V. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. "Flux": = da i. "Force": = -Â g a ik k = X i. Â J i X i (7.
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1 Themodynamcs and Knetcs of Solds 71 V. Pncples of Ievesble Themodynamcs 5. Onsage s Teatment s = S - S 0 = s( a 1, a 2,...) a n = A g - A n (7.6) Equlbum themodynamcs detemnes the paametes of an equlbum and ndcates whethe a eacton s themodynamcally possble o not. Clausus Equaton ds dq/t (7.1) Closed System ds 0 (7.2) >: spontaneous eacton, = : evesble eacton Entopy poducton: Equlbum: s = 0 Außehalb des Glechgewchts: s < 0 Taylo Sees Development aound the equlbum (a 1 = a 2 =... = 0): because of s (0, 0,...) = 0. The 0. pat of the sees dssapeas. Because of the maxmum of s n the equlbum state all fst dffeental quotents ( s/ a ) a = 0 dsappea j+ and accodngly the lnea tems of the Taylo sees. Only the tems of the second ode (a 2,a, a j,...) eman. (In the followng hghe tems ae neglected and the coeffcents of the tems of second ode ae defned as def g : s = dds dt (7.3) s = g a, a (7.7), 7.1. Entopypoducton n the case of heat conducton. Closed system: Entopypoducton: - 1 2, g [ a ( da / dt) ] + a ( da / dt) = - g,, a ( da / dt) (7.8) "Flux": = da dt = J Abb Two patal systems ae connected by a heat conducto. It s assumed that T 1 > T 2, and the amount of heat q n a cetan peod of tme fom system 1 to system 2. The entopy changes n the systems ae -q/t 1 and +q/t 2. ds = dq s = q T T ( T 2 - T 1 1 ) (7.4) ( ) = 7.2 Fluxes and Foces. [( ) / T 1 T 2 ] (7.5) q T 1 - T 2 Entopy of a system: S = S (A 1, A 2,...) Equlbum state: S 0 = S 0 (A 0 1, A 0 2,...) "Foce": = - g a = X Entopypoducton: J X (7.9) Dmenson: Enegy / tempeatue tme. The fluxes and foces have to esult n the coect entopy poducton of the pocess Phenomenologcal equatons In the case of heat conducton, we have
2 72 Themodynamcs and Knetcs of Solds J = q und X = ( 1 T 2-1 T 1 ) The foces have to geneate fluxes; thee s accodngly elatonshp J = f(x ). Fo lnea pocesses the elatonshp has to be: J = LX (7.10) (phenomenologcal equaton of heat conducton) L: Phenomenologcal co-effcent (evesble themodynamcs of lnea pocesses) Example of the dependance of flux J n of seveal foces: Themodffuson: Themal foce esults n a flux of matte (even f the coespondng foce to the flux J n s 0). L : phenomenologcal (o Onsage-) coeffcents Entopy poducton of Dffuson Closed System: Futhe examples: - Fc s Dffuson law - Rate laws fo eacton of 1. and 2. ode - Vscosty - Ohm s law I = 1 R U (7.11) I U : Powe (Dmenson enegy/tme, whle the entopy poducton has the dmenson enege/tempeatue tme. Equaton (7.11) s not a penomenologcal equaton, because the poducts of fluxes and foces do not esult n the entopy poducton. It s the tas to ndcate fomula fo the entopy poducton of the pocesses and to dentfy n these fomulas the coespondng fluxes and foces. Ohm s law: p = q = IU (7.12) s = q /T = ( U/T) (7.13) The coespondng foce to the flux I s U/t. In geneal, the entopy depends of seveal ndependant paametes a and thee ae seveal paametes a whch contbute to the foce X. Analogously, a flux J s not only dependant on a sngle foce, but also depends on othe foces X j (j. Fo n fluxes J n a coespondng foce X n may be elated to each flux. Phenomenologcal system of equatons: J 1 = L 11 X 1 + L 12 X L 1n X n : J n = L n1 X 1 + L n2 X L nn X n TdS = -m dn (7.15) Moton of fom 1 Æ 2: ds = -dn ( m,2 / T - m,1 / T) (7.16) Entopy poducton: s = ds dt = - dn ˆ Á m Ë dt,2 / T - m,1 / T Accodngly the fluxes and foces ae ( ) (5.17) J = n (7.18) X = - m,2 T - m,1 ˆ Á = -D m ˆ Á (7.19) Ë T Ë T Dvng foces fo the dffuson: D( m / T) (nstead of D c) 7.5. Entopypoducton of chemcal eactons Affnty: A = - G ˆ Á Ë x T,p x: Numbe of Reactons Reacton Equaton (7.20) 0 = n D (7.21) : Reacton : Component
3 Themodynamcs and Knetcs of Solds 73 n : stochometc Coeffcent of the component n the - th eacton equaton Affnty of the -th eacton: A = - n m (7.22) Change n the amount of mateal = some of the changes of all eactons dn = n fom (7.22) and (7.23) esults dx (7.23) A dx = -m dn (7.24) Fom TdS = du + pdv + entopy poducton ( A / T) dx dt Summay Coespondng foces and fluxes Heat Conducton Expanson A dx esults fo the Dffuson Chem. Reacton (7.25) elect. Cuent Foces D (1/T) D (p/t) -D (m /T) A /T U/T Fluxes Q o V n 7.6. Ievesble Themodynamcs of Lnea Pocesses x I( q el ) Lneae PocessA lnea elatonshp exsts between foces and fluxes. Recpocal condtons: Thomson: Pelte-Effect q = P AB I (7.26) (Coolng by Heat Cuent whch s coupled to the electcal cuent) and Seebec-Effect E = e AB DT (7.27) Abb Two patal systems contan only one gas: they ae connected n such a way that a heat flux J u and a flux of matte J n may flow. By ths model, e.g. the themomolecula pessue dffeence may be calculated. P AB = Te AB (7.28) The equaton s a ecpocal elatonshp snce two ecpocal phenomena (Heat cuent Æ electcal cuent, electcal cuent Æ Heat cuent) ae quanttvely elated to each othe. Onsage: L mn = L nm (7.29) Matx s symmetc. Example: Closed 1-Component System wth heat cuent J u and flux of matte J n J u = Q, X u = D( 1 / T) (7.30) J n = n, X n = -D( m / T) (7.31) Phenomenologcal Equatons: J n = L 11 X n + L 12 X u (7.32) J n = L 21 X n + L 22 X u (7.33) Fo the 1-Component system holds X n : X n = -D G ˆ Á = - DG Ë T T + G T DT = S T DT - V T + H T 2 DT - S T DT = - V T + H 2 DT (7.34) T (Geneaton of an EMF n tempeatue gadents): Accodngly, the followng equatons hold fo the fluxes:
4 74 Themodynamcs and Knetcs of Solds J n = - L 11 V T + L 11 H - L 12 T 2 DT (7.35) J u = -L 21 V T + L 21 H - L 22 T 2 DT (7.36) In steady state holds (I n =0): DT = H - L 12 / L 11 (7.37) L nn X n L mn n m< n Dffeentaton afte X l : X mx n (7.44) s = 2L X ll X l + 2 L ln l = 2 L lnx n = 2J l (7.45) n 2 Steady state (J l = 0) : mnmum entopy poducton) n s = 0 (Theoem of the X l (themomolecula pessue dffeence) Wth L 12 = L 21 esults unde the assumpton DT=0, 0: J u J n = L 21 L 11 (7.38) and accodngly DT = H - J u / J n (7.39) Tansfeence enegy U * : = J u J n Tansfeence heat Q * : = U * H DT = - Q* (7.40) 7.7. Steady State Equlba: J n = 0, n = 1, 2,..., n (7.41) Steady State: X = const ( = 1,..., n) (7.42) Some fluxes ae dffeent fom 0. Fom the phenomenologcal equatons J 1 = L 11 X 1 + L 12 X L 1n X n : J n = L n1 X 1 + L n2 X L nn X n (7.43) esults unde consdeng the Onsage elatons fo the entopy poducton
5 Themodynamcs and Knetcs of Solds 75
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