MOLECULAR ATTRACTION. XIII. RELATIVE T O AN ARTICLE BY A. P. MATHEWS1 B Y T.
E. MILLS
It is always with regret that I publish an article only to discuss the work of another. But a recent article by A. P. Mathews contains so much of value that the article will receive attention, and I therefore feel compelled to point out the very erroneous ideas that he advances in explaining my own equation and the equation of Dieterici. First.-It is obvious that two equations which are simultaneous and true can be combined, and the resulting combined equation will also be true. Consequently the several equations that I have from time to time derived, and several others that I have investigated, have been combined with other equations in various ways. But I have consistently refused to combine these equations with the equation of van der Waals, I.
(P
+ ):
(V--’b)
=
RT,
because that equation, as is well known, is not correct. I n the above equation a / v z is the term added in order to allow for an effect produced by the cohesive or molecular forces, and is usually termed internal pressure. I have calculated and published an expression for the internal pressure derived theoretically upon the assumption of an inverse square law of force, and have accumulated considerable evidence to show that my expression is correct. Now hfathews, in a round about w a y , as will be shown later, simply substitutes my ( I think correct) expression Jor the internal pressure Jor uan der Waals’ incorrect expression, and thereby i s enabled to calculate “a” of van der W a a l s ’ equation correctly at the critical temperature. He is pleased with the result (of that I am glad) and, by one of those oversights which i t is so difficult to avoid, thinks that this proves the Jour. Phys. Chem.,
20,
554 (1916).
J . E. Mills
102
efficiency of van der Waals' equation and furnishes an explanation of my own work on molecular attraction. As a matter of fact the substitution proves nothing a t all except that the substituted expression was correct if the results obtained are correct. Since the substituted expression was my own, I a t least am glad if correct results were obtained. The correctness or incorrectness of the original equation of van der Waals cannot of course be determined by such a substitution, and some of Mathews' admiration of that equation for the results he obtained ought in reason to have been distributed elsewhere. Mathews makes exactly the same mistake with regard to an equation derived by Dieterici and investigated and extended by myself and others. Needless to say, had Mathews reached his conclusions directly he would not have fallen into the errors he made. Probably he overlooked an article1 by the author on internal pressure. In that article, page 263, Equation IO, there is given for the internal pressure p in millimeters of mercury the expression,
where V is the volume of one gram in cubic centimeters. (So far as the critical temperature is concerned the calculation had been published several years before.) If we let u/zl2 of van der Waals equal this expression at the critical temperature, we have, since V of van der Waals' equation is the volume of a gram molecular weight, 3.
- a- 3&, MV,2 3V,"3
o*
a
= 31414 pLIV;/3Mz
3
Changing from volume to density, and from pressure in millimeters of mercury to calories we have the equation in the form given by Mathews as his Equation 26, 4. 1
Jour. Phys. Chem., 19,257 ( 1 9 1 5 ) .
Molecular Attractioa
‘03
Ij “a” i s calculated according to this equation and substitated in vait der Waals’ equation the operation i s oj course identically the same operation as rubbixg a l v 2 out of van der Waals’ equation and writing in its place the value given in Equation 2 . Y e t f r o m this operation ivathews claims to have discovered an explanation of my equation, calculating p’ of my equation ifi terms o j (‘a’’ and failing entirely to realize that the operation he performed was the substitution of my equation j o r the internal pressure j o r v a n der Waals’ expression in toto. Kor was any attempt made t o show that “a” thus calculated a t the critical temperature was a constant at other temperatures or volumes. As regards Dieterici’s equation I also showed in the paper cited,l page 269, Equation 17, that the internal pressure could be exmessed as, p = -CRT 5. V ’ an equation probably exactly true only a t the critical temperature. Mathews similarly substitutes this expression for van der Waals’ a l v 2 and finds in consequence also an “explanation” of Dieterici’s equation. See Mathews’ Equations 14and 15. - =U C R T ~ or , a = CRT,V,. 6. M2VC2 V, Had Mathews taken a third expression for the internal pressure a t the critical temperature given in the paper above
he could in an exactly similar manner have derived a most interesting (‘explanation’’ of the Clausius-Clapeyron thermodynamical equation. Then finding the mine so rich he would undoubtedly have suspected that it was salted. Secoqzd .--Mathews compares the “a” calculated by Equation 4 with the “a” calculated by Equation 6. Of course 1
Jour. Phys. Chem., 19,2 5 7 (1915).
J . E. Mills
I04
in comparing two functions one can multiply or divide them by any desired quantity but such multiplication adds nothing new to a comparison, and the relation between p’ and C for 30 substances is shown in Table 2, page 1 1 2 0 . ~Also the equation dP
8.
2R
a=v,
and the values of p’ calculated by its aid are there shown. lktathews’ Tables 111 avtd I V add absolutely nothing new. Third .-The fact that 9.
34d = energy necessary to overcome molecular attraction 2 X kinetic energy of molecules 3RT,
c = p’ -c
was pointed out by the author in 1 9 0 9 . ~ Compare this with Mathews’ statement at the bottom of page 575. Fourth.-Mathews, in Equation 3 8 , page 591, and Table VI, gives the relation between the internal and the external critical pressures also as something new brought to light by his discussion. This Equation 38 is the same as my Equation 2 I with the term “S”transposed from one side of the Equation to the other. Fijth.-Mathews refers to Dieterici’s equation for the internal heat of vaporization as being theoretically grounded and says that my own expression is an empirically discovered relation. I have already tried to convince Mathews4 that my equation was theoretically derived. I t was entirely so derived. N o experiwental evidence whatever was examined until the equatiotz. was complete. Dieterici, on the other hand, has this to say5 concerning his relation: “Die Beziehung, IO.
V
X = CRTln-
V I
ist als richtig zu betrachten; das Resultat ist ohne jede
5
Jour. Am. Chem. Soc., 31, 1099 (1909). Ibid., 31, I I 17 (1909). Jour. Phys. Chem., 19, 655 (1915). Ibid.; 18, 104 (1914). Drude’s Ann., 25, 574 (1908).
LZlolecular Attraction
105
hypothetische Annahme allein durch Berechnung gewonnen, die sich unmittelbar auf das Beobachtungsmaterial stiitzt ; ist also als ein reines Beobachtungsergebnis zu betrachten.” Sixth.-On page 567 Mathews calculates the energy required to get one molecule into the surface layer and then multiplies this value by the number of molecules to get the total energy required. This procedure is contradictory to the law of force he gives on page 589. If his argument on page 567 is correct the force must vary as the mass and not as the square of the mass. The same mistake is repeated on pages 568 and 570. (Mathematically in my own work, insofar as the mass is concerned, I can proceed as Mathews does on page 567, since the law of attraction that I have found to be true does require the force to be proportional to the mass and not to the square-of the mass.) I make the above criticism humbly as I fell into identically the same error in my first paper. Seventh.-On page 563 Mathews says, “since only the vertical component of the molecular motions tends to carry the molecules from the liquid to the vapor, the increase will be proportional to the cube root of the temperature interval from the critical.” See also page 571. If V’ is the velocity in any direction and x is the vertical component, it is usually TT’Z
supposed that
x2. In investigating the work of Goldhammer and Eotvos and in attempting to find a simple relation between gravitational, molecular, and chemical forces, Mathews’ work is of real value. While I do not agree perhaps with all of his arguments, I particularly hope that his work in the latter direction will prove of real significance and I do not see why the errors pointed out in this article should affect the real value o€ that work. Summary It is shown that Mathews in his recent discussions of equations by Dieterici, Mills, and van der Waals, simply substitutes for the internal pressure term in the last equation 3
=
106
J . E . Mills
the more correct expressions previously derived and published by the author from the first two equations. Consequently his conclusions as regards these three equations are wrong. Other mistakes are mentioned. The essential part of his work on the relations of Goldhammer and Eotvos and on the attractive forces is not vitiated by these mistakes.” University of South Carolina Columbia, S. C. Oct. 14, 1916
