I
Leonard K. Nash Harvard University Cambridge, Massachusetts 02138
Applicability of the Equation: dE = TdS - PdV
For closed systems that involve (1) only P d V work, (2) no tensile effects, and (3) no field gradients, the first and second principles of thermodynamics are powerfully combined in the familiar equation
dE = TdS -PdV where P denotes the pressure of the system. This is variously described as "the basic equation" or "one of the fundamental equations" of thermodynamics, hut accounts of its applicability are strangely divergent. Some authors indicate that, apart from the above-noted exclusions, the equation is generally applicable to all changes, reversible and irreversible alike. What is here presumably a tacit reservation ought surely to be made explicit: as such, this equation cannot be applied to irreversible chemical reactions; e.g., in an isothermal closed system of constant volume, such a reaction does not ordinarily proceed with A S = AEIT. Other authors restrict application of the equation to strictly reversible changes, and suggest that for irreversible changes one must turn to the related inequality, T d S > d E PdV. This is too severe: the equality applies perfectly well t o the highly irreversible Joule expansion, and also to the irreversible equalization of pressure andlor temperature within a rigidly bounded and externally insulated body made up of parts that stand initially a t different temperatures andlor pressures.' Most numerous of all are the authors (among whom has fallen the present writer) who fail adequately to define the applicability of this "basic" or "fundamental" equation. The best such definition seems to have been given by Denbigh,2 who in essence suggests that, for closed systems limited to P d V work, the equation applies alike to reversible and irreversible changes, excludingonly irreversible changes of composition such as arise from chemical reactions. Consider the three closely related equations
+
dE=6q+6w
(1)
dE=TdS-PdV (2) dG = -SdT + VdP (3) All apply alike to closed systems, but eqn. (1) applies to all changes, chemical and physical, reversible and irreversible. Equation (Z), on the other hand, does not apply to irreversible changes of chemical composition. However, eqn. (3) often supplies a point of departure for calculations on chemical reactions that may in practice proceed either reversibly or irreversibly. And now one may well wonder (a) in deriving eqn. (2) from eqn. (I), bow has the exclusion of irreversible chemical changes crept in; and (b) in deriving eqn. (3) from eqn. (2), how has the exclusion of irreversible chemical changes apparently crept out? Query (a) is easily answered. For a reversible change in a closed system involving only P d V work, we draw on the equalities 6q,, = T d S and hw,., = -PdV to obtain eqn. (2) by simplesubstitution in eqn. (1). For an irreversible progress of the same change, dE must remain the same-whence i t follows that d E = 6q + 6w = 6q,., + 6w, Substituting as before for 6q, and 6 w , we may seem to have established the applicability of eqn. (2) to all irreversible
chanees as well. But this involves the overlv hasty assumption s t make 6w that i n y irreversibility making 69 < ~ d ~ n malso > -PdV to exactlv the same extent. T o be sure, the above equation will require that if 6q = 69, - p (where p is a meap. sure of the irreversibility) then certainly 6w = 6w.., Though it does then follow that 69 = TdS - p, it emphatically does not follow further that 6w = -PdV p, and for a very simple reason. Under the usual conditions of constant temperature and constant total pressure, an irreversible chemical reaction yields a -PdV that does not a t all represent the quantity 6w,,. For we well know that in these circumstances 6 w . = dA = dG - PdV. Under the specified conditions this means 6w,,, = -PdV &dni. A familiar van't-Hoff-box analysis3 demonstrates that the entire 6w, may be comprehended within a -ZPdVterm when the reaction proceeds reversibly, but it is not so comprehended in -PdV when the reaction proceeds irreversibly at constant temperature and pressure. For this irreversible reaction we must then write
+
+
+
And when thelast equation is truncated to eqn. (2). obviously this simpler equation cannot be applied as such to irreversible changes of chemical composition. In that truncation we thus find a readilv intellieible oriein for vreciselv the exclusion we have found to attacK to eqn:(2) b i t not tdeqn. (1). Turning then to query (b), we see that in passing from eqn. (2) to eqn. (3) we introduce no more than the definition C = E + PI' TS. This. when differentiated and added to e m . (2), at once yields eq&.(3). Obviously eqn. (3) can contain "no new knowledae" and. moreover. the same exclusion that delimits the applicability of eqn. (2) must equally delimit the applicability of eqn. (3). Yet this exclusion is rarely attached to eqn. (3) with the emphasis given the same exclusion when (however imperfectly stated) i t is attached to eqn. (2). Why? The answer seems to be that, in an amply familiar treatment of chemical reactions, the shortcomings of eqn. (3) are inconspicuously made good by supplementing (or supplanting) it with a further equation which, though immediately derivative from eqn. (3), is free from the restrictions appertaining thereto. So rarely (if ever) encountered is the perfectly analogous exploitation of eqn. (2) that one may thus be brought t o quite different perceptions of equations actually identical in their notential annlicabilitv to chemical reactions-as shown inLhat follo& The requisite supplemental equations are in both cases
-
When such differencesof intensive propeniev exist, the equation must of course be separately written for each part so differing,and theoverall effect will then be establrshd bv simple summation over all the part equations. Denbigh, K. G., "Principles of Chemical Equilibrium," 3rd Ed., 'Cambridge University Press, London, 1971, p. 45. See for example Kauzmann, W., "Thermodynamics and StatisDD. 292295. tics." Benismin. New Yark.. 1967.... 'Ref. (i), p. i s . Volume 54, Number 7, July 1977 1 409
perfectly routine. T o the isothermal expansion of one mole of pure ideal gas, proceeding without change of composition, we may apply eqn. (3) which, on integration, yields
G=GO+RTInP
tire mixture stands a t equilibrium, we can surely apply these equations to a virtual displacement (d[) which must then proceed reversibly. Starting from eqn. (3), dG = VdP SdT, we substitute from eqn. (9) to find for this isothermal system
-
(4)
where Go symbolizes the standard molar free energy a t the given temperature and a specified unit pressure. Given that G is an extensiue (i.e., first-order homogeneous) function, we may write for n moles of gas
since the sum of all the changes of partial pressure must equal the total pressure change (dP). We call now on eqn. (5)to write
G=nGO+nRTlnP (5) This apparently trivial through-multiplication will be found to invest eqn. (5) with an applicability surprisingly outranging that of the parent eqn. (3). Starting next from eqn. (2). the corresponding development leads, by way of the remark that d E = 0 in the isothermal expansion of an idealgas, to the result
+
xd(nMGMO nMRT in PM)= V Z ~ P M
Under isothermal conditions this means
+
X G M O ~+ ~ ERT M in PMd n ~~ ~ M ~R P MTI P = MVEdPM Since ~ M R T / P M will in every case yield the common total volume V
S=SO-RInP where Sosymbolizes the standard molar entropy. A similarly significant through-multiplication yields for n moles of gas
Cancel the last two terms, and substitute for d n from ~ eqn. (8)
S=nS0-nRhP (6) All these simple equations are now applied to a general reaction of ideal gases formulated as
The common multiplier d[ may be factored and dropped. For the first summation we then suhstitute from eqn. (101, and rearrange to
vnA+veB=vzZ+~yY
The stoichiometric coefficients (UM)are taken to be negatiue for the reactants A and B and positive for the products Z and Y. By ( (between 0 and 1) we symbolize the fractional advancement of the indicated reaction. For each component (M), at any stage of the reaction the number of moles then present ( n ~ is) uniformly related to the number of moles present originally ( n ~ , )by the equation nM = nMo + V
M ~
RT In n PMUM = -AGO Having stipulated all the PM'S to be equilibrium pressures, we can identify IIPM"Mwith Kp, the constancy of which is guaranteed by the invariance of AGO under isothermal conditions. And so we arrive a t In K , = -AG"IRT (11) Starting now from eqn. (2), d E = TdS - PdV, we call on eqns. (9) and (6) to find for this isothermal coustant-volume system
(7)
so that in any infinitesimal advance of the reaction d n =~V M ~ F
(8)
We now defme an ideal gas mixture as one in which GMand
Observe that with an ideal gas we may replace by Fh10,a constant under the specified isothermal conditions. Differentiation then yields
SMfor each component are, respectively, as given by equs. (5) and (6) when for P we substitute the partial pressure (PM)of that component. For the whole of such a homogeneous mixture, Kremer has demonstrated5 that the total value of any extensive function of state (Y) is given by Y= ~ ~ M T M
For this isothermal constant-volume mixture of ideal gases
(9)
E ~ dM In PM= x n d In ~ (~MRTIV) = Z ~ dM In nM = Z d n =~ X w d I = An dt
In an isothermal system of oceanic dimensions, where one round of reaction g a y proceed without effective change of composition; each YMstays substantially constant. For such a reaction we may then write AY= Y- Yo= ~ n M ~ M - ~ n M o ' M= ~ ( ~ M - ~ M o ) ~ For one full round of reaction under the specified conditions, with [ = 1substitution from eqn. (7)will yield AY = ~ Y M ~ M (10) Equlvalence of Equations (2) and (3) For our general reaction of ideal gases, the equivalent applicability of eqns. (2) and (3) can easily be shown by parallel analyses exploiting the van't-Hoff equilibrium box, but one mav well nrefer a less contrived examde. Consider then a mixture oiall reactant and product species in a thermostated constant-volume enclosure that excludes aU uerformance of work. Were any significant amount of reaction to occur in these circumstances, it would be an irreversible change of composition to which application of eqns. (2) and (3) is certainly excluded. However, if we suppose that the original partial pressure of each component (PM)is such that the en-
Kremer, M. L., J. CHEM.EDUC.,42,649 (1965) 410 1 Journal of Chemical Education
M
since 2 v is~nothing but the net change in mole number (An) consequent to one round of the reaction as written. Inserting this result in the preceding equation, we further substitute for d n from ~ eqn. (a), finding
ZUMEMO dl = T E Y M S ~d< O- RTEYMIn PMdt - (An)RTdE The common multi~lierdE mav be factored and d r o ~ n e d . Noting thar in this r i m i o n o f ideal gases (AnJRT= A ~ P v ) , we further note that eon. (10)establishrssim~le . eauivalences . for the first two summations. By rearrangement we then ohtain RT In IIPMYM = -(AEO + APV - TASo) Calling now on thedefinitions of Kr. and H, weobtain for this isothermal reaction a relation fully equivalent to eqn. (11). namely In K , = -(AH0 - TASo)/RT
Setting out from eqn. (2) as formerly we set out from eqn. (3), we have thus demonstrated that thesame method yields the same end-result with the same ease. The analysis taking departure from eqn. (2) is noteworthy in that, prior even to introduction of the concept of free energy, such an analysis
yields an attractively simple and direct definition of the essential condition for chemical eauilibrium. But. bevond this. we aspire to a delinition of the thermodynamic changes associated with one full round ofan isothermal chemical reaction that may proceed either reversibly or irreversibly. Such a definition is easily extracted from either of the eqns. (5) and (6) we have, respectively, derived from eqns. (3) and ( 3 , which are themselves here inapplicable. Equivalence of Equations (5) and ( 6 )
Imagine an isothermal ocean of an inhibited mixture in which each reactant and ~ r o d u csoecies t amears a t anv nartial pressure PM.~ e m o ; i n ~the inhihito; (or introducing a catalyst) for a suitable ~ e r i o dwe . brina about one full round of reaction-which wili in these circu&stances leave all the partial pressures effectively unchanged. Starting then from eqn. (5), we replace nM by YM; i.e., just what we would have obtained from eqn. (4) by a through-multiplication with UM rather than nM. Summation over all components yields = ZYMCMO + RT In n P M " ~ EYMCM Calling next on eqn. (lo), and symbolizing by Q , the function ~ P M which " M here retains the form hut not the value of the equilibrium constant, we arrive a t
In the corresponding derivation setting out from eqn. (€4, we again replace nM by VM, and add over all components t o find
EYMSM = &,,SMO - R In n P M Y ~ Making the corresponding substitutions, and multiplying through by -T, we obtain -TAS = -TASO + RT In Q, In this isothermal reaction of ideal gases, moreover AH= AH0 Adding that to the immediately preceding equation, we arrive a t the full equivalent of eqn. (12), namely
+
AH - TAS = (AHo- TASo) RT In Q,
Setting out from eqn. (6) as formerly we set out from (5). we duly find that the same method yields the same end-result with the same ease. Clearly (6)and (5) are as fully equivalent as the eqns. (2) and (3) from which they have, respectively, been derived. Once satisfied of this equivalence, we can extend to eqns. (2) and (6)the conclusions now drawn from exploration of the relation between eqns. (3) and (5). Nonequlvalence of Equations (3) and (5)
In production of the equilibrium expression set forth in eqn. ( l l ) , the rble of eqn. (3) is not only minimal but wholly replaceable. Fur eqn.,ll) is quite as easily elicited from the more general ean. (12). to thederivationof whicheqn. (3) makesno direct contribution. To exploit this possibility, one need only observe that, with an isothermal isobaric system, one may certainly write A S ILS >IT. As the equilibrium condition, in which alone the equality applies, one then finds AG = AH TAS = 0,which is all one needs to pass from (12) to (11). Thus one sees clearly that in these analyses the vital relation is not equ. (3) as such hut only the derivative eqn. (5). T o discover just how this comes about, we need only invert the argument that originally produced (5) by integration of (3). For a single pure ideal gas, eqn. (5) gives G=nCo+nRTlnP Allowing for possible changes of temperature, pressure, and quantity, we obtain as the total differential d G = n d C 0 + C o d n + R T ( I n P ) d n + n R ( I n P ) d T + n R TdP P Since Gorefers to one mole of gas a t an invariant unit pressure,
dG0 can represent n o s o r e than the change with temperature given by eqn. (3) as dGo = -So dT. Making this substitution, and collecting terms, one then finds d C = VdP- ( d o - n R l n P ) d T + ( G O + R T I n P ) d n The first parenthesized function is shown by eqn. (6) to be nothing but the entropy (S), and the second parenthesized function is precisely that we ordinarily symbolize as fi. These substitutions at once produce dG = VdP - SdT + pdn Going one step further, we can easily pinpoint the origin of this chemical-potential term. Repeating exactly the same analysis, but starting now from eqn. (4) rather than (5), we obtain as the terminal equation just dG = VdP - SdT Thus from eqn. (4) we recover only the here appropriate form of (3), while from (5) we recover an expression more powerful than (3). Hence this crucial difference derives solely from the through-multiplication by n that produced eqn. (5) from (4). For the effect of that apparently trivial operation is actually to introduce n as a variable and, through this extension of the definition of G, eqn. (5) acquires a realm of applicability far surpassing that of eqns. (3) and (4) as such. T o display the full margin of superiority of eqn. (5) over eqn. (3), let us treat a finite chemical reaction occurring irreversibly in a finite ideal-gas mixture which, for simplicity, we now assume constant both in temperature and in total pressure (P). As such, eqn. (3) is certainly inapplicable to such an irreversible change of composition, hut eqn. (5) proves altogether competent. Consequent to any advancement (() of the reaction, the difference between final and initial values of the free energy (Gc and Go, respectively) is given by eqn. (9) as
- Go = X ~ M G-MXnMoGMo On substitution for GMand G Mfrom ~ eqn. (5),and takingPM = PXM, a perfectly straightfoward development correctly yields
+
RT In P ) Gt - G,, = €ZVM(CMO
+ RTXnM In XM-
R T Z ~ MIn, X M ~ where for each component X M and X M ~respectively, , symbolize the mole fractions in the final and original mixtures. Though of little practical utility, this equation is illuminatingly interpretable6, and offers an easy calculation of conceptually vivid U-shaped plots of G versus (. Immanence of the Chemical Potential
How can it be that eqn. (5) so exceeds in power the eqn. (3) from which it was derived? The only possible answer is that the same power must be immanent in (3),just as it was in (4) before the through-multiplicat~ionthat converted it to eqn. (5). Thus prompted, let us perform that through-multiplication directly on eqn. (3). For one mole of any pure substance eqn. (3) gives dC=Vdp-S~T and for n moles ndC = nVdP - nSdT Following theline expLnitej in integrafiokby-parts, we_ohserve that d(nC1 = n dG + L'dn. Hencen d(; = d \ n C , - Gdn, on substitution of which we find d(nC) - Gdn = niidP - nSdT d(nG) = nVdP - nSdT + pdn meaning simply d(nG) = nvdP - nSdT + pdn Ref. (2).pp. 1367. Volume 54, Number 7. July 1977 1 411
which is the untruncated form of eqn. (3). For any homogeneous mixture we can write the penultimate equation for each component separately, and by summation then obtain d Z ( n ~ i % d= , Z ( ~ M V M dP) - ~ ( ~ M SdT M+)ZI~M d n ~
Calling now on eqn. (9),we find for any homogeneous mixture (gaseous or liquid) that d c = Vd?'
- SdT + Z
p ~ d n ~
Taking the extensive thermodvnamic functions of state to be homo&eous functions of the first degree in number of moles, we thus elicit from eun. (3) the complete expression directly applicable to chemical reactions. In one last demonstration of the complete equivalence of eqns. ( 2 ) and (31, we observe that the chemical-potential term is immanent in the former just as in the latter. Starting now from eqn. (2), exactly the same analysis here proceeds =T ~ S pdii
n& = nTdS - nPdV d(nF) - Fdn = Td (nS)- TSdn
- Pd(nV) + PVdn
or for any homogeneous mixture dE = TdS - PdV + Z
PM~~M
Thus one readily understands why eqns. (5) and (6).resoectivelv. vields results unobtainable from ems. (3) and (2) us such. i'& eqns. (5)and (6,exploit powersaiready latent in (3) and (2). whence they are elicited by theahwe analbsis that renders them explicit. Though i t complicates judgment of the intrinsic applicability of eqns. (2) and (3), the essential triviality of this analysis may gladden teachers of freshmen who seek access to the chemical potential by a route free from all the terrors of partial derivatives. Acknowledgment
No thermodynamic friend has been spared exposure to some version of this essay. All have kindly offered assistance, and particularly helpful critical commentaries were supplied by Henry A. Bent, R. Stephen Berry, William H. Eberhardt, and Martin Goldstein.
