The Carnot Cycle Revisited Edwin F. Meyer DePaul University, Chicago IL 60614
Thoughit seems to have a lowprofile nowadays (as indeed does classical thermodynamics in general), the Carnot cycle remains a rich source bf practica~applicationsof the basic ideas presented in introductory thermodynamics lectures, nrovidine insieht relevant to the first and second laws.,cvclic . processes and their graphical interpretation, and of course the extreme inefficiencv of our ~ r i m a r vmeans of Dower generation. The followine brief discussion. which assumes prior exoosure to the ~ & o t cycle and the concept of entrbpy, represents a slightly different approach to the idea of a heat engine and introduces a somewhat paradoxical situation whose resolution may lead students to a better understanding of the thermodynamic principles upon which it is based. We start by acknowledging the "driving force" associated with a difference in temperature; i.e., given a means of conduction, heat will flow spontaneously from a high-temperature reservoir to one a t a lower temperature. I t is common in introductory courses to calculate the entropy change associated with such a flow as an illustration of the second law. If the process occurs irreversibly, with no change in the surroundings,
- -
and A L '
0
(all symbols have their usual meanings). Consider allowing the heat, q, to flow reversibly between the same two reservoirs, using an ideal gas in the usual piston-cylinder arrangement as the boundary to the surroundings (i.e., the pressure of the gas is opposed by an external force that allows "reversible" exoansion or compression). The reversible process is achievLd in three steps: (1) . . Place the eas in thermal contact with the hieh-temnerature reservoir, and allow an isothermal expansion to occur until q units of heat are absorbed by the gas; (2) isolate the piston-cylinder arrangement, and allow an adiabatic expansion to occur until the temperature has dropped from TH to TL;(3)perform an isothermal compression of the gas while it is in thermal contact with the low-temperature reservoir until an identical q units of heat are transferred from the gas to the reservoir. All three steps are carried out reversibly. At this noint the reversible flow of heat has been accomplished; t i e entropy change of the reservoirs is the same as given by eq 1, of course, and that of the gas is simply the negative of that quantity (since the entire process was carried out reversihlv). But now we change our point of view, focus on the gas, and ask, what will be its final state if we now use a fourth step to compress it adiabatically and reversibly until irs tempirature rises bark to TI!? Is theentire proressa cvcle?'l'hat is, is the final state the same as the initial? It isreadily shown, using the familiar ideas associated with the Carnot cycle, that the total q and the total w for the four steps described are both zero; thus no change in internal energy has occurred (indeed, if the gas is ideal, this follows directly from the stipulation that ATfor the process is zerosince (aE1dP)~ = 0.
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MDdlfled Cernot cycle for 1 moi of monatomic Ideal gas operating reversibly between reservoirs at 400 K and 300 K. In m e isomerma1 compression between states 3 and 4, all the heat removed from the first reservoir is deposited in the second. A subsequent adiabatic compression until T = 400 K puts the gas into state 5, at which point q = w = A E = 0, but m e gar is now capable of doing work in returning to its original state. 1.
The fact is, the four steps do not compose a cycle because using the same familiar ideas associated with the Carnot cycle, we find that the gas is at a higher pressure finally than initially. I t now lies at a higher point (5 in the figure) on the P V isotherm associated 4 t h T H , and we have-apparently achieved a reversible, isothermal compression of an ideal gas without doing any work! This observation may cause initial skepticism, hut it is perfectly valid. Although the internal energy of the gas is no higher after the four steps, we must recognize that the reversihility of the entire process requires that no "degradation" can be associated with it. Hence the "driving force" associated with a units of heat at TH relative to Tr. cannot have been lost d;ring the four steps.-1t is in fact "st&ed" in the pas, which is capable of more "useful" work when at a higher pressure, even though its internal energy is the same. In rerms of entropy, the increase thereof associated with the flow of heat to slower temperature is exactly offset by the decrease in the entropy of the gas associated with its compression. (Since AH,,, = 0 in the isothermal process, AG,, = -THA&~,> 0; that is, the Gibhs function, or "free" energy, is the property of the gas that has been increased by the process in question (since ASp, < 0). If the Gibhs function Volume 65 Number 10 October 1988
873
actually flows between reservoirs, and the latter reflects the (maximum) amount of work associated with the "driving force" stored in the gas as a result of that spontaneous flow of heat. Thus the efficiency, e, is given by
has not yet been introduced when this material is covered, this discussion might he an appropriate vehicle to anticipate the usefulness of a thermodynamic function which provides a measure of the work that might he done when a system undergoes a certain change in state.) If we want to take advantage of the "driving force" associnow stored in ated with q units of heat a t THrelative to TI,, the gas, we let it expand reversibly and isothermally along TH until its initial state is achieved, absorbing qr units of heat from the high-temperature reservoir, and converting i t completely to work. At this point, the gas has been through a complete cycle (all its thermodynamic properties are unchanged from their original values), and i t has produced quantity of work as a result of the flow of a quantity of heat from high to low temperature. The gas-plus-pistonxylinder arrangement is thus referred to as a "heat engine". The efficiency of a heat engine, a quantity of great practical interest, is traditionally defined as the work done divided by the total amount of heat taken from the high-temperature reservoir in one cycle. T h e reversible nature of the process under discussion makes its efficiency the maximum attainable, i.e., it represents an upper limit to the efficiency of any real heat engine. In the present case, the total heat 4,. T h e former is the amount that absorbed a t TH is q
We can convert this ratio into a more familiar (and more practical) form by recognizing that AS, for the gas in the final expansion must he equal and opposite in sign to AS,., for the previous four steps (AS,,,,, = 0). AS,, is of course equal and opposite in sign to the entropy change of the reservoirs given by eq 1. Thus we have AS, = -AS,
= q , l T ~= q(1ITL - 1ITd
Substitution of this ratio into the above expression for the efficiency provides the famous Carnot limit to the efficiency of any heat engine,
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Journal of Chemical Education: Soff ware Second Apple II Issue Announced JCE: Software announces its second issue in Series A (for the Apple I1 family of computers). Volume IA, Number 2 is entitled "The Computer-Based Laboratory" and was written by Daniel C. Krause. It contains printed and software materials needed to interfaceApple I1 computers to simple laboratory devices. Directions for building these devices and six student experiments, ready to be reproduced and banded out, are also included. The issue is described in detail on page 875. Volume 1.4, Number 2, costs $35 ($37 foreign) and will be shipped in November 1988. To reserve a copy, complete the form below (photocopy is acceptable); (2) make a check payable to JCESoftware; (3) send both to JCESoftware, c/o Project SERAPHIM, Department of Chemistry, Eastern Michigan University, Ypsilanti, MI 48197. Payment must be made in US. funds drawn on a US.bank or by international money order or magnetically encoded check. In addition to this new issue of JCE;Softwnre, the previous two issues are still available. Volume IA, Number 1is "The One-Computer Classroom" and contains five programs that are suitable for classroom demonstrations using just one Apple I1 computer. The issue was abstracted on page 388 of the May 1988 issue of this Journal. Volume 18, Number 1, is "KC? Discoverernandisan easytouse data base of facts about the elements, designed for MS-DOScompatibleeomputers. It allows the user to find all elements that have a particular property or range of property values, to graph one numeric property against another,to list up to seven properties simultaneously,to sort properties into numeric or alphabetic order, or to use the oeriodie table to eraob group. was described on page 695 of the August 1988issue of . or sort by. .period or . - The program ~
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