March ii, 1909J 



NA TURE 



39 



Number of Molecules in Unit Volume of a Gas. 



The following is an attempt to calculate the number of 

 molecules of helium in unit volume direct!)', that is, with- 

 out assuming Avogadro's hypothesis and the known value 

 of the number of molecules of any gas per unit volume. 



V'arious methods can be adopted leading to the five 

 results given below : — 



(a) Meyer, in his " Kinetic Theory of Gases," has 

 calculated the molecular free path of helium at o" and 

 atmospheric pressure to be L = 24Xio-' cm. ("Kinetic 

 Theory of Gases," p. 194, second edition). 



Now let Q = sum of the diametral sections of the spheres 

 of all the molecules contained in unit volume, then, by 

 the formula 



Q = i/4v'2L (I) 



we calculate Q for helium = 7402 sq. cm. nearly. 



It is also proved in treatises on the kinetic theory that 

 S = 6v'2cL, where S = radius of the sphere of action of the 

 molecule, »■= space actually occupied by the molecules con- 

 tained in unit volume or the coefficient of condensation. 

 Now this coefficient of condensation is always less than 

 the ratio E of the densities of the substance in the gaseous 

 and liquid states, so that, putting E for v in the above 

 formula, we shall get the superior limit to the value of S. 

 We have then EHtiinm = A/5, where A, 5 are the densities 

 of helium in the gaseous and liquid states. Putting 

 S = o-i5 (Onncs), we have 



SHe = 2'4x 10"' cm (2) 



(6) This value is probably high, and Meyer has shown 

 that a method involving deviations of the gas from Boyle's 

 law is more accurate. Onnes gives the value O'Oooy for 

 h in van der Waals's equation for helium (quoted in N.iTURE, 

 October 22, igo8, p. 635). .Accordingly, assuming Max- 

 well's relation b = 4i(, where ti = co-volume or volume 

 actually occupied by the molecules per unit volume, we get 

 the known formula S = (3/v/2)/)bL, where /> = pressure in 

 metres of mercury, for which the value of L which is 

 employed holds good. We have, then, 



SHe = 2'5 X lo"* cm. nearly . ... (3) 



(c) Adopting the relation h^^-Jiu instead of Maxwell's, 

 wc get 



Sue = 176 X io~* cm. nearly . ... (4) 

 This calculation has, it seems, a claim to greater accuracy, 

 since the relation fc = 4v/2« has been confirmed experiment- 

 ally by Holborn (Exner's Report, 1891, xxvii., p. 369) and 

 by Sydney (Chem. News, 1898, Ixxviii., p. 200). 



(d) Calculating Shc by another method, which has been 

 found, in many cases, to give an inferior limit to the value 

 of S from the formula g = {ft.- — i)l(i/.' + 2), where ^ = fraction 

 of the volume containing the gas which its molecules 

 actually occupy, and ;ii = index of refraction. Hence, re- 

 placing v by ^ in Loschmidt's formula, we have 



S = 6v/2^L. 

 Taking data from the paper by Cuthbcrtson and Metcalfe 

 (Phil. Trans., .'V, vol. ccvii., p. 138, 1907), we get 



nearly, whence 



Silt = 0'5 X 10"'^ cm. nearly ... (5) 



(c) Lord Kelvin found that there must be from 200 to 

 600 molecules in the volume of one wave-leng'th of the light 

 <>mitted by the body. Taking the lower limit, 200, the 

 mean distance between molecules is found to be 



jr = 5 X 10-^x5876 X 10"* cm, ... (6) 

 If N = number of molecules per unit volume, we find 

 NHe=0-oi7 X lo'' nearly from (i) and (2) 

 = 1-4x10"' ,, (ijand(3) 



= 2-8 X 10" „ (I) and (4) 



= 36x10"-' „ (I) and (5) 



=4x10"' ,,Equ. (6) . Nx^=i. 



Comparing these values of N for helium, we see that 

 the value obtained from the density of liquid helium is 

 very low, whereas the refractivity method gives a 

 very high value. Greater accuracy seems to belong 



NO. 2054, VOL. 80] 



to those obtained from van der Waals's equation; 

 of these two, the latter value obtained from b = ^^2 is 

 probably the best. Hence, no doubt, it is safest to adopt 

 2-8xio" as the value of N. It is interesting to compare 

 with this the value of N as obtained by Rutherford and 

 Geiger from their recent counting experiment (Nature, 

 November 5, 1908, p. 14), viz. N = 2-72x10"'. 



P. Ghose. 

 Physical Laboratory, Presidency College, 

 Calcutta, January 14. 



An Electromagnetic Problem. 



I AM glad to get Mr. Campbell's views (Nature, 

 January 21) on the electromagnetic problem which I sub- 

 mitted (Nature, November 19, 1908). His method of 

 going back to fundamental definitions is, of course, in 

 general the only safe way where any doubt may enter. 



His remarks, however, considered as an answer to my 

 question, are not quite to the point. .As I carefully stated 

 in the original letter, I am not desirous of setting up a 

 conservation of energy paradox, but merely wish to show 

 that apparently the ordinary expression for the energy of 

 any electromagnetic field is, in the present case, not in 

 harmony with the first law of energy. The accepted ex- 

 pression for the energy in any electromagnetic field is 



^.f(^:-"= 



K, 



where E is the force in dynes on a unit stationary test 

 charge, H the force in dynes on a unit stationary " mag- 

 netic pole," and dr is the element of volume. The test 

 charge and pole are not parts of the system. 



This expression for the energy does not appear to re- 

 main constant while the sphere of electricity is allowed to 

 expand under the mutual repulsion of its parts, for the 

 magnetic force on the test pole is obviously always zero, 

 while the region of integration for E' is constantly 

 diminishing. The difficulty, then, is with this generally 

 accepted expression for the energy, and this is the only 

 difficulty to which I refer. D. F. Comstock. 



Institute of Technology, Boston, Mass., February 10. 



I am sorry if I have misunderstood Prof. Comstock, but 

 niany others besides myself thought that he maintained 

 that the difficulty vanished in some way if the distribu- 

 tion of the electrification on the sphere was discontinuous. 

 My letter was directed against this contention. 



I do not know how the integral expression for th« 

 energy is " generally " interpreted, but if it is interpreted 

 with intelligence it will give perfectly accurate results. 

 The system by which Prof. Comstock measures the electro- 

 static energy is a uniform distribution of " test points " 

 throughout space. When the sphere expands the " region 

 of integration " diminishes, since some of the points pass 

 within the sphere ; but the loss of energy, as calculated by 

 the integral due to this cause, is balanced by the amount 

 of work which the sphere does in passing over these points. 

 If we do not neglect to consider this work, the ordinary 

 integral gives the amount of electrostatic energy whether 

 the distribution on the sphere is continuous or not. 



Norman R. Campbell. 



Trinity College, Cambridge, February 24. 



The Production of Prolonged Apncea in Man. 

 In- Nature of March 4 Mr. Royal-Dawson recalls a 

 statement of Faraday to the efl'ect that Mr. Brunei, jun., 

 and a companion were able to stay under water about 

 twice as long as usual if they had previously been breath- 

 ing air at double the normal atmospheric pressure, and 

 he inquires whether a similar relationship might not hold 

 after forced breathing and o.xygen inhalation, and so 

 enable the maximum time of 8m. 13s. for which I could 

 hold my breath under such conditions to be doubled. 

 Increased pressure would, as a matter of fact, have scarcely 

 any influence. .As I pointed out in my letter of February 18, 

 the essential conditions of prolonged apncea are a previous 

 removal of as much carbon dioxide as possible from the 



