The Enthalpy of Vaporization of a Small Drop Arthur W. Adamson University of Southern ~alifornia.Los Angeles, CA 90089 Milton Manes Kent State University, Kent, OH 44242
In the usual physical chemistry course, the student is (or should he!) given a brief exposition of the effect of curvature on vapor pressure. That is, a droplet of liquid has a higherthan-normal vapor pressure; and the vapor pressure above a meniscus is less than the normal value. The relationship is known as the Kelvin equation (1.2): Here AG is the difference between the molar free energy for the liquid under the curved surface, Gs, and that of liquid having a plane surface, GP. The quantitiesPOf and Poare the respective vapor pressures, y is the surface tension, V is the molar volume, and r is the radius of curvature. It is assumed here that the surface is a section of a sphere, so that the two radii of curvature that in general exist for a curved surface are each equal t o r . The Kelvin effect plays a central role in supersaturation in phenomena and, of course, the associated theory of nucleation (3).There has been a recent experimental confirmation of the Kelvin eauation (4). The purpose of'this article is to describe a related effect that seems not to he mentioned in ohvsical chemistrv references. . Just as curvature affects the molar free energy of a liquid, so should it also affect the molar energy or enthalpy. There should, in other words, he a curvature effect on the heat of vaporization of a liquid (or solid). How might the required relationship he obtained? A simple approach is the following. We differentiate eqn. (1) to ohtain
-
(the vapor enthalpies cancel). It is interesting that eqn. (5) has exactly the same form as the Kelvin equation, eqn. (1). How big is this effect of curvature on the enthalpy of vaporization? Its magnitude depends, of course, on the radius of the droplet. As an example, we can turn to nucleation theory and the case of water vapor at 25 'C having a supersaturation ratio, PO'IPO, set a t 4. There will he a critical droplet radius, r,, below which a droplet placed in the supersaturated vapor will evaporate, and ahove which the droplet will grow indefinitely. This is just the droplet radius from equ. (1) (6),and is about 7.5 A for the case in question. The surface energy of water is about 110 erg and substitution into eqn. (5) gives A(AHvO)= (2 X 18 X 11017.5 X lo@) (1/4.18 X 107) = 1260 kcal mole-'. This amounts to a 12% decrease in the enthalpy of vaporization. There is an interesting additional question, illustrated in the figure. Keeping to the level of simple physical chemistry, the Clausius-Clapeyron plot of in PO versus 1/T should be linear. Consider a droplet a t some temperature TI. By eqn. (1) its vapor pressure is greater than that ahove a plane liquid surface. By eqn. (51, however, the slope of the vapor pressure line should he less. There should therefore be some temperature T, a t which the two vapor pressures become eaual and ahove which the vapor ofthe drop should he lkss than that ahove the olane surface! A simole resolution of this anparent p a r a d ~ ~ f o l l o wWe s . can write eqn. (2) in the form' ~
+
R I n ( P I P ) = 2HSVIrT (2Vlr)dyldT
~~
~
(7)
If we set ln(PO'IPO)= 0 a t T,, eqn. (6)reduces to
(RlnP-RInP)+RTdinP0'IdT-RTdinPIdT = (2Vlr)dyldT (2)
assuming that molar volume varies negligibly with temperature. We use eqn. (1) to replace the terms in parentheses by 2y VlrT, and the Clausius-Clapeyron equation to replace the next two terms on the left by AHV0'ITand AH,O/T, respectively. Here, AHvO'is the enthalpy of vaporization from the curved surface, and AH,O, the usual value. We thus obtain A(AHV0)= 2yVlr - (2yVTlr)dyldT
A semi-empirical equation given by Guggenheim (7) states that y
(3)
where the first term on the left is (AHvo- AH,O'). We next invoke the thermodynamic statement that (5) Hs = y - TdyldT
Finally, the use of eqn. (4), eqn. (8) becomes
(4)
= yU(l- TIT,)"
where T, denotes critical temperature and the exponent n is often around unity. Equation (10) with n = 1 states that d y l d T is a constant and that y goes to zero a t T,. Returning
This is a surface-chemical parallel to the general second law relationship H = G - T(dGIaT),. Equation (4) results if we replace H by HS,the surface enthalpy (often equated to the surface energy, E"), and G by CS,that is, by y . With this substitution eqn. (3) reduces to A(AHvo)= 2HSVIr
(5)
Since AHV0and AHV0'can also he written as the difference between the vapor and liquid enthalpies, eqn. (5) may alternatively he written AH = HS- HP = 2HBVIr
(6)
when HP is the molar enthalpy of liquid having a plane surface 590
Journal of Chemical Education
(10)
I
'/T
Clausius-Clapeyron plot of in of vapor pressure versus I I T .
to eqn. (9), we see that it is satisfied at T, = T,, that is, at y = O, The crossing point of the figure is thus of trivial importance: it occurs at the critical temperature and the paradoxical rexioh above T, has no physicalmeaning. Literature Cited (1) Adsm9on.A. W , " A TexfbaokofPhyaical Chemi.~y," 2nd ed., Academic Prear. New
Y W ~ 187% . p. 27s. (2) Piker'. K. S., and Brewer, L., "Thermodynamics," McGraw-Hill, New York. 1 9 6 1 , ~ . 482. 13) s e e ~ d a m s a n . AW . . . " ~ h ~ ~ h y s icehs el m i s t r y o l s u r f a ~ ewiiey, ~ ~ NPW ~ 0 1 k . 4 t hd. (41 pisher, 1882,L. p. R., 320.and isradaehvi~i,J. N., ~ o t u r e , ~ ? ? , 5 411979). 8 (5) Ref. (3). p. 50. ( 6 ) Soo Re6 (3). p. 320. (7) Guggenheim. E.A,. J. Cham.Phys., 13,253 11915).
Volume 61
Number 7
July 1984
591
