An Important Elementary Theorem in Thermodynamics Verner Schomaker Arthur Amos Noyes Laboratory of Chemical Physics,' California Institute of Technology, Pasadena, CA 91 125 Jiirg Waser 6120 Terryhill Drive, La Jolla, CA 92037 The student comes to understand that thermodynamics relates work and heat to change in state. involves three kev added quantities in cornparis& to mechanics (heat itsell, temperature, and entropy), and sometimes answers such questions as Q1: What is the greatest amount of work that can possibly be realized from a given amount of heat? and Q2: What is the least possible amount ofwork needed to achieve a given change in state?
hoth regardless of any details about the process to he used and indeed without any specification or knowledge of any such details for even one particular possihle process. But for the second question it is not made clear either (a) under what conditions an answer should he forthcoming or (h) how to find it if it can he found. Good problem examples occur in many of the textbooks, hut the student is expected todiscover his or her own ineenious a~olication of the First and .. Second i.aws of Thermodynamirs in rvery rase. In this article a com~rehensiveoutline of all vossihlc situations covers item (ai, and a simple, elementary theorem that covers (h) is derived and illustrated. It is stated in the next paragraph. Finally, the theorem is applied to constantvolume, nominally isothermal processes and to nominally isobaric-isothermalprocesses, i.e., to processes with the temperature of the surroundings held constant for the duration of the orocess. and to nrocesses with hoth the temoerature and pressure bf surroundings held similarly constant. The result is that the Helmholtz function. A = U - TS. and the P, V, T , a n d ; ~ a r ethe Gibhs function, G = U + P V - T S (u, internal energy, pressure, volume, temperature, and entropy of the system), are shown to he applicable to these processes even though the usual derivations are defective. They are based on the assumptions of truly isothermal and of truly isothermal-isobaric conditions, whereas no assurance in the irreversible case is possihle, except for times before the process starts and after i t ends, that the respective system variables are constant and uniform. The Monothermoslatlc Theorem Stated As derived later (see relation 41, if just one heat source or sink, of temperature T,, is available, the least possible work to produce a specified change in the state of a system is equal to the increase in internal energy .. of. thesvstem minus the monothermostatic temperature, T,, times-the increase in entropy of thesystem. Here thermodynamics appears in a characteristic role, providing not a definite result, hut only a limit, a least amount of work to attain the stated end. There is no need to consider the details of a possihle process hut only to assume that the surroundings, with which thermal contact may he either continuous or only occasional, are at a = T, whenever single and constant temperature T,,,.de, thermal contact occurs and that the initial and final states are thermodynamically definite, so that AUand A S are well defined. Note that the initial and final temperatures need not he equal to T, nor even to each other.
' Contribution No. 7392
Baslcs We take the first and second laws of thermodynamics in the form
Here AS and AU are the changes in entropy and internal energy of a system in consequence of a change in state accomplished hy some process, q is the energy added to the system by the mechanism of heat flow (thermal transfer) in consequence of differences in temperature between the system and its surroundings, w is the amount of work done on the system, and TsurroundingJ is the temperature of the surroundings, assumed to he well defined and constant during any stage of the process in which heat flow occurs. In relation 2 it is assumed that TsurroUndings is the same in every such stage; if it is different in different stapes, - the chanee in entropy has of cuurse to he expr~ssedas a sum ofterms. The i~tequrtlityin relation 2 applies tvhen the Drucesses that coon in the system and lead to its change in state are irreuers;'ble; if these processes are reuersible, the equality applies, and the temperature T of the system is well defined. Among the processes in question is heat flow from or to the outside, = T. The equality reversihility of which requires T.,,,,,.~i,,,, part of relation 2 can therefore also he stated as AS = q/T. (Note that the eaualitv does not reauire that the heat flow in the surroundin& be ;eversible; required is only that the temperature of the part of the surroundines in contact with the system be nnifoLm.) When Do 01 and 0 2 Have Answers? These questions reflect the traditional preoccupation of thermodynamics with work. A natural sequknce of situations comes into consideration: I. Isolated system: w = q = AU = 0; AS t 0. Adiabatic system: q = 0. w = AU; AS 2 0. Monothermostatic process: There is a single temperature of surroundings, T,, at which heat exchange occurs. Our "Elementary Theorem" concerns monothermostatic processes. IV. Polythermostatic process: Heat exchange occurs with the sur-
11. 111.
roundings at different temperatures during different stages of the process. The simplest polythermostatic processes are dithermostatic and eyelie, namely, Carnat cycles. "How much work must he expended to achieve a given change in state?"The answers for I and I1 are trivial: zero for I and w = AU for 11, as already noted. (The latter is just an expression of the equivalence of work and energy inmechanics, extended, to he sure, to include the new dimension temperature in the specification of state.) For 111, as shown in the next section, there is a simple, nontrivial answer. For general polythermostatic svstems (IV) neither Q1 nor Q2 is any longer sensible: there is too much freedom of fhoice of conditions. Q1 does remain sensible for the dithermostatic, cyclic system. Indeed, it motivated the development of thermodynamics, beginning with Carnot. The answer is eiven by the Carnot efficiency, Volume 63 Number 11 November 1986
935
where T, and Tl(T, > TI) are the temperatures of the heat reservoirs and q, is the heat transferred to the system from the reservoir a t the higher temperature. But Q2 is no longer proper, because the answer would he none, or even --. With the two (presumably infinite) reservoirs we could generate any power that might be needed and sell an unlimited excess to the electrical utilities besides. The Monothermostatic Theorem Derived For I11 we have w = AU - q (first law) and -q 2 -T,AS (second law). Now add the equality and inequality:
the following simple assumptions: the molar ratio of Nz to 0 2 in air is 41; air is an ideal gas mixture; Cp for Ndg) is constant a t 712 R; the work recoverable from the atmosphere is given by 298 X R, i.e., the final volume of the system is negligible. The boiling point and enthalpy of vaporization of N2 are2 Tb = 77.34 K and AH, = 1333 callmol. With R = 1.987 cal mol-I K-' the calculation for five moles of air then runs AS/(cal/K) = 4 X 1.987 In 0.8
AUPxcal = 4 X 4.97(77.34 - 298)11000 - 4 X 1.333 + 1.987 X
wkeal t -9.563 This is precisely the theorem enunciated earlier. Example 1 "What is the least possible amount of work that must he done to convert 2 mol of carbon dioxide at 298 K and 1atm and 1mol of steam a t 400 K and 1atm into 1molof acetylene a t 1atm and 298 K and 512 mol of oxygen a t 20 atm and 400 K, the only available heat source (or sink) being a t 373 K?" The chemical reaction and specified conditions,
-
C,H, (g, 298 K, 1 atm)
+ 512 O2 (g, 400 K, 20 atm)
have been chosen, in disregard of any question of practicality, so that the necessary data could easily he got from a standard compilation. Stull, Westrum, and Sinke2 give the following data for the ideal-gas approximation, which we therefore adopt also:
+ 1.987 In02 + 4 X 6.94 In (77,341298)
- 1333177.34 - 1.987 X in 20 = -65.68
0.0774 = -9.564
+ 0.373 X 65.683 = 14.937
Neglecting the final volumes, we then have w'kcal > 14.937 - 5 X 1.987 X 0.298 = 11.977 In kilowatt hours per pound w'is surprisingly small, (12.014) X 4.18 X (454/28)13600 = 0.565). Evidently the thermodynamic limit for air separation has very little to do with the actual cost of producing and distributing liquid nitrogen. Formal Illustrations These numerical examples are good in their way, illustrating how the theorem can he used "in the real world", hut they are perhaps not enough. The student should solve (or better invent) formal illustrations, too. Illustration: How much work is needed to bring 1molof an ideal gas of constant C,from T, to T2,a t constant volume VI, with T, = TI? The theorem says w
r CJT, - T,) - T,C,ln (TJT,)
The possible reversible path of isothermal expansion to \" followed by reversible adiabatic rompression to \',,'/'? - confirms the minimum amount of work as w = RT, In (V'IV,)
+ Cv(T2- T,)= -C,T,
in (T21T,) + C,(T2 - TI)
the adiabatic compression satisfying the familiar relation
5
*Thecalculation then runs
Another illustration: The theorem can easily be applied to the Carnot Engine and the Heat Pump. Take as the system of the theorem the Carnot "upper" reservoir or the room to he heated or cooled. Suppose this system to be ideal, with very large heat capacity as compared to q. The relevant increments in U and S are then q and qIT, where T is the virtually constant temperature of the system. If T, is the effective ambient temperature for the heat pump or the temperature TI of the Carnot engine, the result for w follows immediately as
wkcal t 300.46 - 11.22 X 0.373 = 296.28 There is no question here of how to achieve this limit for such an unlikely reaction, of course, but only the firm statement that no process could ever possibly be devised that would violate it. Example 2 "Estimate the minimum work required to produce, from air a t 1atm and 25 OC, 1mol of liquid nitrogen a t 77 K and 1 atm and the corresponding amount of oxygen a t 20 atm and 25 "C, the effective available heat sink being a t 100 'C." If we looked up the precise data on the properties of air, nitrogen, and oxygen that doubtless exist, in view of the great industrial importance of these gases, we could work this exercise in the same way as we have worked the first one, largely or entirely from the measured properties of the snbstances. Here, however, we shall make estimates based on 938
Journal of Chemical Education
With q > 0 and T > T, or q < 0 and T < T, this describes the heat pump; with q < 0 and T > t,, the Carnot engine. Note that opposite sign conventions apply to q, in relation 3 for the Carnot efficiency, which is valid only when q, is positive, and to q in the present relation, which is valid for either sign of q. We donot suggest that the student should be excused from the requirement to be able to derive these results directly from the First and Second Laws. We do suggest that the derivation given here is simpler and less likely to be bungled if thereis hardlytime to think or if paper
Stull, D. R.; Westrum, E. F.; Sinke. G. S. "The Chemical Thermodynamics of Organic Compounds"; Wiley: New York, 1969.
and pencil are not a t hand. The sense of the inequality is always the same, and the answer to the memorv-vexineaues.. . tion d r r hirh Tshould he in the denominator isgiven almost nutumaticalls hq its association with o to define AS for the entity chosen adthe system for the dis&sion. Connections wllh A and G
While questions concerning work are important enough to make relation 4 a central issue in chemical thermodynamics, the use of that relation as a test of thermodynamic possihility for supposed processes is just as important. Consider an ordinary fluid system undergoing some process while held a t constant volume and temperature. I t is customary to introduce the Helmholtz function, A = U - TS, and to derive as the condition that the imagined process be (thermodynamically) possible, with no further restrictions, or else
constant and equal to P, during any changes in the volume of the system. With the additional restriction of zero net work (w' = O), criterion 9 becomes AG 5 0
(12)
The usual derivation of these criteria (relations 9 and 12) assumes a genuinely isothermal, isobaric process, hut in the irreversible case, as explained and illustrated above, there can nu Kuarantee either that the nominally isorhermnl process is nrtually isothermal or that the nominally runstant system pressure is actually uniform and constant. Accordingly, criteria 9 and 12 are not justified for their usual application by their usual derivation. However, just as before, the first and second laws can be combined once and for all in a form precisely suited t o the monothermostatic, monobarostatic situation and the definition of w', simply by adding relations 4,10, and 11 to get w ' 2 A U - T,AS+P,AV
(13)
and, for w' = 0, with the additional restriction w = 0, which follows if the only possible work is P,V work. Both criteria (and especially 6)are then -aenerallv ~ o l i e dalso to situations that are onlv . a .. nominally isothermal. An example is the explosion of a mixture of hvdroaen and oxvaen in a constant-volume homh. where the system temperikre is by no means constant even though the final temperature is forced to he equal to the initiqYtemperature by keeping the homh in a thermostat throughout. Such applications are not justified by the usual derivation, although, as will now he shown, the conclusion is correct. Relation 4 is obviously equivalent to the criterion
Relations 5 and 9 imply the sense in which, for isothermal and for isothermal-isobaric processes, A and G are sometimes called3 Free Energies or even Amounts of Available Work. We would emphasize that relations 7 and 13 are more generally applicable than the AA and (see below) AG criteria and are instructiw in showing how under other ronditions a temperature and the differences in hoth Uand S nmtribute separately to the determination of the limiting amounts of work. Again, when the initial and final states are well deflned, with pressures equal to P, and temperatures equal to T,, relation 13 reverts to relation 9, and, for w' = 0, relation 14 or, with the restriction w = 0, reverts to relation 12. Again, a thermodynamic function of state, here G, is properly shown t o be the touchstone for thermodynamic possibility, as long as all heat exchanges with the surroundings occur with the surroundin~sa t the without any further restrictions such as T = const. or V = same temperature (the process is monothermostatic) and all const. The temperature need not euen be nominally convolume changes occur when the surroundinns exert the same stant, just as long as the monothermostatic and work conditions are satisfied. Moreover, for Ti,,ti.t = T f i ,= Tsurrounding. pressure on the system (the process is also monoharostatic). Many textbooks are incomplete hoth as to justifying the = T,, the criteria revert to relations 5 and 6. This argument application of G t o processes that are not truly isothermal conveniently shows that AA 5 0 is valid (after all) for proand isobaric, but only nominally so, and as to pointing up the cesses at constant volume that are onlv nominallv isotherconditions under which this touchstone may be applied to mal, provided, of course, that all heat exchanges oEcur when processes that are nominally but not genuinely isothermal the surroundings are at the same temperature, T,. This is and isobaric. convenient because A is a single function of the state of the In our opinion any teaching of thermodynamics should system, whereas relations 7 and 8 depend separatelv on AS. feature relation 4 and the criteria 7.8.13, and 14 that follow AU, and T,. from it, and withenoughexamples to ensure comprehension. Of much more frequent application in chemistry is the Any other course would appear to he as unwise as teachina criterion elementary algebra withoutthe quadratic formula. AG 5 w' (9) To our knowledge the only chemical text that approaches the treatment here is Berrv. Rice. and R o d ., and it. in our ~~, for the thermodynamic possibility of an isothermal, isobaric view, ron~plicates the presentation t ~ yemphasizing Romeprocess, where what detailed and rmfusina considerations of hvnotherical arrangements for achieving reversihility when the system temperature has to be varied, rather than emphasizing how is the net work, that is, all work except the pressure-volume such details, however important for conceiving of possible work, wp,". This P , V work is the integral of -Psurroundines dV; actual processes, are irrelevant for the aeneral thermodvit is given by the expression namic results. ~~
~~~~~
~~~
".
-wp," = P,AV
(11)
if the pressure Psurroundings exerted by the surroundings is
Berry, R. S.: Rice, S. A,; Ross, J. "Physical Chemistry": Wiley: New York, 1980.
Volume 63
Number 11
November 1986
937
