2. Introduction to Thermodynamics

Transcription

1 . Introducton to hermodynamcs.a..b..c..d..e..f..g..h. Introductory Remarks State Varables and Exact Dfferentals Some Mechancal Equatons of State he Laws of hermodynamcs Fundamental Equaton of hermodynamcs hermodynamc Potentals Response Functons Stablty of the Equlbrum State

2 .A. Introductory Remarks

3 .B. State Varables and Exact Dfferentals hermal equlbrum state: no hysteress -> functon of state varables only. State varables: path ndependent (exact dfferentals) Conugate pars of state varables: Extensve (generalzed dsplacement) Intensve (generalzed force) Examples: V, P M, H L, J A, P, E S, N, ' Varous chemcal potentals: ' per partcle μ per mole (molar) per unt mass (specfc) Response functons: C heat capacty κ compressblty χ magnetc susceptblty hermodynamc potentals: U nternal energy H enthalpy A Helmholtz free energy G Gbbs free energy Ω grand potental.b.1. Exact Dfferentals.B.. Maxwell Relatons.B.3. Exercse.1

4 .B.1. Exact Dfferentals Let F F x, x 1, we have If F and F F df dx dx x F x 1 1 x x x1 are contnuous and F F x x x x 1 x x 1 x x 1 1 then df s an exact dfferental. (.1) (.) Let c x, x 1 1 F x1 x c x, x 1 F x x1 c1, c are called conugate varables wth respect to F. Alternatve defntons: 1. he ntegral I c dx c dx B A 1 1 s path ndependent,.e., B A I df F B F A. c dx c dx 1 1 hus, df determnes F up to a constant. Generalzng to n varables, we have F F x,, 1 xn df n F dx 1 x (.3) x ;

5 c x,, x 1 n F x x ; so that df s exact f and only f c c x x xk ; k x ; k k (.4) By defnton, dfferentals of all state varables are exact.

6 .B.. Maxwell Relatons x, y, z F x, y, z Wrtng z z x, y, we have z z dz dx dy x y y x z z dx dy z z x y y x 1 1 x dx dy z z y y dy dx x z z z z z z y x x y 1 x z z y x y z y x x z y 1 y x y z x z x x y, z x x dx dy dz y z y z

7 dx x dy x dz dw y dw z y dw z dx x dy dw z y dw z z x x y, w x x dx dy dw y w y w w w y, z w w dw dy dz y z y z x x w w dx dy dy dz y w w y y z z y x x w x w dy y w y w z w y z y y x x y, z dz x x x w y y w y z w y z

8 .B.3. Exercse.1 Consder d x y dx xdy (a) a. d s exact From (a), we have x y x y y x x Hence 1 y x x y y x x y and d s exact. b. Integrate along paths 1 and Along path 1: xb, ya xb, yb I1 d d x, y x, y Along path :.e., A A B A xb yb dx x ya dyxb xa ya xb xa ya xb xa xb yb ya 1 x x x y x y 3 y y y y x x x x 3 3 B A B B A A A B A A B A y y B A y ya x xa xb xa y y y x y x x x x x x B A A B B A B A B A

9 dy y x B B y x A A dx y y y x y x x x x x d x B A x A B B A dx B A B A xb yb ya yaxb ybx A I x x dx x x B xa xb x A A yb ya x x x x y x y x 3 x x B A B A A B B A B A 1 x x x y x y B A B B A A c. Integrate usng ndefnte ntegrals From x y x y y x x we have 1 dx x y dx x x 3 yx c y y x 3 1 dy xdy xy c x y o reconcle these expressons, we must have so that 1 c x x c 3 3 c1 y c x yx c and the ntegral from A to B gves I B A x x y x x y 3 3 B B B A A A

10 .C. Some Mechancal Equatons of State Ideal Gas PV n R (.1) R Vral Expanson J / mol K nr n n P 1 B B3 (.11) V V V B B Also, B for low for hgh for quantum deal gases. Vander Waals Equaton an P V nb nr V (.1) Hence an nr nb P 1 V V V 1 an nr nb V V m V m m nr n a n 1 b b V V R mv m so that B VW a b R B VW m m 1 b for m 3 Solds 1 V 1 v P V v P P co-effcent of thermal expanson 1 V 1 v P sothermal compressblty V P v P

11 where V v n molar volume For solds, 1 1 / Pa 4 P 1 / K herefore, t's a good approxmaton to wrte v v v, P v P P v P P 1 P where v v P Elastc Rod Hooke's law: where,. J A L L (.15) 1 A A A For most substances, A1. Surface enson t 1 t ' n (.16) where t' s the crtcal temperature n C and n 1. Electrc Polarzaton D E P (.17) b P a E (.18)

12 Cure's Law B H M (.19) nd M H (.)

13 .D. he Laws of hermodynamcs hermodynamc equlbrum: mechancal varables tme ndependent. no macroscopc flow. Isolated system: No heat and no matter exchange wth external system. Closed system: Heat but no matter exchange wth external system. Open system: Heat and matter exchange wth external system. Reversble (quas-statc) changes: System remans nfntesmally close to equlbrum at all tmes. Orgnal state can be restored wthout affect state of external system..d.1..d...d.3..d.4. th Law 1 st Law nd Law 3 rd Law

14 .D.1. th Law bodes n separate thermal equlbrum wth a 3 rd body are themselves n mutual thermal equlbrum. hs leads to the concept of temperature and the constructon of thermometers.

15 .D.. 1 st Law Energy s conserved,.e. du dq dw (.1) where so that dw YdX PdV JdL da E dp H dm de ' dn (.) du dq Y dx ' dn (.3)

16 .D.3. nd Law here are many ways to state the nd law. A few are lsted below: 1. Heat flows spontaneously from hgh to low temperatures.. Reversng a spontaneous process generates heat. 3. In a cyclc process, heat cannot be transformed entrely nto work. 4. For an solated system, S for all processes..d.3a. Carnot Engne (Cycle).D.3b. Effcency.D.3c. homson And Kelvn Scales.D.3d. Exercse..D.3e. Entropy

17 .D.3a. Carnot Engne (Cycle) he Carnot engne s a heat engne consstng of only reversble processes. We shall use Q to denote the heat absorbed sothermally by the engne at temperature τ. Wth reference to fg.., we see that 1 : Heat Q h s absorbed sothermally at temperature h. 3: emperature dropped adabatcally to c. 3 4 : Heat Q c s dumped sothermally at temperature c. 4 1: emperature rased adabatcally to h. he total work W tot done by the engne s ust the shaded area n Fg... he effcency of the engne s defned as otal work done Wtot (.4) Heat absorbed Q h For each complete cycle, the system returns to ts ntal state. Hence, Utot Qtot Wtot (.5) or so that W Q Q Q (.6) tot tot h c Q c 1 1 Q h Q Q c h (.7)

18 .D.3b. Effcency No heat engne can be more effcent than a Carnot engne. Proof s as follows. Consder engnes, A and B that operate between the same heat reservors of temperatures h and c. Let A be an rreversble heat engne and B be a Carnot engne workng as a heat pump (refrgerator) as shown n Fg..3. Note that snce a Carnot cycle s reversble, ts effcency s the same whether t s used as a heat engne or a heat pump. Assume now A s more effcent than B,.e., W Q A tot A h W B tot A B B Q Note that, snce B s a refrgerator, both c B W tot (.9) B and Q c are negatve. Next, adust the parameters so that the output of A s ust enough to drve B,.e., A B W W W (.8) tot tot tot Eq(.9) becomes A Q B Q c h (.3) whch means the combned setup s causng heat to flow from the low temperature reservor to the hgh temperature one wthout any net work done somewhere. hs volates the nd law so that we must have A B, where the equal sgn holds only f A s also reversble. Incdentally, ths also mples all Carnot engnes must have the same effcency. In other words, η of a Carnot engne depends only the temperatures of the heat reservors,.e., Q Q c h f, c h (.33)

19 .D.3c. homson And Kelvn Scales Consder the Carnot cycles shown n Fg..4. Usng (.33), we have ' ', h h Q f Q (.34), ' ' c c Q f Q (.35), c c h h Q f Q (.36) so that,, ' ', c h c h f f f (.37) whch can be satsfed by, c c h h g f g where g s an arbtrary functon. homson scale: g e exp exp c c h h Q Q (.38) so that for 1 c h, we have 1 exp 1 exp c h h h Q e Q Kelvn scale: g c c h h Q Q (.39) hs choce agrees wth the temperature used n the deal gas equaton and s the

20 accepted temperature scale. o use the same spacng as the Celsus scale, we set t C (.4) where t s the Celsus temperture. he trple pont of water s at K. C

21 .D.3d. Exercse. Ideal gas Carnot Engne PV nr Wth reference to fgure, U 3 nr 1 and 3 4: = const du dq dw V nrh Q1 Q h PdV dv V 1 V1 V nr (1) h ln V1 V4 nrc Q34 Q c dv V V3 V nr () 4 c ln V3 3 and 4 1: Q du dw.e., hus 3 nr du nrd PdV dv V 3d dv V 3/ V const V V 3/ 3/ 3 3 V V (3) 3/ 3/ so that 3/ 3/ V V V V V V V V 3 4 h c c h 1 1 Q Q ln V / V Q Q ln V / V c 43 c 3 4 c h 1 h 1 h 1 Q c 1 Q h c h (4)

22

23 .D.3e. Entropy Snce all Carnot engnes has the same effcency 1 Q c 1 Q h c h (.41) we have, for every Carnot cycle, or Q Q h Q h Q h h c c c (.4) c hus, dq (.43) Now, any reversble process can be decomposed nto a seres of adacent Carnot cycles wth common adabatc legs (see Fg..5). hs means (.43) s applcable to any reversble processes so that dq ds (.44) s an exact dfferental and S s a state varable whch wll be called the entropy. Now, replace the reversble processes 1 3 n a Carnot cycle wth another rreversble one 13. [Note that n general, the process 13 cannot be traced n the X-Y plane snce the system may be so far from equlbrum that no state varables can be defned]. Let the heat absorbed durng 13 be Q IR. For the cycle , we have U Q Q W ' IR c where W ' s the total work done. Now, ths cycle s an rreversble heat engne operatng between the temperatures 1 h and 3 c. Its effcency s Q W ' Q c ' 1 1 Q Q Q IR IR IR c

24 Snce no heat engne can be more effcent than a Carnot engne, we have, c Q c Q Q 1 1 Q IR h.e., h Q Q (.45a) IR where Q h s the heat absorbed durng the 1 sothermal leg of the Carnot cycle. Now so that Q S S S S h h h 1 h 3 1 S S Q 3 1 IR whch can be thrown nto a more general notaton applcable to any process connectng equlbrum states: S Q where Q s the heat ntake, S s the entropy ncrease, and s the temperature of the sothermal leg of the sothermal-adabatc process par that connects the states. he equal sgn apples only f the process s reversble. For an solated system, Q and we recover one form of the nd law S

25 .D.4. 3 rd Law Nernst: S as for reversble processes. Alternatve form: It's mpossble to reach va reversble routes n a fnte number of steps. Consder S S Y S Y Y 1 (.5) hermal stablty requres (see Secton.H) so that S Y Y S S Y Consder a process at Y Y dy X X dx Wrtng S S Y, S ds Y dy Wrtng S S X,, we have, we have such that at (.51) S ds X dx at (.5) If the process s reversble, 3 rd law requres ds so that S S Y X (.53-4)

26 .E. Fundamental Equaton of hermodynamcs he entropy S S U X N,, s a functon of entrely extensve varables and s addtve. Hence,,,, S S U X N S U X N (a) so that S S S ds d U d X d N U X N Now S S 1 1 U U S S Y Y X X S S ' ' N N (.58) (.59) (.6) where means evaluatng all extensve varables x at x. hus, y y for any ntensve varable y. 1 Y ' ds d U d X d N 1 Y ' 1 Y ' d U X N du dx dn On the other hand, we also have from (a) that ds d S Sd ds Comparng these expressons gves ds du YdX ' dn (b) whch s ust the combned 1 st and nd law for reversble processes, and S U YX ' N (.61) whch s called the fundamental equaton of thermodynamcs.

27 Combnng the dervatve of (.61) and (b) gves Sd XdY Nd ' (.6) known as the Gbbs-Duhem equaton.

28 .E.1. Sngle Component System For a sngle component system, the fundamental equaton reduces to S U YX ' N (a) For each extensve varable X, we can defne a molar densty x X, whch s an n ntensve varable (ndependent of n). In order to avod confuson, the conugate Y of X s often called a feld. In terms of the denstes, (a) becomes s u Yx (.63) where n ' N. For reversble processes, 1 st and nd law gve du ds Ydx (.64) so that the Gbbs-Duhem equaton reduces to d sd xdy (.65) whch means, Y s Y s a functon of felds only. Furthermore x Y

29 .E.. Exercse.3 he entropy of n moles of a monatomc deal gas s gven by the Sackur- etrode equaton: S 5 V n ln V n 3/ nr he equaton of state s PV nr wth the standard state satsfyng PV n R In terms of molar denstes, we have 3/ 5 v s R ln v (a) Pv R (b) v V P v R and n Compute U From (a), we have s s ds d dv v v where s 3R s v v R v Puttng these nto the combned 1 st - nd laws we have du ds Pdv s s du d P dv v v 3R R d P dv v 3R d [from(b)]

30 hus, u 3 R where u at. Hence 3 U nu nr 3 U n u n R Compute From eq(.65), we have 3/ 5 v s R ln P v R v P P Upon ntegratng, we have 5 P ln P 5/ R 5 P 5R R ln f P 5/ P ln and ln P P 5/ R f P R ln P g so that 5/ R ln P P () Fnd the Fundamental Eq Usng

31 nu nu (see part a.) the Sackur- etrode equaton becomes the fundamental equaton S V n U 5/ 3/ 5 V n ln U nr 3/ 3/ 5 VU V ln U nr 5/ ln 5/ n n whch s clearly homogeneous n the 1 st order for n, V and U. (3)

32 .F. hermodynamc Potentals.F.1..F...F.3..F.4..F.5. Internal Energy Enthalpy Helmholtz Free Energy Gbbs Free Energy Grand Potental

33 .F.1. Internal Energy Consder the fundamental equaton U S YX ' N (.66) where the ndependent varables are the extensve ones, namely, S, X, N. he nd law dq ds combned wth the 1 st law du dq dw gves du ds dw ds YdX ' dn (.67) where the equal sgn holds only for reversble changes whch satsfy U S X, N (.68) Y U X S, N (.69) U ' N S X N,, (.7) Snce du s exact, we have U U X S S X X, N S, N S, N X, N (.71).e., Y X S,, S N X N (.7) For a J component system, there are J equatons n eq(.68-7). We thus have

34 1 1 equatons of the form of eq(.7), namely, J C J J ' N S, X, N S X, N Y ' N S, X, N X S, N ' ' N N S, X, Nk S, X, Nk (.73) (.74) (.75) hese are known as Maxwell relatons.

35 .F.1a. One Component System Swtchng to molar denstes, the equatons of the last secton become u s Yx (.66a) du ds Ydx (.67a) u s x u Y x Y x s s s x (.68a) (.69a) (.7a)

36 .F.1b. U as Free Energy Consder the 1 st law U Q W (.76) he work W done by the system can be wrtten as W W W (.77) X free where nvolves changes of the system's extensve varables and W free WX does not. [See Fg..7 for an example] Eq(.67) thus gves free (.79) U ds YdX ' dn W For a reversble process ( Q ds ) n an solated ( Q ) and closed ( N const ) system wth all extensve varables fxed ( X const ), eq(.76) becomes U S X N W (.78),, free hus, for constant S, X, N, work can be stored and extracted reversbly as changes n the nternal energy,.e., U s the relevant free energy. If the process s rreversble ( Q ds ), eq(.78) becomes U S X N W (.8),, free so that some work s wasted and cannot be recovered later. Combnng both cases gves U S X N W (.81),, free For a completely solated and closed system wth no nteracton wth ts envronment, all possble processes must satsfy U (.8) S, X, N

37 hus, when equlbrum s fnally reached so that no further spontaneous changes are possble, U must be a mnmum.

38 .F.. Enthalpy he enthalpy H s the Legendre transform of U that replaces X wth Y. hus, H S, Y, N U XY (.83) S ' N so that the combned 1 st & nd law gves dh ds XdY ' dn (.84) hus, H S Y, N (.85) X H Y S, N (.86) H ' N S Y N,, (.87) he Maxwell relatons are X Y S,, S N Y N (.88) ' N S, Y, N S Y, N X ' N S, Y, N Y S, N ' ' N N S, Y, Nk S, Y, Nk (.89) (.9) (.91)

39 .F.a. One Component System Swtchng to molar denstes, the equatons of the last secton become h s (.83a) dh ds XdY (.84a) h s Y u x Y x Y s s s Y (.85a) (.86a) (.88a)

40 .F.b. H as Free Energy From eq(.84), the analog of (.79) s free (.9) H ds XdY ' dn W whch means H S Y N W (.93),, free When no work s done H (.94) S, Y, N hus, when equlbrum s fnally reached so that no further spontaneous changes are possble, H must be a mnmum.

41 .F.c. Exercse.4 he entropy of n moles of a monatomc deal gas s gven by the Sackur- etrode equaton: S 5 V n ln V n 3/ nr he equaton of state s PV nr wth the standard state satsfyng PV n R In terms of molar denstes, we have 3/ 5 v s R ln v 5 P ln P 5/ R where s 5/ s R ln P P 5 R Pv R v V P v R and n (b) (a) Compute H From (a), we have s s ds d dp P P where s 5R s R P P P Puttng these nto

42 we have hus, dh ds vdp s s dh d v dp P P h 5R R d v dp P 5 5R d R where h at. 5 H nr [from(b)] Another way s to start wth h s h v P P s and wrte them n terms of varables s and P. Now, from (a), we have so that 5/ P s s exp P R (c) /5 P s s P R exp 5 /5 P s s exp P s v /5 R P s s exp P P s (d) Integratng (c) wth the help of (d) gves /5 P s s h exp ds P s and /5 P s s s exp f P P s /5 1 exp s s 3/5 h R P dp P s

43 /5 1 s s 5 5/ R exp P g s P s /5 /5 P 1 exp s s s R exp s s g s P s P s Comparson of the expressons gves f g and /5 P s s h s exp P s s 5 R

44 .F.x. Legendre ransform Consder a functon df ydx f x and ts dervatve df y dx so that he Legendre transform of f s defned as so that df g x f xy f dx dg dg xdy ydx f xdy dy dy whch means g s a functon of ndependent varable y.

45 .F.3. Helmholtz Free Energy he Helmholtz free energy A s the Legendre transform of U that replaces S wth. hus, A, X, N U S (.95) YX ' N so that the combned 1 st & nd law gves hus, da Sd YdX ' dn (.96) A S X, N (.97) Y A X, N (.98) A ' N X N,, (.99) he Maxwell relatons are S Y X,, N X N (.1) S ' N, X, N X, N Y ' N, X, N X, N ' ' N N, X, Nk, X, Nk (.11) (.1) (.13)

46 .F.3a. One Component System Swtchng to molar denstes, the equatons of the last secton become a Yx (.95a) da sd Ydx (.96a) a s a Y x s Y x x x (.97a) (.98a) (.1a)

47 .F.3b. A as Free Energy From eq(.96), the analog of (.79) s free (.14) A Sd YdX ' dn W whch means A X N W (.15),, free When no work s done A (.16), X, N hus, when equlbrum s fnally reached so that no further spontaneous changes are possble, A must be a mnmum.

48 .F.3c. Exercse.5 he entropy of n moles of a monatomc deal gas s gven by the Sackur- etrode equaton: S 5 V n ln V n 3/ nr he equaton of state s PV nr wth the standard state satsfyng PV n R In terms of molar denstes, we have where 3/ 5 v s R ln v Pv R v V P v R and n Compute A Snce we have da sd Pdv (a) (b) 3/ a 5 v s R ln v v a R P v v Integratng gves 3/ 5 v 1 3 a R ln R lnd v 3/ 5 v 1 3 R ln R f v v ln

49 and 3/ R 1 ln v v a R ln v g f v Comparson of the expressons gves f and 3/ a R 1 ln v v 3/ A nr 1 ln v v 3/ nr 1 ln V n V n

50 .F.4. Gbbs Free Energy he Gbbs free energy G s the Legendre transform of U that replaces S wth and X wth Y. hus, G, Y, N U S YX (.17) ' N so that the combned 1 st & nd law gves hus, dg Sd XdY ' dn (.18) G S Y, N (.19) X G Y, N (.11) G ' N Y N,, (.111) he Maxwell relatons are S X Y,, N Y N (.11) S ' N, Y, N Y, N X ' N, Y, N Y, N ' ' N N, Y, Nk, Y, Nk (.113) (.114) (.115)

51 .F.4a. One Component System Swtchng to molar denstes, the equatons of the last secton become g (.17a) dg sd xdy (.18a) g s Y g x Y s x Y Y (.19a) (.11a) (.111a)

52 .F.4b. G as Free Energy From eq(.18), the analog of (.79) s free (.116) G Sd XdY ' dn W whch means G Y N W (.117),, free When no work s done G (.118), Y, N hus, when equlbrum s fnally reached so that no further spontaneous changes are possble, G must be a mnmum.

53 .F.4c. Exercse.6 Let dw YdX dw ' (1) dq ds ds () G U XY S (3) Show that ' dg dw d S Y Answer From (3), we have dg du XdY YdX ds Sd (4) Combnng 1 st law du dq dw wth (1) and (), we have ' du ds d S YdX dw so that (4) becomes Hence dg d S XdY Sd dw ' dg d S dw ' Y

54 .F.4d. Mxtures For a PV system of a mxture of m types of partcles, the total Gbbs free energy s N G N N N m m ' ' A 1 1 A m m G n n (.119) 1 1 n, P, n where, for type partcles, n s the number of moles and G n, P, n the molar chemcal potental. Accordng to the Gbbs-Duhem equaton, Hence, G s a 1 st order homogeneous functon of depends only on ntensve varables. n. hs means consdered as the partal molar Gbbs free energy for type partcles. can be By defnton, the extensve varables V m nv 1 m S ns 1 are 1 st order homogeneous functons of quanttes n so that we have the partal molar v s V n S n, P, n, P, n (.1) (.11) hus, the enthalpy m H G S n s n h 1 1 s a 1 st order homogeneous relaton of m n and defnes the partal molar enthalpy h H n, P, n

55 .F.5. Grand Potental he grand potental Ω s the Legendre transform of U that replaces S wth and wth '. hus,, X, ' U S ' N (.1) YX so that the combned 1 st & nd law gves hus, d Sd YdX N d ' (.13) N S X, ' (.14) Y X, ' (.15) N ', X, ' (.16) he Maxwell relatons are S Y X, ' X, ' (.17) S N ', X, ' X, Y N ', X, ' X, ' N N ' ', X, ', X, k ' k (.18) (.19) (.13)

56 .F.5b. Ω as Free Energy From eq(.13), the analog of (.79) s free (.131) Sd YdX N d ' W whch means,, ' (.13) X W free When no work s done (.133), X, ' hus, when equlbrum s fnally reached so that no further spontaneous changes are possble, Ω must be a mnmum.

57 .G. Response Functons.G.1. hermal Response Functons (Heat Capacty).G.. Mechancal Response Functons.G.3. Exercse.8

58 .G.1. hermal Response Functons (Heat Capacty).G.1a..G.1b..G.1c..G.1d..G.1e..G.1f. Constant X Constant Y Sngle Component System Constant X, agan Constant Y, agan Sngle Component System, agan

59 .G.1a. Constant X o obtan C, we use X, and N X, N as ndependent varables. Usng U U U du d dx dn X, N X, N N, X, N the 1 st law dq du YdX ' dn becomes U U dq d Y dx X, N X, N U ' dn (.134) N, X,N Hence, CX, N dq U d X, X, N N (.135)

60 .G.1b. Constant Y o obtan C, we use Y, and N Y, N as ndependent varables. Usng X X X dx d dy dn Y, N Y, N N, Y, N eq(.134) becomes U U X dq Y d X, N X, N Y, N U X Y dy X, N Y, N so that CY, N U U X ' Y dn N X, X,, N N N, Y, N dq d Y, N U U X Y X, N X, N Y, N U X CX, N Y X, N Y, N (.138)

61 .G.1c. Sngle Component System For a sngle component system, we can replace all extensve varables wth molar denstes. hus, the molar heat capactes are c 1 u x CX, n n x (.135a) 1 u x cy CY, n cx Y n x Y (.138a)

62 .G.1d. Constant X, agan Usng X, and N as ndependent varables. we have S S S ds d dx dn X, N X, N N (.139), X, N so that CX, N dq S d X, X, N N A X, N (.14) (.141) where we've used A S X, N

63 .G.1e. Constant Y, agan Usng Y, and N as ndependent varables, we have X X X dx d dy dn Y, N Y, N N, Y, N so that (.139) becomes S S X ds d X, N X, N Y, N S X dy X, N Y, N and CY, N S S X dn (.143) N X, X,, N N N, Y, N dq S d Y, Y, N N S S X X, N X, N Y, N S X CX, N X, N Y, N (.144) Usng G S we can also wrte C, Y N Y, N G, Y N Comparng (.138) wth (.144), we have

64 ,, 1 N N S U Y X X (.145) Now, the Maxwell relaton,, N X N S Y X (.1) gves,,, N X N X N Y S X,, X N N S X,, 1 X N N C X (.146)

65 .G.1f. Sngle Component System Agan In terms of molar denstes, the one component verson of the equatons n secton.g.1d-e. become 1 a c C x X, n n Y (.141a) 1 g c C Y Y, n n Y (.144a) s 1 u x x s Y x x Y (.145a) (.1a) Y 1 cx x x (.146a)

66 .G.. Mechancal Response Functons Isothermal susceptblty:, N X G Y, N Y, N (.147) Adabatc susceptblty: S, N X H Y S, N Y S, N (.148) hermal expansvty: Y, N X Y, N (.149) Poof of the followng useful relatons s left as an exercse C,, C N Y N X, N (.15) Y, N C Y, N, N S, N (.151) Y, N C C (.15) Y, N, N X, N S, N For a PV system, we have Isothermal compressblty:, N 1 V 1 G V P, N V P, N (.153) Adabatc compressblty: S, N 1 V 1 H V P S, N V P S, N (.154) hermal expansvty:, P N 1 V V, P N (.155)

67 .G.3. Exercse.8 Compute the molar response functons deal gas usng 3/ 5 v s R R ln v c v, c P,, S, and and Pv R P for a monatomc Answer Molar Heat Capacty c v s 3 R 3 cv R v Molar Heat Capacty c P c P s P 1 v 3 R v P From the equaton of state, we have so that v R P c P P R vp c v 5 R cv R Isothermal Compressblty κ From the equaton of state, we have so that v R v P P P 1 v 1 v P P Adabatc Compressblty κ S o evaluate S 1 v, we begn wth the Maxwell relaton v P S v s s P P v 1 S v P

68 Now s 3 R P P v v 3 R v 3v R s 1 3 R v P v v P 1 3 P 5R R v R v so that 1 3v v 3 v 3 S v 5R 5 R 5P hermal Expansvty α P From the equaton of state, we have so that v R P P 1 v R 1 P v vp P

69 .H. Stablty of the Equlbrum State.H.1..H...H.3. Condtons for Local Equlbrum n a PV Systems Condtons for Local Stablty n a PV Systems Implcatons of the Stablty Requreemnts for the Free Energes

70 .H.1. Condtons for Local Equlbrum n a PV Systems Consder a mxture of m types of partcles n an solated box of volume V dvded nto parts A and B by a conductng, porous, movable wall. (see Fg..8) he extensve varables can be wrtten as U V (.156) U A, B (.157) V A, B N (.158) A, B N S (.159) S A, B where the subscrpt denotes quanttes of the whole system. Consder now spontaneous fluctuatons subect to the contrants U V N (.16) so that U A U B, etc. In the absence of chemcal reactons, the entropy change can be wrtten as S S S U V A, B U V N V UN a a m S N 1 N UV N, S S Ua V A, B U VN V UN m S N (.161) 1 N VN, where the superscrpt α denotes evaluaton at cell α. Usng eq(.58-6), we have m 1 P ' S Ua Va N A, B 1 a

71 m 1 1 P A ' B ' A P B U A VA N A A B A B 1 A B where we've used (.16). (.16) For a system n equlbrum, S s a maxmum. hus, any spontaneous changes must satsfy S. However, U A, VA and N A can be of ether sgn. herefore, the condtons for local equlbrum are (.163) P A A A B P (.164) B ' ' 1,, m (.165) B

72 .H.. Condtons for Local Stablty n a PV Systems Consder a closed and solated m-component PV system dvded nternally nto K cells by conductng, porous, and movable parttons. the system dscussed n secton.h.1 to 1,, K ] [hs s ust a generalzaton of.h.a. aylor Expanson of S.H.b., P, N as Independent Varables.H.c. Stablty Condtons.H.d. Quadratures.H.e. Extras.H.f. Exercse.9

73 .H.a. aylor Expanson of S Consder the aylor expanson of a functon f x about ts maxmum at x,.e., 1 f f x f x xx x x, where x x x and the matrx g f x x s symmetrc and postve defnte. Settng y f x, we have x x f f f 1 y xx 1 y x x, where y y x x When applyng the above to S U, V, N f so that S x U, V, N, we set,,,, S S U V N S U V N m 1 S S S U V N U V N V UN 1 N V N, 1 1 U V N m P ' 1 Now, wth all sub- and super- scrpts suppressed for clarty, we have 1 1 U S P V ' N ' S V N 1 P

74 so that P 1 P V P V 1 P P V V ' 1 ' N ' N 1 ' ' N N m 1 S S P V ' N 1 Summng over all cells gves K m 1 S ' S P V N (.171) 1 1

75 .H.b., P, N as Independent Varables Choosng, P, N as ndependent varables, we have S S S m S S S P N PN P 1 N N (.17) PN, m V V V V P N PN P 1 N N (.173) PN, ' ' m ' ' P Nk PN P N k1 N (.174) k PN, k Next, we substtute these nto K m 1 S ' S P V N 1 1 (.171) and collectng terms. he terms n the square bracket [ ] then gves, S S V P PN P N PN m S ' N 1 N PN, PN m V ' V P P N P N 1 P N N PN, m m ' N N N 1 1 PNk, k whch, together wth the Maxwell relatons S V P P N reduce eq(.171) to S V N P N N P

76 K 1 S V S P 1 PN PN m m V ' P N N (.175) P N 1 1 N PNk, k

77 .H.c. Stablty Condtons We now apply the stablty condton S to eq(.175). Allowng n turn only one of, V and N to be non-vanshng, we have P N S V N V P, ' k P N k N Snce these apply to all cells, we can drop the superscrpt α. Wth all dervatves understood to be evaluated at equlbrum, the superscrpt can also be dropped. hus, we have,, P N P N S C (hermal stablty),, 1 N N V V P (Mechancal stablty) (.179),, ' ' k P N k N (Chemcal stablty) hese nequaltes are n fact a realzaton of the Le Chatelers' prncple, whch states that If a system s n stable equlbrum, any spontaneous change n ts parameters must brng about processes that tend to restore the system to equlbrum.

78 .H.d. Quadratures Consder the real quadrature x x q axx A, where the matrx A a s symmetrc,.e., a a. We wsh to fnd the condtons on A such that q for all vectors x x (a) A necessary condton for (a) can be obtaned by consderng the specal cases where x has only 1 non-vanshng component xm 1,.e., x m. hus, q for all x a for all (b) Note that (b) was used n the text n the dscusson followng (.183). Snce A s real and symmetrc, there exsts an orthogonal transformaton that dagonalze A,.e., OAO D wth d d omamno n where the d 's are the (real) egenvalues of A. Hence, q x O DOx y Dy d y where y Ox hus, mn d q y, x the matrx A s then sad to be postve. defnte. If the equal sgn s excluded, A s postve Note that, coupled wth (b), ths means that f the egenvalues of a real symmetrc matrx are all non-negatve, all the dagonal elements must be non-negatve.

79 .H.e. Extras In order to obtan an nequalty for C V, N, we start wth the Maxwell relaton (.8), S S S V V P V P S P V V V P S P V V V P [dagram used] [(.6) used] (.176) whch,usng the defntons (.135), (.153), and (.155), can be wrtten as C C P N V N N P N V From the mechancal stablty condton [see (.179)] N we have C C P N VN (.18a) Next, treatng, V, N as ndependent varables, we have m P P P P V N V N V N 1 N VN, so that, for all N fxed, or P P P N N V N VN V N V V P V N P N N P N P N V N V V P P N N PN N (.178) Now, strppng all rrelevant sub- and super- scrpts, we have

80 V V V V P P V P P P P or P V V V P P V P P P V V P Wth reference to eq(.175), we see that the contrbutons to fxed, nvolve, S V V A P P P Usng (a), we have P P S (a), when all N are S P V P A V P V P V S P V V V [(.176) used] Restorng the full notatons, (.175) becomes K 1 S P S V 1 VN V N N k 1 1 P N, k m m ' N N (.177) whch n turn provdes the desred thermal stablty condton C C [(.18a) used] (.18) P N V N Smlarly, an nequalty for S, N can be obtaned by treatng S, P, N as ndependent varables. We lst here only the fnal result,, (.181) N S N

81 .H.f. Exercse.9 Let nanb G naa P, nbb P, RnA ln xa RnB ln xb n Plot the regon of thermodynamc nstablty n the plane. Answer For chemcal stablty, the symmetrc matrx M A, A A, B B, A B, B xa must be postve sem-defnte..e., hs means all the prncpal mnors are nonnegatve,,,,,, (1) A, A B B A A B B A B B A Now, usng n n A n A n B n A n n B B 1 x A n A n n n n A n A A B B nb 1 na n B n n n xb n we have x B n B n n n n A n A A B B nb n B n xb n ln x A 1 x A na xa na nb nb xb x n A nb n n A ln x B 1 x B na xb na nb nb xb x n B 1 n nb nb n ln x n ln x ln x ln x na n n A A B B A A nb n n n n n na n n n A B B A B nb n B n x x B 1 A

82 so that G P, R ln x 1 x A A A A na PnB () A, A A n A PnB xb x R x n n A B Makng the swtch A B gves P, R ln x 1 x B B B B x A B, B R x B n x n A so that 1 xaxb B, A R A, B (3) n n he stablty condtons (1) thus mply and xb xb R x n n A xa xa R x n n xb xb xa xa 1 xaxb R R R xan n xbn n n n B he left sde of the last condton s dentcally so that t s automatcally satsfed. he 1 st two are equvalent to R x x A B A B A A (4) R R or x x x 1 x whch s always satsfed f C. For C, an unstable regon develops R under the (coexstence) curve x 1 x A A, [see shaded area n Fgure]. A R pont nsde the unstable regon thus represents a mxture, or segregaton, of phases gven by the ntercepts of the coexstence curve wth a horzontal lne passng through the pont. Relatve proporton of the phases are gven by the lever rule.

83 .H.3. Implcatons of the Stablty Requrements for the Free Energes.H.3a..H.3b..H.3c. Convex And Concave Functons Helmholtz Free Energy Gbbs Free Energy

84 .H.3a. Convex And Concave Functons A functon f x s convex n an nterval I f d f dx x I hs means [see Fg.9], 1. For any ponts x1 x n I, the chord onng f x and 1 f x always les above the curve f x.. Any tangent lne of f x x I always les below t. A functon f x s concave n an nterval I f f x s convex there.

85 .H.3b. Helmholtz Free Energy Usng A A S P V N V N the stablty condtons (.18-1) become S CV N A V N V N.e., V P A V N P N V N V N 1 1 A A V V N N (.184) (.185) hus, A s concave wrt and convex wrt V n the vcnty of equlbrum.

86 .H.3c. Gbbs Free Energy Usng G G S V P N P N the stablty condtons (.18-1) become.e., S CP N G PN PN V G V N P N P N G G P P N N (.186) (.187) hus, G s concave wrt both and P n the vcnty of equlbrum.