Thermodynamics of Irreversible Processes across a Boundary: Elementary Principles

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1 Thermodynamcs of Irreversble Processes across a Boundary: Elementary Prncples Jurgen M. Hong Department of Chemstry, Purdue Unversty 560 Oval Drve, West Lafayette, IN, , USA Tel: E-mal: jmh@purdue.edu Receved: August 17, 2011 Accepted: September 6, 2011 Publshed: November 1, 2011 do: /apr.v3n2p3 URL: Abstract The commonly used thermodynamc varables are generalzed so as to take account of rreversble processes across a thn boundary separatng a system from ts surroundngs. Ths permts one to relate heat transfer and work performance n rreversble processes to those carred out reversbly. The commonly used functons of state have been generalzed to nclude the contrbuton of rreversble changes to the entropy of the system. These contrbutons are specfed n terms of expermental observables. Keywords: Quasstatc rreversble processes, System-surroundng nteractons across a thn boundary, Generalzed thermodynamc functons of state, Specfcaton of entropy assocated wth rreversblty 1. Basc Commentary The magnfcent structure of thermodynamcs has been of mmense utlty n unfyng our understandng of physcal phenomena for over 170 years. The early developments that were based on the fundamental laws of thermodynamcs nvolved the concept of equlbrum and reversblty: systems were assumed to be ether n a statonary state, or all processes were assumed to proceed nfntely slowly so that the system never devated sgnfcantly from equlbrum condtons. Begnnng around the turn of the 20 th century, nvestgators extended the orgnal theores so as to nclude processes that nvolved frst order departures from the equlbrum state, by allowng the propertes of nterest to become local functons of poston and tme. Ths opened up a plethora of new effects and applcatons under the headng of nonequlbum thermodynamcs or extended thermodynamcs, whch are well documented n many revews, a selecton of whch s provded below (Mexner, 1954) (de Groot and Mazur, 1962) (Prgogne, 1967) (Gyarmat, 1970) (Haase, 1990) (Mueller and Rugger, 1996) (Jou, 1996) (Kondepud and Prgogne, 1998) (Hong, 2007). In ths connecton t may be of nterest to draw attenton to a recent complementary approach (Hong and Ben Amotz, 2005, 2008) (Ben Amotz and Hong, 2006), n whch the thermodynamc functons of state are enlarged to accommodate the contrbuton of rreversble processes to the entropy of the system. Ths s acheved by replacng the customary nequaltes n elementary treatments by equaltes. Such an approach fnds ts prncpal applcaton n dealng wth the transfer of matter, energy, and entropy across thn boundares, by forcng relevant thermodynamc varables to adhere to a strct predetermned tme dependence. As a prelude to ths approach, detaled n earler papers (Hoehn and Hong, 2011) (Hong and Hoehn, 2011), t was necessary to augment the conventonal approach to the frst and second laws through ncluson of rreversble processes. The am of the present paper s to revew ths partcular methodology. The theory, outlned below, certanly has antecedents cted n de Groot and Mazur (1962) and Gyarmat (1970), but those theoretcal studes focused more on what s now termed extended thermodynamcs, n whch emphass s placed on the dentfcaton of fluxes and thermodynamc forces n relaton to steady state condtons, rather than on the tme dependent phenomena studed n our earler publcatons. Contrbutons n the lterature drectly relevant to the current nvestgaton wll be referred to at approprate places. The author s not aware of any pror unfed dscusson of the materal covered n the present revew. 2. Prelmnares The unverse under study conssts of a system that s characterzed by temperature, pressure, and chemcal potentals that dffer from the correspondng ntensve varables of a reservor to whch the system s anchored va a thn nterlayer, as depcted n Fg. 1. For purposes of llustraton the dagram depcts the temperature profle Publshed by Canadan Center of Scence and Educaton 3

2 under consderaton: all processes are assumed to occur suffcently slowly that the system temperature remans unform throughout; the temperature T then changes smoothly but abruptly over the extenson of the separaton barrer untl t assumes the unform value T 0 of the system. Ths unformty assumpton restrcts us to the consderaton of quasstatc rreversble (QSI) phenomena,.e., condtons not too far removed from reversblty, so as to avod excesses such as turbulence, vortex formaton, and the lke. However, there s no restrcton on the magntude of the temperature dfference. Smlar consderatons apply to other ntensve varables, such as the pressure, or the varous chemcal potentals. The reservor s assumed to be of such mmense sze and to reman so well mxed that ts ntensve varables reman unaltered n any of the processes consdered below. Moreover, n conformty wth standard practce, we assume that all events wthn the reservor take place reversbly. The above unformty restrctons on ntensve varables may certanly be relaxed, but the resultng theory then becomes much more complex and obscures the fundamental prncples developed below. 3. Entropy Changes We begn wth a study of entropy changes durng an nfntesmal step of a partcular process, whch s frst carred out reversbly. Desgnate the entropy change of the unverse as ds u, and let t be wrtten as a sum of the entropy change ds of the system and the entropy change d a S 0 of the surroundngs. Then, accordng to the Second Law no change s ncurred n the entropy of the unverse; thus d a S u ds + d a S 0 = 0. (1a) Next, let the same step n the process wthn the system be carred out rreversbly. Snce entropy s a functon of state of the system, ds s the same as before; however, the entropy change n the surroundngs, desgnated by d b S 0, s now dfferent. The entropy change of the unverse s then governed by the nequalty d b S u ds + d b S 0 > 0. (1b) The use of nequaltes s at best awkward and carres no nformaton on the actual value of the varable of nterest. Hence t s apposte to ntroduce an entropy defct functon đθ > 0 for the express purpose of convertng Eq. (1b) nto an equalty, whereby t becomes possble to extend the standard methodology to rreversble phenomena: d b S u ds + d b S 0 - đθ = 0. (1c) At ths stage the quantty đθ s smply a bookkeepng devce. However, t becomes a meanngful quantty when we subtract (1a) from (1c) to obtan d b S 0 - d a S 0 = đθ, (1d) whch ndcates that đθ represents the entropy transferred between the system and the surroundngs when the nfntesmal step of nterest n the system s carred out rreversbly as opposed to reversbly. Hence, a study of đθ becomes mportant n ts own rght. We frst restrct attenton to heat exchanges takng place as a result of the temperature dfference between the system and the reservor. For ease of vsualzaton we assume that T 0 > T, so that n an nfntesmal step n the solated unt of system + surroundngs, any heat (đq) ganed by the system operatng at temperature T s lost by the surroundngs (đq 0 ) operatng at temperature T 0, whence đq = - đq 0. Durng a reversble process (r), where the surroundngs are at temperature T 0 nfntesmally above T, the correspondng entropy change of the system s gven by ds d r S = đ r Q/T, whence, by (1a), the entropy change n the surroundngs s gven by - d a S 0 = ds = đ r Q/T. (2a) For the same step executed rreversbly (), wth the temperature of the system at T and that of the surroundngs at T 0, the resultng heat exchange - đ b Q 0 = đ Q produces an entropy change n the surroundngs specfed by - d b S 0 = - đ b Q 0 /T 0 = đ Q/T 0, (2b) where we nvoked the reversblty of all processes n the reservor. On nsertng Eqs. (2a) and (2b) nto Eq. (1d) we obtan the fundamental relaton đ Q/T 0 = đ r Q/T - đθ. (3a) or ts equvalent đ Q = (T 0 /T) đ r Q - T 0 đθ = T 0 (ds đθ) (3b) 4 ISSN E-ISSN

3 đ r Q + (T 0 /T - 1) đ r Q - T 0 đ (3c) whch now relates the heat transferred rreversbly n the nfntesmal step to that transferred reversbly. Eq. (3b) duplcates the expresson derved by a dfferent method (Tolman and Fne 1948). Eq. (3a) clearly shows that đ Q < (T 0 /T) đ r Q = T 0 ds, (3d) and n the event that the reservor temperature approaches that of the system we obtan a more restrctve nequalty đ Q < đ r Q = TdS, (3e) whch s the famous Clausus nequalty. It apples when a reversble heat transfer trggers rreversble processes totally wth the system; Eq. (3d) s ts generalzaton, arsng from the double nequalty đ Q < đ r Q < (T 0 /T) đ r Q. In vew of the requrement (3e) whch automatcally allows for (3d) - the sum of the last two terms n Eq. (3c) must be negatve: (T 0 /T - 1) đ r Q - T 0 đθ < 0. (3f) Careful examnaton shows that the above relatons stll apply when T > T 0, whereby heat s transferred from the system to the reservor: n ths event đ Q s more negatve than đ r Q, whch s more negatve than (T 0 /T)đ r Q. Furthermore, the nequalty (3f) remans unaltered: when đ r Q < 0 the factor (T 0 /T - 1) s also negatve. Eq. (3b) may be turned around to read ds = đ Q/T 0 + đθ > đ Q/T 0, (3g) Thus, as s well known, n rreversble phenomena the total entropy change n the system s only partally compensated for by the transfer of heat across ts boundares, the remander beng attrbuted to the rreversblty effects totally wthn the system. Observe that t s the well-defned temperature of the reservor that enters ths relaton. Ths should be contrasted wth the usual expresson ds = đ r Q /T, where the entropy change s tracked by the reversble heat transfer at the common temperature T. However, Eq. (3g) becomes useful only after methods are developed below that specfy đθ n terms of expermentally measurable quanttes, as shown below. It s noteworthy that for processes that occur n solaton Eq. (3g) reduces to δs = đθ > 0, (3h) where the δ symbol serves as a remnder of the specal condtons that preval. As s well establshed, ths equaton shows that all processes proceedng wthout outsde nterventon (after release of a constrant) ncrease the entropy of the system. Such changes take place rreversbly snce they are unaffected by outsde manpulatons. Successve ncrements produce a monotonc entropy ncrease untl a state of maxmum entropy s reached when the spontaneous process has run ts course, at whch pont δs = 0. Ths s the content of the famous dctum by Clausus that the entropy of the world (nowadays, we would say, of our unverse) tends toward a maxmum. 4. Performance of Work At ths pont we ntroduce the Frst Law of Thermodynamcs accordng to whch the energy E s a functon of state. Then de = đ r Q + đ r W = đ Q + đ W, (4a) where W s the work performed durng the process. On elmnatng đ Q between (3c) and (4a) we may solve for đ W = đ r W - (T 0 /T - 1) đ r Q + T 0 đθ = đ r W - (T 0 T)dS + T 0 đθ. (4b) The last two terms n Eq. (4b) dffer n sgn from those cted n Eq. (3f). It follows that đ W > đ r W. (4c) When the process s carred out under condtons where T and T 0 dffer only nfntesmally, đ W = đ r W + T 0 đθ, (4d) so that T 0 đθ may then also be nterpreted as the nfntesmal dfference n work performance under reversble, as opposed to rreversble condtons, where addtonal phenomena take place as a result of the rreversble heat transfer. Publshed by Canadan Center of Scence and Educaton 5

4 The above results are well known and accord wth ntuton. In any (nfntesmal) step the work performance s ndependent of the mode of operaton. That s, n a gven change de n energy of the system t does not matter whether ths process s executed n terms of mechancal, electrc, magnetc, or as any other type of work. Takng gravtatonal work as an example, t s ntutvely clear that n rreversbly lftng a weght aganst the force of gravty over a gven dstance, more work s expended than n dong ths reversbly. Hence, more generally n accord wth Eq. (4d), for any gven process we may set đ W > đ r W no matter what type of work s performed. Ths nequalty holds even when both quanttes are negatve: the system performs less useful work n an rreversble than n a reversble step (.e., đ W s less negatve than đ r W). In vew of (4c) or (3f), Eq. (4b) leads to an mportant nequalty: đθ > (1 - T/T 0 )ds, (4e) Here the rght hand sde s always postve, as explaned earler. Thus, đθ has as ts lower nontrval bound the quantty on the rght, whch exceeds zero when the temperature of the system dffers from that of the surroundngs. Eq. (4e) provdes useful nformaton on đθ n terms of expermentally determnable varables T, T 0, and ds. 5. Specfcaton of the Defct Functon A very sgnfcant feature of the present analyss s manfested by solvng Eq. (4b) for đθ = (đ W - đ r W)/T 0 + (1 T/T 0 )ds. (5a) Ths relaton allows one to determne the contrbuton of rreversble phenomena to the nfntesmal entropy change of any process (subject to the constrants ntroduced earler) n terms of quanttes that may be measured or calculated. Ths requres montorng the change n entropy of the system whle ncrementally carryng out a gven process, takng due account of the change n temperature T; heat capacty measurements may be used for ths purpose. Also, one needs to determne the total work requred for executng the same change reversbly and rreversbly, equvalent to notng by how much a gven weght changes heght n the earth s gravtatonal feld n the two processes. We note that n the absence of work performance Eq. (5a) reduces to đθ = (1/T 1/T 0 ) đ r Q, (5b) whch was derved elsewhere by dfferent means (Kestn, 1966). The frst term n Eq. (5a) was obtaned by a dfferent approach (Bejan, 1997). Eq. (5a) further provdes a thermodynamc background for the theoretcal work of Jarzynsk (1997) who related the average of repeated executons of rreversble mcroscopc processes to thermodynamc equlbrum processes. Ths work was confned to processes executed at constant temperature; the above relaton shows how ths analyss may be extended. We next consder alternatve methods for specfyng đθ. 6. Functons of State So far we have dscussed the propertes of path-dependent quanttes. We now turn to the development of functons of state under non-equlbrum condtons. Ths generalzes the conventonal development of state functons. 6.1 The Energy We begn wth a consderaton of the energy of a system. For ths purpose we frst apply the Frst Law of Thermodynamcs to the surroundngs, whose propertes are all desgnated by the subscrpt zero. For an open system the energy dfferental under the assumed reversble operatng condtons s specfed by de 0 = T 0 ds 0 - P 0 dv 0 + Σ μ 0 dn 0, (6a) where P 0 s the prevalng pressure, V 0 the volume, μ 0 the chemcal potental of speces, and n 0 s the correspondng mole number, all n the surroundngs. Let the closed unt of system + surroundngs be mantaned at a constant volume. In that event, when the solated unt undergoes rreversble processes, the system s subject to the constrants de = - de 0, ds 0 d b S 0 = - ds + đθ (See Eq. (1c)), dv 0 = - dv, and dn 0 = - dn, so that the energy dfferental of the system s gven as de = T 0 ds - P 0 dv + Σ μ 0 dn - T 0 đθ. (6b) Ths relaton, nvolvng dfferent arguments, was derved by Kestn (1966). The ntensve varables appearng here are those of the surroundngs and are therefore well defned, even when processes wthn the system are 6 ISSN E-ISSN

5 rreversble. The extensve quanttes ds, dv, dn are the control varables for the energy of the system; đθ s determned as shown n Eq. (5) or by the expressons developed below. Hence, de s well defned n the QSI procedure even under non-equlbrum operatng condtons. If work other than mechancal s nvolved, approprate terms must be added to Eq. (6b) as a product of ntensve varables approprate to the relevant work reservor, and correspondng extensve varables for the system proper. It s nstructve to rewrte Eq. (6b) n the form de = (T 0 T)dS - (P 0 P)dV + Σ (μ 0 μ )dn + TdS - PdV + Σ μ dn - T 0 đθ, (6c) where the varables lackng the subscrpt zero refer to the propertes of the system. Eq. (6c) s the generalzed verson of the conventonal energy dfferental. Two specal stuatons are of nterest: Case (a). The nfntesmal step s executed reversbly, wth T = T 0, P = P 0, μ 0 = μ and đθ = 0; then (6c) reduces to the standard form de = TdS - PdV. + Σ μ dn. (6d) Case (b). Here one removes the system slghtly but rreversbly from ts equlbrum state wth ds = dv = dn = 0 (all ) by external nterventon, then removng the nterference and allowng the system to respond under these constrants. Ths process s characterzed va the relaton (6b): δe = - T 0 đθ, (6e) where the δ symbol serves as a remnder of the specal condtons under whch ths probng process s carred out. Thus, n the absence of normal drvng processes any ongong process occurs purely nternally, whle not beng subject to expermental control. Thus, wth đθ > 0 the energy of the system wll spontaneously dmnsh n ths step. On generalzng to fnte processes, the energy of the system wll contnue to decrease untl a state s reached where the spontaneous process ceases: then δe = 0, and the energy of the system has reached a mnmum consstent wth the appled constrants. Ths s a statement that the energy of a system undergong spontaneous processes wll decrease untl an equlbrum confguraton has been reached. Lastly snce E s a functon of state we may subtract (6d) from (6c) to obtan T 0 đθ = (T 0 T)dS - (P 0 P)dV + Σ (μ 0 μ )dn. (6f) Snce the quanttes on the rght are expermentally accessble the above equaton permts the determnaton of đθ n terms of S, V, and n as the control varables. However, the above quanttes are not generally the ones used n experments; we next develop alternatve expressons for đθ n terms of more readly accessble control parameters. 6.2 The Helmholtz Energy In a smlar manner we can construct the Helmholtz energy by ntroducng the defnton A = E TS, whch sensbly nvolves the temperature T of the system. Ths transforms the ndependent varable from S to T. Usng Eq. (6c) we then fnd that da = (T 0 T)dS - (P 0 P)dV + Σ (μ 0 μ )dn - SdT - PdV + Σ μ dn - T 0 đθ. (7a) However, snce A = A(T,V,n ), the control varables for the Helmholtz energy are supposed to be temperature, volume, and composton, whereas Eq. (7a) nvolves a mx of varables. As a remedy we express the entropy n terms of the applcable control varables by settng S = S(T,V,n ). Ths yelds the entropy n the approprate dfferental form Substtuton n (7a) then produces the expresson: ds = ( S/ T) V, n dt + ( S/ V) T, n dv + Σ ( S/ n ) T,V, n dn j (7b) da = (T 0 T)[( S/ T) V, n dt + ( S/ V) T, n dv + Σ ( S/ n ) T,V, n dn ] - (P 0 P)dV + Σ (μ 0 μ )dn - SdT - PdV + Σ μ dn - T 0 đθ. (7c) The partal dfferentals n the above relaton may be rewrtten n the followng manner: ( S/ T) V, n = C V /T, and ( S/ V) T, n = ( P/ T) V, n, where C V s the heat capacty at constant volume and composton, and where the j Publshed by Canadan Center of Scence and Educaton 7

6 approprate Maxwell relaton has been ntroduced. Also, we use the mathematcal dentty P/ T V,n V / T P,n / V / P T, n and then replace the numerator and denomnator by - αv and by - βv, where α and β are the sobarc coeffcent of expanson and the sothermal compressblty respectvely. We n S S / as the dfferental entropy at constant temperature, volume and composton of also set T, V, n j speces. In ths notaton, da = (T 0 T)[(C V /T)dT + (α/β) dv + Σ S dn ] -(P 0 P)dV + Σ (μ 0 μ )dn - SdT - PdV + Σ μ dn - T 0 đθ (7d) The above equaton specfes the dfferental of the Helmholtz functon under non-equlbrum condtons n terms of T, V, and the n. The assocated coeffcents nvolve measurable quanttes. Under constrant (a) n whch a reversble process s carred out wth T = T 0, P = P 0, μ = μ 0, and đθ = 0, the Helmholtz functon reduces to the standard form da = - SdT - PdV + Σ μ dn. (7e) Under constrant (b) we deform the system by external nterventon under the condtons dt = dv = dn = 0 to an adjacent state, and then release the system under these condtons. By analogy to the precedng dscusson the resultng process wll agan be spontaneous and subject to the relaton δa = - T 0 đθ, (7f) whch shows that n a spontaneous process the Helmholtz energy dmnshes and reaches a mnmum at equlbrum, consstent wth the ndcated constrants. Snce A s a functon of state we may subtract (7e) from (7d) to fnd T 0 đθ = (T 0 T)[(C V /T)dT + (α/β) dv + Σ S dn ] - (P 0 P)dV + Σ (μ 0 μ )dn, (7g) whch provdes a second scheme for determnng the entropy defct functon, here n terms of T, V, and n as relevant expermental varables; all the coeffcents nvolve measurable quanttes. 6.3 The Gbbs Energy Usng by now famlar methodology, we defne the Gbbs energy by the relaton G = E TS + PV. When converted to dfferental form, and on nserton of Eq. (7a), we obtan dg = (T 0 T)dS - (P 0 P)dV + Σ (μ 0 μ )dn - SdT + VdP + Σ μ dn - T 0 đθ (8a) as the expresson that holds under non-equlbrum condtons. However, the control varables n present crcumstances should be T, P, and composton. Therefore, followng the above recpe, we must express the entropy n the form S = S(T,P,n ) and the volume as V = V(T,P,n ), wth correspondng dfferentals for ds and dv. On nsertng these n (8a) we obtan dg = (T 0 T)[( S/ T) P, n dt + ( S/ P) T, n dp + Σ ( S/ n ) T,P, n dn j ] - (P 0 P)[( V/ T) P, n dt + ( V/ P) T, n dp + Σ ( V/ n ) T,P, n j dn ] + Σ (μ 0 μ )dn - SdT + VdP + Σ μ dn - T 0 đθ. (8b) We next ntroduce the relatons ( S/ T) P, n = C P/T and the Maxwell relaton ( S/ P) T, n = - ( V/ T) P, n. As before, the partal dervatves - ( V/ P) T, n and ( V/ T) P, n are specfed by βv and by αv respectvely. ( S/ n ) T,P, n and ( V/ n j ) T,P, n represent partal molal entropes S j and volumes V. We then rewrte Eq. (8b) n the less unweldy form dg = (T 0 T)[(C P /T)dT - αvdp + Σ S dn ] - (P 0 P)[ αvdt 8 ISSN E-ISSN

7 - βvdp + Σ V dn ] + Σ (μ 0 μ )dn - SdT + VdP + Σ μ dn - T 0 đθ. (8c) Eq. (8c) s a generalzaton of the conventonal Gbbs functon whch holds under non-equlbrum condtons. We once more consder two cases. Under alternatve (a) we obtan the standard form dg = - SdT + VdP + Σ μ dn. (8d) Under alternatve (b) we contemplate a dsplacement of the system n the famlar manner from ts equlbrum value to an adjacent state under the constrants dt = dp = dn = 0, and then allow the system to undergo a spontaneous process back to the equlbrum state. Ths leads to the relaton δg = - T 0 đθ, (8e) whch shows that the Gbbs energy at equlbrum s a mnmum under the ndcated constrants. Note further that when (8d) s subtracted from (8c) we obtan T 0 đθ = (T 0 T)[(C P /T)dT - αvdp + Σ S dn ] - (P 0 P)[αVdT - βvdp + Σ V dn ] + Σ (μ 0 μ )dn, (8f) whch shows how one may determne the defct functon when T, P, and composton are the ndependent varables. 6.4 The Enthalpy Lastly, we turn to the enthalpy H = E + PV. By the customary technque we develop the dfferental form dh = de + PdV + VdP and then nsert Eq. (6c). We obtan dh = (T 0 T)dS - (P 0 P)dV + Σ (μ 0 μ )dn + TdS + VdP + Σ μ dn T 0 đθ. (9a) Snce H = H(S,P,n ), t s S, P, and composton that are regarded as control varables. We therefore consder the volume frst n the form V = V(P,T,n ). We next ntroduce the entropy as a functon of the same varables: S = S(P,T.n ), whch functon we nvert to read T = T(S,P,n ). Lastly, we nsert ths expresson nto the equaton of state: thus, V = V(P,T(S,P,n ),n ) V(S,P,n ). On takng the dfferental of ths latter relaton we fnd dv = ( V/ S) P, n ds + ( V/ P) S, n dp + ( V/ n ) S,P, n dn j. (9b) Then, substtutng (9b) nto (9a) we obtan dh = (T 0 T)dS - (P 0 P)[( V/ S) P, n ds + ( V/ P) S, n dp + ( V/ n ) S,P, n dn ] + Σ (μ 0 μ )dn + TdS + VdP + Σ μ dn - T 0 đθ. (9c) Ths specfes the enthalpy change n an nfntesmal step under nonequlbrum condtons. Under constrant (a) Eq. (9b) reduces to the standard form dh = TdS + VdP + Σ μ dn, (9d) whle under constrant (b), wth dp = ds = dn = 0 we obtan the result δh = - T 0 đθ, (9e) where agan we noted that the enthalpy change n such a spontaneous process s a mnmum under equlbrum condtons, subject to the relevant constrants. When (9d) s subtracted from (9c) we fnd that j Publshed by Canadan Center of Scence and Educaton 9

8 T 0 đθ = (T 0 T)dS - (P 0 P)[( V/ S) P, n ds + ( V/ P) S, n dp + Σ ( V/ n ) S,P, n dn ], + Σ (μ 0 μ )dn, (9f) whch yelds yet another formulaton for the defct functon that holds when S, P, and n are the varables of nterest. The mnma acheved by the above functons of state should be contrasted wth the maxmum that characterzes the entropy of an solated system undergong spontaneous processes. 7. Concludng Remarks Complementary to the standard methodology of rreversble thermodynamcs the above approach focuses on extendng the conventonal functons of state so as to nclude contrbutons arsng from rreversble processes. In partcular, ther contrbuton to the entropy of a system has been specfed n terms of applcable control varables. Ths serves as a bass for dealng wth exchange of matter, energy, and entropy across a narrow nterface between a system and ts surroundngs under non-equlbrum condtons. As shown elsewhere (Hoehn and Hong, 2011; Hong and Hoehn, 2011), the actual determnaton of the entropy s carred out by specfyng the tme dependence of all applcable control varables and then ntegratng the relevant relaton nvolvng đθ, namely (6f), (7g), (8f), or (9f). The prncpal restrcton, n common wth standard procedures, s that the changes must occur suffcently slowly that all ntensve quanttes n the system change unformly, but there s no lmt set on the dfference between these quanttes and those of the reservor to whch the system s anchored. Explct relatons have been provded, loc. ct., whch allow one to determne the contrbuton of rreversble processes to the entropy change of a system under prescrbed condtons, based on the use of Eqs. (8f) and (7g). It s hoped that the present treatment provdes further nsghts on how to deal wth rreversble phenomena wthn the framework of equlbrum thermodynamcs. Acknowledgments The author wshes to acknowledge wth grattude the many frutful conversatons wth hs collaborator, Professor Dor Ben Amotz, on the topcs dealt wth n ths artcle. References Bejan A. (1997). Advanced Engneerng Thermodynamcs, 2nd Ed. Wley, New York, p 135. Ben Amotz D. & Hong J.M. (2006). The Rectfed Second Law of Thermodynamcs. J. Phys. Chem. B, 110, pp Ben Amotz D. & Hong J.M. (2006). Average Entropy Dsspaton n Irreversble Mesoscopc Processes. Phys. Rev. Lett., 96, pp / de Groot S.R. & Mazur P. (1962). Non-Equlbrum Thermodynamcs. North Holland, Amsterdam. Gyarmat I. (1970). Non-equlbrum Thermodynamcs, Feld Theory and Varatonal Prncples. Sprnger Verlag, New York. Haase R. (1990). Thermodynamcs of Irreversble Processes. Dover, New York. Hoehn R. and Hong J.M. (2011). Analyss of Irreversble Processes across Narrow Junctons. Acta Phys. Polon. A, 119, pp Hong J.M. (2008). Thermodynamcs, Prncples Characterzng Chemcal and Physcal Processes, 3rd Ed., Academc Press, Amsterdam, Chap. 6. Hong J.M. & Ben Amotz D. (2005). The Analyss of Spontaneous Processes Usng Equlbrum Thermodynamcs. J. Chem. Ed., 83, pp Hong J.M. & Ben Amotz D. (2008). Rectfcaton of Thermodynamc Inequaltes as a Means of Characterzng Irreversble Phenomena. Chem. Educator, 13, pp Hong J.M. & Hoehn R. (2011). Entropy of Irreversble Processes across a Boundary. Open J. Chem. Eng., 5, pp Jarzynsk C. (1997). Non-Equlbrum Equalty for Free Energy Dfferences. Phys. Rev. Lett., 78 pp Jou D. (1996). Extended Irreversble Thermodynamcs. Sprnger Verlag, New York. j 10 ISSN E-ISSN

9 Kestn, J. (1966). A Course n Thermodynamcs. Blasdell, Waltham, MA, Chap. 13. Kondepud D & Prgogne I. (1998). Modern Thermodynamcs, from Heat Engnes to Dsspatve Structures, Wley, Chchester. Chaps Mexner J. (1954). Thermodynamk der Irreversblen Prozesse. Mezner, Aachen. Mueller I. & Rugger T. (1993). Extended Thermodynamcs, Sprnger Verlag, New York. Prgogne I. (1967). Introducton to Thermodynamcs of Irreversble Processes. Wley, New York. Tolman R.C. & Fne, P.C. (1948). On the Irreversble Producton of Entropy. Rev. Mod. Phys., 20, pp Fgure 1. Temperature Profle for a System at temperature T (left) Attached to a Reservor at Temperature T 0 (rght); the Temperature Dfference s confned to a Narrow Regon of Area A and Length l. Publshed by Canadan Center of Scence and Educaton 11