Vemer k h o m a k e r Universitv of Washinoton. Seattle. WA 98195 and Arthur Amos Noyes Laboratory of Chemicai Physics1, California Institute of ~ e c h n o l i ~~ya. s a d e & ,CA 91125 Jiirg Waser 6120 Terryhlii Drive, La Jolla, CA 92037 The instructive elementary prohlem in chemical thermodynamics to he discussed here goes somewhat as follows: A liquid sample of, say, ether is contained, with some of its vanor. in a thin-walled rieid hulb that is nlaced in a lareer evicusted rigid glass vessel immersed in thermostat h L h held at temnerature T,. The laree vessel is shaken to break the hulb, a i d some hutnot all ofihe liquid evaporates. With T I = Tz = T , and PI = Pz = P,, one readily sees that AGether is zero. But the process is clearly irreversible. Why is AG.,h,, not less than zero? We were told of this prohlem by James L. Dye of Michigan State University, who enjoys using i t in his classes, and we recognized it as one used by Stuart J. Bates a t Caltech some 50 years ago. Bates was one of those grand teachers of thermodvnamics (as were Roscoe G. Dickinson and Richard M. ~ a d i e r who ) educated generations of Caltech undergraduates land eraduate students. too) in the wavs of loeical - and precise thinking that are so urgently required in any venture into this taxing discipline. His course was heavily based on well-thought-out problems. One of his scientific achievements was a precise measurement2 of the Faraday, by iodine coulometry as well as silver coulometry, that was not bettered for many decades. Dye may have gotten this problem (hut says he does not remember) from either Robert E. Rundle of Iowa State University or Max T. Rogers of Michigan State University. Both were graduate students a t Caltech almost 50 years ago, both were fine teachers and did notable research, and hoth unfortunately are deceased. Bates expected the student to answer that the process described is actually neither isothermal nor isoharic, and that the criterion AG < 0, in view of its standard deviation, therefore does not apply. We cannot he sure that he did not also hope the really good student would answer in some of the greater detail that we presume to give in this note. I t must be recalled that the term isothermal, whenapplied to a process, does not only require that the beginning and final temperatures must he equal, but rather that the system temperature must remain uniform and constant during the entire sequence of events under consideration. This concerns many situations, like the one of our prohlem, to which one might wish to apply the Gihhs criterion, AG 5 0, or the Helmholtz criterion, AA 5 0, for whether a process is thermodynamically possible. However, as shown elsewhere,3 these criteria can also be applied to processes that satisfy only a weaker, two-part condition: (1) that they must he monothermostatic, i.e., have heat transfers to or from the surroundings only when the surroundings are at a single temperature, T,, and (2) that the beginning and final temperatures are uniform and the same, and equal to T,. In short, the criteria apply to a monothermostatic process with T I = T z = T,, no matter whether the temperature in the
a
' Contribution no. 7860.
Bates. S. J.; Vinal. G. W. J. Am. Chem. Soc. 1914,36,916-937. Schomaker,V.; Waser, J. J. Chem. Educ. 1986,63,935-937.
988
Journal of Chemical Education
system throughout the process is constant or even definable. &ch monothkrmostatk processes we shall call isothermostatic. For the Gibhs criterion, similar conditions apply to the measure also: the nrocess must he monobarostatic and have PI = P? = P,. Such a monobarostatic process we shall call isoboroslatic. While the term rsvhorrc imulies that the system pressure remains constant throughout the process under consideration, the condition here is only that the process he monoharostatic (that the external pressure in any step of the process in which there is a change in volume should have the same value, P,, i.e., that the work done on the svstem is equal to -P,AV, where AV is the sum of the volume changeiin these steps) and that the initial pressure, P I , final pressure, P2, and monobarostatic pressure, P,, all he eaua1.-That is. with resDect to the oressure. the nrocess is only;equired to be isoh&static; it is not required that the svstem nressure he uniform and constant throuehout the process. In tackling this problem we must he clear as to what is to be included In the;hermodynamic system and what is to be relegated to thesurroundings. It'only theetherissplem, the is neither isothermal nor isobaric: the evaporation does not occur a t uniform, constant temperature and uniform. constant nressure. Instead. the nressure suddenlv falls rises back to P,.,(T,), while the" temand then perature. es~eciallva t the liauid-vapor interface. falls and -it he no great then risds back ti T,. ~ c c o ; d i n ~ l ~ , should surprise that the Gihbs criterion fails. But is the process isothermostatic and isobarostatic? If so, the criterion should apply after all. One can imagine the vessel to he a perfect thermal conductor, so that the process, though non-isothermal, is isothermostatic, but i t is still neither isoharic nor isobarostatic. The pressure exerted by the vessel on the ether system a t first falls to zero a t precisely the moment when the volume that has to be ascribed to the system increases by AV,.,, this being the volume difference between the inside of the glass vessel and the outside of the thinwalled bulb. The pressure then slowly rises back to P,, at constant volume. Most importantly, rather than the work -P,AV, no work is done on the system. So the proper, less severe conditions of being hoth isothermostatic and isobarostatic are not satisfied either, and the criterion AG 5 0 still is not applicable. However. if the entire vessel and contents is reearded as the thermddynamic system, then the isothermos~atic-isoharostatic condition for the Gibbs criterion is satisfied, at least if it is assumed (as i t "always"is until you get to work in the lahoratorv) that the thermostat fluid has a ~erfectlv constant temperature and exerts a perfectly constant pressure. For the total system vessel plus ether AG < 0 must therefore hold. We shall return to this a t the end of this note.
~.~~
~
Three Treatments of the Problem By AS. It is most straightforward simply to look at the change in entropy of the ether in relation to T,, the temperature of the thermostat, and q, the beattransferred to the
ether: is A S > qlT, satisfied for the irreversible process? With w = 0, we have
and q,
= T,AS = A U
+ P,,AV,,
which immediately yield the required result:
statement of the problem. And for the glass AG is just -P,,AV,.,, corresponding to its loss of nominal volume. (Releasing the strain of withstanding the difference in pressure between the evacuated soace and the interior of the ampule implies asmall further change. We neglect it, assuming the glass to be ideally rigid. We also neglect the work done in breaking the ampule and the consequent changes in enerm and entropv of the tube and the -alass of the ampule.) All s z d , we thenh&e AGbw = -P,,AV,.,
P,,,AV,, being a positive quantity. By AA. Since the process is isothermostatic (see above) with zero work, the Helmholtz criterion for our irreversible process, AA < 0, is applicable:
The criterion is satisfied. By AG. Here we return to the implied real problem: to discuss the process in terms of the Gibbs energy. For AG@,,i we have AGwul = AGether AGglas.. In a moment we shall show that AGto,,l does not depend on PeXt, the external pressure, hut to simplify the problem we first suppose that PeXt, as exerted by the thermostatic fluid, is adjusted to P,. The two parts of the total system-glass and ether-are a t the same uniform temnerature T, -. both initiallv and finally. They also are subjected to the same pressureP,, by t h i r respective surroundinas both initiallv and finallv. This is ob;ious for the ether; but requires possibly infamiliar notion for the alass: that the initially evacuated space, of of its volume AV,,,, is part of the gloss system, not surroundings. (Later it is part of the space occupied by the ether system.) For the ether AG is zeroras guaranteed in the
+
a
-PvqAU, i.e.. to0 A&., = A&,.,= > -P,, AV.
+
Volume 65
Number 11 November 1968
969
