Jeffrey S. Wicken Biochemistry Department Penn State University The Behrend College Erie, Pennsylvania 16510
The Entropy Gradient: A Heuristic Approach to Chemical Equilibrium
The second law of thermodynamics states that all physico-chemical processes occurring within an isolated reaction system under conditions of constant energy and volume must increase the entropy of that system until some, final, maximum value is attained, which corresponds to a state of equilibrium. This has been stated succinctly as follows dS>O
(1)
Thus, "natural" processes are those that generate entropy. According to its formulation in statistical thermodynamics, the entropy of a particular state of such a chemical system is given by the Planck-Boltzmann equation S=klnW
(2)
where W is the numher of microstates or "complexions" of the system consistent with this macroscopic state. Each such complexion is delineated by a particular spatial configuration of molecules and the assignment of these molecules to a particular set of energy levels. Since the numher of complexions that contribute to a given chemical state reflect the extent to which the system's matter and energy can be randomized by that state, entropy can he interpreted as a measure of statistical disorder. The second law had its origins in empirical generalizations regarding the irreversibility of certain natural processes: heat "flowed" in the direction of decreasing temperatures, never the reverse; mechanical energy could be transformed without residue to heat, hut not conversely. With the formulation of statistical thermodynamics, entropic maximization was shown to he required by probabilistic considerations. According to eqn. (2), reactions which increase the entropy of an isolated system do so by increasing the number of microstates through which its total matter-energy can he randomized. Since all possible microstates in an adiabatic system have the same energy, each occurs with equal probability. Therefore, reactions that increase the numher of microstates occupied by the system result in macrostates of increased probability.
Chemical reactions orovide the means for movine the svstem from states of low prLhahility to states of high progahilit;; that is.. thev. .orovide the means for its exoansion in orohahilitvspace. At equilibrium, the system's volume in probability mace has been maximized. In spite of its cognitive clarity, the principle of entropic maximization in chemical reactions is usuallv awkward to apply. Most chemical processes occur unde; energetically open, rather than adiabatic, conditions. It is therefore common procedure to divide the large adiabatic system in which the reaction takes place (the universe) into the limited system of interest and a heat reservoir of infinite capacity with which this system can exchange energy. Thus d S = d S ( s ) + d S ( r )> 0
(3)
The designations (s) and (r) refer to system and reservoir, respectively. Since the entropy of this limited system may either increase or decrease without necessarily violating the second law, "free-energy" functions are used to provide reaction criteria for these systems. Under isothermal, isobaric reaction conditions, for example, thermal entropy transferred to the resenroir is provided by a reduction in the system's euthalpy, and the second law becomes d S = d S ( s ) - dHIT 2 0
or dG50
(4)
where d H and dG refer to the infinitesimal chances - in enthalpy and Gibhs free-energy, respectively. This formulation has areat com~utationalutility, since the trw-mer~n. change for a given reaction can be directly related r u itsequilibriumconstant. On thttother hnnd, thervisadihtinct cognitive loss suffered when entropy is displaced by free-energy from its eminent position in reaction theory. While the necessity for entropic maximization is immediately demonstrable on probabilistic grounds, the necessity for free-
Volume 55. Number 11. November 1978 / 701
gradient can now be separated into configurational thermal terms as follow
e n q y minimization is no1 evident unless one sorts out all the entropic contrihutions that are buried within this pnrameter. This paper will c,xplore an alternative appn~nchto the ani~lvsis of chtmical reactions; hopetl~lly,this approach umil add intuitive clarity to thc cmcept of chemical equilibrium.
(9) S'= S',/+ Sfth(s)+ S l t h ( r ) The Configurational Entropy Gradient The values of each of the above terms can be calculated a t any point along the &axis, and the relative contribution of each term to the progress of the reaction a t that point can be assessed. This can he illustrated by considering the reaction system represented by eqn. (6). The numbet of molecular configurations for this system can be calculated from the equation
The Entropy Gradient
All transformational processes occur along certain "entropy gradients," which may he generally defined by the equation (5) S' = dS1dZ where Z represents any coordinate axis along which entropic increases can occur (e.g., distance, temperature, volume, composition), and S' represents the entropy gradient. Irreversible transformations all occur over vositive entropy - gra~ dients. In chemical systems we are most often interested in entropic changes owurring over certain composiiional ranges, so that the entropy gradient will he expressed in terms of some comoositionnl v:iriirhle. It will he cunvenient here to formulate S' 4 t h respect to the "extent of reaction" parameter, 5, which is defined as the fractional extent to which a reaction has proceeded to completion.' This can be illustrated by the general reaction shown below
aA+bB-cC+dD (6) If a. and bo represent the number of moles of A and B initially present in the system, and if A is taken as the limiting reactant, the number of moles of these molecular species can be expressed as
wbere $ can assume a continuous range of values between 0 and 1. The entropy gradient will be defined for this system as the differential change in entropy that accompanies an infinitesimal change in 8 that is, S' = dSId6 Thus defined, S' provides a specific measure of a reaction's thermodynamic tendency to proceed in a given direction. (A similarly defined function in thermodvnamics is referred to as the "reaction potential." 2, The entropy gradient is due to entropic changes occurring both within the system and in its surroundings as a result of compositional changes occurring within the system. This can be expressed as follows S' = S'(s) + S'(r) (8) S1(r)is the thermal entropy gradient generated in the reservoir by an infinitesimal change in t. I t will therefore he referred to as Stth(r)in the treatment below. According to energy conservation, St,(?-) must be compensated by a potential energy gradient occurring within the reaction system. Under isothermal. isobaric conditions. this takes the form of an enthalpy gradient, and S'h(r) = - ( d ~ l d a I ~ . S'(s) reflects changes in the randomness of both matter and e n e r g occurring wikin the system as a result of this change in t. I t therefore contains both configurational and thermal contributions. For ideal systems, each molecular configuration contributing to a chemical state is energetically equivalent; that is. the densitv of thermal auantum states is the same for each configuration. For such cases, the number of microstates associated with each state can be exvressed as a ~ r o d u c of t configurational and thermal factors ( W = w ~ w ~ and ~ )one , may separately define configurational and thermal entropy as SCf = k lu Wcrand Sth(s) = k In Wth.=The overall entropy
Dickerson, R., Gray, H., and Haight, G., "Chemical Principles," W. A. Benjamin, Inc., New York, 1970, p. 639. Wall., F... "Chemical Thermodvnamics." 2nd Ed., W. H. Freeman and Co., Ssn Francisco, 1965, p. i00 702 1 Journal of Chemical Education
W,,= N!INA!NB!NC!N~!
.
(10) wbere N is the total number of molecules in the svstem. If this expression i-.insertcd in eqn. (21,using the appropriate substitutions from ean (71alone with Stirlinr's ap~roximation for In N!, and then differentiated with respect to 5, one obtains (aoRla) In [(I+ bolao + rflaP(1 - fP(b01ao- b Z l ~ ) ~ l (c€laP(dClaPl (11) where r = (c d - a - b ) . This expression for SIcisimplifies greatly for specific reactions, as will be considered presently. Still, it is evident from this general equation that the value of S',f will he infinite when 6 = 0 and negatiuely infinite when ( = 1.I t can be concluded from this that S',/ will always promote the generation of some product molecules, regardless of how energetically unfavorable the reaction might be, and will always require the retention of some reactant molecules, regardless of how energetically favorable it might be. These characteristics give Src, a unique role in establishing positions of chemical equilibrium, as will be discussed presently. S',/
=
+
The Thermal Entropy Gradient
According to the previous discussion, the overall thermal entropy gradient for a reaction can be expressed as S'th = Slth(s) Slth(r).In isothermal, isobaric processes, Slth(r) = -(dHldt)lT. For ideal systems, these terms can be readily calculated from the molar entropies and enthalpies of the reactants and products
+
- ~ S B= oa~S/a ) Snlh(s)= (uo/a)(cSc+ dSn - OSA Srfh(r)= -(aolaT)(ciTc + dRo - aRa - bfig) = aoAl7loT (12) In these equations ASand & represent the partial molar entropy and enthalpy of reaction, respectively. The thermal entropy gradient can now be written as Srth= ( o o / a ) ( ~-S &IT)
(13)
A positive value for Sffhindicates that the reaction is favorable on energetic grounds; for each infinitesimal quantity of product formed a certain amount of chemical potential energy will be converted into thermal energy. Discussion The sign of S' a t any point along the €-axis for a chemical reaction determines the direction of the reaction a t that ~ .o i n.t : the magnitude of S' provides a quantitative measure of its thermodvnamic reaction tendencv or "reaction votential" at From eqns. (11) andU(l3),it is evident that the that configurational and thermal components of the entropy gradient contribute to the progress of a reaction in very different ways. For ideal systems, the value of S J t h is independent of composition; this parameter will therefore make a constant contribution to the overall entropy gradient for a given reaction. An energetically favorable reaction is one with a positive thermal entropy gradient. The behavior of the configurational entropy gradient is far more interesting. A plot of S',/ versus [ will have the following characteristics: (1) The value of S',/ will cross the €-axis at some point t,, where 0 < 5, < 1; (2) the curve will have an inflection point at, or in the vicinity of, $1; (3) the value of S',f
C . .
.
.
.2 .4 .6 .8 EXTENT OF REACTION Figure 1. Plots of SGr and S',,vwsus extent of reaction for A 0
Entropy units are expressed in caiories/maledegree.
+B
-+ C
EXTENT 0.
OF
REACTION
Figure 2. Graphical dnsrmimtion of the extent of reanion at equilibrium for 3H2 + N2 2NH3 by intersection of curves for S',,/ao and -S'm/ao.See text far details. +
Properties of, : S
Reaction A+B-C+D A+B-C A+B-C A-C+D
for Selected Reactions
Initial Conditions ao=bo a. = bo a, = bo/lOOO
ao
s6f aoR in ((1 aoR In ((1 aoR In ((1 a& in ((1
- 2.)*/E2) - €)2/2.(2- 2.)) - ()I€) - (2)/(2)
I? 0.50
0.29 0.50 0.71
will always he infinitp at E = 0 and negarive!\. inlinite at E = 1. S',,thcn:fore represents an plastic thermodynamic reaction
tendencv whose value is zero a t the woint of maximum configurational randomness ((I) and heEomes exceedingly large for states of the system that are remote from this point. This "elastic" composition-dependency exhihited by S',/ assures that the overall entropv eradient for anv reaction must become zero a t some poiniaiong its (-axis; regardless of whether the reaction is enerzeticallv favorable. Were it not for the particular composition-dependency of this parameter, the value of S' would either be negative or positive along the entire €-axis. accordinr t o the value of S',;. In this event. a given reactioh would eGher he thermodyna&ically forbidden, or i t would wroceed t o comoletion: there would he no intermediate gnnmd. Thns, s',,is nsponsihlc. for rhephenomc,non of chemical equilihrium, all bough S',,, profoundly influences its ~wsilimon the (-axis. l'he extent of reaction at equilihrium ( & I occurs at the point whrrr S',r = - S ' , h . For ~!ner~oti(!aIls f&orahle this requires {hat te >&; for energeticall; unfavorable processes, (, < h. The properties of STcf are summarized in the tahle for some special cases of eqn. (6). The values of El obtained for these reactions indicate the fractional extent each reaction will tend to progress on the basis of configurational (matter-randomizind ... considerations alone. These values are comoletelv in aced with wlr intuitive expermrions. For the first reartion in thii tahle. S,i and S',, arc each dotted as n function of E in Figure 1. F& this reaction, (I = 0.50. For the second (associative) reaction, there is a net reduction in molecular number as the reaction proceeds, which requires that (1 < 0.50. The reverse is true of the fourth (dissociative) reaction. The third reaction was included to illustrate the law of mass action: if one reactant is present in high excess, the reaction will he
pushed to a higher degree of completion than ifboth reactants are oresent in eaual amounts. A; an illustra&m of the way in which the configurational and thermal entropy gradients jointly establish the eqwilihrium position for a chemical reaction, owe might consider the production of ammonia from its elements: 3Ho N7 2NH1. From the standard entropies of these gases and the enthalp; of formation of ammonia? it can he readilv shown that S'+hlao = 8.89 calorieslmole-degree. Also, from kqn. (11), s',~[& = R ln ( 9 0 - [)3/16(2(1 - (/3))/3. Since S'&o and S't,,lao become equal a t equilihrium, 5. can he obta~nedgraphically from the intersection of these curves. as shown in Fieure 2. I t is evident from this pranh that Srd makes a n e g l i g i ~ econtribution to the reac&nain the comoositional reeion about the inflection ooint. For states that are remote from?,, however, this parameter assumes greater auantitative importance. - Even though these conclusions were obtained with the assumption of ideal behavior, thev have considerable oualitative generality for non-ideal systems as well. For such systems, there are significant energetic differences between molecular configurations that preclude the factorization of W and the explicit definition of configurational and thermal entropy. Nevertheless.. the " eeneration of entronv still ~roceedsin these systems through the randomization of matter and energy, and both of these sources of statistical disorder contribute to the overall entropy gradient for the reaction. The euergy-randomizing contribution to S' favors the eeneration of the most stable possible molecular species from The reactants, hecause the formation of these molecules transforms the maximum amount of chemical potential energy to random thermal euergy. As with ideal systems, however, the need to preserve so& c d i g ~ ~ r a r i mrandomness al in the system is rAponsi hle for .'"crossing the E-axisat some point prior U,completion and rstahlishing a distinct position of' equilibrium.
+
-
."
Denhigh, K., "The Principles of Chemical Equilibrium," 4th Ed., The Cambridge University Press, 1964, p. 353. Glasstone, S., "Thermodynamics for Chemists," van Nostrand Co., lnc., New York, 1964, pp. 506507.
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Volume 55, Number 11, November 1978 / 703
