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2 24 Thermodyamics ad Reactio Rates Miloslav Pekař Bro Uiversity of Techology Czech Republic 1. Itroductio Thermodyamics has established i chemistry pricipally as a sciece determiig possibility ad directio of chemical trasformatios ad givig coditios for their fial, equilibrium state. Thermodyamics is usually thought to tell othig about rates of these processes, their velocity of approachig equilibrium. Rates of chemical reactios belog to the domai of chemical kietics. However, as thermodyamics gives some restrictio o the course of chemical reactios, similar restrictios o their rates are cotiuously looked for. Similarly, because thermodyamic potetials are ofte formulated as drivig forces for various processes, a thermodyamic drivig force for reactios rates is searched for. Two such approaches will be discussed i this article. The first oe are restrictios put by thermodyamics o values of rate costats i mass actio rate equatios. The secod oe is the use of the chemical potetial as a geeral drivig force for chemical reactios ad also directly i rate equatios. These two problems are i fact coected ad are related to expressig reactio rate as a fuctio of pertiet idepedet variables. Relatioships betwee chemical thermodyamics ad kietics traditioally emerge from the ways that both disciplies use to describe equilibrium state of chemical reactios (chemically reactig systems or mixtures i geeral). Equilibrium is the mai domai of classical, equilibrium, thermodyamics that has elaborated elegat criteria (or, perhaps, defiitios) of equilibria ad has show how they aturally lead to the well kow equilibrium costat. O the other had, kietics describes the way to equilibrium, i.e. the oequilibrium state of chemical reactios, but also gives a clear idea o reactio equilibrium. Combiig these two views various results o compatibility betwee thermodyamics ad kietics, o thermodyamic restrictios to kietics etc. were published. The mai idea ca be illustrated o the trivial example of decompositio reactio AB = A + B with rate (kietic) equatio r kcab kcacb where r is the reactio rate, kk, are the forward ad reverse rate costats, ad c are the cocetratios. I equilibrium, the reactio rate is zero, cosequetly k/ k cacb/ cab. Because the right had side eq correspods to the thermodyamic equilibrium costat (K) it is cocluded that K k / k. However, this is simplified approach ot takig ito accout coceptual differeces betwee the true thermodyamic equilibrium costat ad the ratio of rate costats that is called here the kietic equilibrium costat. This discrepacy is sometimes to be removed by restrictig this approach to ideal systems of elemetary reactios but eve the some questios remai.

3 674 Thermodyamics Iteractio Studies Solids, Liquids ad Gases Chemical potetial () is itroduced ito chemical kietics by similar straightforward way (Qia & Beard, 2005). If it is expressed by RT lc, multiplied by stoichiometric coefficiets, summed ad compared with rate equatio it is obtaied for the give example that: cacb AB A B RT l RT l r / r Kc AB (1) (ote that the equivalece of thermodyamic ad kietic equilibrium costats is supposed agai; rr, are the forward ad reverse rates). Equatio (1) used to be iterpreted as determiig the (stoichiometric) sum of chemical potetials () to be some (thermodyamic) drivig force for reactio rates. I fact, there is o kietics, o kietic variables i the fial expressio RT l r / r ad reactio rates are directly determied by chemical potetials what is questioable ad calls for experimetal verificatio. 2. Restrictios put by thermodyamics o values of rate costats 2.1 Basic thermodyamic restrictios o rate costats comig from equilibrium Perhaps the oly oe work which clearly distiguishes kietic ad thermodyamic equilibrium costat is the kietic textbook by Eckert ad coworkers (Eckert et al., 1986); the former is i it called the empirical equilibrium costat. This book stresses differet approaches of thermodyamics ad kietics to equilibrium. I thermodyamics, equilibrium is defied as a state of miimum free eergy (Gibbs eergy) ad its descriptio is based o stoichiometric equatio ad thermodyamic equilibrium costat cotaiig activities. Differet stoichiometric equatios of the same chemical equatio ca give differet values of thermodyamic equilibrium costat, however, equilibrium compositio is idepedet o selected stoichiometric equatio. Kietic descriptio of equilibrium is based o zero overall reactio rate, o supposed reactio mechaism or etwork (reactio scheme) ad correspodig kietic (rate) equatio. Kietic equilibrium costat usually cotais cocetratios. Accordig to that book, thermodyamic equilibrium data should be itroduced ito kietic equatios idirectly as show i the Scheme 1. Simple example reveals basic problems. Decompositio of carbo mooxide occurs (at the pressure p) accordig to the followig stoichiometric equatio: 2 CO = CO 2 + C (R1) Stadard state of gaseous compoets is selected as the ideal gas at 101 kpa ad for solid compoet as the pure compoet at the actual pressure (due to egligible effects of pressure o behavior of solid compoets, the depedece of the stadard state o pressure ca be eglected here). Ideal behavior is supposed. The a = p /p = p rel / for = CO, CO 2, where p rel = p/p, ad a C = 1; a is the activity, p is the partial pressure, p the stadard pressure, is the umber of moles, ad the total umber of moles. Thermodyamic equilibrium costat is the give by ( c c ) c K p p c CO CO CO CO 2 2 rel CO eq rel CO eq (2)

4 Thermodyamics ad Reactio Rates 675 Thermodyamic data (e.g. ΔG f ) Equilibrium compositio (a eq, c eq) Substitutio ito kietic equatio r(c eq) = 0 Kietic equilibrium costat Kietic data (rate coefficiets/costats) Scheme 1. Coectig thermodyamics ad kietics correctly (Eckert et al., 1986) O cotrary, the ratio of rate costats is give by k cco cc 2 k c CO 2 (3) eq It is clear that thermodyamic ad kietic equilibrium costats eed ot be equivalet eve i ideal systems. For example, the former does ot cotai cocetratio of carbo ad though this could be remedied by statig that carbo amout does ot affect reactio rate ad its cocetratio is icluded i the reverse rate costat, eve the the kietic equilibrium costat could deped o carbo amout i cotrast to the thermodyamic equilibrium costat. Some discrepacies could ot be remedied by restrictig o elemetary reactios oly i this example the presece of p rel ad of the total molar amout, geerally, the presece of quatities trasformig compositio variables ito stadard state-related (activity-related) variables, ad, of course, discrepacy i dimesioalities of the two equilibrium costats. Let us use the same example to illustrate the procedure suggested by Eckert et al. (1986). At 1300 K ad 202 kpa the molar stadard Gibbs eergies are (Novák et al., 1999): G m (CO) kj/mol, Gm (CO 2 ) kj/mol, G m(c) kj/mol ad from them the value of thermodyamic equilibrium costat is calculated: K = Equilibrium molar balace gives ( CO ) 2 eq ( C ) eq x, ( CO ) eq 1 2x, 1 x. The from (2) follows x = (Novák et al., 1999). Equilibrium compositio is substituted ito (3): eq

5 676 Thermodyamics Iteractio Studies Solids, Liquids ad Gases k k (4) eq ad this is real ad true result of thermodyamic restrictio o values of rate costats valid at give temperature. More precisely, this is a restrictio put o the ratio of rate costats, values of which are supposed to be idepedet o equilibrium, i other words, depedet o temperature (ad perhaps o pressure) oly ad therefore this restrictio is valid also out of equilibrium at give temperature. The umerical value of this restrictio is depedet o temperature ad should be recalculated at every temperature usig the value of equilibrium costat at that temperature. Thus, simple ad safe way how to relate thermodyamics ad kietics, thermodyamic ad kietic equilibrium costats, ad rate costats is that show i Scheme 1. However, it gives o geeral equatios ad should be applied specifically for each specific reactio (reactig system) ad reactio coditios (temperature, at least). There are also works that try to resolve relatioship betwee the two types of equilibrium costat more geerally ad, i the same time, correctly ad cosistetly. They were reviewed previously ad oly mai results are preseted here, i the ext sectio. But before doig so, let us ote that kietic equilibrium costat ca be used as a useful idicator of the distace of actual state of reactig mixture from equilibrium ad to follow its approach to equilibrium. I the 2 previous example, actual value of the fractio cco c 2 C / c CO ca be compared with the value of the ratio k/ k ad relative distace from equilibrium calculated, for more details ad other examples see our previous work (Pekař & Koubek, 1997, 1999, 2000). 2.2 Geeral thermodyamic restrictios o rate costats As oted i the precedig sectio there are several works that do ot rely o simple idetificatio of thermodyamic ad kietic equilibrium costats. Holligsworth (1952a, 1952b) geeralized restrictio o the ratio of forward ad reverse reactio rates (f) defied by f ( c, T) f ( c, T)/ f ( c, T) r / r Holligsworth showed that sufficiet coditio for cosistet kietic ad thermodyamic descriptio of equilibrium is F(Q r, T) = (Q r /K) ad (1) = 1 (6) where F is the fuctio f with trasformed variables, FQ (, T) f( c, T), ad Q r is the well kow reactio quotiet. The first equality i (6) says that fuctio F should be expressible as a fuctio of Q r /K. This is too geeral coditio sayig explicitly othig about rate costats. Idetifyig kietic equilibrium costat with thermodyamic oe, coditio (6) is specialized to (Q r /K) = (Q r /K) z (7) where z is a positive costat. Equatio (7) is a geeralizatio of simple idetity K k / k from itroductio. Holligsworth also derived the ecessary cosistecy coditio: r (5)

6 Thermodyamics ad Reactio Rates 677 f 1 = (Q r /K 1) (c, T, u j ) (8) i the eighbourhood of Q r /K = 1 (i.e., of equilibrium); u j stads for a set of othermodyamic variables. Example of practical applicatio of Holligsworth s approach i a ideal system is give by Boyd (Boyd, 1977). Blum (Blum & Luus, 1964) cosidered a geeral mass actio rate law formulated as follows: r k a k a m m 1 1 (9) where is some fuctio of activities, a, of reactig species, ad are coefficiets which may differ from the stoichiometric coefficiets ( ), i fact, reactio orders. Supposig that both the equilibrium costat ad the ratio of the rate costats are depedet oly o temperature, they proved that where k/ k K z (10) z ( )/ ; 1,, (11) Geeral law (9) is rarely used i chemical kietics, i reactios of ios it probably does ot work (Laidler, 1965; Boudart, 1968). It ca be trasformed, particularly simply i ideal systems, to cocetratios. Samohýl (persoal commuicatio) poited out that criteria (11) may be problematic, especially for practically irreversible reactios. For example, reactio orders for reactio 4 NH NO = 5 N H 2 O were determied as follows: NH 1, 3 NO 0.5, N 2 H2O 0. Orders for reversed directio are ukow, probably because of practically irreversible ature of the reactio. Natural selectio could be, e.g., NO 0 (reactio is ot ihibited by reactat), the z = 1/12 ad from this follows NH 2/3 3 which seems to be improbable (rather strog ihibitio by reactat). 2.3 Idepedece of reactios, Wegscheider coditios Wegscheider coditios belog also amog thermodyamic restrictios o rate costats ad have bee itroduced more tha oe hudred years ago (Wegscheider, 1902). I fact, they are also based o equivalece betwee thermodyamic ad kietic equilibrium costats disputed i previous sectios. Recetly, matrix algebra approaches to fid these coditios were described (Vlad & Ross, 2009). Essetial part of them is to fid (i)depedet chemical reactios. Problem of idepedet ad depedet reactios is a iterestig issue sometimes foud also i studies o kietics ad thermodyamics of reactig mixtures. As a rule, a reactio scheme, i.e. a set of stoichiometric equatios (whether elemetary or oelemetary), is proposed, stoichiometric coefficiets are arraged ito stoichiometric matrix ad liear (matrix) algebra is applied to fid its rak which determies the umber of liearly (stoichiometrically) idepedet reactios; all other reactios ca be obtaied as liear combiatios of idepedet oes. This procedure ca be viewed as a a posteriori aalysis of the proposed reactio mechaism or etwork. Bowe has show (Bowe, 1968) that usig ot oly matrix but also vector algebra iterestig results ca be obtaied o the basis of kowig oly compoets of reactig

7 678 Thermodyamics Iteractio Studies Solids, Liquids ad Gases mixture, i.e. with o reactio scheme. This is a priori type of aalysis ad is used i cotiuum oequilibrium (ratioal) thermodyamics. Because Bowe s results are importat for this article they are briefly reviewed ow for reader s coveiece. Let a reactig mixture be composed from compoets (compouds) which are formed by z differet atoms. Atomic compositio of each compoet is described by umbers T that idicate the umber of atoms (= 1, 2,..., z) i compoet (= 1, 2,..., ). Atomic masses M i combiatio with these umbers determie the molar masses M : a M z M T (12) 1 a Although compouds are destroyed or created i chemical reactios the atoms are preserved. If J deotes the umber of moles of the compoet formed or reacted per uit time i uit volume, i.e. the reactio rate for the compoet (compoet rate i short), the the persistece of atoms ca be formulated i the form T J 0; 1,2,, z (13) 1 This result expresses, i other words, the mass coservatio. Atomic umbers ca be arraged i matrix T of dimesio z. Chemical reactios are possible if its rak (h) is smaller tha the umber of compoets (), otherwise the system (13) has oly trivial solutio, i.e. is valid oly for zero compoet rates. If h < z the a ew h matrix S with rak h ca be costructed from the origial matrix T ad used istead of it: S J 0; 1,2,, h (14) 1 I this way oly liearly idepedet relatios from (13) are retaied ad from the chemical poit of view it meas that istead of (some) atoms with masses M a oly some their liear combiatios with masses compoets: M e should be cosidered as elemetary buildig uits of M h M S (15) 1 e Example. Mixture of NO 2 ad N 2 O 4 has the matrix T of dimesio 2 2 ad rak 1; the matrix S is of dimesio 1 2 ad ca be selected as 1 2 which meas that the N O elemetary buildig uit is NO 2 ad M M 2M M 2M. e a a a a M a Multiplyig each of the z relatios (13) by correspodig all it follows that 1 M J ad summig the results for 0. This fact ca be much more effectively formulated i vector form because further importat implicatios tha follow. The last equality idicates

8 Thermodyamics ad Reactio Rates 679 that compoet molar masses ad rates should form two perpedicular vectors, i.e. vectors with vaishig scalar product. Let us itroduce -dimesioal vector space, called the compoet space ad deoted by U, with base vectors e ad reciprocal base vectors e ( = 1, 2,..., ). The the vector of molar masses M ad the vector of reactio rates J are defied i this space as follows: M, 1 1 M e J J e (16) To proceed further we use relatios (14) ad (15) because i cotrast to relatios (12) ad (13) the matrix S is of full rak (does ot cotai liearly depedet rows). The product of the two vectors ca be the expressed i the followig form: h h α α MJ. MeSe. J e Me Se. J e 0 (17) where the latter equality follows usig (14). Because the matrix S has rak h, the vectors α f S e ; 1,2,, h (18) 1 that appear i (17) are liearly idepedet ad thus form a basis of a h-dimesioal subspace W of the space U (remember that h < ). This subspace uambiguously determies complemetary orthogoal subspace V (of dimesio h), i.e. U = V W, V W. From (17) follows: h M e 1 M f (19) which shows that M ca be expressed i the basis of the subspace W or M W. From (14) ad (16) 2 follows: J.f 0; 1,2,,h (20) which meas that J is perpedicular to all basis vectors of the subspace W, cosequetly, J lies i the complemetary orthogoal subspace V, J V. Let us ow select basis vectors i the subspace V ad deote them d p, p = 1, 2,..., h. Of course, these vectors lie also i the (origial) space U ad ca be expressed usig its basis vectors aalogically to (16): p p d P e (21) 1 Because of orthogoality of subspaces V ad W, their bases coform to equatio p f.d S 0 (22) 1 p P

9 680 Thermodyamics Iteractio Studies Solids, Liquids ad Gases which ca be alteratively writte i matrix form as P p S T = 0 (23) Meaig of the matrix P p ca be deduced from two cosequeces. First, because the reactio vector J lies i the subspace V, it ca be expressed also usig its basis vectors, h p J Jpd. Substitutig for J from (16) 2 ad for d p from (21), it follows: p1 h p J JpP ; 1, 2,, (24) p1 Secod, because the vector of molar masses M is i the subspace W, it is perpedicular to all vectors d p ad thus p 0 d.m P M ; p 1, 2,, h (25) 1 p as follows after substitutio from (19), (21), (220. Eq. (25) shows that matrix P p eables to express compoet rates i h quatities J p which are, i fact, rates of h idepedet reactios show by (25) if istead of molar masses M the correspodig chemical symbols are used. I other words P p is the matrix of stoichiometric coefficiets of compoet i (idepedet) reactio p. Vector algebra thus shows that chemical trasformatios fulfillig persistece of atoms (mass coservatio) ca be equivaletly described either by compoet reactio rates or by rates of idepedet reactios. The umber of the former is equal to the umber of compoets () whereas the umber of the latter is lower ( h) which could decrease the dimesioality of the problem of descriptio of reactio rates. I kietic practice, however, chages i compoet cocetratios (amouts) are measured, i.e. data o compoet rates ad ot o rates of idividual reactios are collected. Reactios, i the form of reactio schemes, are suggested a posteriori o the basis of detected compoets, their cocetratios chagig i time ad chemical isight. The depedecies betwee reactios ca be searched. Vector aalysis offers rather differet procedure outlied i Scheme 2. Depedecies are revealed at the begiig ad the oly idepedet reactios are icluded i the (kietic) aalysis. Vector aalysis also shows how to trasform (measured) compoet rates ito (suggested, selected) rates of idepedet reactios. This trasformatio is made by stadard procedure for iterchage betwee vector bases or betwee vector coordiates i differet bases. First, the cotravariat metric tesor with compoets d rp = d r.d p is costructed ad the its iversio (covariat metric tesor) with compoets d rp is foud. From h J J p1 p pd it follows that h p r Jp r Jr p1 J.d d.d. Usig i the latter equatio the well kow relatioship betwee metric tesors ad correspodig base vectors ad the defiitio of base vectors (21) it fially follows: h r Jp J P drp; p 1, 2,, h 1 r 1 (26)

10 Thermodyamics ad Reactio Rates 681 Of course, so far we have see oly relatioships betwee reactio rates ad o explicit equatios for them like, e.g., the kietic mass actio law. Aalysis based oly o permaece of atoms caot give such equatios they belog to the domai of chemical kietics although they ca also be devised by thermodyamics, see Sectio 4. Simple example o Wegscheider coditios was preseted by Vlad ad Ross (Vlad & Ross, 2009) isomerizatio takig place i two ways: A = B, 2A = A + B (R2) Fid out compoets of reactig mixture Costruct the matrix T ad determie its rak Costruct the matrix S Select the stoichiometric matrix P pα fulfillig (23) h idepedet reactios ad their rates to describe chemical trasformatios Use the method of Sectio 4 Fid compoet rates from (24) Scheme 2. Alterative procedure to fid reactio rates Vlad ad Ross ote that if the (thermodyamic) equilibrium costat is K c / c if kietic equatios are expressed e.g. r 1 k 1 c A k 1 c B ad B A eq the the cosistecy betwee

11 682 Thermodyamics Iteractio Studies Solids, Liquids ad Gases thermodyamic ad kietic descriptio of equilibrium is achieved oly if the followig (Wegscheider) coditio holds: k / k k / k K It ca be easily checked that i this mixture of oe kid of atom ad two compoets the rak of the matrix T (dimesio 1 2) is 1 ad there is oly oe idepedet reactio. The matrix S ca be selected as equal to the matrix T ad the the stoichiometric matrix ca be selected as 1 1 which correspods to the first reactio (A = B) selected as the idepedet reactio. There is oe base vector d 1 e1 e givig oe compoet 2 cotravariat tesor d 11 = 2 ad correspodig compoet of covariat tesor d 11 = 1/2. Cosequetly, the rate of the idepedet reactio is related to compoet reactio rates by: (27) J A = J 1 P 11 = J 1, J B = J 1 P 12 = J 1 (28) ad J A = J B which follows also from (14). Kietics of trasformatios i a mixture of two isomers ca be thus fully described by oe reactio rate oly either from the two compoet rates ca be measured ad used for this purpose, the other compoet rate is the determied by it, ca be calculated from it. At this stage of aalysis there is o idicatio that two reactios should be cosidered ad this should be viewed as some kid of exteral iformatio comig perhaps from experimets. At the same time this aalysis does ot provide ay explicit expressio for reactio rate ad its depedece o cocetratio this is aother type of exteral iformatio comig usually from kietics. Let us therefore suppose the two isomerizatio processes give above ad their rates formulated i the form of kietic mass actio law: r kc kc r kc kc c A 1 B, 2 2 A 2 A B The the oly oe idepedet reactio rate is i the form J 1 = r 1 + r 2. Note, that although the first reactio has bee selected as the idepedet reactio, the rate of idepedet reactio is ot equal to (its mass actio rate) r 1. This iterestig fidig has probably o specific practical implicatio. However, idividual traditioal rates (r i ) should ot be idepedet. Let us suppose that r 2 is depedet o r 1, i.e. ca be expressed through it: r 2 = br 1 ; the bk k c c bk k c c 1 2 A A 1 2 A B 0 should be valid for ay cocetratios. Sufficiet coditios for this are b k c / k k c / k ad from them follows: 2 A 1 2 A 1 kk kk i.e. kietic part of Wegscheider coditio (27). Substitutig derived expressios for b ito br 1 it ca be easily checked that r 2 really results. Although the derivatio is rather straightforward ad is ot based o liear depedecy with costat coefficiets it poits to assumptio that Wegscheider coditios are ot coditios for cosistecy of kietics with thermodyamics but results of depedecies amog reactio rates. Moreover, this derivatio eed ot suppose equality of thermodyamic ad kietic equilibrium costat. (29) (30) (31)

12 Thermodyamics ad Reactio Rates 683 There is a thermodyamic method givig kietic descriptio i terms of idepedet reactios as oted i Scheme 2, see Sectio 4. More complex reactio mixture ad scheme was discussed by Ederer ad Gilles (Ederer & Gilles, 2007). Their mixture was composed from six formal compoets (A, B, C, AB, BC, ABC) formed by three atoms (A, B, C). Three idepedet reactios are possible i this mixture while four reactios were cosidered by Ederer ad Gilles (Ederer & Gilles, 2007) r 4 = b 1 r 1 + b 2 r 2 + b 3 r 3 with followig mass actio rate equatios: r k c c k c, r k c c k c, r k c c k c, r k c c k c 1 1 A B 1 AB 2 2 AB C 2 ABC 3 3 B C 3 BC 4 4 A BC 4 ABC Let us suppose that the fourth reactio rate ca be expressed through the other three rates: b 1 r 1 + b 2 r 2 + b 3 r 3. By similar procedure as i the precedig example we arrive at coditios b2 k 4 / k 2, b3 k 4cA / k 3, ad b1 b2k2cc / k1 b3k3cc / k1ca from which it follows that kkkk (32) (33) kkkk i.e., Wegscheider coditio derived i (Ederer & Gilles, 2007) from equilibrium cosideratios. Thus also here Wegscheider coditio seems to be a result of mutual depedece of reactio rates ad ot a ecessary cosistecy coditio betwee thermodyamics ad kietics. If reactios A + B = AB, AB + C = ABC, ad B + C = BC are selected as idepedet oes the (24) gives J A = J 1, J B = J 1 J 3, J C = J 2 J 3, J AB = J 1 J 2, J BC = J 3, J ABC = J 2 (34) Remember that, e.g., J 1 r 1 but that the relatioships betwee rates of idepedet reactios ad mass actio rates (32) follow from (34): J 1 = r 1 + r 4, J 2 = r 2 + r 4, J 3 = r 3 r 4 (35) Eq. (26) gives more complex expressios for idepedet rates, e.g. J 1 = J A /2 J B /4 + J AB /4 J BC /4 + J ABC /4, whereas from (24), i.e. (34), simply follow: J 1 = J A, J 2 = J ABC, J 3 = J BC. This is because the rates are cosidered as vector compoets compoets J of six dimesioal space are trasformed to compoets J i i three dimesioal subspace. Cosequetly, i practical applicatios (24) should be preferred i favor of (26) also to express J i i terms of J. Message from the aalysis of idepedece of reactios i this example is that it is sufficiet to measure three compoet rates oly (J A, J ABC, J BC ); the remaiig three compoet rates are determied by them. Although cocetratios, i.e. compoet rates, are measured i kietic experimets, results are fially expressed i reactio rates, rates of reactios occurrig i suggested reactio scheme. Compoet rates are simply ot sufficiet i kietic aalysis ad they are (perhaps always) traslated ito rates of reactio steps. However, from the three idepedet rates there caot be uambiguously determied rates of four reactios i suggested reactio schemes as (35) demostrates (three equatios for four ukow r i ). Oe equatio more is eeded ad this is the above equatio relatig r 4 to the remaiig three rates. Equatios cotaiig r i are too geeral ad i practice are replaced by mass actio expressios show i (32) eight parameters (rate costats) are thus

13 684 Thermodyamics Iteractio Studies Solids, Liquids ad Gases itroduced i this example. They ca be i priciple determied from three equatios (35) with the three measured idepedet reactios, four equatios relatig equilibrium compositio (or thermodyamic equilibrium costat) ad kietic equilibrium costat ad oe Wegscheider coditio (33), i.e. eight equatios i total. Alterative thermodyamic method is described i Sectio 4. Algebraically more rigorous is this aalysis i the case of first order reactios as was illustrated o a mixture of three isomers ad their triagular reactio scheme which is traditioal example used to discuss cosistecy betwee thermodyamics ad kietics. Here, Wegscheider relatios are cosequeces of liear depedece of traditioal mass actio reactio rates (Pekař, 2007). 2.4 Note o stadard states Precedig sectios demostrated that oe of the mai problems to be solved whe relatig thermodyamics ad kietics is the trasformatio betwee activities ad cocetratio variables. This is closely related to the selectio of stadard state (importat ad ofte overlooked aspect of relatig thermodyamic ad kietic equilibrium costats) ad to chemical potetial. Stadard states are therefore briefly reviewed i this sectio ad chemical potetial is subject of the followig sectio. Rates of chemical reactios are mostly expressed i terms of cocetratios. Amog stadard states itroduced ad commoly used i thermodyamics there is oly oe based o cocetratio the stadard state of oelectrolyte solute o cocetratio basis. Oly this stadard state ca be directly used i kietic equatios. Stadard state i gaseous phase or mixture is defied through (partial) pressure or fugacity. As show above eve i mixture of ideal gases it is impossible to simply use this stadard state i cocetratio based kietic equatios. Although kietic equatios could be reformulated ito partial pressures there still remais problem with the fact that stadard pressure is fixed (at 1 atm or, owadays, at 10 5 Pa) ad its recalculatio to actual pressure i reactig mixture may cause icompatibility of thermodyamic ad kietic equilibrium costats (see the factor p rel i the example above i Sectio 2.1). This opes aother problem the very selectio of stadard state, particularly i relatio to activity discussed i subsequet sectio. I priciple, it ca be selected arbitrarily, as depedet oly o temperature or o temperature ad pressure. Stadard states strictly based o the (fixed) stadard pressure are of the former type ad oly such will be cosidered i this article. All other states, icludig states depedet also o pressure, will be called the referece state; the same approach is used, e.g. by de Voe (de Voe, 2001). The value of thermodyamic equilibrium costat ad its depedece or idepedece o pressure is thus depedet o the selected stadard (or referece) state. This is quite ucommo i chemical kietics where the depedece of rate costats is ot a matter of selectio of stadard states but result of experimetal evidece or some theory of reactio rates. As a rule, rate costat is always fuctio of temperature. Sometimes also the depedece o pressure is cosidered but this is usually the case of oelemetary reactios. Cosequetly, attempts to relate thermodyamic ad kietic equilibrium costats should select stadard state cosistetly with fuctioal depedece of rate costats. O the other had, the method of Scheme 1 is self-cosistet i this aspect because equilibrium compositio is idepedet of the selectio of stadard state.

14 Thermodyamics ad Reactio Rates Chemical potetial ad activity revise Chemical potetial is used i discussios o thermodyamic implicatios o reactio rates, particularly i the form of (stoichiometric) differece betwee chemical potetials of reactio products ad reactats ad through its explicit relatioship to cocetratios (activities, i geeral). Before goig ito this type of aalysis basic iformatio is recapitulated. Chemical potetial is i classical, equilibrium thermodyamics defied as a partial derivative of Gibbs eergy (G): G/,, (36) Tp j Although aother defiitios through aother thermodyamic quatities are possible (ad equivalet with this oe), the defiitio usig the Gibbs eergy is the most useful for chemical thermodyamics. Chemical potetial expresses the effect of compositio ad this effect is also essetial i chemical kietics. To make the mathematical defiitio of the chemical potetial applicable i practice its relatioship to compositio (cocetratio) should be stated explicitly. Practical chemical thermodyamics suggests that this is a easy task but we must be very careful ad bear all (tacit) presumptios i mid to arrive at proper coclusios. Geerally the explicit relatioship betwee chemical compositio ad chemical potetial is stated defiig the activity of a compoet : a exp RT (37) which ca be trasformed to RT l a (38) but this still lacks direct itercoectio/likage to measurable cocetratios. Just this is the mai problem of applyig chemical potetial (ad activities) i rate equatios which systematically use molar cocetratios. Eve whe reactio rates would be expressed usig activities i place of cocetratios the activities should be properly calculated from the measured cocetratios, i other words, the cocetratios should be correctly trasformed to the activities. Activity is very easily related to measurable compositio variable i the case of mixture of ideal gases. Providig that Gibbs eergy is a fuctio of temperature, pressure ad molar amouts, followig relatio is well kow from thermodyamics for the partial molar volume: V / p T,. I a mixture of ideal j gases partial molar volumes are equal to the molar volume of the mixture, V m (Silbey et al., 2005). Because V m = RT/p we ca write:,g,g RT / p / p p / p x / p (39) ad,g RT / p / p (40)

15 686 Thermodyamics Iteractio Studies Solids, Liquids ad Gases Itegratio from the stadard state to some actual state the yields,g,g RT l p / p (41) Comparig with the defiitio of activity it follows a p / p (mixture of ideal gases) (42) Applicatio of this relatioship was illustrated i the example give above. Note that (42) was ot derived from the defiitio of activity but comparig the properties of chemical potetial i the ideal gas mixture (41) with the defiitio of activity. Note also that the partial derivative i the origial defiitio of chemical potetial is i geeral a fuctio of molar amouts (cotets) of all compoets but eq. (42) states that the chemical potetial of a compoet is a fuctio oly of the cotet of that compoet. I a real gas mixture, o-idealities should be take ito accout, usually by substitutig fugacity (f ) for the partial pressure:,g,g RT l f / p The fugacity ca be elimiated i favor of directly measurable quatities usig the fugacity coefficiet f (43) p (44) ad its relatioship to the partial molar volume ad the total pressure (de Voe, 2001): p,g,g 0 RT l p / p V RT / p d p It should be stressed that i derivatio of the expressio for the fugacity coefficiet it was assumed that the Gibbs eergy is a fuctio of (oly) temperature, pressure, ad molar amouts of all compoets. Comparig with the defiitio of activity we have (45) a f / p (mixture of gases) (46) If kietic equatios for mixture of real gases are writte i partial pressures the thermodyamic ad kietic equilibrium costats are icompatible due to the presece of fugacity coefficiet or the itegral i eq. (45). Kietic equatios for mixture of real gases could be formulated i terms of fugacities istead of cocetratios (or partial pressures) to achieve compatibility betwee thermodyamic ad kietic equilibrium costats but eve tha the same problem remais with the presece of the stadard pressure i thermodyamic relatios. Kietic equatios formulated i fugacities are really rare some success i this way was demostrated by Eckert ad Boudart (Eckert & Boudart, 1963) while Maso (Maso, 1965) showed, usig the same data, that fugacities eed ot remedy the whole situatio. Similar derivatio for liquid state (solutios) has differet basis. It stems from the equilibrium betwee liquid ad gaseous phase i which the followig idetity holds:,g =,l. Itroducig expressio (41) or (43) ad usig either Raoult s or Hery s law for the

16 Thermodyamics ad Reactio Rates 687 relatioship betwee compositios of equilibrated liquid ad gaseous phases fial form of,l depedece o the compositio of liquid is obtaied. For example, with Raoult s law p = x p * ad ideal gas phase we have this equatio ref,l,g RT l x p / p RT l x (47) which has, i fact, ispired the defiitio of a ideal (liquid, solid, or gas) mixture as a mixture with the chemical potetial defied, at a give T ad p, as ref RT l x where ref is a fuctio of both T ad p. This defiitio, as well as the idetity i (47), ca be simply related to the defiitio of activity oly if the stadard state is selected cosistetly with the referece state, i.e. if the former is a fuctio of both T ad p. If the stadard state is selected as depedet o temperature, as it should be, tha the pressure factor ( ) should be itroduced (see, e.g., de Voe, 2001) ref exp RT The the activity of a (o-electrolyte) compoet i real solutio is writte as a (48) x ref where is the activity coefficiet itroduced by the equatio RT l x. Itroducig activities i place of cocetratios meas i this case to kow the pressure factor ad to trasform molar fractios ito molar cocetratios to be cosistet with thermodyamics. The mai problems with usig activities defied for liquid systems ca be summarized as follows. Activity is based o molar fractios whereas kietic uses cocetratios. Although there are formulas for the coversio of these variables they do ot allow direct substitutio, they itroduce other variables (e.g., solutio desity) ad lead to rather complex expressio of thermodyamic equilibrium costat i cocetratios. Whereas cocetratios of all species are idepedet (variables) this is ot true for molar fractios value of oe from them is uambiguously determied by values of remaiig oes. Chemical potetial i liquid ad activity based o it are itroduced o the basis of (liquid-gas) equilibrium while kietics essetially works with reactios out of equilibrium. Applicability of equilibriumbased formulated i fugacities are really rare i oequilibrium states deserves further study. The problem with molar fractios ca be resolved by the use of molar cocetratio ref based Hery s law givig for ideal-dilute solutio,l, c RT l c / c, however, rate equatios should be formulated with the stadard cocetratio. Sometimes followig ref relatioship is used:,l, c RT l c / c (Ederer & Gilles, 2007) where c is the sum of all cocetratios. I this case, the ivertibility for c is problematic because it is icluded i c ; reactio rates should be the formulated i c /c istead of cocetratios that is quite uusual. Of course, the value of activity is depedet o the selected stadard state, ayway. All attempts to relate thermodyamic ad kietic equilibrium costats should pay great attetio to the selectio of stadard state ad its cosequeces to be really rigorous ad correct. It is clear from this basic overview that chemical potetial, activity ad their iterrelatio are i priciple equilibrium quatities which, i kietic applicatios, are to be used for

17 688 Thermodyamics Iteractio Studies Solids, Liquids ad Gases o-equilibrium situatios. Let us ow trace oe relatively simple o-equilibrium approach to descriptio of chemically reactig systems ad its results regardig the chemical potetial. Samohýl has developed ratioal thermodyamic approach for chemically reactig fluids with liear trasport properties (heceforth called briefly liear fluids) ad these fluids seem to iclude may (o-electrolyte) systems ecoutered i chemistry (Samohýl & Malijevský, 1976; Samohýl, 1982, 1987). This is a cotiuum mechaics based approach workig with desities of quatities ad specific quatities (cosidered locally, i other words, as fields but this is ot crucial for the preset text) therefore it primarily uses desities of compoets (more precisely, the desity of compoet mass) istead of their molar cocetratios or fractios that are commo i chemistry. This desity, i fact, is kow i chemistry as a mass cocetratio with dimesio of mass per (uit) volume ad ca be thus easily recalculated to cocetratio quatities more commo i chemistry. Chemical potetial of a reactig compoet is defied i this theory as follows: g f / (49) Here is the desity of mixture, i.e. the sum of all compoet desities, ad f is the specific free eergy of (reactig) mixture as a fuctio of relevat idepedet variables (the value of this fuctio is deoted by f). Ispiratio for this defiitio came from the etropic iequality (the secod law of thermodyamics) as formulated i ratioal thermodyamics geerally for mixtures ad from the fact that this defiitio eabled to derive classical (equilibrium) thermodyamic relatios i the special case that is covered by classical theory. The chemical potetial g thus has the dimesios of eergy per mass. The product f essetially trasforms the specific quatity to its desity ad the defiitio (49) ca be viewed as a geeralizatio of the classical defiitio (36) partial derivative of mixture free eergy (as a fuctio) with respect to a idepedet variable expressig the amout of a compoet. The specific free eergy f is fuctio of various (mostly kiematic ad thermal) variables but here it is sufficiet to ote that compoet desities are amog them, of course. I the case of liear fluids it ca be proved that free eergy is fuctio of desities ad temperature oly, f f 1, 2,,, T. The same result is proved also for chemical potetials g ad also for reactio rates expressed as compoet mass created or destroyed by chemical reactios at a give place ad time i uit volume, r r 1, 2,,, T. These rates ca be easily trasformed to molar basis much more commo i chemistry usig the molar mass M : J = r /M. Compoet desities are directly related to molar cocetratio by a similar equatio: c = /M. I this way, the well kow kietic empirical law the law of mass actio is derived theoretically i the form: J J c1, c2,, c, T. Apparetly, activities could be itroduced ito this fuctio as idepedet variables cotrollig reactio rates by meas of relatios as a c / c but this is ot rigorous because these relatios are cosequeces of chemical potetial ad its explicit depedece o mixture compositio ad ot defiitios per se. Therefore, chemical potetials should be itroduced as idepedet variables at first. This could be doe providig that compoet desities ca be expressed as fuctios of chemical

18 Thermodyamics ad Reactio Rates 689 potetial, i.e. providig that fuctios g g,,,, T are ivertible (with respect to desities). This ivertibility is ot self-evidet ad the best way would be to prove it. Samohýl has proved (Samohýl, 1982, 1987) that if mixture of liear fluids fulfils Gibbs stability coditios the the matrix with elemets g / (, = 1,..., ) is regular which esures the ivertibility. This stability is a stadard requiremet for reasoable behavior of may reactig systems of chemist s iterest, cosequetly the ivertibility ca be cosidered to be guarateed ad we ca trasform the rate fuctios as follows: 1 2 J J T J g g g T J T (50),,,,,,,,,,,, where the last trasformatio was made usig the followig trasformatio of (specific) chemical potetial ito the traditioal chemical potetial (which will be called the molar chemical potetial heceforth): = g M. Usig the defiitio of activity (37) aother trasformatio, to activities, ca be made providig that the stadard state is a fuctio of temperature oly: J T J a a a T (51),,,,,,,, It should be stressed that chemical potetial of compoet as defied by (49) is a fuctio of desities of all compoets, i.e. of, = 1,...,, therefore also the molar chemical potetial is followig fuctio of compositio: c1, c2,, c, T. Note that geerally ay rate of formatio or destructio (J ) is a fuctio of desities, or chemical potetials, or activities, etc. of all compoets. Although the fuctios (depedecies) give above were derived for specific case of liear fluids they are still too geeral. Yet simpler fluid model is the simple mixture of fluids which is defied as mixture of liear fluids costitutive (state) equatios of which are idepedet o desity gradiets. The it ca be show (Samohýl, 1982, 1987) that ad, cosequetly, also that g g, T f / 0 for ;, 1,, (52), i.e. the chemical potetial of ay compoet is a fuctio of desity of this compoet oly (ad of temperature). Mixture of ideal gases is defied as a simple mixture with additioal requiremet that partial iteral eergy ad ethalpy are depedet o temperature oly. The it ca be proved (Samohýl, 1982, 1987) that chemical potetial is give by g g ( T) R Tl p / p (53) that is slightly more geeral tha the commo model of ideal gas for which R = R/M. Thus the expressio (41) is proved also at oequilibrium coditios ad this is probably oly oe mixture model for which explicit expressio for the depedece of chemical potetial o compositio out of equilibrium is derived. There is o idicatio for other cases while the fuctio g g, T should be just of the logarithmic form like (47). Let us check coformity of the traditioal ideal mixture model with the defiitio of simple mixture. For solute i a ideal-dilute solutio followig cocetratio-based expressio is used:

19 690 Thermodyamics Iteractio Studies Solids, Liquids ad Gases ref RT l c / c (54) ref where icludes (amog other) the gas stadard state ad cocetratio-based Hery s costat. Chagig to specific quatities ad desities we obtai: g ref / M RT / M l / M c which looks like a fuctio of ad T oly, i.e. the simple mixture fuctio g g, T. However, the referetial state is a fuctio of pressure so this is ot such fuctio rigorously. Except ideal gases there is probably o proof of applicability of classical expressios for depedece of chemical potetial o compositio out of equilibrium ad o proof of its logarithmic poit. There are probably also o experimetal data that could help i resolvig this problem. 4. Solutio offered by ratioal thermodyamics Ratioal thermodyamics offers certai solutio to problems preseted so far. It should be stressed that this is by o meas totally geeral theory resolvig all possible cases. But it clearly states assumptios ad models, i. e. scope of its potetial applicatio. The first assumptio, besides stadard balaces ad etropic iequality (see, e.g., Samohýl, 1982, 1987), or model is the mixture of liear fluids i which the fuctioal form of reactio rates was proved: J J c1, c2,, c, T (Samohýl & Malijevský, 1976; Samohýl, 1982, 1987). Oly idepedet reactio rates are sufficiet that ca be easily obtaied from compoet rates, cf. (26) from which further follows that they are fuctio of the same variables. This fuctio, Ji Jic1, c2,, c, T, is approximated by a polyomial of suitable degree (Samohýl & Malijevský, 1976; Samohýl, 1982, 1987). Equilibrium costat is defied for each idepedet reactio as follows: (54) p RT l Kp P ; p 1, 2,, h (55) 1 Activity (37) is supposed to be equal to molar cocetratios (divided by uit stadard cocetratio), which is possible for ideal gases, at least (Samohýl, 1982, 1987). Combiig this defiitio of activity with the proved fact that i equilibrium (Samohýl, 1982, 1987) it follows 1 p eq P ( ) 0 Kp p P ( c ) eq (56) 1 Some equilibrium cocetratios ca be thus expressed usig the others ad (56) ad substituted i the approximatig polyomial that equals zero i equilibrium. Equilibrium polyomial should vaish for ay cocetratios what leads to vaishig of some of its coefficiets. Because the coefficiets are idepedet of equilibrium these results are valid

20 Thermodyamics ad Reactio Rates 691 also out of it ad the fial simplified approximatig polyomial, called thermodyamic polyomial, follows ad represets rate equatio of mass actio type. More details o this method ca be foud elsewhere (Samohýl & Malijevský, 1976; Pekař, 2009, 2010). Here it is illustrated o two examples relevat for this article. First example is the mixture of two isomers discussed i Sectio Rate of the oly oe idepedet reactio, selected as A = B, is approximated by a polyomial of the secod degree: A 01 B 20 A 02 B 11 A B J k k c k c k c k c k c c (57) The cocetratio of B is expressed from the equilibrium costat, (c B ) eq = K(c A ) eq ad substituted ito (57) with J 1 = 0. Followig form of the polyomial i equilibrium is obtaied: A eq A eq 0 k k Kk ( c ) k K k Kk ( c ) (58) Eq. (58) should be valid for ay values of equilibrium cocetratios, cosequetly ; 10 01; k k Kk k K k Kk (59) Substitutig (59) ito (57) the fial thermodyamic polyomial (of the secod degree) results: J k Kc c k K c c k Kc c c (60) 1 10 A B 02 A B 11 A A B Note, that coefficiets k ij are fuctios of temperature oly ad ca be iterpreted as mass actio rate costats (there is o coditio o their sig, if some k ij is egative the traditioal rate costat is k ij with opposite sig). Although oly the reactio A = B has bee selected as the idepedet reactio, its rate as give by (60) cotais more tha just traditioal mass actio term for this reactio. Remember that compoet rates are give by (28). Selectig k 02 = 0 two terms remai i (60) ad they correspod to the traditioal mass actio terms just for the two reactios supposed i (R2). Although oly oe reactio has bee selected to describe kietics, eq. (60) shows that thermodyamic polyomial does ot exclude other (depedet) reactios from kietic effects ad relatioship very close to J 1 = r 1 + r 2, see also (29), aturally follows. No Wegscheider coditios are ecessary because there are o reverse rate costats. O cotrary, thermodyamic equilibrium costat is directly ivolved i rate equatio; it should be stressed that because o reverse costat are cosidered this is ot achieved by simple substitutio of K for k j from (27). Eq. (60) also exteds the scheme (R2) ad icludes also bimolecular isomerizatio path: 2A = 2B. This example illustrated how thermodyamics ca be cosistetly coected to kietics cosiderig oly idepedet reactios ad results of oequilibrium thermodyamics with o eed of additioal cosistecy coditios. Example of simple combiatio reactio A + B = AB will illustrate the use of molar chemical potetial i rate equatios. I this mixture of three compoets composed from two atoms oly oe idepedet reactio is possible. Just the give reactio ca be selected with l K /( RT) ad equal to equilibrium costat defied by (55): A B AB

21 692 Thermodyamics Iteractio Studies Solids, Liquids ad Gases K cab / cacb, cf. (56). The secod degree thermodyamic polyomial results i this eq case i followig rate equatio: A B AB J k ( c c K c ) (61) that represets the fuctio J 1 J 1 ( T, c A, c B, c AB ). Its trasformatio to the fuctio J J ( T,,, ) gives: 1 1 A B AB J A B A B AB 1 k 110 exp exp exp RT RT RT This is thermodyamically correct expressio (for the supposed thermodyamic model) of the fuctio J discussed i Sectio 3 ad i cotrast to (1). It is clear that proper thermodyamic drivig force for reactio rate is ot simple (stoichiometric) differece i molar chemical potetials of products ad reactats. The expressio i square brackets ca be cosidered as this drivig force. Equatio (62) also lucidly shows that high molar chemical potetial of reactats i combiatio with low molar chemical potetial of products ca aturally lead to high reactio rate as could be expected. O the other had, this is achieved i other approaches, based o ii, due to arbitrary selectio of sigs of stoichiometric coefficiets. I cotrast to this straightforward approach illustrated i itroductio, also kietic variable (k 110 ) is still preset i eq. (62), explaiig why some thermodyamically highly forced reactios may ot practically occur due to very low reactio rate. Equatio (62) icludes also explicit depedece of reactio rate o stadard state selectio (cf. the presece of stadard chemical potetials). This is ievitable cosequece of usig thermodyamic variables i kietic equatios. Because also the molar chemical potetial is depedet o stadard state selectio, it ca be perhaps assumed that these depedeces are cacelled i the fial value of reactio rate. Ratioal thermodyamics thus provides efficiet coectio to reactio kietics. However, eve this is ot totally uiversal theory; o the other had, presumptios are clearly stated. First, the procedure applies to liear fluids oly. Secod, as preseted here it is restricted to mixtures of ideal gases. This restrictio ca be easily removed, if activities are used istead of cocetratios, i.e. if fuctios J are used i place of fuctios J all equatios remai uchaged except the symbol a replacig the symbol c. But the still remais the problem how to fid explicit relatioship betwee activities ad cocetratios valid at o equilibrium coditios. Nevertheless, this method seems to be the most carefully elaborated thermodyamic approach to chemical kietics. (62) 5. Coclusio Two approaches relatig thermodyamics ad chemical kietics were discussed i this article. The first oe were restrictios put by thermodyamics o the values of rate costats i mass actio rate equatios. This ca be also formulated as a problem of relatio, or eve equivalece, betwee the true thermodyamic equilibrium costat ad the ratio of forward ad reversed rate costats. The secod discussed approach was the use of chemical potetial as a geeral drivig force for chemical reactio ad directly i rate equatios.

22 Thermodyamics ad Reactio Rates 693 Both approaches are closely coected through the questio of usig activities, that are commo i thermodyamics, i place of cocetratios i kietic equatios ad the problem of expressig activities as fuctio of cocetratios. Thermodyamic equilibrium costat ad the ratio of forward ad reversed rate costats are coceptually differet ad caot be idetified. Restrictios followig from the former o values of rate costats should be foud idirectly as show i Scheme 1. Direct itroductio of chemical potetial ito traditioal mass actio rate equatios is icorrect due to icompatibility of cocetratios ad activities ad is problematic eve i ideal systems. Ratioal thermodyamic treatmet of chemically reactig mixtures of fluids with liear trasport properties offers some solutio to these problems wheever its clearly stated assumptios are met i real reactig systems of iterest. No compatibility coditios, o Wegscheider relatios (that have bee show to be results of depedece amog reactios) are the ecessary, thermodyamic equilibrium costats appear i rate equatios, thermodyamics ad kietics are coected quite aturally. The role of ( thermodyamically ) idepedet reactios i formulatig rate equatios ad i kietics i geeral is clarified. Future research should focus attetio o the applicability of depedeces of chemical potetial o cocetratios kow from equilibrium thermodyamics i oequilibrium states, or o the related problem of cosistet use of activities ad correspodig stadard states i rate equatios. Though practical chemical kietics has bee successfully survivig without special icorporatio of thermodyamic requiremets, except perhaps equilibrium results, tighter coectio of kietics with thermodyamics is desirable ot oly from the theoretical poit of view but may be of practical importace cosiderig icreasig iterest i aalyzig of complex biochemical etwork or icreasig computatioal capabilities for correct modelig of complex reactio systems. The latter whe combied with proper thermodyamic requiremets might cotribute to more effective practical, idustrial exploitatio of chemical processes. 6. Ackowledgmet The author is with the Cetre of Materials Research at the Faculty of Chemistry, Bro Uiversity of Techology; the Cetre is supported by project No. CZ.1.05/2.1.00/ from ERDF. The author is idebted to Iva Samohýl for may valuable discussios o ratioal thermodyamics. 7. Refereces Blum, L.H. & Luus, R. (1964). Thermodyamic Cosistecy of Reactio Rate Expressios. Chemical Egieerig Sciece, Vol.19, No.4, pp , ISSN Boudart, M. (1968). Kietics of Chemical Processes, Pretice-Hall, Eglewood Cliffs, USA Bowe, R.M. (1968). O the Stoichiometry of Chemically Reactig Systems. Archive for Ratioal Mechaics ad Aalysis, Vol.29, No.2, pp , ISSN Boyd, R.K. (1977). Macroscopic ad Microscopic Restrictios o Chemical Kietics. Chemical Reviews, Vol.77, No.1, pp , ISSN

23 694 Thermodyamics Iteractio Studies Solids, Liquids ad Gases De Voe, H. (2001). Thermodyamics ad Chemistry, Pretice Hall, ISBN , Upper Saddle River, USA Eckert, C.A. & Boudart, M. (1963). Use of Fugacities i Gas Kietics. Chemical Egieerig Sciece, Vol.18, No.2, , ISSN Eckert, E.; Horák, J.; Jiráček, F. & Marek, M. (1986). Applied Chemical Kietics, SNTL, Prague, Czechoslovakia (i Czech) Ederer, M. & Gilles, E.D. (2007). Thermodyamically Feasible Kietic Models of Reactio Networks. Biophysical Joural, Vol.92, No.6, pp , ISSN Holligsworth, C.A. (1952a). Equilibrium ad the Rate Laws for Forward ad Reverse Reactios. Joural of Chemical Physics, Vol.20, No.5, pp , ISSN Holligsworth, C.A. (1952b). Equilibrium ad the Rate Laws. Joural of Chemical Physics, Vol.20, No.10, pp , ISSN Laidler, K.J. (1965). Chemical Kietics, McGraw-Hill, New York, USA Maso, D.M. (1965). Effect of Compositio ad Pressure o Gas Phase Reactio Rate Coefficiet. Chemical Egieerig Sciece, Vol.20, No.12, pp , ISSN Novák, J.; Malijevský, A.; Voňka, P. & Matouš, J. (1999). Physical Chemistry, VŠCHT, ISBN , Prague, Czech Republic (i Czech) Pekař, M. & Koubek, J. (1997). Rate-limitig Step. Does It Exist i the No-Steady State? Chemical Egieerig Sciece, Vol.52, No.14, pp , ISSN Pekař, M. & Koubek, J. (1999). Cocetratio Forcig i the Kietic Research i Heterogeeous Catalysis. Applied Catalysis A, Vol.177, No.1, pp , ISSN X Pekař, M. & Koubek, J. (2000). O the Geeral Priciples of Trasiet Behaviour of Heterogeeous Catalytic Reactios. Applied Catalysis A, Vol.199, No.2, pp , ISSN X Pekař, M. (2007). Detailed Balace i Reactio Kietics Cosequece of Mass Coservatio? Reactio Kietics ad Catalysis Letters, Vol. 90, No. 2, p , ISSN Pekař, M. (2009). Thermodyamic Framework for Desig of Reactio Rate Equatios ad Schemes. Collectio of the Czechoslovak Chemical Commuicatios, Vol.74, No.9, pp , ISSN Pekař, M. (2010). Macroscopic Derivatio of the Kietic Mass-Actio Law. Reactio Kietics, Mechaisms ad Catalysis, Vol.99, No. 1, pp , ISSN Qia, H. & Beard, D.A. (2005). Thermodyamics of Stoichiometric Biochemical Networks i Livig Systems Far From Equilibrium. Biophysical Chemistry, Vol.114, No.3, pp , ISSN Samohýl, I. (1982). Ratioal Thermodyamics of Chemically Reactig Mixtures, Academia, Prague, Czechoslovakia (i Czech) Samohýl, I. (1987). Thermodyamics of Irreversible Processes i Fluid Mixtures, Teuber, Leipzig, Germay Samohýl, I. & Malijevský, A. (1976). Pheomeological Derivatio of the Mass Actio LAw of homogeeous chemical kietics. Collectio of the Czechoslovak Chemical Commuicatios, Vol.41, No.8, pp , ISSN Silbey, R.J.; Alberty, R.A. & Bawedi M.G. (2005). Physical Chemistry, 4 th editio, J.Wiley, ISBN X, Hoboke, USA Vlad, M.O. & Ross, J. (2009). Thermodyamically Based Costraits for Rate Coefficiets of Large Biochemical Networks. WIREs Systems Biology ad Medicie, Vol.1, No.3, pp , ISSN Wegscheider, R. (1902). Über simultae Gleichgewichte ud die Beziehuge zwische Thermodyamik ud Reaktioskietik. Zeitschrift für physikalische Chemie, Vol. XXXIX, pp

24 Thermodyamics - Iteractio Studies - Solids, Liquids ad Gases Edited by Dr. Jua Carlos Moreo Pirajà ISBN Hard cover, 918 pages Publisher ITech Published olie 02, November, 2011 Published i prit editio November, 2011 Thermodyamics is oe of the most excitig braches of physical chemistry which has greatly cotributed to the moder sciece. Beig cocetrated o a wide rage of applicatios of thermodyamics, this book gathers a series of cotributios by the fiest scietists i the world, gathered i a orderly maer. It ca be used i post-graduate courses for studets ad as a referece book, as it is writte i a laguage pleasig to the reader. It ca also serve as a referece material for researchers to whom the thermodyamics is oe of the area of iterest. How to referece I order to correctly referece this scholarly work, feel free to copy ad paste the followig: Miloslav Pekař (2011). Thermodyamics ad Reactio Rates, Thermodyamics - Iteractio Studies - Solids, Liquids ad Gases, Dr. Jua Carlos Moreo Pirajà (Ed.), ISBN: , ITech, Available from: ITech Europe Uiversity Campus STeP Ri Slavka Krautzeka 83/A Rijeka, Croatia Phoe: +385 (51) Fax: +385 (51) ITech Chia Uit 405, Office Block, Hotel Equatorial Shaghai No.65, Ya A Road (West), Shaghai, , Chia Phoe: Fax: