P34 - REIEW NOTES Capter 1 Energy n Termal Pyss termal equlbrum & relaxaton tme temperature & termometry: fxed ponts, absolute temperature sale P = nrt deal gas law: ( ) ( T ) ( / n) C( T ) ( ) + / n vral expanson: P = nrt 1+ + L Equpartton teorem, quadrat degrees of freedom 1 o U = NfkT frst law of termodynams: U = Q + W o P( ) f W d s work done ON system = o Q s eat absorbed by system reversble (quasstat) and rreversble proesses sotermal proesses o for deal gas, sotermal proess: U = 1 Nfk T = adabat proesses f / o for deal gas, adabat proess: T = onstant and ( f + ) / f P = onstant U U eat apates: C = and CP = + P P P o for deal gas: C = C + Nk = C nr P + latent eat entalpy and onstant pressure proesses o H = U + P H o CP = P dfferental sannng alormetry transport oeffents: termal ondutvty, vsosty, dffuson onstant 1 o mean free pat: l 4πr N Q dt o termal ondutvty: = kt t dx Fx du x o vsosty: = η dz dn o dffuson: J x = D dx
Capter Te seond law of termodynams mrostate, marostate, & multplty of a marostate multplty of a system of two-state partles (.e. paramagnet) N! o Ω ( N, n) = n! ( N n)! multplty of a system wt q unts of energy parttoned among N quantum partles (.e. Ensten sold) o ( ) ( q + N 1 )! Ω N, q = q! ( N 1)! Multplty of te marostate of a omposte system onsstng of weakly nteratng oupled systems o Ω total = Ω Ω Fundamental ssumpton of Statstal Means o all aessble mrostates of an solated system n termal equlbrum are equally lkely. Seond law of termodynams o for spontaneous ange of an solated system, total entropy must nrease Large systems o Strlng s pproxmaton o example: multplty for a large Ensten sold (g & low T lmts) o Sarpness of multplty for a large omposte system peak wdt 1/ N multplty for an deal gas of N ndstngusable atoms 3N / o Ω ( U,, N ) = f ( N ) N U Entropy n terms of multplty: S = k ln Ω 3 / Entropy of an Ideal Monatom Gas: 4πmU 5 S = NK ln + N 3N o anges n entropy for sotermal expanson and free expanson Entropy of Mxng Reversble and Irreversble Proesses
Capter 3 Interatons For nteratng systems, total multplty, Ω total = Ω Ω, s maxmum at termal equlbrum leads to defnton of temperature 1 S o = T U N, o sowed onssteny wt equpartton for large Ensten sold and monatom deal gas Temperature dependene of entropy: o S ( T f ) S( T ) = T T f C T dt Trd law of termodynams: lm S( T ) = o mples lm ( T ) = C T T o resdual entropy: nulear degrees of freedom Spn-1/ paramagnet U = µ N N = µ N N o ( ) ( ) U o M = µ ( N N ) = o onept of negatve temperature n systems wt bounded energy µ o M = µ tan kt o Heat apaty: C U = = Nk os ( µ / kt ) ( / kt ) N, µ Meanal equlbrum and pressure S o P = T U, N Termodynam dentty: du = TdS Pd S Dffusve equlbrum and emal potental: µ = T N U, o emal potental for monatom deal gas Generalzed termodynam dentty: du = TdS Pd + µ dn
Capter 4 Engnes and Refrgerators Heat Engnes: Cylal operaton; eat extrated from ot reservor and onverted to work and waste eat delvered to old reservor. o Q = W + Q W s work Y engne; Q s eat FROM ot reservor, TO old reservor Q / T Q / T equalty for reversble (quasstat operaton) nequalty f entropy reated n yle e = W / Q = 1 Q / Q o ( ) ( ) Q s eat o effeny: ( ) ( ) e 1 ( T / T ) o Carnot yle: maxmum effeny for yle operatng between gven par of reservor temperatures sotermal expanson; eat absorbed from ot reservor adabat expanson; work done by workng flud sotermal ompresson; waste eat dumped to old reservor adabat ompresson; work done on workng flud Refrgerator: ylal operaton; work used to extrat eat from old reservor and delver to ot reservor. o Q = W + Q W s work ON engne; Q s eat FROM old reservor, TO reservor Q / T Q / T equalty for reversble (quasstat operaton) nequalty f entropy reated n yle COP = Q / W = Q / Q Q o ( ) ( ) o Coeffent of performane: ( ) ( ) COP (( T / T ) 1) 1 Q s eat Heat pump Steam Engne (Rankne yle): workng substane undergoes pase transton Real Refrgerator: o oolng ours n trottlng step (Joule-Tomson expanson) Joule Tomson Proess: onstant entalpy o results n oolng of flud IF ntal temperature s below te nverson temperature Lquefaton of gases: trottlng, Strlng yle very low temperature: o pumpng on 4 He or 3 He o elum dluton refrgerator o magnet oolng o laser oolng
Capter 5 Free Energy Helmoltz free energy: F U TS o for onstant T proess: F U T S and F W were W s all work ON system Gbbs free energy: G U TS + P = H TS o for onstant T and onstant P proess: G Woter were W oter s all work ON system EXCEPT expanson/ompresson work Calulaton of G for emal reatons o make use of tabulated values Termodynam denttes: o du = TdS Pd + µ dn o dh = TdS + dp + µ dn o df = SdT Pd + µ dn o G f F F S =, P =, N,, N dg = SdT + dp + µ dn µ = F N G G G S =, =, µ = P, N P, N N, P Maxwell relatons: from mxed dervatves of termodynam potentals T P o from du = TdS Pd + µ dn get = S S T o from dh = TdS + dp + µ dn get = P S S P S P o from df = SdT Pd + µ dn get = S o from dg = SdT + dp + µ dn get = P P Free energes and equlbrum o for fxed T,, N, spontaneous proesses ave df o for fxed T, P, N, spontaneous proesses ave dg G Gbbs free energy and emal equlbrum: µ = N o for a mxture of partlesg = µ o for deal gas: µ ( T, P) = µ ( T ) + kt ln P o N P T, P T,
Pase transformatons: Pure substanes o Pressure-Temperature pase dagram: In gven regon of PT spae, stable pase s one wt lowest G boundares are lnes n PT spae along w oexstng pases ave same Gbbs free energy: separate regons n w dfferent pases are stable rtal pont trple pont o Clausus-Clapeyron Relaton: slope of boundary between regons of pase dagram dp S = dt S s ange n entropy per unt amount on rossng pase boundary s ange n volume per unt amount on rossng pase dp dt boundary L = T L s latent eat per unt amount of transton at boundary For transton between ondensed pase and gas pase, an often neglet volume of ondensed pase van der Waals model: pproxmate equaton of state for nteratng partles wt fnte volume an NkT an o P + ( Nb) = NkT or P = Nb o P vs soterms for T > T, P dereases monotonally wt nreasng. No transton. dp d P for T = T, rtal pont dentfed by = and d d = for T T <, range of Pn w tree values of orrespond to gven P Transton for P at w gest and lowest solutons gve same G Identfy wt Maxwell Construton.
Free Energes of mxtures o for N moleules of type, N moleules of type, so tat mole N fraton of s x = N + N o IDEL MIXING: U and unanged by mxng. and partles ave no preferene for lke or unlke negbours: free energy of IDEL MIXTURE at spefed T n spefed pase (gas, lqud, sold) s G = (1 x) G + xg + RT x ln x + 1 x ln 1 x [ ( ( ))] G and pase at spefed T Plot G versus x for IDEL MIXTURE frst two terms gve stragt lne from x = 1 G are free energy of pure substanes n same G at x = to G at last term omes from entropy of mxng. G vs x s onave up. for gven x, lowest G s always tem omogeneous mxture o NON-IDEL MIXING: nteratons between lke partles more attratve tan nteratons between unlke partles: U (onave down) mxng Plot G versus x for IDEL MIXTURE at g T, entropy term domnates and G vs x s onave up. o for gven x, lowest free energy s omogeneous mxture wt tat omposton at lower T, ompetton between entropy and U mxng o G vs x wll be onave up for g and low x and onave down for ntermedate x. o unque lne an be drawn to be tangent to G vs x at two ponts o between tose ponts, system lowers G by separatng nto oexstng mxtures wt ompostons gven by x at te two tangent ponts. o Gves boundares of two-pase oexstene regon for tat T s T s redued toward K, ompostons of oexstng pases approa x = and x = 1. regon of two-pase oexstene s solublty gap o If pure substanes at T are solds wt dfferent rystal strutures ( α, β ) G α and G β are dfferent urves between ompostons were ommon tangent toues bot urves, separate α and β pases oexst wt ompostons orrespondng to tangent ponts
Pase anges of Msble Mxtures: o If and are msble n bot lqud and gas pases For g T, G gas < Glqud for all x. Homogeneous gas pase For low T, G gas > Glqud for all x. Homogeneous lqud pase G lqud x may ross. For ntermedate T, G gas and ( ) between ompostons were ommon tangent toues G gas and G lqud, mxture separates nto a gas pase and a lqud pase wt ompostons gven by tangent ponts. Pase anges n Eutet Systems o sold pases of and ave dfferent rystal strutures ( α, β ) but and are msble n lqud pase For g T, G lqud < G α, β for all x. Homogeneous lqud pase s T lowered, G lqud wll ross G α or ommon tangent gves ompostons of oexstng lqud and sold pases x x x s T lowered furter, G lqud ( ) wll ross bot G α ( ) and G β ( ) an draw two ommon tangents from G lqud to G α and G lqud to G β ranges of omposton were α stable, were α and lqud oexst, were lqud as lowest free energy, were lqud and β oexst, and were β as lowest free energy. t eutet temperature, sngle lne s tangent to G lqud, and G β lowest temperature at w lqud s stable elow eutet temperature, ommon tangent gves ompostons of oexstng α and β sold pases G β G α,