Entropy generaton n a chemcal reacton E Mranda Área de Cencas Exactas COICET CCT Mendoza 5500 Mendoza, rgentna and Departamento de Físca Unversdad aconal de San Lus 5700 San Lus, rgentna bstract: Entropy generaton n a chemcal reacton s analyzed wthout usng the general formalsm of non-ulbrum thermodynamcs at a level aduate for advanced undergraduates In a frst approach to the problem, the phenomenologcal knetc uaton of an elementary frst order reacton s used to show that entropy producton s always postve second approach assumes that the reacton s near ulbrum to prove that the entropy generated s always greater than zero, wthout any reference to the knetcs of the reacton Fnally, t s shown that entropy generaton s related to fluctuatons n the number of partcles at ulbrum, e t s assocated to a mcroscopc process PCS: 0570, 860, 0540
I - Introducton Teachng some topcs of non-ulbrum thermodynamcs to undergraduates s not an easy task The usual thermodynamcs courses for scence students emphasze systems at ulbrum [] and do not pay attenton to entropy generaton n common phenomena such as heat conducton or a chemcal reacton Moreover, the textbooks that deal wth nonulbrum thermodynamcs see for example Ref [, ] ntroduce the usual formalsm n terms of generalzed fluxes and forces before studyng those phenomena nd a teacher may be nterested n explanng some non-ulbrum concepts wthout usng that formalsm The heat conducton problem has been analyzed n that way [3] In ths artcle entropy generaton n a chemcal reacton s studed wthout mentonng the general formalsm The author teaches a thermodynamcs course for physcs and chemstry students followng the well-known textbook by tkns [4] ddtonally a short ntroducton to chemcal knetcs s gven, and at ths pont the entropy generaton n a chemcal reacton s explaned n smple terms For those students famlar wth the statstcal descrpton of matter (or those especally enthusastc) entropy generaton s related to fluctuatons n the number of partcles at ulbrum, e t s related to mcroscopc propertes of the system Consuently, the am of ths artcle s twofold ) To evaluate the entropy producton n a chemcal reacton wthout mentonng the general formalsm of non-ulbrum thermodynamcs; ) To show that entropy producton s related at a mcroscopc level wth fluctuatons n the number of partcles nd t may be used n two ways: ) To close a thermodynamcs course wth an ntroducton to chemcal knetcs ) To show the relatonshp between mcroscopc and macroscopc propertes n a statstcal mechancs course that ncludes processes out of ulbrum
I I - Macroscopc analyss chemcal reacton s descrbed by ν + ν ν C C + ν D D The ν are the stochometrc coeffcents that are postve for the product (C and D) and negatve for the reactants ( and ) The startng pont of the analyss s Gbbs uaton: du TdS PdV dn () s usual U, T, S, P, V, μ y n are the nternal energy, temperature, entropy, pressure, volume, chemcal potental and mole number of The Frst Prncple states that du = dq + dw, where dq s energy transferred as heat to the system and dw the work done on a system; assumng that there s only expanson work, e dw = -pdv, uaton () can be wrtten as: ds dq dn () T T It should be remembered that the extent of reacton ξ s related to the number of moles n and of partcles by: d dn, V k d R s usual, V s the vogadro constant, R the gas constant and k the oltzman constant d Introducng the tme dfferental dt n () and callng ds dt Q d T T dt, Q dq / dt one gets: The frst term n (4) s the entropy producton per unt tme due to the heat exchange wth the surroundngs whle the second term s the entropy generaton assocated wth the chemcal reacton tself The affnty of a chemcal reacton s: (3) (4)
Α (5) It s zero n ulbrum because the chemcal potentals of reactants and products are ual postve value of the affnty means that the chemcal potentals of the reactants are greater that those of the products, and the reacton stll goes forward If S ext s the entropy generated due to the nteracton wth the surroundngs and S nt s that generated nsde the system, one may rewrte (4) as surroundng s ds dt ecause the entropy producton rate S (6a) ext S nt S ext due to the nteractons wth the Q /T, (6b) and usng the defnton (5), t follows that the producton rate of entropy system s S ext S nt nsde the d S nt (6c) T dt Equatons (6) are a central result; they show how the entropy changes n the system and clearly dstngushes the contrbuton of the chemcal reacton tself smple example clarfes ths pont n elementary frst order reacton s consdered:, and the reacton velocty w s gven by w = k n, where k s a phenomenologcal constant greater than zero lthough n chemcal knetcs the reacton velocty s defned n terms of concentratons, n ths paper t s assumed that the volume remans constant and the velocty s wrtten n terms of the mole number: w dn dt d, dt k n, (7) From (6c) and (7) one gets:
S nt k n 0 (8) T Ths result shows that the entropy producton due to the reacton s always postve as demanded by the Second Prncple (remember that >0 f the system s not yet n ulbrum) smlar calculaton could be carred out for reactons of hgher order, but the concluson s the same It has been shown that the entropy producton n a system where a chemcal reacton takes place can be wrtten as the sum of two contrbutons (6) nd for ths partcular example a frst order elementary reacton t s explctly shown that entropy s always generated by the reacton tself Ths concluson s reached by usng thermodynamcal consderatons and a phenomenologcal constant, e ths s a purely macroscopc result n alternatve approach, wthout any reference to the knetcs, s gven n the next secton III - more detaled analyss The am of ths secton s to ntroduce chemcal potentals n the analyss of the entropy generaton and to fnd an expresson for t wthout reference to the knetcs The chemcal potental of an deal soluton can be wrtten n dfferent ways [5] convenent one s: c ( T, p) RT ln (9) c In ths uaton c s the concentraton expressed as the number of moles n per unt mass The chemcal potental always refers to a standard state desgnated wth the symbol θ Rememberng the relatons between the mole n and partcle numbers, (9) can be wrtten as: ( T, p) RT ln n, ( T, p) RT ln η and η are functons that do not depend on the concentraton of the chemcal speces To express the chemcal potental n terms of the partcle number s usual n statstcal mechancs textbooks for physcs undergraduates [6, 7] Therefore, the second lne of (0)
(0) s famlar to physcs students whle those n chemstry would prefer to start the analyss from (9) The symbol s used to desgnate a physcal magntude n ulbrum; snce the affnty s zero n ulbrum, one can wrte: Α RT ln, () The next step evaluates the entropy generated when the chemcal reacton goes from a state characterzed by the values 0 and ξ 0 to the ulbrum state wth and ξ From s (3), (6) and (),we fnd: S nt d dt, T dt ln R 0 R 0 ln d d, () The ntegral can be evaluated and the result rewrtten n terms of the number of partcles: 0 0 ln S nt / k (3) Up to ths pont the results are completely general, but a new assumpton has to be made to proceed It s assumed that the system s close to ulbrum; the rght sde of (3) can be expanded as a power seres and only the most relevant contrbuton kept Ths yelds: 0 S nt / k (4a) Ths expresson may be rewrtten n terms of easly measurable quanttes: 0 S nt / R n n (4b) n Eq (4b) s preferred by chemstry students because all the quanttes on the rght hand sde are macroscopc and measurable However, physcs students are more nterested
n the relaton of entropy generaton wth the mcroscopc vew of matter For them t s convenent to ntroduce a parameter and rewrte the above expresson as: S nt n / R, 0 (4c) It s obvous that s always postve and entropy s always generated by the reacton as rured by the Second Prncple otce that s a macroscopc quantty t can be evaluated just by knowng the stochometrc coeffcents and the ulbrum concentratons but ts mcroscopc nterpretaton wll come out n the next secton IV - Mcroscopc analyss The results gven by (4) are vald for any reacton close to ulbrum However, to understand the meanng of a smple reacton of the knd s analyzed For ths partcular reacton, the total number of partcles remans constant: = 0 + 0 = + t a mcroscopc level an molecule has two optons: t remans as an molecule wth probablty p or t becomes a molecule wth probablty (-p) Ths means the partcle number follows the well-known bnomal dstrbuton From elementary probablstc theory [8, 9] t s known that the average numbers of and molecules n ulbrum are: p, ( p) For a bnomal dstrbuton [8, 9], the varance σ s: (5) p( p) (6a) So, for the system descrbed by (5), the varance σ at ulbrum can be wrtten as: (6b) Usng the expresson of gven above, t fnally results that: (6c)
From (4c) and consderng that ν = ν = for ths partcular reacton, the value of can be evaluated: n n V (7) nd comparng t wth (6c) one gets: V (8) Thus, from the second lne of (4c) t comes out that the total entropy produced by the elementary reacton consdered n ths secton s: 0 R S V nt (9) esde a numercal factor, the produced entropy s related to the fluctuatons of the partcle number at ulbrum, e the mcroscopc orgn of entropy s clearly shown n (9) V - Concluson The producton of entropy n a chemcal reacton has been studed at a level aduate to advanced undergraduate students Startng from the Gbbs relaton (), t has been shown that the rate of entropy producton n a chemcal reacton has two contrbutons: one of them assocated wth the heat nterchanged wth the surroundngs and the other orgnated by the reacton tself (6) For an elementary frst order reacton t has been proved that the entropy produced by the reacton s always postve (8) To get ths result the knetcs of the reacton has to be explctly known n alternatve approach developed n Secton III gves the total entropy generated by the reacton n terms of macroscopc measurable magntudes (4b) Fnally a mcroscopc analyss of the problem was carred out and t comes out that the entropy producton s assocated wth the fluctuatons of the partcle number (9) Once agan statstcal physcs shed lght on
the orgn of entropy lthough the calculaton was performed for the elementary reacton prevously consdered, t could be generalzed for any reacton; the detals, however, become cumbersome and nothng new s learned cknowlegment: The author thanks the atonal Scentfc and Technologcal Research Councl (COICET) of rgentna for fnancal support
References [] Callen H 985 Thermodynamcs and an Introducton to Thermostatstcs (ew York: John Wley & Sons) [] de Groot S R Mazur P 984 on-ulbrum Thermodynamcs (ew York: Dover) [3] Lure D Wagesberg J 980 m J Phys 48 868 [4] tkns P W 998 Physcal Chemstry (Oxford: Oxford Unversty Press) [5] Levne I 00 Physcal Chemstry (ew York: McGraw Hll) [6] Ref F 965 Fundamentals of Statstcal and Thermal Physcs (ew York: McGraw- Hll) [7] Huang K 987 Statstcal Mechancs (ew York: John Wley & Sons) [8] Hsu H 996 Schaum s Outlne on Probablty, Random Varables and Random Processes (ew York: McGraw-Hll) [9] Feller W 968 n Introducton to Probablty Theory and ts pplcatons (ew York: John Wley & Sons)