THE ISTERS-U, PRESSURES O F LIQUIDS BY h L B E R T P. M.ITHE\X'S
The fund;Lmental significance of the constant "(z" of \.a11 der \\-:tals' eciuation makes its exact determinatioii important. T-arious methods 1iax.e been proposed for the determination of this constant. but none of them are entirely satisfactory. In-a recent paper' attentioil \vas drawn to a method by ivhich it coulcl be determinect froni the surface tension ljy the use of Thomas \-ouny's formula, T = ),E;3 , combined \vith the laiv of Ramsay m d Shields. and the values thus computed ivere shon-11 to be closely similar for most substances to the \-alues compiited by \-an der \\*sals' method from the critical temperature arid pressure, but it1 some cases they deviated considerably from his x.alues. I found. also, that the \-slues of the constant " t i " obtained by this method \vere simple functions of the prod~ictsof the molecular iveight and the nuinber of \-&leiices in the molecule and that they could be computed from these 1-alues Iiiasmuch as the method used in computing "tz ' from the surface tension in\-ol\-etl the \-slue of the density a t absolute zero, ivhich \\-ascomputed from Cailletet anti JIal.hias' l a i r - of the rectilinear diameter, and in\-olved. therefore, some uncertainty and \\-;is certainly too high, i t \\-as desirable to find a method of corripiitiiig " a " directly from the surface- tensioii measurements. The desire of finding such a method ivas stimulated by the present great uncertainty of the value of the internal pressures of liquids. Trauhe' has within the past fen- years computed the internal pressure for many liquids, but the results he has obtained are, in my opinion, unreliable because his method of computation in\-olves the use of' the value "b" the real molecular \-olume or co-volume, a L-ery doubtful \ d u e . The values he finds for the internal pressure a t zero degrees are. also. \videl? different from those computed by the use of * ' u " found ~~~~~
~~~~
1Iatheir.s: Jour. Phys. Cliem.. 1 7 , 154 (1913). Traube: & i t . phys. Chem., 68, 291 (190~)).
604
Albert P . Mathews
a t the critical temperature; and in some cases they are not more than half those calculated recently by Lewis' from the latent heats of expansion of liquids ITalden,' also, has recently calculated the value of " a " arid the internal pressure from the surface tension. His calculation is, honever, almost m-holly empirical. It is based, first, on Stefan's conclusion' that it takes onehalf the work to move a particle into the surface \\-hich is required to carry it all the \l-dy to the vapor; and, second. upon an empirical relationship found by n'alden betn een the surface tension and the molecular latent heat a t the boilinq point I t is, hon-ever, by no means certain that Stefan's conclusions are correct and his reasoning does not carry ~0x1victioii There is, also, probably an error in the assumption that the molecules do not change in size on pasqing from the liquid to the vapor and that the latent heat of vaporization represents only the work done in overcominq molecular cohesion Finally ITalden's values for " a " resemble Traube's " a " is aln-ays much less than n-hen computed by van der ITaals' method from the critical data and much less than the values of Lewis The values n-hich Walden has obtained are about tn-o-thirds the values qix-en in this paper. T-alues still smaller have been computed by Davies4 from the latent heat of vaporization. The values he obtains are only about one-third those of Lewis. JVintherj has still other results. Many modifications of van der JT-aals' equation have heen proposed in m-hich "a" was considered \-ariable with the ternperature and "h" more, or less, constant. These attempts have not been fruitful. I t is far more probable t h a t .'b," the volume correction, varies with temperature and volume and that " a , " which is the "mass" factor of the cohesion, is Lewis: Phil. Mag., [ 6 ] 25, 61 (1912). Ivalden: Zeit. phys. Chem., 66, j 8 j ( 1 9 0 9 ) . Stefan: Ivied. h n n . , 2 9 , 6 j j (1886). Da\-ies: Phil. Mag., [6] 24, 4 2 2 (1912). j \Tinther: Zeit. phys. Chem., 60, 603 (190;) l
constant. "il," indeed, as van der Waal's has repeatedly shown, should be considered constant unless association, or quasi-association, occurs. " a " contains the factor S', X being the number of molecules in the volume, J7, hence any association will lon-er ' a" by this factor. The wide divergence of these various values proposed for the internal pressure is khon-n in Table I expressing the internal pressure in atmospheres a t zero degrees, except in the case of 11-alden where the values are for the boiling points and Davies for 15' C. TABLEI Davies 15
Benzene Toluene Cymene
1102
118s 66 I
Ether
7;s
cc1,
1076 165;
CS, Et acetate
--
9
Traube C"
\\.alden b 11.
x-. d \\ aals 0"
1380 I IS0
15 7 0
2494
1,340
222s
__
._
990
\I1501 [ r21o\
-
Len.is oo
2630 2S47
\T.iiitlier
1792 -
271s
-
I-31
ry32
I220
I~SO
,-,)
1305
1490
_ ? 7-OJ-
251s
19So
2170
3363
2917
__
j12SO1
-
2q50
14S6
22 I O
II34('\
In this paper are gix-en the values of "a" obtained in several quite different trays, all of which yield closely a g r e e i q results. I . The first method is a computation from the surface tension. The assumption in\-olved in this method is the depth of the surface film expressed in the number of molecular layers. That the assumption is correct is proved by the outcome. 1. The second method is a modification of Thomas 1-oung's method combined with the lan- of Eotvos as developed in my former paper, but with certain corrections. 3 . In the third method . ' a " is computed from van der \\-sals' ecitiation a t the critical temperature, the assumption being made that in all normal substances 0,. 217 -
Ethyl alcohol
1.746
132.4
21j
IIethyl iiobut!.rate
,719
I.;?.;
'
I
I .7 2 0
T,
2. Derivation of "a" from t h e Surface Tension by Eotvos
Law By the lan- of Eotvos the surface-tension energy TI-; is equal to C(T,-T) The surface-tension energy is a linear function of the temperature counting downward from the critical temperature The derivation of TIT: and S'3TT: has already been given on page 60;. a %.'hat is C of EotT-os? Bot\-os found t h a t C varied betlveen 2 2 ; and 2 34 as is shown in Table but he believed the variation to be accidental and t h a t C should be constant for all substances. It has since been shon-n thdt C is not constant. Ji7hat C is may be shon-n as follons Since the surface energy decreases uniformly XT ith an increase of the kinetic energy of the molecules, and is accordingly a linear function of the temperature, one of the constitutents of C must be the gas constant R ; and since there are o n l y y " molecules in the surface, where S is the number in a qram mol, R must be R for a gram mol divided by 3''".Theremainder of C should be the r'itio of the internal to the external pressure a t the critical temperature as that is the point of departure I-ouIig has shon n that the surface-tension pressure is the cohesive pressure, and the qreater the external pressure the lesi important the internal pressure nil1 be Hence C . I thought, must equal K,R 3P,S n'hile the foregoing redsoning TI as not entirelj. con\ inciiiy the result turned out to be correct, I believe as \Till preiently be sho\fn I hdt\-e uniformly taken S as 0 2 1 K 10;' and R as S 321 > 10: 7
(b)
Since Kc
f
= ti
1
7
~
=
I;cwrc 'r) -
?pc\
1.: \\e have
io)
I I-,
If)
(I
= i2Rj'I'c -
'I?) i\,PcS
I
