176 



NATURE 



[Dec. 25, 1879 



LETTERS TO THE EDITOR 



Whe Editor does not hold himself responsible for opinions expressed 

 by his correspondents. Neither can he undertake to return, or 

 to correspond with the writers of, rejected manuscripts. No 

 notice is taken of anonymous communications. 



[The Editor urgently requests correspondents to keep their letters as 

 short as possible. The pressure on his space is so great that it 

 is impossible otherwise to ensure the appearance even of com- 

 munications containing interesting and novel facts.'] 



The Temperature of the Air at Various Levels 



In a treatise recently published at Prague, 1 the author, Mr. 

 Schlemuller, proposes to establish a formula, by which the 

 temperature of the atmosphere at any level above the surface of 

 the earth could be calculated, a similar calculation giving also the 

 height of the atmosphere. 



Mr. Schlemuller's train of reasoning is about this : — 



The temperature of a gas is dependent on the vis viva of the 

 motion of its molecules. Now, the molecules of the air moving 

 Upwards are gradually losing their vis viva by the action of 

 gravity, whereas, in moving downwards they gain velocity by the 

 same action. It is, therefore, evident that the molecules must 

 have more vis viva in the lower strata of the atmosphere than at 

 higher levels, that is to say, the temperature of the atmosphere 

 must decrease as the height increases. 



If we know the velocity of an air-molecule at the surface of 

 the earth, we can easily calculate the maximum height to which 

 it can move when going up vertically. This height is the height 

 of the atmosphere. At the upper limit of the atmosphere the 

 molecules have no velocity at all, the temperature is there at the 

 absolute zero. (It must be remembered that the author treats of 

 an atmosphere not exposed to radiation.) 



Now, these ideas are not new, as the author himself admits on 

 page 9 of his treatise. He has, however, added to them two 

 new suppositions of his own, and to these we shall confine our 

 attention. 



First, the author supposes that at any temperature of a gas the 

 molecules have a certain velocity, which is equal for all of them, 

 that is to say, the molecules move in all possible directions, but 

 altogether at the same speed. This is, of course, a hypothesis, 

 which can neither be proved nor refuted ; it is, however, admis- 

 sible. The other supposition of the author, however, is quite 

 erroneous, and so the results arrived at by means of it are also 

 valueless. Mr. Schlemuller supposes that the molecular velocity 

 of gases has not been calculated rightly as yet, and he therefore 

 proposes to correct the error. His own words are as fol- 

 lows : — 



' ' Let M be a point of the wall P Q inclosing the gas. The mole- 

 cules will strike this point in all directions, each of them having 

 a mass m, and moving at a certain speed V. All the striking 

 forces form, therefore, a hemisphere, whose radius is equal to 



M 



in V, the wall PQ being the basis of it. The acting componen 

 of the striking force is evidently MN = wFcosa. _ All the 

 possible components wf'cos a represent, therefore, ordinate* of 

 the hemispherical surface mentioned before, taking PQa< a 

 basis." 



As there is no preference for any of the directions, the mean 

 striking force acting on the wall PQ will be measured by the 

 mean value of all m Kcos o, viz., by the ordinate of the centre 

 of gravity of the hemispherical surface. As, however, this centre 

 of gravity is situated at half the length of the radius M K from 



P Q, the mean value of the striking force will be m u = ■ or 



u = _ and V = 211, that is to say, the mean component of the 

 2 



1 Der Zusammenhang zwischeJQ HQhenuQterschied, Temperalur und Druck 

 in einer ruhenden nicht bestrahlten Atmo^phare, sowie die Hohe der -Atmo 

 sphare. Von W. Schlemuller. (Prag : Dominicus, l88o.) 



2 Ey a misprint the original has — - 



molecular velocity taken at a right angle to the wall PQ is equal 

 to half the actual velocity." 



"Considering all this, we shall be able to establish a relation 

 between the molecular velocity, the volume, and the mass of a 

 gas inclosed in a cubical vessel. We shall follow the method in- 

 dicated by Joule, 1 and introducing into the calculations through- 

 out the mean value — , instead of Kwe get for Tdouble the 

 ordinary value, viz. : — 



V= 2V3gP o r (i + at) 

 g being the acceleration of gravity, P a the normal pressure, V Q 

 the volume of one kilogramme of the gas at o° C. (32° F.), 

 a = 0, 00365 the coefficient of dilatation, t the temperature in 

 Centigrades above the freezing-point." 



It seems that Mr. Schlemuller is not aware of the fact that 

 Clausius fully twenty-two years ago published a very elabo- 

 rate treatise,- in which he calculated the molecular velocity, sup- 

 posing the molecules to have equal velocities, but to move in all 

 possible directions. Now these are exactly the conditions sup- 

 posed also by Mr. Schlemuller, and yet Clausius has found, just 

 as Kronig before him — 



V= *]ZgPJ T o{i + °.t) 

 instead of Mr. Schlemuller's double value. 



In another way Briot 3 has found the same result, whereas 

 according to the theory published by the late Prof. Maxwell, 1 

 the molecular velocity is — 



V! 



gP.VJl +at). 



There are thus pretty many calculations published already, 

 all of them, according to Mr. Schlemiiller, being wrong, and 

 even very muck wrong (viz., by loo per cent.). 



It can be shown, however, that the fault is Mr. Schlemuller's, 

 and not Kronig's, Clausius's, Briot's, or Maxwell's. Mr. Schle- 

 miiller, according to his own statement, accepts the calcula- 

 tion given by Kronig (which he ascribes to Joule), simply 



replacing the value V by — . Now, in Kronig's final formula 



2 

 the value V- occurs, and this value is arrived at by a double 

 step. First, it is shown that the force with which a molecule 

 strikes the wall is proportionate to its velocity V ; secondly, the 

 number of strokes occurring in one second is shown to be also 

 proportionate to the value V. Thus the final result is found to 

 contain the value V". If the molecules are supposed to move 

 in all possible directions, it might perhaps be admissible to make 

 the mean striking force of a molecule proportionate to the mean 



normal component m — , (being the mean value of all m Fcos o) 



but it is quite wrong to replace V simply by — , when the num- 

 ber of strokes is calculated. If a molecule of a gas contained 

 in a cubical vessel is moving in the direction of one side of the 



y 

 vessel, it will strike one of the ..walls — times per second, V 



la 

 being the velocity and a the length of the vessel's side. If, 

 however, the molecules move in all possible directions, it would 

 be quite erroneous to suppose that the mean number of strokes 



per second will be *- , viz., that V can be replaced simply by 

 2a 



— . But that is exactly what Mr. Schlemuller does. The problem 



2 



is not very easy indeed, and certainly not so simple as Mr. 



Schlemiiller seems to. think. The elaborate calculations of 



Clausius and Maxwell are a sufficient proof of that. 



Mr. Schlemuller, having thus found his value of V, proceeds 

 to calculate the decrease of vis viva of a moving molecule corre- 

 sponding to a given increase of elevation above the surface of 

 the earth, or, in other words, he calculates the decrease of tem- 

 perature towards the higher regions of the atmosphere. The 

 result found by him is a fall in temperature of 1° Centigrade to 

 every 175-611 m. or 1° F. to 1067 yards. Calculating further . 

 the height of the atmosphere, viz., the height which can be reached 

 by a molecule starting at a given speed from the surface of the 



* For we ought to know the formula for the molecular velocity was first 

 given by Kronig. 



2 This paper was also published in the Phil. Mag., 4'h series, vol. xiv. 

 p. 108. 



3 "Theorie mecanique de la Chaleur," chap. vs.. § 141. 



4 Phil. Mag., 4th series, vol. xix. p. 22. 



