi: and of LDNZ by lhe inlrotluction of tbe zero point enel"gy, from those of LIlNZ mOleOver by the lact (§ !2), that fOI" lhe magnitude of lhe energy elements % kv IS accepted.

- 3 - 229 the molecular rotations by EINSTEIN and STERN 1) fol' the explanation of the behaviour of the specific heat of hydrogen. The supposition of a zero point energy corresponds to the newest ideas of PLANCK, according to which an electrical oscillator ahsorbs energy from radiation continuously, but emits energy by whole quanta at once. According to the fundamental hypothesis underlying tbis paper, a si mil ar behaviour should therefore be made also fol' the ideal gaR. I In order to be able to write down an expres sion for the total energy of the whole system we need the knowledge of the "spectrum" of the gas. We wi1l suppose, that this spectrum can be determined in a way corresponding to that in which the spectrum of a solid is determined by DEBm;. With these data the equation of state of an ideal monatomic gas can be deduced, and then all thermodynamic quantities for such a gas can be derived. The pressure is found 2) to be greater than the "equipartition yalue" p = RT/v ; at high temperatm'es it deviates little fr om this value, the ratio of th is deviation to the value itself approaching ultimately indefinitely to 0; at low temperatures, if the gas may then still be treated as an ideal one, the pressure approaches to a value which depends upon the molecular weight and upon the density, but does not depend upon the temperature. That value we call the zel'O point pressw'e. For a gas like helium at normal \density and say 0° C. the devi ation ti'om the equipartition value appears to be stiJl smalI. It has, ho wever, already such a value that for instance in determining the VAN DER WAAI.S quantities a w and b", or in tbe discussion of the virial coefficients, as was done for the second one in Suppl. N°·. 25 (Sept. '12) and N°. 26 (Oct. '12), that deviation has to be taken into account. 'rhe zero point pressUl'e for helium at normal density is found to be 1/4 mmo At greater densities deviations from the equipartition laws become more important. For the specific heat at constant volume of a gas like helium (supposed to be ideal) at normal density an appreciable de, iation from the constant equipartition value is found only at extremely ~o_w te~~el'atul'es 3). Ultima,tely the specific heat also decl'eases and 1) A. EINSTEIN and O. STERN. Ann. d. Phys. (4) 40 {1913), p.551. S.A.CKUR, l.c., assumes the zero point energy also, as is done in this paper, for the molecular translations. \ S) Cf. O. S.A.CKUR, l.c. S) That the speciflc heat of monatomlC gases dE'viates from the equipartition value m an appl'eciable measure only at extremely low temperatures was already prcdicted by NCRNST and LINDEAlANN, ZS. f. Electrochem. 17 (1911), p. 826, note 1; NERNST, Physik. ZS. 13 (1912), p. 1066. [Note added in the translation.]

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appt'oaehes to O. The suppositions underlying the deductions- of this paper imply that also for ideal gases NERNST'S heat theorem is valid. The l'esults derived in 1his paper for an ideal gas mayalso be of \Talue fol' the theory of fl'ee electl'ons in metals. For this we l'efer ,to the next paper.

§ 2. Tlte energy elements. We may imagine the equilibrium 1 'between radiation alld molecular translatory motion in an idt"al ) monatomic gas enclosed in a given vessel 10 be brought about in the following way: let the vessel which contains the gas be sur I'ounded total1y or partially by a vessel whieh eontains radiation. The walls of the latter vessel are thought to be perfeetly refleeting on the inner side. In the wall whieh is eommon to the two vessels a eylindl'ical hole is made, in which. moves a piston (refleeting on the side of the radiation). This piston is held. e.g. by a suitable construeted spring, in such a way th at under the action of the pressure of those rays which have a frequency v' it is foreed to vibrate. We may interpret the newest theory of PLANCK so that exchange of energy (absorption as weU as emission) takes plaee only by whole quanta at ouee, if we take care to add the zero point energy to tbe valne of the energy at equilibrium of an oscillator derived on that supposition. Sa we may suppose that those rays ean give their ener15y to the piston only by quanta ot magmtude !tv'. The pressure of radiation, being proportional to the product of electrie and magnetic force, has the fl'equeney v = 2v'. -The piston is fOl'ced to vibrare with the same frequency linder the aetion of tee pressl1l'e. _ W. e suppose v' to be chosen in sueh a way that v is a principal frequency of the gas, The motion of th'e piston wiII th en exeite vibrations in the gas of the same frequenry v. We will suppose that the piston ean 1ransfer the quanta hv' immediately to the gas in the form of energy of rays with its fi.'equency v (wbether the piston transfers all the quanta, which it receives, in this form, or only part of them, land perhaps l'emits another part to the radiation, is immaterial). A mode of vibl'ation v of the gas then receives energy by quanta ltv' = 1/2ltV. The l'evel'se, viz. transfer of energy of a mode of vibration v in the gas by the aid of the above-mentioned piston to the mode of vibration v' in the radiation must also be assumed, in faet in the (case of equilibrium to the same amount per unit of time. We could

1) By this I underslund in this paper a gas such, thut the volume of the molecules themselves and their mutlla] attraction need not be considered . .,. ~ I • j " ..

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imagine that the energy transferred to the gas in the first pl'ocess is remitted to the radiation by another way, e.g. with another frequency. In this case enel'gy would, however, move continuously in a cycle. Tbis one will not feel inclined to accept. In the way indicl1ted above, say by the application of a sufficien nnmber of similar pistons, the equilibrium with the radiation cau be brought about for all principal frequencies of the gas. If there are still other ways in which energy can be transfel'red from radiation to the gas molecules or vice versa, the nature of the equilibrium win presumably not be changed thereby. The result of these considerations is that we shaH admit - in so far as the above argument is not considered suf:fieiently cogent we will put as a bypothesis which may be justified or not in its con sequences - th at in the equilibrium between the molecular trans latory motion in a gas and l'adiatioJl eneJ'gy elements of ~ magnitude t hv play a part, if v is a principal frequency of the gas. , \

~ 3. The enel'gy ancl the entl'Opy of an icleal rnonatomic gas. In calculating the mean 1) energy and entropy to be ascribed to a definite mode of dbration with frequency v of the gas, we follow the 'reasoning which PJ,ANCK, Wäl'meatrahlung 2tc Aufl. ~ 135-143 follows for ideal linear electrical oscillators. Considering that for the gas according to ~ 2 we have to do with energy elements ~!tv we obtain (cf. PLANCK l.c. equation (22)): 2UJ (2UY (2UY (2UY t Sy =k ( -+-1) In -+-1) -- ---1) in ----1) . (1) 1 !tv 2 !tv 2 'LV 2 'LV 2 w here Sy and Uv represent the mean entropy and enel'gy fol' the mode of vibration considered. The tempel'ature T is determined by (:~J=~. In this differen tiation at constant volume the wave-lellgth À remains constant; v is connected with .\ by the relation v = ciM c l'epresenting the velo city of pl'opagatiol1. In ~ 4 it will appeal' tbat in the gas, when in thermodynamic equilibrium, on the suppositions for which the simp Ie ia ws of propagation of sound hold, c is pl'oportional to U1/2, also when equipal'tition does not hold, where U is the total energy of molecular tl'anslatory motion of the gas. We will now assume thai I for each mode of vibration we may put c -:: U /2 • This hypothesis is inconsistent with observations fol' vibrations which we can observe as sound, e.g. fol' vibrations with small amplitude, if they are

1) Numerical mean if we think the gas to be repeated many times.

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considel'ed isolatecl. Fo!' vibl'ations with "et'y small wave-Iengths, wbich are here the most important, it may, howevel', be assumed to hold, in connection with their relation to heat motion, at least as an appl'oximate hypothesis for states which diffel' only little from the state of equilibrium, It then follows 1) that

llV 1 h" -+ 2 !tv . (2) kT e -1 As the different modes of yibration, which are possible in a gas enclosed in a given vessel, must be supposed to have the same -T in the state of equilibrium, equation (2) at the same time represents the part which each mode of vibl'ation in the gas in the state of equilibrium at the temperature T contributes to the whole energy. We now assume that we obtain an approximately correct value for the \ whole energy if, in a way corresponding to that which DEBIJE follows for a solid, we suppose the number of different principal modes of vibration w hich are contained in the region determined by the frequencies v and v + dv to be equal to ')

(3) and cut off the so detel'mined "spectrum" at VIII' given by putting the total 11l1mbel' of modes of vibration equal to the numbel' of degl'ees of freedom 3N. V represenrs the volume of the gramme molecule of the gas, Nis the number of AVOGADRO. We then obtain: vm !tv 1 I 9N \ 2 (4) u- 3 !!:.- + 2 hv v dv, - 2vm I I e kT - 1 I oS where Vm is determined by (5)

For the tótal entropy an expression can now al80 be easily given.

1) As SOMMERFELD loc. cito observes, the hypothesis mentioned above causes as it were automatically that at high temperatures the mean energy per degree of freedom becomes = 1/2 kT, as it must be for the molecular translatol'Y motion. For the difference of equation (2) from the corresponding one of LENZ ef. p. 228 no te 5. 2) In aeeordanee with aremark by TETRODE, l.c. this expression ean be easily dedueed for a eubieal vessel from the formula for the wave·lengths oecurring in it: RAYLEIGH, Theory of Sound lI, 2nd ed. London 1896 p. 71.

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§ J. Tlte pressul'e. (1. We will assume that the relation 2U p = 3V' .,. . . . .'. (6) as depending on the isotropy Of molecular motion and on the elemenLat'y fllndamental law of dynamics regarding the connection between force and mO\TIentlllIJ, reUlains generally valid. As TETRODE 1. c. shows, we have then also in genera! 10 U ()~=9" M' ... (7) .LW being the molecular weight. By (4), (5), (6) and (7) the equation of state of the ideal monatomic gas is given. It is easily demonstrated, that the expres sion fol' the entropy dedueed from (1) is consistent with this equation 1). 1f we introduce hv _§ hVm _ , kT - 'kl' - IV, • ...... (8) we obtain x U=~Nk T l~lV + ~Jg3 d§l '... (9) 2 8 ,7: 8 e;-1 \ o By introdueing thc "charaeteristie tempel'ature" 8, determined by _ hVm _'t (9 N)l/S (10 U )1/2 ' f) - -k- - k . 4n- V . 9" M ,.... (10) we ean write

,v (11) = -T ...... b. High tempemtw'es. Developing fol' high temperatm'es '), we obtain 3 l 3 BI 3 B~ 3 Ba I U= 2 Nk T ~ 1 + 521,v 2 -7 4! a;4 + 96ïa;6 .... \ . . . . (12) 1- 1 1 where BI = 6"' B 2 = 30' BB = 42" are the BERNOUJILIAN roefficients Ol'

u = ~ Nk T \ 1 + 2.-,v 2 __I_.v 1 + _I_.v6 • " (13) 2 1 20 1680 90720 Limiting olll'selves to the two fil'st tel'lllS the pressure beeomes Nk T \ 1 f)2 I P = ----v I 1 + 20 T~ ~...... - (14) and aeem'ding to (10), substituting ~for U only t11e first term of(13):

1) Cf. H. TETRODE, 1. c. 2) According to DEBIJE 1. c. tbis development is suitable fr om x = 0 to x = 2.

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(15a) •

-2a A - Taking N= 6,85. 10 , - = 4,86. 10-1I, k= 1,21.10-16, we obtain k 1 8(T»6) = 45,1 . JI1- h V-1fa Tl/~ . (156) fI(T»O) is eonneeted with flo to be introduced in c by the relation

8eT> >0) = (~ fI ° T}/2 . . (15c) Far helium at 0° C. and 1 atm. we find 8 = 13.2. Fl'om (14) follows a deviation from BOYJ.E'S law to an amount of 0.12°;;0' Tbis deviation is in the direction found by experiment and has sueh an amOl1nt that wJth reasonable suppositions about the VAN DER WAAJ.S constants it is not in conflict with the valne experimentally observed 1) :

0.512% 0 , On the othel' hand thel'e would have been contradiction if in the expression (2) for the energy the zero point enel'gy had I not been taken up. This, and a similar result eoncel'ning the pressure coefficient of helium between 0° and 100° C. were the l'easons which led Prof. KAMERUNGR ONNES and me in Our cornmunication ,to the W OLFSKEHL-congress to the hypothesis of the zero point energy for an ideal gas. Although ,the deviation from BOYLE'S law, which follows from the application of the quantnm-theory in the way explained in this paper, is still small in the special case discussed above, it ne\'el'tpeless appears (and for greater densities tbis is even more the case), th at in discussing the eql1ation of state this deviation has to be taken into acrount. Further discussion in this direction has to be postponed, ho wever, to a later communication. c. Low tempemtul'es. For low temperatures the following develop ment 2) is more appropriate: • U=~ l~kTl~.7] + 3f:4 ~_a;1l2(1J e-nx (~+ _3_+_6_+_6_)1' (16) 2 8 15.7]3 n=l n.7] n2a;2 n3a;a n4a;' with ,v according to (11) and (10), The first two terms give (T << 8): - 9 I 16.7!'4 1'4 I 1 U = 16 Nk8 ° 1 + -1"5 ij0 4 ! ' . (17) where _~ (~N)%~Nk fI ° - k2' 43l V . 8 M . (18a) or (cf. b) 2 8 0 = 761 . M-l V- /3 . (18b) 1) H. KAMERLINGH ONNES, Camm. N0. 102a, Dec. 1907. 2) Accarding la DEBYE, l.c., suitable from X = oe to X = 2. )

1 1

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Fo!' the pressure 'P is found: 4 -2 NkO o \ 1 16:r ~I p-S V l +'-15804\' . (19)

For helium at normal density_ 8 0 = 0.239. Rence formula (17) 'ould be valid for helium at normal density at extremely low mperatures onIy. At these low temperatures helium at that density LU no longer exist as an ideal gas. If we call ze1'O point pressw'e, ,. the value which p according to (1.9) assumes for T = 0, Po is a lantity which immediately enables us to get an estimate of the ~viations from the OHART.ES'S law which are to be expected at lew :mperatures. Fot' helium at normal density the zero point pl'essure found to be 1) 332 baryes = 0.25 oom. If from this we deduced

.e temperatm'e according to p -= R1fv, a temperatUl'e ~ 8 0 = 0°.09 ould correspond to it. The error in the reading of sllch a thermo eter, which would occur according to, the above application of the lantum-theory, 'will remain below this amount. For the purpose of the tbeory of free electrons in rnetals we rite (19) in the form ') p = a V_5/3 + b T4 V,...... (20) bere a and bare constants whose values can be easily del1ived. It is easily verified that the th'st term of (20) does not cause Iy decrease of temperature at adiabatie expansion: extern al work dorre at T= 0 at the expense of the zero point energy. § 5. Tlte specijic heat. a. From (4) the specific heat at constant ,lllme ean be derived: fol' this purpose it is to be taken into nsideration that VIII depends according to (5) and l7) on the tem rature 3). We will write down only a few terms of the two cor ,ponding developments. L) The result obtained here differs from that obtained by SACKUR l.c., although : above deductions in many respects run parallel to his. ij Tbe OCCUl'l'ence of the positive power of V gives a waming that the range validity of th is formula at larger V extends to correspondingly lower T only. ,) An expression is found, which c,m be brought to the fo!lowing form:

- (~) + ~ IV eX + 1 Cv _ ~ (~). CrIJ sol 2 eX - 1 Cua:; - 5 \. Ca:; sul ( C ) 3 eX + l' - +-IV-- I CrIJ sol 10 eX - } ~or the values of (~) ,i.e. the ratio of the specific heat for asolid belonging Ca:; sol x to its Iimiting value at high temperatures see DEBTJE l.c. p. 803, or NERN~T, ~lin Sitz.-Ber. 5 Dec. 1912.

\ '

- 10 - b. High temperatures. For high temperatm'es (T>\> 0) we find: ~ C = 3Nk _.!_~) (21) v 2 (1 525 T4 . . . Hence the deviation in the specific hearis of a smaller order of magnitude than that in the pressure. The temperature at which Cv deviates 1 0/00 trom the constant equipartition value, is deter\llined by .1] = 0,85. For helium at norm al density this temperature is found to be T = 0,9; hence except at a considerably largel' densitr a deviation of the specific heat from the equipartition value could not be observed experimentally.

c. Low temperatures. For the theory of free electrons in metals it is of interest to develop the formula for the specific heat at low temperatures. Fl'om (17) we find immediately (T" ( &0) : 4 4 Cv= 12.n Nk (T)3 = 8.:rr • Cva> (T)3. .. (22) 5 &0 5 tJo CvrtJ , the value to which Cv approaches at high T, being one half of the corresponding vahie for asolid the multiplicand of

(~J is equal to that for asolid: cf. DEBIJE, l.c. p. 800. From (20) iu connection with (22) it follows on account of the weIl known thermodynamic relation between Cp and Cv that their ratio % approaches to the value 1 as T approaches to O.

Physics. - "On the theory of free elect?'ons in metals" . By Dr.W. H. KEESO.i\l. Supplement N°. 30b to the Communications from the Pbysical Laboratory at Leiden. (Communicate? by Prpf. H. KAMERLINGH ONNES). (Communicated in the meeting of May 31, 1913).

~ 1. Jntroduction..; SU1n11la1'Y. It seems natural to transfer to the theory of free electrons 1) in metals the considerations of the former paper l'egarding the application of the quantum-theory to the equation of state of an ideal monatomic gas. The frequencies in an electron

1) PLANCK, Berlill Sltz.-Ber. April 3, 1913, has recently treated the equilibrium between oscillators, free electrons and radiation on very special suppositions, which for the free electrolls lead to the equipartition laws. The mode of treatment folIo wed in this paper may in somè measure be considel'ed as the reverse of that by LORENTZ, These Proceedings April 1903, where ft'om the motions of thc free electrons (in the case of equipartition) he deduced the law of radiation (which bolds 'in that case).

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