Leonard K. Nash Harvord University Cornbridge, Mossochusetts 02138
Chemical Equilibrium as a State of Maximal Entropy
The most vivid expression of the second principle of thermodynamics is undoubtedly the statement that in an isolated system all spontaneous changes are characterized by an increase of entropy. Equilihrium can then be attained only when the entropy of the system has reached a maximum. These statements are summarized in the expression AS 2 0 or, better, dS/d~: 0, where 2: measures the extent of completion of a chemical reaction taking place in an isolated system. However, this simple and perspicuous expression is rarely, if ever, applied directly to the determination of the position of equilibrium in a chemical reaction-for a t least three distinct reasons. First, and most often stressed ( I ) , is the fact that chemical reactions are not ordinarily run in isolated systems. Second, it might be added, is the mathematical laboriousness of this analysis of chemical equilibrium. Third, as will be seen in what follows, this analysis yields an equilibrium expression for only one particular temperaturewhich is not preselected as such but, rather, determined in an unforeseeable fashion by one's choice of the original condition of the isolated system involved. These considerations generally persuade one to choose a different pedagogic approach to the concept of chemical equilibrium-proceeding by way of a more or less elaborate development of the free-energy functions. The immediate relevance of the fundamental entropy concept to the position of chemical equilibrium is then all too easily overlooked, and the student may fail entirely to grasp how the cessation of chemical reaction in an isolated system is associated with attainment of a maximal "randomness" or "spread." For he now comes to grips with the general problem of chemical equilibrium only after mastering the not inconsiderable difficulties associated with additional (free-energy) concepts less notable for their ready interpretability than for their convenience as accounting devices. These non-trivial disadvantages of the conventional approach may prompt one to reconsider the possibility of a direct exploitation of the simple entropy criterion in the discussion of chemical equilibrium. I n a recent publication (d), Everdell has shown how the entropy criterion can be deployed in a determination of the equilibrium vapor pressure of water a t some one temperature. Though this analysis necessarily pivots on consideration of an isolated system, the determined vapor pressure is of course generally characteristic of pure water a t that temperature. In exactly the same way, moreover, the analysis of a chemical reaction in an isolated system yields an equilibrium constant generally characteristic of the reaction at the indicated temperature. Thus the first objection noted above is seen to be specious and irrelevant. However,
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noting the substantial laboriousness of even the calculation chosen by Everdell for its comparative simplicity, and the much greater laboriousness of the calculation of the position of the entropy maximum in a chemically reactive system, one must acknowledge that the second objection is of some weight. But this weight is reduced to nothing by the increasing accessibility of computers to undergraduates. For, using a stored program, the student can promptly acquire a complete analysis of the chemical system that concerns him. Moreover, the program is simple enough not merely to be used but also to be written de novo by a novice chemist, and to be executed by a computer with far less capability than the IBM-1620 now available on so many campuses. Thus the second objection is today far from decisive. And the third objection is easily sidestepped by a slight extension of the same computational program, which then yields equilibrium constants for as many temperatures as one likes. We turn now to the thermodynamic analysis on which this program is founded. Determinotion of the Entropy Maximum
The primary analysis will refer to homogeneous gasphase reactions in which all components of the gas mixture are reasonably approximated as ideal gases. The final result can easily be generalized to heterogeneous reactions involving, in addition to the gas phase, one or more pure solid phases. Though presumably perfectly feasible, the related analysis of systems involving ideal solutions bas not been attempted. Let a general homogeneous gas-phase reaction be represented as aA(g)
+ bB(g)
=
zZ(g)
+ yY(g)
where the small letters symbolize the stoichiometric coefficients. Let the reaction proceed in an isolated system-adiabatic and of constant volume-wherein a moles of pure A are brought together with b moles of pure B at an initial temperature T,. How will the final temperature (T,) depend on the fractional advancement of the reaction ($)? To establish how T, varies with Z:, which can take on any value between zero (no reaction) and one (complete reaction), we need only adapt an argument familiar in its application to determination of the peak temperature attained in an adiabatic explosion. In Figure 1, let the line-segment I represent the actual course of the increase of temperature as the reaction advances to some extent Z:. An alternative route between the same initial and final states is represented by the line-segments I1 and 111. In segment I1 the reaction is conceived as advancing isoVolume 47, Number 5, May 1970
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Having in hand this expression for T, as f(8,we seek now to establish the total change of entropy (AS) accompanying any fractional advancement of the actual reaction along the line-segment I. Since entropy is also a function of state
Figure 1.
How does the entropy of each gaseous species depend on its partial pressure in an ideal-gas mixture? The familiar relation AS = nR In P ' / P n can be rewritten in the form
Vmriation of TI with advanconrent of the reoelion.
thermally, a t the original temperature T,, to the extent f . In segment I11 the resultant mixture of reactants and products is then conceived to be warmed, without further reaction, until it reaches the same final temperature attained in the course of the actual reaction. We have of course made the tacit assumption that the reaction in question is exothermic. Were it endothermic, T, would fall below T,, and segment I11 would represent a cooling operation; but all the analytical expressions then obtained remain identical with those now to be derived for the exothermic reaction. Energy being a function of state ABII + AEIII = aE, Since the isolated system is both adiabatic and constant in volume, AEI = q, = 0, whence it follows that Ahr
+ Ah11 = 0
(1)
For the reaction, as written, one can easily determine from tabulated data the value of AE,", symbolizing the standard energy change that occurs when the reaction proceeds to completion a t the original temperature Tcwhich will ordinarily be taken as 298.15°K. And then, for any fractional advancement of the reaction AEII = CAE.'
Here SO represents the standard molar entropy of the species at the initial temperature and a pressure of 1 atm. And S then represents the entropy of n moles of the species at a partial pressure of P atm in an ideal mixture at the same temperature. If PA* is taken to symbolize the partial pressure of species A in the original stoichiometric mixture, the partial pressure of B in that mixture will be (b/a)PA*. Thus the entropy (8')of the original mixture of reactants can be formulated as
where the term a / a has been inserted only to preserve symmetry. When the reaction has advanced to the extent f , the moles of A, B, Z, and Y present will be a(1 - f ) , b ( l - t), zf, and yf, respectively. And, still at the initial temperature, the corresponding partial pressures will be ( a / a ) ( l - UP**, ( b / a ) ( l - €)PA*,( z / a ) € P ~ * , and ( y / a ) f P ~ *respectively. , For the entropy (S") of the reaction mixture produced when the reaction has advanced isothermally along the line-segment 11, one then obtains
If we simplify by assuming effectively constant heat capacities over the temperature range T,to T,, the energy change in the constant-volume heating represented by segment I11 will be AEIII=
n,Cv,(T,
- T.)
Here the n,'s represent the numbers of moles of each species present when the reaction has advanced to the extent f , the C,,'s represent the corresponding constantvolume heat capacities, and the summation is to extend over all the j species present. Having started with a moles of A and b moles of B, one then finds AEIII= [(I - I)aCn (1 - € ) ~ C Y B € C v z EYCVYIX (TI - T,) = [aCn b C v ~ IACVI(TI- TJ where as usual ACT represents the diierence between the total heat capacity of the pure products collectively and the total heat capacity of the pure reactants collectively. Substitution for AEII and AEIIIin eqn. ( 1 ) now yields [aCv* + bCva €ACvI(T1- T.) + WE,* = 0 (2)
+
+
+
+
+
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Henceforth AS," will be used to symbolize the function within the first set of square brackets, for this function clearly does represent the change of entropy (easily obtained from tabulated data) when a moles of A and b moles of B, each a t unit pressure, are isothermally converted to z moles of Z and y moles of Y , each also at unit pressure. Now when the isothermal reaction advances along line-segment 11, to the extent f , the net change of entropy will be
When, without further change of composition, this reaction mixture traverses the line-segment 111, as it is heated a t constant volume from Ttto T , , the corresponding entropy change will be
and our earlier evaluation of this summation then yields AS,,, = (oCm ~ C I B FACV)In (T,/Ti)
+
+
Substitution in eqn. (4) now establishes what was sought-the functional expression for the total entropy change that accompanies the reaction in the isolated system
AS
+ b Cvs + (ACT)ln ( T t l T i ) + €AS;' -
= ( a CVA
+
Figure 2. Entmpy change with advancement of reaction. A, NlIg) 3 Hdg)=2 NHaI.1, at Psi,*= 1 0 atm; B, some reaction, at Pnir* = 1 d m ; C, HzObl COdgl NaKOsirl = 2 NoHCOslsI, a t P , u * = 0.5 d m .
+
+
Strictly for .computational convenience, this equation may better be written
+
where An symbolizes the difference [z y - a - b], representing the change in the number of moles of gas consequent to completion of the reaction. Derived for homogeneous gas-phase reactions, this equation can he applied without difficulty to heterogeneous reactionsby the simple expedient of dropping from all the squarebracketed functions any term that refers to a nongaseous species. The foundation is now laid for a determination of the equilibrium state of a chemical reaction in an isolated system. For each of a series of values of & one determines from eqn. (3) the value of the final temperature T I when the reaction proceeds to that extent. And, on substituting in the last equation the paired values of $ and T I , one finds the net increase of entropy corresponding to any fractional advancement of the reaction. This tediously repetitive calculation is easily performed by a computational program which systematically changes $. from 0.1 to 1.0 in steps of 0.1. When the general zone of maximal entropy increases has been established in this fashion, the program permits acceptance of instructions to vary $ in steps of, say, 0.005 across this zone-thus establishing the exact composition of the (equilibrium) reaction mixture a t the entropy maximum. Results so obtained for typical examples are depicted in Figure 2. From the value of 5 associated with the peak of each such curve, one can readily calculate the equilibrium constant of the reaction in question a t the coordinated temperature T,. Thus with P,,,*= 1 atm, the reaction Na(g) 3 Hz(g) = 2 NH3(g) attains its entropymaximum when $ = 0.2298, and the corresponding temperature is 563.0°1 q, clearly A S > q/T. The second quantitative difference, scarcely less obvious but usually much less emphasized, arises whenever the operative circumstances ensure that the quantity of work performed shall he independent of the degree to which the specified change is conducted reversibly or irreversibly. In such circumstances q can he equated to q,,,, but of course the reversible and irreversible changes remain associated with quite different values of AS. Though the q's are equal, the values of A S still reflect quantitative differences in the temperatures of any heat sources or sinks involved. Precisely how the variation in AS arises from these temperature differences is illustrated in the following analysis of a very simple specific case: the cooling (or warming) of an object in a manner that is, to any desired degree, reversible or irreversible. This analysis is conceived as demonstrating for transfers of heat just what Eberhardt's experiments demonstrate for work transfers.
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Cooling in Geometric Series
By successive contact with some number (n) of large heat reservoirs or baths, let an object be cooled from some initial temperature TI to some final temperature T,. Over the temperature range T, to T,,let the object have an effectively invariant heat capacity of C cal/grepresenting either C , or CV, according to the conditions of the experiment. Whether the cooling is couducted reversibly or irreversibly, with entropy established as a function of state we know that the entropy change (AS.,) of the object will remain constant at AS, = C In (T,/T,) = - C In (TJT,) (1) What is the corresponding entropy change (ASB)in the n large baths used to cool the object from T, to T,? Let the baths be numbered in order of temperatures decreasing from T,: namely, in the order T,, Volume 47, Number 5, May 1970
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the IBM-1620, the author is much beholden to James A. Van Zee. Literature Cited ~
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(1) NAB", L. K.. "Elements of Chemical Thermodynamics" (2nd ed.). Addison-Wesley, Rezding. Mass.. 1970, p. 92.
(21 EVERDBLL. M . H., J. CXEM.EDUC., 46.107 (19691. (31 Lmsolr. W.D.,nmo Dooor. R. F.. J . Am. Chem. Soc., 45, 2918 (19231. (41 SMITH,N. o.,J. CXEU.EDYC., 42, 654 (19651. (51 "International Critical Tables," MoGraw-Hill, New York, 1930,vol. vii,
. , -. .30s .. . .
(6) DENDICH. K.. "The Principles of Chemical Esullibrium" (2nd 4.1. Cambridge Univ. Press, 1966, pp. 48-56.
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