RALPHK. BIRDWHISTELL
textbook forum
Univenity of West Florida Pensacola. FL32504
Thermodynamics of a Process in a Rigid Container Richard M. Noyes University of Oregon, Eugene, OR 97403 Statement of the Problem Schomaker and Waser (1)recently followed a more general paper (2) with a discussion of an irreversible process in a system whose initial state is illustrated in the figure. Two rigid flasks, called flask 1 and flask 2, can be isolated by a stopcock. Flask 1 contains both liqh i d and aaseous ether a t PSY.. vacuum the equilibrium vapor presT sure P,,, and flask 2 is initially evacuated. The system consisting of the two flasks and their contents is surrounded by a thermobarostat (not shown), which maintains an exterior temperature T and an exterior pressure P,, on the outer surfaces of the flasks. (No subscript i s used to designate T because both the initial state and the final state of the system are at the same uniform temperature.) The process is initiated by turning the stopcock, so Initial state of the system. Flask 1 contains a mixture of ether liquid ether vapor can enter flask state, flask and vapor. If the stopcock is 2. In the turned, vapor from flask 1 enters contains a smaller-but flask 2 in an irreversible Drocess. nonzero-amount of liquid The regions occupied by ihe three in addition to vapor, and subsystems%P, and yare desig- flask 2 contains ether vapor at pressure P,,. nated by arrows.
,,,,,,
The Conflicting Solutions Schomaker and Waser ( I ) maintained that for the total system as shown
AG,d= -PvapVz where V s is the volume of the interior of flask 2. Tykodi (3) subsequently disagreed with their analysis and claimed that AGbhI = 0 Peterson (4) has independently expressed concerns about the published analysis, and Schomaker and Waser (5)have responded to both Tykodi and Peterson. In the following sections, I shall present arguments leading to each of these alternative determinations of AGb,,l, and then I shall present my resolution of the apparent paradox.
470
Journal of Chemical Education
Argument Based on the Pressure upon the System When the stopcock is turned to initiate the irreversible process being considered, the amount of heat that enters the system from the thermostat isqi,. During the irreversible process, the total system does no work on the surroundings or vice versa; therefore w, = 0.We can then write eq 1. = 4i" + "in = P*
(1)
We can also devise a reversible path to accomplish the same change of state. The value of w, for that path will be the negative of the minimum work necessary to force the ether in flask 2 back into flask 1; that minimum work is P,.,V2. That minimum work is also the negative of A A w for our process of interest, and we can write eq 2. "rev
= A.4,w
= -P&
(2)
If the pressure of the system were uniform in both the initial and final states, we could use standard eq 3.
AG = A.4 + A(PW
(3)
The system we are considering involves the rigid flasks and their contents. Thus,
A V ~ =I0 If the P appropriate for eq 3 is considered to be P.,, 0 also, and A ( P V = 0 in eq 3. We then obtain eq 4. AGtota1 = -PvapVz This is the conclusion of ~chomakerand Waser ( I ) .
AP = (4)
Argument Based on the Extensiveness of the Gibbs Free Energy All textbooks agree that the Gibbs free energy G is an extensive property. Therefore, if the total system is divided into subsystems by appropriate placement of boundaries, the value of A G b a for any process will be the sum of the values of A G for all of the subsystems. Editor's Note A series of articles about the free expansion of ether into a vacuum has been published in this Journal. The above article by Richard M. Noyes is the culmination of his efforts to resolve the differences put forward by the series. There is a lesson to be learned from the series by both students and instructors of thermodynamics: Before conclusions of value can be reached from
thermodynamics or other disciplines, the system or the problem must be clearly and completely defined. We-the editors and all the referees involvecChope that instructors will use this or similar controversies in the futureto demonstrate the importance of defining the system as the initial step in any problem-solving activity.
The Subsystems
Arrows in the figure designate three closed subsystems-+, p, and y. In cornbination they constitute the total system being considered. Subsystem a is bounded by the outer and inner surfaces of flask 1 and bv the surface of the sto~cock:it consists solely of the matkrial used to construct fiask 1. Similarly, subsystem p consists solely of the material used to construct flask 2. Subsystem y owupies the combined interior volumes of flasks 1 and 2. The matter in this subsvstem consists only of ether molecules in liquid or gaseous state. The Approximate Calculation
In the initial state. flask 2 is evacuated and all of the ether is confined to flask 1. In the final state, the ether is distributed throuchout both flasks. Due to the extensiveness of G, we can-write eq 5. AGtdal = AGO+ AGp+ AG, (5) In the initial and fmal states, all ether is present either as liquid or as vapor at temperature T and pressure P,,. The chemical potentials of liquid and vapor are equal under such conditions. Therefore, we can write eq 6. AG,=O
(6)
In both the initial and final states, flask 1is subject to an external pressure of P,, and to an internal pressure of P,,: Therefore, the initial and final states of subsystem a are identical, and we can write eq 7. AG, = 0
(7)
The auantitative evaluation of AGn presents some Droblema. 1; the initial state, the pressuAon the extemai surface is P..., and the Dressure on the internal surface is 0. In the final state, the pressure on the external surface is still P..,. but the pressure on the internal surface has increased-top,.,. Tvkodi (6)has considered the evaluation of AG for an isothermal change of state by a rigid solid that is subject to different pressure distributions in the initial and final states. ~ u e an h evaluation would seem to require integration of stresses and strains throughout the entire solid. The necessary information will not generally be available for a real system. All we can really say for the problem as stated is that AGD will be positive and much smaller in magnitude than the P,,V2 magnitude of AG1, as claimed by Schomaker and Waser in the preceding section. Then we can write eq 8. AGp= 0 (8)
.
Equations &E lead to eq 9. AGml = 0
(9)
This is Tykodi's conclusion (3). A Resolution of the Apparent Paradox Equation 4 was derived assuming that the pressure "of"
the system was always P-;
it concluded that
AGtot, = -PwpVz
Equation 9 was derived by the argument that AG, = 0 and AG, = 0. It concluded that AGtotalis an extremely small positive quantity. The two arguments can not possibly both be correct! The reasons for the failure of at least one argument involve a topic that I do not recall ever finding in any textbook. Defining a Process in a Ridgid Container
Let a process of interest take place inside a rigid container that is thermostated and subject to a constant exter-
nal pressure. Let us also assert that no electrical circuits connect the inside and outside of the container and that no mechanisms exist for phvsicd work on or bv the surround. ings of the container. Our total system consists of the container and its contents. These two subsystems can be separated for discussing the change of any extensive thermodynamic property during any process that the system undergoes. For such a process, we can write eq 10. =tot,
= =mn,,er
+ =-tents
(10)
Equation 10 can be used to calculate AX%l when X is any of the properties U,S,or A. If the distributions of pressures between the subsystem contents and the inside surface of the subsystem container remain unchanged between the initial and the final states, then eq 10 will also be valid for the properties H and G. Furthermore, if we know the pressures on the subsystem contents, we can talk meaningfully about AX-,, for any property including H and G. A Problem of Ambiguity and Definition
However, there will be ambiguities about trying to define AH,.;, or AGmtai., if the pressure distribution on the inside of that subsystem changes during a process of interest. The definition of H or G for any chemical system includes a P V term in which P is the "pressure of the system". If a system contains a continuousbody offluid (liquid or eas or both) that is in Dhvsical contact with all of the bolndanes of the system, t'h& the pressure ofthat body of fluid can be defined unarnb~rmouslv.However. fa svstem consists of a rigid solid that 7s subiected to differen; pressures on different portions of its surface, then it will he difficult to define the pressure "of' that solid system in an unambiguous manner. ~ c h o i a k e and r Waser attempted to calculate AG for a total system that included a rigid flask whose internal pressure changed during the process of interest. Since the pressure "OFthat flask could not be defmed simolv. calcuiating AG for that portion of the system was a&&uous. They were not justified simply to set the pressure of that flask equal to P,,, and the conclusion of eq 4 is invalid. Tykodi has not proposed a way to calculate AGp rigorously, and the conclusion of eqs 8 and 9 must be left as an a~~roximation. However. for anv real container of elass or & & I thicker than about 1m m ~ ~ ~can ~ ,be , ~set. ~ equal to 0 with an uncertainty that could not be measured exoerimentally. Conclusions
The general conclusion that I reach is that if a chemical process takes place within a ricid container, and if the pressure distribution on the inner surface of the container changes due to that process, then it is not useful to attempt to measure or to calculate AH or AG either for the container or for the total system. Ifthe system originally proposed by Sehomaker and Waser is restricted to the contents of the flasks in the figure shown here, we obtain AGmt,ts
=0
which is in virtual agreement with eq 9. Although all ether is at the same pressure in both the initial and final states, the evacuatedhrtion of subsystem y is at a different pressure in the initial state. The process in subsystem y takes place at constant volume rather than constant pressure. Eauation 2 shows that AA is neeative for our spontaneous process at constant volume, jist as thermodynamics tells us it must be. However, since the process does not take place at constant pressure everywhere in subsystem y, the sign of AG, does not provide any Volume 69 Number 6 June 1992
471
criterion for the direction of the change. There is no violation of thermodynamics when we fmd that AGy= 0
Acknowledgment
The ore~arationof this manuscri~twas SuDDorted in part by' th; National Science ~oundakonunde;~rant No. CHE-8717791. It was only aRer an extensive and sometimes acrimonious correspondence with Schomaker, Waser, and Tykodi that I came t o understand why generally accepted thermodynamic procedures sometimes lead to different conclu-
472
Journal of Chemical Education
sions for the same change of state. Rubin Battino and Scott Wood also contributed to the correspondence about this problem. I appreciate the concurrence with my conclusion that was given by Dr. James Dye of Michigan State University, who apparently suggested this problem to Schomaker and Waser in the f r s t place. Literature Cited
1.Sehomaker.V: Waser.J. J. Chem. Edue. 1988.65.968-969.
