Leonard K. Nash Haward University. Cambridge, MA 02138 At the normal boiling temperature ( T h )vaporizatjun proceeds with standard molar changes of enthalpy (Atlhn)and entropy ( A S h n ) for which Trouwn's rule gives APb" cal ar88- joule &O , = -.- 21Tb male "K mole OK or, in a form happily independent of units ASb'VR = ARbolRTb
-
11
That rule works best for nonpolar, quasi-spherical molecules, but as a first approximation is as remarkat,le for its generality as for its simplicity. T o begin this review at the centenary of the rule's disruverv. we recall that the ohvsicist Frederick Trouton (1863-19i2) was Dublin horn and bred and is the subject of a good article in the "Dictionary of Scientific Biography" (which lists other sources). Remembered also for the Trouton-Noble exoeriment. Trouton himself attached little impurtance to his r&, which he discovered while still an undereraduate "as the result of an afternoon spent playing with n&nhers." This statement of a memorialist ( I ) is probably too extreme, for Trouton published two closely related papers (2) in only the second of which (a year after the fust) do we find Trouton's rule as such. Let X symbolize the latent heat of vaporization (in callg), and M the molecular weight of the vapor (in glmole). In his second paper Trouton declares the apprdximateeonstancy of the ratio MXITh and documents his claim by tabular citation of emp i r i d d a t a for 32 substances (one quarter of which comprise water, alcohols, and carboxylic acids we recognize as infelicitous exemplars of Trouton's rule). Trouton is no douht the effective discoverer of his rule. T o he sure, that relation had earlier been recognized by Raoul Pictet. But, in a wholly theoretical paper (3), Pictet gives no stress to an emoirical relation he values mainlv as (eauivocal) evidence for tde theoretical hypothesis "thacthe cohesion of liquids is constant for all." In Trouton's brief paper, on the other hand, his rule and its empirical documentation is the sole messaee. Where less mieht leave one skevtical. and more might obscure the message, Trouton's deilaration comes across utterlv clear and convincina. No reference to e n t r o ~ y appears in what has (or to therm;dynarnics more the flavor of a vurelv factual statement. Indeed, Trouton's "molecular weights" &e too factual expressions of relative gas densities referred to hydrogen taken as unity. Hence (like Pictet before him) Trouton finds for the numerical value of his quotient only half the modem value (-21). But thisisonly a de&l with no real bearing on his message: ". . .the molecular latent heat [is] directly proportional to the absolute temperature of the point of ebullition"-which Trouton finds strikingly reminiscent of Dulong and Petit's law. In the same veriod asTrouton's discoverv. there a o r ~ a r e d also a number bf related generalizations wgich, whe;.'joined with the Clausius-Claoevron eouation. entail (and are entailed g (4), by) Trouton's rule. Some, like the ~ a m s a y : ~ o u nlaw imolv Trouton's rule while leaving its constant indeterminate: others, like Crafts' rule (5), yizd Trouton's rule with nu: merical constant.' Van der Waals' conceot of corresvondine states ~uggestedthe possibility of ration&zing andiefinink Trouton's rule, hut contemvorarv efforts (7) in this direction were unimpressive. Yet the need for some refinement of
Trouton's rule could early he recognized in the systematic increase of the Trouton auotient with hoiline temoerature (e.g., from 17 for nitrogen doiling a t 77'K, to 7.3101 z&c boiling a t 1180°K). This trend prompted many efforts to replace the Trouton constant by some purely empirical function of Tb. Of these endeavors v e r h a ~ the s most successful was that ; t o he confused with his distinof V. A. ~ i s t i a k o w s k(not guished nephew, the late G. B. Kistiakowsky). A quasLtheoretical approach (8)suggested to Kistiakowsky that AHbnITb = R In RTh. Inside the In term, where the RTh term represents the molar ;olume of the vapor at the normalhoiling point, he setsR = 82 ml atmlmole O K . With that suhstitution wearrive a t Kistiakowsky's rule, in dimensionless form, as ARbolRTb = 4.40
+ In Tb
This appears (9)a substantial improvement on Trouton's rule. Unfortunately, however, the theoretical derivation of Kistiakowsky's rule is unpersuasive, and that rule was completely overshadowed by an antecedent proposal to repair the shortcomings of Trouton's rule not by analytical refinement hut, rather, by graphical construction. Hildebrand's Rule While exolorine the use of Trouton's rule as a measure of association-in liqiids, in 1915 the indefatigable Joel Hildebrand came (10) to rewrite the Clausius-Clapeyron equation in the form
For substances ranging from nitrogen to zinc, Hildebrand olotted v a ~ o r - ~ r e s s udata r e on axes chosen for convenience m versus log T P K . On such a plot let us draw log ~ l i Hg a horizontal line revresenting loe P = log 760 = 2.88. Where this line intersects the vapor-pre&ure curves the slope of each = curve is d log P l d log T = d In P l d In T = AH,,IRT AS,,IR for the corresponding substance; but Hildehrand ohserves that (Trouton notwithstanding) these slopes are not acceptably equal. At their intercepts with the log P = 2.88 horizontal, Hildehrand continues, the vapor-pressure curves of his very diverse suhstances display a slope that
.. . increases reeularlv with the ~ , loearithm of the temoerature. Therefim the entropy of vaporilarion for different sul,stanw cannot be the same at equal pressures, bur rather at prersurev that increase that the tangentato insome way, with the temperature. It wapf
