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 XXXI. Intelligence and Miscellaneous Articles. 



ON SOME PROBLEMS OF THE MECHANICAL THEORY OF HEAT. 

 BY PROFESSOR LUDWIG BOLTZMANN. 

 rPHE first section of the memoir has for its subject the relation 

 -*- between the Second Proposition and the calculation of probabili- 

 ties; the second, the thermal equilibrium of a heavy gas. To these 

 the author adds the following communication: — In the Beibldtter to 

 Wiedemann's Annalen der Physih, Band II. Stuck 5, is a treatise 

 by S. Tolver Preston, in which the diffusion of gases is brought 

 into relation with the Second Proposition of the mechanical theory 

 of heat. This cannot (as, according to the notice in the Beibldtter, 

 the author seems to think) serve for the refutation of that proposi- 

 tion ; but it may well be a new and interesting application of it. 

 There is in the same Stuck of the Beibldtter a notice of a memoir 

 by M. Clausius on this subject. Xow, knowing nothing more of 

 the contents of the memoir than what may be gathered from this 

 notice, and not in the least wishing to forestall it, I will merely 

 mention that, as it appears to me, the most essential problem that 

 here comes into question, namely the calculation of how much heat 

 can be transformed into work without any other compensation than 

 the mixture of two dissimilar gases — or, to use the terminology of 

 M. Clausius, the calculation of the transformation-value of the 

 mixture of two dissimilar gases — is only a special case of some cal- 

 culations which I have carried out in my treatise " On the Relation 

 between the Second Proposition of the Mechanical Theory of Heat 

 and the Calculation of Probabilities as regards the Propositions re- 

 specting Thermal Equilibrium." I have, namely, there considered 

 quite generally the case that in any mixture of substances, whether 

 it be the mixture or the distribution of velocities or of directions of 

 velocities, the entropy does not perfectly correspond to the definitive 

 final state, and have shown that then it must always be less than 

 in the definitive final state, so that consequently, on the transition 

 into the latter, heat may be converted into work ; and I have given 

 a formula (formula 51) by which the difference of entropy, and con- 

 sequently also the amount of the convertible heat, can be calculated. 

 From this formula (51) it immediately follows that the entropy 

 of a mixture of several gases is exactly equal to the sum of the en- 

 tropies which would belong to the individual gases if, at the same 

 temperature and under the same partial pressure, each were alone 

 present in space. If V is the volume, T the absolute temperature, 

 % the weight of the gas, c and c' its two specific heats, then its en- 



J7Ci 

 — , dQ being = the heat intro- 

 duced) is 7 fcdT . , , , v, T7 

 |~T~ + / < c - c ) i °g y - 



We will now consider two cases : first, two different gases are 

 present in two different spaces V 1 and V 2 , under equal pressure p 



