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

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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 This PDF was made on 24 September 2010, from the 'Digital Library' of the Dutch History of Science Web Center (www.dwc.knaw.nl) > 'Digital Library > Proceedings of the Royal Netherlands Academy of Arts and Sciences (KNAW), http://www.digitallibrary.nl' - 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. \1 - 2 - / 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.] - 4 - 230 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.
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