DO MOLECULES ATTRACT COHESII’@L\7 INT7ERSEI,U AS T H E SQU-ARE O F T H E DISTANCI’:? BY ALBERT P. MATHEJVS
In a very interesting and valuable rcccnt paper in this journal by Mills, the conclusion 11 as drawn that the cohesional attraction of molecules varied inversely as the square of the distance. Besides this conclusion, 11 hich was founded on the interesting discovery that the internal latent heat of vaporization divided by the difference of the cube roots of the densities of the liquid and vapor \\as a constant, a most valuable part of the paper was the reopening of the question whether thc. field of molecular attraction is delimited by the surrounding molecules, or whether it owes its small size to the very rapi(1 decrease of the attraction with the distance This qiiestioli raised a century ago by Laplace,’ was answered by him in the latter sense nithout any convincing reason for his conclusion, and has not been reopened since, the opinion being almost universal that the shortness of the radius of action is due to the attraction diminishing n i t h the distance a t a rate far more rapid than the square; the fourth, fifth, seventh and even higher pon-ers having been suggested In thus reopening the question Mills has rendered a valuahlc service. That his conclusion is correct, that the molecular field is delimited by the surrounding molecules, is clearly indicated by Einstein’s’ calculation of the radius of action, shox-ing that the radius is proportional to the distance bctween the molecular centers. The conclusion that the attraction is inversely as the square of the distance, h o ever, ~ I believe to be erroneott5 for thc rc.av)ii\ \\liicli \\ill 1,r prc. sentetl in this paper.
’ Mills: 2
Jour. Phys. Chctn., 15, -11 j ( 1 0 1 1 ) . 1,aplact.: “’l‘rait4 dt. rii6c:iniyue cblestt,” Siipp. Einsteiii: Urudr’s Ann., 34, 16j ( 1 0 1I ) .
:III
T,i\re IO, 13, ,i.ii
Do illolrcules Attract, Etc.
521
Nillsl discovered the empirical relationship that the quotient of the internal latent heat of vaporization divided by the difference of the cube roots of the densities of the liquid and vapor \!as a constant, except in the neighborhood of the critical temperature. If I, is the total latent heat, and E is the part of it used in doing external work, then I, - E would be the part of the heat used in doing internal u-ork. He found that (I, -- E);.(d: - D’’J) = p’. He assumed that this internal heat was all used in overcoming molecular-cohesion, and he ascribed the fall of the constant near the critical temperature to the inaccuracy of the data. Ne then reasoned that s i n x lor U : - bv the expression I I-; - 1 , q J might be substituted, the molecules miist attract each other inversely as the square of the distance; iince it is only 011 such a supposition that the diflerence in potential energy of the molecules in the liquid and the vapor c u i be given by an expression of this kind. The similarity - 1/1’;’’) to Helmof his expression: L - E = ,Y’(I holtz’s formula for the heat given off by the contraction of the sun seemed significant, the Helmholtz formula being i V =- 3, ,RX2KL(~, R - I , CR). From this similarity Rlills reasoned that molecular attraction, like gravitational, must follow the inverse square law. Since it is impossible that molecules should attrzct each other cohesively according to this law, if the cohesional attraction penetrated matter, he concluded that cohesion did not penetrate matter, but was clelimited by the surrounding molecules. There is, however, another relationship expressing the Intent heat consumed in overcoming molecular cohesion or the internal pressure, which has been given by van der Waals :tnd is derived from his expression for cohesive pressure of 11 This relationship is: L - E = cI(1, IYl - I, where
1’7
ZIi11$ Jour P h j s . C‘hem, 15, 417 ( I ~ I I ) , Comptes rendus, 153, 1 0 ; Phil. M a g , [C,] 2 2 , 8 1 ( I ( J I ~ ) , [O] 23, 48.1. ( 1 9 1 2 ) This formula \hould, in my opinion, he nritten I, - E - X = i L ( I ’i’,i,’Y,) \\here X rqireients h e i t riicd in any other internal n o r k than the 5epnr:itl(J1l of thv nioltctilei Sce \ a n dc,r iV‘~11i “ C o i i d v n ~ a t i i mof h s e i , ” Ellcycii). Uritdnlllca, x i ctfitiirn. A l w S~~tlic.rlautlPhil. Mag, 151 2 2 , 83 ( 1 3 8 h ) . ( I ( ~ II ) ,
Albert P. Mathews
Do Molecules Attract, Etc
523
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Albest P. Matltew,r
524
" a " is van der Waals' constant. -According to Sutherland' the latter expression indicates that the attraction of the molecules is inversely as the fourth power; whereas Ilills has interpreted the former as meaning that it is inversely as the square. To show ho\y constant Mills' constant is, I have given, in Table I , the results of the calculations by his formula of a number of substances from Young's data. The figures represent ergs for gram molecular quantities. I t will be seen that the constancy is good for a considerable range of temperature, but that in all cases there is a more or less pronounced drop close to the critical temperature, and in some, as in ether and ethyl acetate, there is a pretty steady fall in the constant throughout. The fall near the critical teniperature might be ascribed to errors of observation, or calculation. There is no doubt, therefore, that for most substances the expression (L- E) ('\ d , - '\ U,) closely approximates a constant except near the critical temperature, as Xills has pointed out. The conclusion that this relationship shows that the molecules attract inversely as the square of the distance is, I believe, sound, if the premise is correct. The premise, or assumption, is that the internal latent heat of vaporization, or L - E, represents only the work clone in separating the molecules against their molecular cohesion. While hlills' states in a recent paper that not all the internal heat may be used in doing this work, and attempts to show that this is not incompatible with this conclusion, the conclusion nevertheless depends on the assumption that it is so used and that there is no change in the internal energy of the molecules on passing from the liquid to the vapor. I t is clear that if this premise be not true, then the conclusion does not follow. This premise I believe to be certainly erroneoils I t could only be true if the molecules remainecl of the sanic' ~ _ _ I
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Mills
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size in liquitl aiid vapor, or do not iii other I\ ays gain energy.’ I believe the internal latent heat of vaporizatioii consists of at least three parts, not two as is oftell stated, these three parts are : ( I ) the heat consumed in expanding against external pressure, or E; ( 2 ) ) ihe heat consumed in overcoming molecular cohesion, or A ; (3), heat consumed in increasing internal molecular potential energy by expanding the molecule, or increasing its energy of rotation, or I. This last factor is often overlooked. If L is the total latent heat of vaporization the expression should be: L - E - I = A. And if molecules attract inversely as the square of the distance we should Iiave (I, - E - I),/ 32./--de - ’ ~ 5 = ) ,ut. There are two principal reasons why it cannot be assumed that all the internal latent heat of vaporizatioii goes to iiicreasing the potential energy by separating the molecules against the force of their molecular cohesions. The first of these reasons is that the value “ b , ” of van der Waals’ equation, has to be taken larger in the vapor than in the liquid for some distance belon- the critical point. ,ind there are good reasons for thinking that “ b ’ ’ represents the real volume of the molecules. The second reason is the value 0 representing cohesion in van der JVaals ’ equation. X third reason has been given by Tyrer. To show that the molecules actually do expand in passiiig from the liquid to the vapor, I have calculated the value of “ h ” for pentane, and benzene using Young’s data. I have also calculated several others of his substances, but as the result is similar in them to that in these two, I give only the latter in Table 2 , n-hich s h o w the value of b in cc in the liyuitl and vapor for gram molecular quantities b = J7 - R’I‘, ( P f CL’v2).2 -4similar objection t o Mills’ conclusion has been raised by Profcssor Tyrcr Phil M a g , [6] z j , 1 1 2 ( 1 0 1 2 ) Since sending this paper to the publisher I h a l e found t h a t the value of “ a ” is larger than assumed here This change requires be and b , both to be larger. .it 150’ b i should be about 1 2 7 cc and a t 40’ only -11 The relation betnecn be and bv is not yreatly changed, b u t the difference between them becomes greater.
1 0 0 . 6 -I 40 106.j - 6 4 . 1 114.6 79.9 124.2 140.3 130.8 Ij I . 9 137'2 155.2 144.5 353.3 149.4 149.4
21,420 4 , 4 2 8 00 1,j12.60 790.0j j 6 7 36 447.5 3j!, I O 309.94 < .
I . I j 0.04 5.80 I 09 15.54 8.72 2j.46 31.96 29.61 61 90 31.91 99.55 3 2 . 9 0 154.70
33.03
2 0 7 . j0
Benzene
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1
be -~
80 O I20 I 60 2 00
240 2 60 2 70 2 80
288. j
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bt
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82.3 70 84 6 -170 8741-22 90 8 52 1 96 2 89 100. j ' 110.3 103 6 120 108 2 127 123.8 123 8 1 ~
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(critical) -
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28,j70 10,160 j ,428 2 ,200
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751
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606 470 2j6
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Table 2 shows that, in pentane, b, is larger than b, from the critical temperature to about 160'. Below this point v L falls rapidly and apparently soon becomes negative. The reason for this apparent fall is undoubtedly the association, or quasi-association, occurring in the vapor as the temperature falls, as van der Waals suggests, the result being that the number of the molecules in the space does not remain constant and hence R does not remain constant. The effect of reducing R to its real value, were we able to correct for the association, would be to make b, larger. In benzene b, falls below b, sooner than in pentane, from which we may infer that the association in benzene is a little larger than in pentane. The apparently negative value of b, is found closer to the critical temperature in the esters which are known to associate slightly. Since association produces an
a1)pareiit decrease ill b,, it is practically certaiii that the differences betiveen 0, and 1)' are actually larger than those indicated. The main fact is then established that be is actually larger than b , for some degrees below the critical point in spite of the association which tends to mask the actual molecular expansion and which, a t lower temperatures, conceals it entirely. It will be seen, also, that, as one would expect, 11, actually decreases close to the critical temperature, owing to the compression of the molecules due to the great increase in internal u/T") and external pressure. This increase of pressure ( P is indicated in Table 2 in the case of pentane, the pressures being given in atmospheres per sq. cm. I t is of interest, in this connection, to compute ivhat is possibly the real value of b,, making van der TVaals' assumptioii that in the vapor, a t low temperatures, b, is equal to 2b1. Supposing that this is the case a t absolute zero we may write the rectilinear diameter formula : b, + b, = b, ii3T, + 7') 2T,). This assumes that at absolute zero the molecules are so compressed that in the solid their volume is one-half of what it would be in the vapor a t the same temperature] and that the volume of the vapor molecules a t absolute zero is that of b , . 0,is very nearly T', 2 . Table 3 b , = b,(igT, + T),2T,)) b,.
+
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-
~-
.
b, (taken from Table ____
2)
-140 - 64.1 79.9 140.3 151 . y 155.2 153.3 149.4
Fig.
The parabola represented by these figures is given in I . If there is a,ny association in the liquid the effect
of correcting for it \ioultl be to retlucc tlic \ d u e o f h ,
‘it1(1
increase b, and so to make tlic par.tbola flatter. I 11a\c‘ computed 11, assuming that there is iio associatioti i i i thc. liquid. The increase of iiL lvith the temperature is seen to 1)c very slight. From abiolute zero to the maximum value a t 100’ the increase is a t the rate of o 074 cc per degree for gram mol quantities. On the other halid ‘’ b changes markedly with the pressure. There are several reasons for believing that “ b ” is the real volume of the molecules arid not four times the volume as was originally suggested. One is that “ b ” a t the critical ”
l:ig I--Volunie\ of iiioleculci ( b for gram mol quantitie5 in cc ) lor Iwiit‘uic b, calculated from lormula b , = Y,-RT/IP a0’;) assuming no d\\O ciation b, calculated by formula 6, = bc(i3T, 1. T) ’ ~ 7 ’ ~b), ) b,’ apparent volume of 6, by formula b,’ = Y%RJ!/(P IL ’V:) assuming no association.
+
+
temperature is very nearly Vc 2 and this is just twice the volume at absolute zero. I t is unlikely that the molecules do not expand in passing from absolute zero to the critical temperature, since a t the former temperature they are under a pressure of 3 5 7 2 atmospheres in pentane, whereas a t the critical temperature the pressure is only 240 atmosphere\. It would take very little separation of the atoms to double the volume. The latent heat sho\vs, also, that heat is absorbed by the molecule roughly proportional t o the number of atoms in the molecule This would mean that the atoms vibrated (or expanded) and they must hence take up more space as the vigor of vibration increases. This would seem to be sufficient
to accoiiiit for doubling the volume of h betwceii o" .lbs.
317d
T,. \'an der ltraals' himself only assumed the constancy of the molecular volume 2, ' ' for simplicity, and has now definitely adopted the idea of a change in volume of the molecules. He has obtained, by niakiiig certain assumptions, the follo\\-ing expression for the change of " 0, the molecular volume: f b -- b
