7. Equilibrium Statistical Mechanics

Transcription

1 7. Equlbru Statstcal Mechacs 7.A C. 7.D. 7.E. 7.F. 7.G. 7.H. Itroducto he Mcrocaocal Eseble Este's Fluctuato heory he Caocal Eseble Heat Capacty of A Debye Sold Order-Dsorder rastos he Grad Caocal Eseble Ideal Quatu Gases

2 7.A. Itroducto A closed ad solated syste teds to a statoary state that s charactersed by a few te-depedet state varables. Such a state s sad to be therodyacal equlbru. Accordg to the st law, ts total (teral) eergy s fxed. Accordg to the d law, ts etropy s at a axu sce all etropy-creasg spotaeous processes have ceased. For a syste wth degrees of freedo, ts states wth fxed teral eergy of, say, U E, for a 6 hyper-surface the phase space. he phase space trajectory of a closed ad solated syste s restrcted o ths costat eergy surface. Cosder ow a fte uber of such systes, all detcal ad prepared equlbru wth U solated systes. E. I short, we have a equlbru eseble of closed Accordg to the postulate of equal probablty, we assg a equal probablty to each state pot o the costat eergy surface so that the probablty desty s gve by, E E H X X E for (7.) otherwse X q p s the 6-desoal phase space vector. E where, s called the structure fucto ad deotes the area of the costat eergy surface. ote that as show chapter 6, ths postulate s always satsfed for ergodc systes. he etropy s defed by Gbbs as l X X X (7.) S d C where C s a costat serted to gve the proper desos. For quatu systes, the Gbbs etropy (7.) s geeralzed to S l r (7.) where s the desty operator ad the trace s over ay coplete orthooral set of bass states.

3 7.. he Mcrocaocal Eseble Quatu Verso Exercse 7. (Este Sold) Classcal Verso Exercse 7. (Ideal Gas)

4 7... Quatu Verso Cosder a closed solated syste of volue V, partcle uber, ad eergy E. Let, E, wth,, E haltoa H wth egevalue E,.e., H E, E E, E E, E, I, be a coplete set of orthooral egestates of the E, E, where I s the detty operator. he probablty of fdg the syste state E, s P E, E,. he Gbbs etropy (7.) ca be wrtte ters of P as follows. Frst, we wrte S r l E E, l E, E E, E, E, l E, (7.4a), Usg E, E, E, E, P E, l E, l E, E, l P eq(7.4a) becoes E S P l P (7.4) (7.4b) Proof of (7.4b) A aalytc fucto f of the operator s defed by f d f! d where s obtaed by treatg as a ordary varable. herefore, d f E, f E, E, E,! d

5 d f P! d f P Maxzato Of S he equlbru values of P are those that axze the etropy subject to the costrat E r P (a) hs s equvalet to the varatoal proble S r S r where α s a Lagrage ultpler. Wth the help of (7.4), we have E E P l P P l P P l P P exp cost oralzato (a) the requres P (7.7) E whch s sply the postulate of equal probablty. A collecto of closed ad solated systes dstrbuted accordg to (7.7) s called a crocaocal eseble. he etropy (7.4) becoes S E l l E E E (7.8) Wth the detfcato of E as the therodyac teral eergy U, all other therodyac quattes ca be obtaed through the usual therodyac relatos.

6 7... Exercse 7. (Este Sold) A Este sold s a lattce of stes, each of whch s edowed wth depedet quatu haroc oscllators of the sae frequecy ω. he total eergy s where E M s the uber of quata o the th oscllator ad uber of quata preset. (a) What s the total uber of croscopc states for a gve E? (b) Fd the etropy S,. (c) Fd the heat capacty C. Aswer (a) he proble s to fd the uber of ways W to put quata to dstgushable stes. M E M the total detcal ow, a gve cofgurato ca be represeted as a le of M quata separated to parttos by walls. adjacet postos represet a ste wth o quata.] [ote: walls at hus, all possble cofguratos ca be geerated by perutatos of the cobed M+ quata ad walls "objects". [ote that the parttos, or stes, are dstgushable by vrtue of the order of ther postos the le.] Sce the quata ad walls theselves are dstgushable, we have M! W M!! () (b) Usg (7.8), we have S lw l M! M!! () For large M ad, we ca use the Strlg forula l! l l to wrte

7 S l M M M M l M M M M M M M l M M M M l M M () ow, settg du ds d de dm d we have S S E M Usg d x d l x x l x l x dx dx we have (4a) S M M M l l M M so that (4a) becoes l M l (4) M M exp exp M exp exp exp (5) so that M l l exp

8 M exp exp exp M l l exp M exp M l l exp exp M M l l exp exp Puttg these to () gves exp S l exp exp (6) (c) he teral eergy s M U E exp exp exp (7) Usg exp exp exp the heat capacty becoes C S U exp exp exp exp exp exp (8)

9 7... Classcal Verso Cosder a closed solated syste of volue V ad fxed partcle uber. Let ts eergy be costrated to the shell E E E wth E. he phase space volue of the eergy shell s,,,, E V E V E E, where E, V, s the structure fucto (area of the eergy surface). he equlbru probablty desty X l correspods to the axu of S dx X C X (7.) EH X EE subject to the oralzato costrat EH X EE dx X (7.9) Wth the help of the Lagrage ultpler, ths extreu proble becoes Usg we have dx l C EH X EE C C l l EH X EE (7.a) dx l C (7.) whch, sce s arbtrary, ca be satsfed oly f l C C exp K (7.b) Fro (7.9), we have K E V,,, so that E E V E X,, for E H E E otherwse (7.)

10 Eq(7.) thus becoes or S l C l E E C E E E E, V, S E, V, l (7.) C he costat C caot be detered classcally. Quatu echacally, the ucertaty prcple assgs a phase space volue of h to each state so that we have ad C C h! h for dstgushable partcles for dstgushable partcles where h s the Plac s costat. he serto the factor! oltza coutg. s called the correct It was orgally troduced by had to solve the Gbbs paradox: effect, t reoves the etropy of xg of dstgushable partcles. I a practcal calculato, t s ofte easer to wor wth the volue E, V, stead of the shell,, of thcess where E E E V each. Hece, E, V, E, V,. owards ths ed, we dvde to E / E E (7.) E E E ad EE / E E E E E E. Obvously, the volue of ay shell caot be larger tha the whole volue,.e., E, V, E, V, E shells, ad the outerost shell has the largest volue,.e., E, V, E, V, E E E E, V, E, V, E, V, ag the logarth gves E l E l l l E ow, l, E E Hece, (7.) becoes E E. Hece E E (7.4) (7.5) so that as, (7.5) ples l l. E

11 E, V, S E, V, l (7.6) C

12 7..4. Exercse 7. (Ideal Gas) Fd the etropy ad equato of state for a deal gas. Aswer Quotg the result fro Exercse 6., we have V E / E () For very large ad V, the etropy s gve by (7.6). Usg the dstgushable partcles value of C! h, we have S l E! h V E / l! h () / E l V l! h ow, for large, we have l! l l e (Strlg forula)! e so that l! l l / 5 5 5/ 5 l l l ad / / E 5/ 5 S l V l h / V 4 E 5 l h ()

13 hus, the etropy per partcle s / 4 e 5 s l v h where the lower case letters deote quattes per partcle. ote that the depedece of (4) of s the result of the! factor (4) C. ow, () s the fudaetal equato for the deal gas. VU / l 5/ S S We ca rewrtg t as (5) where S / 4 5 l h For fxed, we have so that dv du ds V U U S V U P V U S U V whch s the equato of state. V U

14 7.C. Este's Fluctuato heory 7.C.. 7.C.. Geeral Dscusso Flud Systes

15 7.C.. Geeral Dscusso Cosder a closed ad solated ergodc syste wth eergy wth shell E, E E. Accordg to (7.8), the etropy s S l E (7.7) where E s the uber of croscopc states shell E, E E. ow, subdvde the syste to cells ad deote the value of a state varable A cell by s therefore A. he probablty of fdg the syste the acrostate E, A,, A P E, A,, E, A,, A A (7.8) E where E A A s the uber of crostates satsfyg E A A Slarly, so that,,,,,, l,,, S E A A E A A (7.9) P E, A,, A exp S E, A,, A E (7.),,,. Accordg to the d law, S s a axu whe the syste s a equlbru state E, A,, A. he aylor expaso of S about ths state s S E A A S E A A g (7.),,,,,, j j, j where the st order ters vash sce the expaso s about a extreu ad A A (7.) g j S g j A A j [S s a axu] (7.) Puttg (7.) to (7.) gves

16 P α P E, A,, A C exp gj j, j C exp α Gα (7.4a) where α,,, G g j G, ad C exp S E, A,, A E Fro the oralzato codto, we have, C d d exp d dp α α Gα where C d exp Γ α Oβ ad O GO dag. Hece C C det G Hece det G P α exp α Gα (7.4) whch s vald oly for sall fluctuatos. by the Laplace trasfor α α h α h exp I D P We ow defe a "characterstc fucto" (7.4a) where hus, Dα d d

17 I Usg α α Gα h α det G D exp α h Oβ δ β h, where δ O h, we have, h α Gα h α ad so that μ Gμ δ Γ δ δ Γ δ h OΓ O h h G h μ μ Gμ det exp G I h exp h G h D exp h G h (7.6) hs s useful for calculatg oets sce fro (7.4a), we have h I D P h h α α ow, wth where I h h h h M M I h h G h, we have, h I M M M I hh j h h j h h j h M h h h G G j j j j j G h j G h j j (7.7a)

18 M h h j G j so that (7.7a) gves for odd j G (7.7) j As wth Ex.4.9, all oets of hgher order ca be expressed as lear cobatos of products of the d oets.

19 7.C.. Flud Systes 7.C... 7.C... Destes Fluctuatos

20 7.C... Destes We shall express all extesve varables by upper case letters, say, X, ad the correspodg destes by lower case letters, say, x. hus, X xv. Oe excepto to ths s the ass desty whch we wrte as M V. Usg dx Vdx xdv () the cobed st ad d law du ds PdV dm () becoes Vdu udv V ds d s P dv Sce ths s vald for arbtrary V ad dv, we ust have du ds d () u s P (4) where () s the cobed law ad (4) s the fudaetal equato wth destes as depedet varables. Cobg () ad (4), we get the Gbbs-Duhe equato sd dp d (5) Slarly, usg () ad collectg ters, we have S PV M V s s P V where (5) was used to get the last equalty. V s (6)

21 7.C... Fluctuatos Cosder a closed ad solated box of flud subdvded to cells. he fluctuatos of the etropy s gve by (.7) as [see Fg.7. for otatos], S S P V M (7.8) Accordg to (7.4), we have det G P S exp S P V M (7.9) where G s to be detered through (7.4) oce the depedet varables are decded. y eq(6) of secto 7.C..., we have S PV M V s (a) S where s ad V varables so that M V are destes. ext, we choose ad as depedet s s s c s (b) s (c) where the Maxwell relato g a f s u Puttg (b,c) to (a) gves s ca be gleaed fro the dagra c V s V where, accurate to order, we've replaced wth. Eq(7.9) thus becoes

22 det G V c P, exp whch, whe coparg wth suggests det G P α exp α Gα (7.4) α,,,,,, g j Vc V for j odd j eve (7.) otherwse so that ad det G g V c V P V c V, V c exp he d oets of the fluctuatos are gve by so that (7.) Vc V j d d j j exp g jj j g [odd tegral vashes] (7.,6) j j

23 V d exp V V (7.4) V c d exp V c V c (7.5) where we've used the Gaussa tegral forula [see, e.g., Ref, Appedx.4] exp dx x ax a

24 7.D. Caocal Eseble 7.D.. 7.D.. 7.D.. Probablty Desty Operator Systes Of Idstgushable Partcles Systes Of Dstgushable Partcles

25 7.D.. Probablty Desty Operator Cosder a closed but ot solated syste that ca exchage eergy but ot atter wth ts surroudgs. A collecto of such detcally prepared systes s called a caocal eseble. As wll be show below, the correspodg probablty dstrbuto descrbes a syste ept at costat teperature. Oce aga, the probablty desty operator s detered by axzg the etropy S. Sce the uber of partcles s fxed, we have the oralzato costrat r (7.7) where s the (fxed) uber of partcles. Furtherore, the free exchage of eergy s expected to stablze, at equlbru, to soe costat average value E,.e., r H E U (7.8) hus, the varatoal proble s S r r H E r l H E r l E H whch ca be satsfed oly f l H (7.4) E exp E H (7.4) he costrats (7.7-8) thus becoe E exp r exp H exp E r exp H H U (a) (b) Eq(a) gves the partto fucto

26 E Z r exp H exp (7.4) so that l Z (c) whle (b) becoes E exp U r H H Z ow, tag the average of (7.4) gves r l H E S U S l Z U E E Coparg wth the fudaetal equato for the Helholtz free eergy we have so that A U S E ad A l Z (7.44) A (c) A (7.4) Z exp r H exp (7.45) (7.4) H exp l Z H exp Z (7.46) whch s the probablty desty operator for the caocal eseble. Oce Z ad hece A s ow, all other therodyac quattes ca be calculated. For exaple, for A A, X,, we have A S Furtherore, A Y X X A ' X

27 U A S A A A A X X A X (e) ow, sce the caocal eseble oly specfes a fxed average E U expect fluctuatos E about ths value. he correspodg varace ca be calculated as follows. Startg wth the oralzato r exp A H we ca dfferetate wth respect to to get (7.47) A exp r A H A H X exp X r H A A H Dog t aga gves r exp A H A A H X X Usg (e) gves U r H E exp A H X U E E X, we.e., E U d U E d X X (7.49) CX sce both U ad E E C E E CX X (7.5) are proportoal to. hus, the therodyac lt (, V ), the relatve devato vashes,.e., the caocal eseble s equvalet to the crocaocal eseble.

28 7.D.. Systes Of Idstgushable Partcles As show Appedx, the states of a syste of dstgushable partcles ust be properly syetrzed wth respect to partcle terchages, aely, syetrc (+) for bosos ad atsyetrc () for feros. he trace of thus becoes r (7.5)! where oetu egestates are assued to tae the drect product for (7.5) a b l a b l whch says partcles,,, have oeta,,,, respectvely. hus, P P a b l,, P,, (7.5) where the su s over all possble perutatos of state dces. 7.D.a. Exercse 7. 7.D.b. Se-Classcal Lt 7.D.c. Exercse 7.4

29 7.D.a. Exercse 7. Copute the partto fucto Z for a deal gas of detcal partcles a cubc volue V L. For coveece, eglect the sp degrees of freedo. What approxatos ca be ade for a hgh ad low desty gas? Aswer For a partcle the box wth haltoa H p, the wave fucto s As xs ys z x x () x y z whch vashes at the walls at x, y, z, L a a provded where a x, y, z ad,,, L a ote: x ad x are learly depedet so that oly postve a eed be couted. he -partcle partto fucto s therefore Z exp exp x y L z x y z () Usg a a, we have, the therodyac lt, L L Z d d d x y z exp x y z L exp dx d y dz x y z L V V V V h (4)

30 where h (5) s the theral wavelegth, whch s a easure of the partcle s coherece legth. For detcal partcles, the haltoa s H p p p wth wave fucto [see Appedx ] s / a,,,,,, ab x x x x x x c a b c s / a where s a a, b, c /!! a, b, c,, P,, a b c a b c P P where s the occupato uber of state. Hece, for a b c,,,,,,,,, a b c a b c b c a c a b,,,,,, (7) b a c c b a a c b s a / a, b, c a, b, c (7')! a, a, b a, b, a b, a, a a, a, b (7a) s a, a, b a, a, b (7a') 6 a, a, a a, a, a (7b) s a, a, a a, a, a 6 (7b') he partto fucto s the, by eq(.), Z a, b, c a, b, c! a b c (6) where

31 exp H H H exp p p p so that for a b c,,,,, exp a b c a b c a b c a, b, c a, b, c exp a b c,,,, exp a a b a a b a b exp a b a, a, b a, a, b,,,, exp a a a a a a a 6exp a a, a, a a, a, a I vew of eqs(7,a,b), the su (6) ust be spltted to parts accordg to so that f abc a b c a b c a b c a c b a b c Z,,,, a b c a b c! a b c f a, a, b a, a, b a, a, a a, a, a a b a hus, for bosos,

32 Z exp a b c! a b c 6 exp a b 6 exp a a b a For feros, ow, Usg Z a! a b c exp a b c a Z exp a b ab a b, we have exp a b exp a b exp a a b ab a Z Z Z Usg a b c abc a b c a b c a c b a b c, we have exp a b c exp a b c a b c abc exp a b exp a a b a Z Z Z Z Z Z Z Z Z Hece, for feros, we have Z Z Z Z Z! For bosos, we have

33 Z Z Z Z Z! 6 Z Z Z 6Z Z Z Z Z! oth cases ca be cobed to gve Z Z Z Z Z! Usg (5), we have Z V V V V / /! V / /! V V (8) I the seclasscal lt (hgh ad large V), Z whch ples Z V (9)! V ()! V s sall so that for a partcle syste. ad resolves the Gbb's paradox. he factor! arses fro the quatu dstgushablty

34 7.D.b. Se-Classcal Lt As show Ex.7., Z should be caculated usg properly syetrzed bass states. However, as copared to the results obtaed fro usyetrzed bases, the ost sgfcat correcto s just the factor!, whch ca be added by had. Other correctos (exchage ad correlato effects) are proportoal to powers of / V. hus, they becoe eglgble the se-classcal lt of hgh ad sall. Hece, H Z,, e,, for hgh, low (7.54)!,, he o-teractg olecules a gas ca possess teral degrees of freedo. For exaple, the haltoa of a typcal olecule ay be wrtte as p H H rot H vb H el H ucl (7.55a) where the successve ters o the rght deote cotrbutos fro the traslatoal, rotatoal, vbratoal, electroc, ad ucleoc degrees of freedo, respectvely. he plct, ad usually vald, assupto of (7.55a) s of course that the varous degrees of freedo are decoupled. hus, for a gas of olecules cotaed fxed volue V, we have Z, exp V r H rot H vb! p H el H ucl (7.55) If the varous degrees of freedo are truly decoupled, the correspodg ters H wll coute wth oe aother. ow, sce A, A A e e e we ca wrte Z V Z Z Z Z Z!, tr rot vb el ucl Z Z Z Z Z! tr rot vb el ucl (7.56) For a se-classcal gas wth o teral degrees of freedo, we have, fro Ex.7.,

35 Z V ev! (7.57), V where the last equalty ade use of the Strlg forula! e vald for large. he Helholtz free eergy s ev A l Z l / V l (7.58) h so that A S V / V l h / 5 V l (7.59) h whch s just the Sacur-etrode equato frst troduced Ex...

36 7.D.c. Exercse 7.4 Cosder a cubc box of volue V L each wth sp / ad agetc oet. (a) Fd Z. (b) Fd U ad C V. (c) Fd M. Aswer cotag a deal gas of detcal atos, A agetc s appled to the syste. (a) Z Followg (7.56), we wrte Z Z Z () tr ag! where, as show Ex.7., V Z tr wth he agetc eergy s so that E s Z ag Hece, () becoes Z h s wth s exp s s V cosh V, cosh! () () ev cosh (b) U, C V Fro the defto (7.45) Z r exp H we have Z r H exp H Z r H Z E Z U so that

37 U Z Z V l Z V ote: as wll be show (c), U U S, so that t s actually a "ethalpy". Usg V l Z l e l l cosh h h we have U sh / cosh / tah tah (4) ad C V U V tah tah sech sech (5) (c) M Aalogous to the Helholtz free eergy (7.58), the agetc free eergy of the syste s gve by (see ote below), l Z

38 wth d Sd M d so that l Z M tah tah (6) ote: Cosder the o-agetc case H H where E U. hs ca be terpreted as the agetc case H H wth. Hece, H H correspods to E U S, M sce U depeds oly o extesve varables. herefore, the haltoa H H averages to E U U M H S,, whch s a "ethalpy". he correspodg caocal eseble thus gves rse to a "Gbbs eergy", l Z.

39 7.D.. Systes Of Dstgushable Partcles For dstgushable partcles, there s o eed for syetrzato. Hece, r,,,, (7.6),, Exercse 7.5 Este Sold Use the caocal eseble to calculate U ad C for a Este sold (see Ex.7.) Aswer Fro Ex.7., we have H () Wth egestates where,,, the partto fucto becoes Z r exp (),, exp,,,, exp exp exp exp exp exp exp exp ()

40 he Helholtz free eergy s, l A Z l exp l exp (4) he etropy s A S exp l exp exp exp l exp exp (5) he teral eergy s (see 7.D.) A U l exp exp exp exp exp exp (6) he heat capacty s U C exp exp

41 exp exp exp exp (7)

42 7.E. Heat Capacty Of A Debye Sold Cosder a sple crystalle sold wth oe ato at each lattce ste. Sce the sze ad shape of the sold are fxed acroscopcally, the oto of the atos ust be restrcted to oscllatos about ther equlbru postos. he degrees of freedo of such vbratos s, where s the uber of atos. If the apltudes of the vbratos are sall eough, the oscllatos becoe haroc,.e., the potetal eergy s quadratc atoc dsplaceets. hus, the atoc vbratos of a sold ca be approxated as a set of coupled haroc oscllators. y eas of a trasforato to the so-called oral coordates, the vbratos ca be de-coupled to depedet oral odes. ypcal easured values of the heat capacty of oatoc solds are show Fg. 7.. At hgh teperatures, CV approaches a costat value of 6 cal/k ole, agreeet wth the classcal theory. For low teperatures, CV ca oly be explaed ters of quatu theory. drops as, whch he Debye theory s a quatu theory of haroc oscllatos a cotuu. Accordg to classcal elastc theory, there are types of waves govered by the wave equatos c c u r, u t t u r, ul L t L t trasverse, doubly degeerate logtudal where c s the phase velocty. c L, ad are called soud waves. he propagatg odes thus obey lear dsperso Let the sold be a rectagular lattce of sdes Lx, Ly, L z. he oral odes are stadg waves whch vashes at the surfaces. vectors are hs eas the allowable wave

43 where where x, y, z ad,,, (a) L L a, wth a beg the lattce spacg, s the uber of stes the th drecto. hus, the total uber of stes s x y z. Wth each ste havg degrees of freedo, the total degrees of freedo s. I the quatzed verso of the theory, the haltoa s H, (7.6) L,, where the su over volves odes for each brach. he partto fucto s Z r exp, L,, exp,, Hece, exp exp sh l Z l sh E l Z cosh sh (7.6) exp exp exp exp

44 exp exp exp (7.64), where the average occupato of the ode,, exp s called the Plac's forula. he su over ca be approxated by a tegral d (7.65) where the desty of states -space,, ca be calculated fro eq(a) as so that L L L V x y z V d where the codto Usg, we have c V f d f V c fro eq(a) restrcts the tegrato to the st quadrat. V d f d f (b) where we've used the fact that tegrato of the agular part over the st quadrat gves st Q d d 8 Sug over the braches gves V f d f For the specal case that c f f, we have

45 where V f d f c V c c c c c L d f o restrct the total uber of odes to, we ust troduce a cutoff (Debye) frequecy so that D V c D d VD (7.68) c D 6 c V / c D (7.69) Defg the desty of states -space by we have g dg V c Eq(7.64) thus becoes 9 (7.7) D D E d g (7.7) where exp Wth the help of (7.7), we get D 9 E d D 4 D 9 D d D 8 exp (7.7) he heat capacty s therefore

46 C 9 D D d 4 exp exp 5 xd 4 dx x D x 9 x e e where x 9 dx D xd 4 x x e x e For low, we have 9 dx D xd xd 4 x x e x e. Usg we get dx e x e x 4 x C 4 5 D (7.74) whch s the faous Debye rule.

47 7.F. Order-Dsorder rastos he trasto fro a dsordered state to a ordered oe ca be studed usg ethods of equlbru statstcs. Cosder a lattce of sp / objects wth agetc oet. betwee objects further tha earest eghbors are assued eglgble. presece of a appled agetc feld, the haltoa of the syste s H s s s where s j j j, j (7.75) Iteractos I the dcates the oretato (up/dow) of the sp at ste alog the z-axs, s the teracto eergy, ad each par of stes couted oce., j deotes a su over earest eghbors wth hs s ow as the Isg odel. If j, the lowest eergy for happes whe all sps are parallel,.e., they are all up or all dow, both cases beg equally probable but ca be chose by a ftesal. hs correspods to ferroagets f j ; otherwse, the syste s ferragetc. Aalogously, the case j correspods to at-ferro or at-ferr- agets. he partto fucto s Z exp jss j s all cofg, j (7.76) where "all cofg" deotes the all cofg s,, s wth all s possble sp cofguratos,.e., I cotrast to the agetc teracto whch teds to alg the sps a orderly fasho, the presece of theral eergy teds to radoze the sp oretatos ad creases the etropy or degree of dsordered. hese copetg forces the leads to a order- dsorder phase trasto. 7.F.. 7.F.. Exact Soluto For A -D Lattce Mea Feld heory For A d-d Lattce

48 7.F.. Exact Soluto For A -D Lattce Cosder a -D lattce of stes wth j ad pose the perodc boudary codtos so that s s. For a gve sp cofgurato, the total eergy s E s s s s (7.77) he partto fucto s therefore, Z, exp ss s s s Uder the perodc boudary codto, we ca wrte s s s s s where we've used s s. Hece, Z, exp ss s s s s (7.78) o proceed, we troduce a trasfer-atrx P wth atrx eleets s P s exp ss s s (7.8) Usg for each ste the bases, P s foud to be P P P P P exp exp exp Puttg (7.8) to (7.78) gves exp, P P P P Z s s s s s s s s (7.79) s s Usg the copleteess relato s s s

49 we have, P r s Z s s P (7.8a) where are the egevalues of the atrx P. Sce P s syetrc, t ca O PO dag, where be dagoalzed by a orthogoal trasforato,.e., are the egevalues of P ad O P O O POO PO O PO O O OO. hus, dag dag.e.,. Solvg the secular equato we get exp exp det exp exp exp exp exp or wth roots exp cosh sh exp cosh exp cosh sh exp cosh cosh exp sh (7.8) so that (7.8a) becoes Z, (7.8) I the therodyac lt, the Gbbs free eergy per ste s g G, l, l l Z, l l l where we've used. Hece,

50 , l cosh cosh exp sh g (7.84) he order paraeter s s g sh sh cosh cosh exp sh cosh cosh exp sh sh cosh exp sh

51 7.F.. Mea Feld heory For A d-d Lattce Cosder the haltoa of a d-d sp lattce of stes ad j H s s s j, j (7.75) j j.. of s s s where.. stads for "earest eghbors" ad the / factor reoves the double coutg of each par of stes the double su. codtos are assued to reove ay spurous surface effects. 7.F.a. 7.F.b. 7.F.c. Mea Feld Heat Capacty Magetc Susceptblty ote that perodc boudary

52 7.F.a. Mea Feld I the ea feld approxato, oe sets j.. of s j s where s the uber of earest eghbors at each ste ad s s s the average sp at a ste. Eq(7.75) thus becoes H s s s where the effectve, or ea, feld eff s s to be detered self-cosstetly. ow, the partto fucto for (7.86) s exp Z r eff s s (7.86) eff r exp eff s expeff s cosh eff s (7.87) wth a Gbbs free eergy per ste the therodyac lt g, l l Z he probablty of havg sp so that l cosh eff exp Z eff s P s s at ste s exp eff s (7.89) cosh eff (7.88) s P s s s cosh eff s s exp s eff sh eff tah eff cosh eff tah s (7.9)

53 whch s the self-cosstet equato for s.

54 7.F.b. Heat Capacty For, we have s tah s tah s tah s (7.9) where C ad C (7.9a) Eq(7.9) ca be solved graphcally as show Fg.7.6. y specto, we have s s for or C C (7.9a) hus, C s the crtcal teperature of the ferroagetc phase trasto. If vewed as a order- dsorder trasto, s stads for the order paraeter. ypcal teperature depedece of s s show Fg.7.7. ote that (7.9a) predcts a fte C for all d ad s thus dsagreeet wth the exact result for d. fact, ea feld theores typcally overestate C for d. [see Chapter 8] Eq(7.9a) gves rse to partto fucto of I Z cosh s s cosh for ad a Gbbs free eergy per ste of [see (7.88)], C C g, l C l cosh s for C C (7.9) he teral eergy s, fro (7.86), U l Z H s s Hece, the heat capacty s s

55 C U s s s s Usg (7.9), we have s s sech s s (7.96) so that C sech s s sech s cosh s cosh s s s (7.97) C s cosh C C s (7.98) As approaches to C fro below,.e., C C or, eq(7.9) splfes s s s eglectg s soluto, we have s s where C hus, (7.98) becoes l C l C cosh

56 l cosh ow, cosh cosh so that l C l C ow, for C, we have s ad C. herefore, C s as show Fg.7.8.

57 7.F.c. Magetc Susceptblty We ow tur to the calculato of the agetc susceptblty M s (7.99) Fro (7.9), we get s s sech s (7.) sech s sech s (7.) cosh s Hece so that (7.) cosh s cosh s C C cosh s C cosh s C C C C (7.) C C cosh s

58 7.G. Grad Caocal Eseble A ope syste allows exchage of both heat ad atter wth ts surroudg. oth of ts eergy ad partcle uber fluctuate about ther equlbru values. A collecto of such detcal systes s called a grad caocal eseble. At equlbru, the Gbbs etropy costrats ad S r l r (7.4) r H E (7.5) r (7.6) s axzed subject to the Wth the help of the Lagrage ultplers, ad E, we have r l E H r l E H (7.7) l H (7.8) E.e., exp E H exp E ZG H (7.8a) where the grad partto fucto s defed as Z G exp exp E r H (7.9) ow, r 7.8 gves S E (7.) E.e., S l Z E G E

59 Coparg wth the fudaetal equato for the grad potetal : U S ' (.4) or ' S U we have l Z (7.) G E (7.a) ' (7.b) so that (7.8a,9) becoes exp H ' Z G ZG exp r exp H ' exp H ' (7.) (7.) Eq(7.) s the fudaetal equato for a ope syste fro whch all other therodyacal propertes ca be obtaed. For exaple, as dscussed secto (.F.5),, X, ' ad hus, d Sd YdX d ' S X ' ' 7.G.. Exercse G.. Fluctuatos Y X ' X

60 7.G.. Exercse 7.6 Cosder the photos equlbru sde a cubc box wth volue V L ad teperature at the walls. he photo eerges are c, where s the wavevector of the th stadg wave. Copute pressure P of ths photo gas. Aswer he allowed photo states are stadg waves that vash at the walls. hus, x, y, z wth j,, L for j x, y, z so that c c x y z L () ad L,, x y z d L d Sce there are trasverse odes for each, the su over odes s, f L, d f,, Whe f s depedet of polarzato, we have, f f d f L Sce ', the grad partto fucto s Z r exp H exp,,, exp, exp exp whch gves a grad potetal, V, l Z l exp, () ()

61 L l exp d Usg we have d f 4 d f 4 c c d f L d l exp c (a) ow, l exp d l exp I d exp l exp d exp (b) he st ter volves the fucto l f x x e x I partcular x x f l x e e x where we've used l x x x ad, by repeated applcato of the L'Hosptal rule, f x f ' l l x a g x x a g ' to get x x x! l x e l l l x x x x x x x e e e Also f l x l x x! l x l x x x!

62 l x l x x l x x! l l l! x x x x x x l x l x l x x x Hece, (b) becoes exp I d exp l x x l x x l x x x e dx x e x (c) ow, J dx x e x dx x e e x x dx x e x x e dx x e x y dy y e!!! where s the Rea zeta fucto. hus, (c) becoes I J! 4! 4 Hece, eq(a) s 4! L 4 L I c c 45 4 L c 45 he pressure s P V 4 c 45 4 (6) where c 5 s the Stefa- oltza costat.

63

64 7.G.. Fluctuatos he fxed quattes specfed a grad caocal eseble are, ', U E, ad hus, t s of terest to fd the fluctuatos E ad. Sce the dervato for the varace E s slar to that for the caocal eseble (see secto 7.D.), we shall cosder oly the case for here. Cobg the oralzato (7.4) wth (7.) gves r exp H ' where, X, '. hus, 7.4 ' (7.4) gves r exp ' ' H X ' X or ' X (7.4a) 7.4 ' whle gves r exp H ' ' ' X X ' ' X X ' ' ' X X X ' X [(7.4a) used] Hece, the devato s ' X ' X (7.5) he fractoal devato s

65 ' X / (7.6) whch vashes as. Hece, the grad caocal eseble approaches the caocal eseble the therodyac lt. I fact, all esebles gve the sae acroscopc (average) physcal quattes the therodyac lt. We ow attept to relate the devato (7.5) to drectly easurable quattes for the case of a PV syste. o ths ed, we start wth V ' ' V V Usg the dagra ' P V we have ' P V so that (a) becoes ' V (b) V V P ' V ' V V ow, the dagra (b) also gves P (a) V P V V ' ' V ' G V (d) P P P where we've used G ' ad dg Sd VdP ' d. Also fro (b), we have P (c) V V ' ' P ' V ' P V Sce ' ', P, we have ' ' ' V P P P V where (d) was used. Hece (c) becoes

66 V ' V V where V V P so that fally, wth the therodyac detfed wth the statstcal, eq(7.5) becoes (7.7) V

67 7.H. Ideal Quatu Gases I quatu echacs, the ucertaty prcple ples that detcal partcles ust be dstgushable sce there s o way to ascerta ther dettes whe the dstace betwee the s saller tha relatve oetu. p, where p s the ucertaty ther As was show the sp-statstcs theore of quatu feld theory, dstgushable partcles -D space ust obey ose-este (Fer-Drac) statstcs f they have tegral (half-tegral) sps. [ote: recet studes the o-tegral quatu hall effects as well as hgh C supercoductors led to the possble exstece -D systes of ayos that ca have arbtrary value of sps.] I geeral, quatu effects are ost otceable whe the syste s ear ts groud state,.e., low for acroscopc systes. For a deal gas of detcal partcles, the haltoa s where where H H H s the -partcle haltoa of the th partcle,.e., H p p s the oetu of the th partcle. If the partcles are cofed a rectagular volue V LxLyLz wth perodc boudary codtos, the allowable egevalues for p are p lx, ly, lz (7.9) Lx Ly L z where l,, for x, y, z. hose for H are (7.) where s s the sp copoet alog the z-axs. z I the -represetato, we have H wth

68 where s the uber of partcles state, exp ' Z V r H r exp ' exp '. he grad partto fucto s (7.8) exp ' For sp depedet haltoas, ths splfes to (7.8a) Z, V exp ' For bosos, there s o restrctos o s so that ZE, V, ' exp ' For feros,,, so that 7.H.. 7.H.. ZFD, V, ' exp ' ose-este Ideal Gases Fer-Drac Ideal Gases s s (7.) (7.)

69 7.H.. ose-este Ideal Gases 7.H.a. ascs 7.H.b. Itegrato 7.H.c. Pressure 7.H.d. uber 7.H.e. herodyac Lt 7.H.f. Heat Capacty 7.H.g. Hgh lt 7.H.h. Exercse 7.7

70 7.H.a. ascs For -sp free bosos, ZE, V, ' exp ' exp ' (7.) where,,, so that Dog the suato, we get. ZE, V, ' whch gves a grad potetal exp ', V, ' l Z, V, ' E E l exp ' he average uber of partcles s E ' V exp ' exp ' exp ' (7.5) (7.6) (7.4) where the s called the occupato uber of state ad exp ' where z exp ' exp s the fugacty. z z (7.7) [ ote: eagful usage of the fugacty as defed abopve requres the plct assupto of. A ore geeral defto s z exp '.] ow, the lowest value of s wth,, ad

71 exp ' z (7.8) z Sce the uber of partcles caot be egatve, we ust have exp ' or ' z whch ples z ad ' hus, addg partcles to the gas actually decreases ts teral eergy. he case ' eas that s o loger fxed. hs apples to photos, phoos, or ay bosos that serve as "carrer" of teractos. I the classcal lt vald for sall, we expect (7.7) to becoe the oltza dstrbuto exp. hs ca be acheved f z exp or, sce, f ad '. ', have o classcal lt. z. hus, the lt z for all, exp ' correspods to Icdetally, ths eas that, carrers of teractos, wth Of partcular terest s the case z for whch. hs eas the groud state s acroscopcally occuped ad we have a ose- Este codesato. Sce the groud state plays a proet role, we expect the stuato apples whe ad '. For exaple, as a helu lqud becoes a superflud, ts checal potetal drops fro a fte value to zero.

72 7.H.b. Itegrato For a acroscopc syste, we ca replace the suato over states wth tegrals,.e., V V d d p However, the possblty of a ose-este codesato eas that the state requres specal atteto. hus, V f ' d p f p where s the volue p-space occuped by the groud state, ad ' V s the p-space volue wth tae out. Sce V p f f d p f ' V f d p f p ' If f s sotropc,.e., f f p, the f f wth, we have 4V f f dp p f p p (7.) where the "radus" p p L wth of the groud state s gve approxately by L / V L / xlylz As a exaple, (7.6) becoes 4V dp p exp ' p z 4V z dp p z p exp p exp p ' z (7.)

73 Slarly, (7.5) becoes, V, ' l exp ' E 4V dp p p p l exp ' 4V l z dp p l zexp p p (7.) Settg x p p p where the theral wavelegth s gve by h (7.5) eq(7.) becoes 4 V f f dx x f x x 4 V f dx x f x x where x p hus, eq(7.) becoes L 4V (7.) E, V, ' l z dx x l z expx Slarly, eq(7.) gves x z 4V z dx x z (7.4) x exp x z

74 7.H.c. Pressure he pressure of the gas ca be obtaed fro (.) ad (7.) as P V E 4 l l exp V z dx x z x x (7.6a) Cosder ow the tegral l exp I z dx x z x z dx x exp x z / g / z z / where we've used the Gaussa tegral forula (7.7a) dx x exp x ad the defto g Hece z / z (7.7) z I z z 5/ 5/ 4 Fro (7.7), we see that, (cf. Fg.7.), g5/ g 5/ 5/ 5.4 (7.4) where s the Rea zeta fucto. ext, 5/ 4 g z

75 a l l exp z dx x z x a dx x z a l l a Hece, (7.6) becoes 4 P l z I z z V ote that sce l z g5/ z V (7.6) z, the cotrbuto of the groud state would have bee lost wthout the specal treatet of (7.).

76 7.H.d. uber he average partcle desty of the gas ca be obtaed fro (7.4) as V z 4 z dx x V z x exp x z (7.9a) Cosder ow the tegral K z dx x exp z x z (7.9b) Usg exp x l z exp x z z exp x exp x we have, fro (7.7a) that di K z z z d z z / dz dz z z / g / (7.9c) Aother, perhaps ore coo, for of (7.9b) ca be obtaed by puttg (7.9b) so that y x K z dy y z y e z dy y y z e (7.9d) ow, fro (7.9c), we have K z g/ z 4 ote that, (cf. Fg.7.), g/ g / /.6 (7.4) where s the Rea zeta fucto. ext, a a z l a exp x z a z dx x z l dx x z

77 Hece, (7.9a) becoes z 4 K z z V z z g V z / whch ca be used to obta '. z (7.9)

78 7.H.e. herodyac Lt We ow cosder the therodyac lts,, V wth V of the expressos P l z g5/ z V (7.6) z g/ z V z Obvously, for z, P g5/ z g/ z (a) (7.9) Hece, we eed oly cosder the case z, whch, as dscussed secto 7.H.a, correspods to the ose- Este codesato lt wth ' ad usually. For z, the st the ter (7.9),.e., l z V z z deotes the desty of partcles occupyg the groud state whe the syste s at teperature. If s ept costat, we have (b) (c) (d) z l V z z (e) Cobg (d) ad (c) turs (7.9) to g / g z / As ca be see Fg.7., for z z (7.45) g/ z s a ootocally creasg fucto of z wth a axu at z. hus, f we eep reducg whle eepg costat, C g / z wll becoe saller tha below a crtcal teperature C gve by

79 C g / / C g / C / g / / (7.48).6 For C, we have g / so that ad we have a ew phase characterzed by a acroscopc occupato of the groud state. he fracto of partcles the codesate s g/ C C / (7.49) Codesato ca also occur f we eep creasg whle eepg costat. the crtcal per partcle volue s gve by v C C (7.47) g / ote that also serves as the order paraeter of the phase trasto. he case thus dcates the coexstece of the "oral" ad "codesed" phases. A plot of vs s show Fg.7.. urg ow to the pressure as gve (7.6), we beg by wrtg (d) as so that hus x l x V x l Vx x or z V x V l l z l l V V V V V z l l V V V where we've used l a l a. If ths s cobed wth the fact that for z, a l l z V V the st ter eq(7.6) ca be dropped so that

80 g 5/ z z P for g 5/ z At the crtcal pot,, ad z, so that PC C g C 5/ Usg (7.47) ad (7.48) gves C (7.44) / PC g g/ vc C g/ v g C C / where we've used v 5/ g 5/ 5/ (7.5) C C. For C, (7.44,7) gves P g 5/ g v g C 5/ / (7.5a) rregardless of the fracto of partcles the codesed phase. hus, treatg (7.5a) as a fucto P P v where v v C gves the coexstece curve the P-v plae [see dotted le Fg.7.4]. For a gve, the rego to the left of the coexstece curve s the coexstece rego where P s a costat gve by (7.5a).

81 7.H.f. Heat Capacty Sce the depedet varables for a grad caocal eseble s, X, ', the etropy per ut volue ay be wrtte as S S s l V V V ' Usg the dagra P P, we have s S V Sce, g 5/ z P g 5/ z for z V '. (7.44) where z exp ' ad, we have ad z ' z V ' V ' ' z d g z g z dz 5/ 5/ V ' where we've used (7.4). hus, for z, ' g / z ' s g 5/ z 4 g5/ z g / z 5 ' g z g z 5/ / 5 ' g5/ z [see (7.45)] 5 g z l z (7.5a) 5/ Settg z, we have so that 5 s g for z s 5/ / as (7.5b), agreeet wth the rd law. ow, the heat capacty at costat desty s

82 c s ow, fro (7.45), we see that for z, g 4 / dg/ z z z dz z g z g z / / z so that / z z g/ g z z (7.5) Hece, for z, 5 c g 4 5/ z 5 g/ z g/ z z z z g/ z z 5 9 g (7.5a) 4 4 / g 5/ z g / z where (7.45) was used. For the case z, eq(7.5b) gves ote that 5 c g 4 g so that 5/ / / c s cotuous at C (7.5b) as show Fg.7.5.

83 7.H.g. Hgh Lt I the hgh lt, z so that becoes 4 g5/ z dx x l z expx (7.7) 4 g5/ z z dx x x exp 4 z z ow, by defto, g so that z z dg z z z g z dz herefore, we have, 5/ / / g z g z g z z as z Hece, (7.45) becoes / z exp ' (7.54) whle (7.44) splfes to z P (7.55) V Fally, the heat capacty (7.5a) becoes 5 z 9 c 4 4 (7.56) V

84 7.H.h. Exercse 7.7 Copute the varace for. Aswer Fro the defto r r exp H ' we have r exp H ' ' ' V V ' V () ow, for, we have z C z V g z Hece, wth z z ' V / z ad so that (7.9) gves () z dg z g z dz eq() becoes z V z g / z ' z V z V V V g / z g / z g / z where we've dropped the V g/ z g / z () ters sce they vashes as.

85 7.H.. Fer-Drac Ideal Gases 7.H.a. ascs 7.H.b. Itegrato 7.H.c. Pressure 7.H.d. uber 7.H.e. Low eperature Lt 7.H.f. Heat Capacty 7.H.g. Exercse H.h. Exercse 7.9

86 7.H.a. ascs For s-sp free feros, where s s ZFD, V, ' exp ' s s sz s exp ' s ad,,, so that we ve assued a sp depedet haltoa. Z FD s, V, ' exp ' s whch gves a grad potetal. Also, Dog the suato gves s exp ', V, ' l Z, V, ' FD FD l exp ' s he average uber of partcles s FD ' V s s exp ' g exp ' exp ' exp ' (7.6) (7.58) (7.59) where g s ad s called the occupato uber of state l ad g exp ' g z exp exp where z exp ' gz s the fugacty. z (7.6) [ ote: eagful usage of fugacty requres the plct assupto of ].

87 ow, (7.6) has o sgularty f all quattes are real. Hece, we ca have ', whch correspods to g. [see Fg.7.6]. At or, we have g for ' ' (7.6a) whch s a step fucto wth dscotuty at ' F. hs zero teperature checal potetal F s also called the Fer eergy. he agtude of the correspodg oetu p s called the Fer oetu. A F F pcturesque way to descrbe (7.6a) s to say that the feros are a Fer sea wth a Fer surface at. hs s a helpful reder that oly partcles ear the F surface are easly excted [see Fg.7.6]. he fact that F eas that addg partcles to the gas s "dffcult" sce t creases the teral eergy of the gas. I the classcal lt vald for sall, we expect (7.6) to becoe the oltza dstrbuto exp. hs ca be acheved f z exp or, sce, f ad '. z. hus, the lt z for all, exp ' correspods to

88 7.H.b. Itegrato For a acroscopc syste, we ca replace the suato over states wth tegrals,.e., V V d d p If f s sotropc,.e., f f p, the 4V f dp p f p (7.6) I practce, t s soetes ore coveet to wor wth eerges. o ths ed, we defe the desty of states by V d d V d ds where S s a space surface o whch the eergy has a costat value. For a free partcle, V d V V so that V 4 / d (7.6a) As a exaple, (7.6) becoes 4V dp p g exp p '

89 4V dp p gz exp p z (7.64) / V d g 4 exp ' / V d gz 4 exp z (7.64a) Slarly, (7.59) becoes FD V g dp p p 4V,, ' l exp ' 4V g dp p l z exp p / V g d l z exp 4 (7.6) (7.6a) Settg x p p p y x where the theral wavelegth s gve by h (7.5) eq(7.6,a) becoes 4 V f dx x f x V 4 dx x f x / V 4 dy y f y

90 V dy y f y (7.6b) ote that dx x f x dy y f y (7.6c) hus, eq(7.6) becoes 4V FD, V, ' g l exp dx x z x V g dy y l z exp y (7.6a) (7.6b) Slarly, eq(7.64) gves 4V g dx x exp V g dy y exp z x z y z z (7.64a) (7.64b)

91 7.H.c. Pressure he pressure of the gas ca be obtaed fro (.) ad (7.6a) as P FD 4 g dx x l z exp x V (7.65a) / g dy y l z exp y (7.65b) Cosder ow the tegral l exp I z dx x z x / dy y l z exp y where f Hece, z z dx x exp x z / f / z z / (7.65b) (7.66) z I z f5/ z 4 whle (7.65a) becoes P g f5/ z (7.65)

92 7.H.d. uber he average partcle desty of the gas ca be obtaed fro (7.64a) as V 4 g dx x exp z x z (7.64b) g dy y exp z y z (7.64c) ote that s a costat sce both ad V are ept costat the grad caocal eseble. Cosder ow the tegral K z dx x exp z x z / dy y exp z y z Usg expx l z exp x z z exp x exp x we have, fro (7.65b) that di z z K z z dz / z so that f z / z f z dy y exp y z (7.64d) or, wth ', f dy y y exp (7.64e) whch s the ore falar defto of the Fer tegral f. Hece K z f/ z 4 Hece, (7.64b) becoes

93 or g f/ z (7.67) z z z (7.67a) / / g ow, the st few coeffcets the verso of a seres x x b y y y y a x x are gve by (see Arfe ) b a b a a Applyg these to (7.67a) gves a b 5 a a a / / z (7.69) g g g whch gves the depedece of z ad hece '. Sce /, we have z as Sce z exp ' z as, ths eas ' as ' as

94 7.H.e. Low eperature Lt We ow tur to the evaluato of the Fer tegral f dy y y exp (7.64e) the lt of low or step fucto so that ts dervatve d exp y dy exp y. hus, y y exp s essetally a vashes everywhere except for y. o tae advatage of ths, we tegrate (7.64e) by part to get f dy exp y exp y y exp dy y y exp y exp y dy y y exp t t exp dt t [ t y exp ]! expt v dt t t!! exp ote that the su over has o upper lt f s ot a teger. For, we ca replace the lower lt of the tegral by so that! expt t f v dt t!! exp! v!! I

95 where I v!! t t exp I dt t (7.7) exp ow, tag t t, we have I d t t t t exp exp t exp t dt t exp exp dt t expt I t Hece, I for odd. For eve, we have j exp I dt t exp t exp t exp j j j dt t t j j j j! j j j whle fro (7.7), we have I. ow, j dt t j t! so that I! [ eve ] Puttg everythg together, we have f v I!!! wth!! 6 v (7.7a)

96 For exaple, (7.67) becoes Usg we have f g / / /!! 6 5!! g 4 ' ' (7.74) 6 / / / whch, for, becoes so that / 4 g ' / ' 4g hus, (7.74) ca be wrtte as so that / ' ' 6 g / / / F 8 / F (7.75) / ' F / ' 8 F / F / ' F / F F / F / F (7.76) F

97 7.H.f. Heat Capacty he teral eergy s gve by U H s V g dy y / exp z y z V 5 g f5/ where we've used (7.6b), (7.64e), Fro (7.7a), we have, ad z exp ' exp y 5 5/ / f5/!! 5 5/ / 5/ f 5 4. U V 5/ / g 5 4 Usg g we have 4 (7.74) 6 / / / g / 4 / / 6 / 4 8 so that / 4 8 U / 5/ / V V 5 5 / 5/ /

98 5 [ V ] Usg ' 5 ' ' F (7.76) F we have U F 5 F F F 5 F (7.77) 5 F so that C V U V F F (7.78) F

99 7.H.g. Exercse 7.8 Copute for. Aswer Fro we have, exp ' Z V r H Z r exp H ' ' V Z V Z r exp H ' ' Z so that Z Z ' V Z Z ' V Z Z Z ' ' V V ' V herefore ' V () For, eq(7.74) splfes to V 4 / ' () g / Hece gv 4 ' 4 gv ' / ()

100 gv ' V ow, () gves ' 4gV so that (4) becoes / / ' (4) / gv ' 4gV V / / g 4 V 4 V (5) Hece, by (), we have / g 4 V 4 V (6)

101 7.H.h. Exercse 7.9 See secto 7.H.b.

102 Suary ose-este Gases P l z g5/ z V z g/ z V z C / g / vc g /