4. More general extremum principles and thermodynamic potentials

Transcription

1 4. More generl etremum prncples nd thermodynmc potentls We hve seen tht mn{u(s, X )} nd m{s(u, X)} mply one nother. Under certn condtons, these prncples re very convenent. For emple, ds = 1 T du T dv + µ T dn m µ dn T mmzes S t constnt U nd V. But wht do we do f we re workng t constnt (T,V) or (T,)? We then hve open systems to del wth, where energy must flow (to keep T constnt), or the volume must chnge (to keep constnt), etc. Ths problem rses for mny lbortory rectons n chemstry. As t turns out, f thermodynmc vrbles other thn U nd V re held constnt, thermodynmc potentls other thn entropy or energy of the open system re etremzed. (Of course, S s stll mmzed for closed system contnng our open system of nterest.) Very convenently, these potentls cn be computed just from the propertes of the open system of nterest lone (e.g. one where T s constnt, nd therefore energy must be llowed to flow n nd out), nd the dded ssumpton tht the correspondng vrble (e.g. T) s lwys constnt n the envronment. The envronment s thus treted s bth or reservor for the ntensve vrble of nterest. Tht s, we ssume the closed system contnng our open system of nterest s so vst tht chnge n etensve vrble (e.g. U) of the smll open system does not ffect the conjugte ntensve vrble n the envronment (e.g. T; the energy deposted or tken from the envronments by the open system s too smll to chnge T n the envronment). Our gol: we wnt potentls where our choce of ntensve vrble s n the dependent vrbles. Let s begn by dscussng n pprently obvous wy of dong ths, whch does not work. Let s sy we hve the fundmentl relton for U, but we wnt to hold T constnt, not S: U = U(S,V,n ) T = U = T(S,V,n S ) V,n solvng for S = S(T,V,N ), nd nsert n U we seem to be ble to get U = U(T,V,n ). Now we cn hold T constnt. The problem wth ths: T s the slope of U(S), nd epressng functon n terms of ts own slope leves the ntercept ndetermnte: U s no longer completely defned, nd we do not hve fundmentl relton to whch the lws of thermodynmcs cn be ppled. We do however wnt n equton n terms of the slope. The soluton s to epress the slope m of y() n terms of the ntercept ϕ (or vce-vers). The ntercept s functon of slope does contn ll the orgnl nformton bout the functon y(). The process of trnsformng functon y() nto ϕ(m) s clled Legendre trnsform. Defnton: The Legendre trnsform of y() s ϕ(m), the ntercept s functon of slope. ϕ(m) nd y() re unquely relted s long s 2 y 0 (no nflecton pont occurs n the 2 d orgnl functon).

2 y() y = + b y(m) b{ y() y = + b ϕ(m) m b{ b m y() } m ϕ(m) } m The procedure s gven mthemtclly below, nd summrzed n the fgure bove. Let m = dy = m() = (m). d If there s no nflecton pont, nd m re unquely relted (but y cn no longer be obtned to better thn shft long the y s). Net we compute ϕ = y() - m = y((m)) - m(m) = ϕ(m). Ths functon cn be used to obtn the orgnl y: dϕ dm = dy dm m d dm = dy d d dm m d dm = (m) Ths yelds m(), thus y() = ϕ(m()) + m() Thus we Legendre trnsform U (or S) to new potentls to obtn fundmentl reltons wth dfferent vrbles.

3 We now consder the most common of these new thermodynmc potentls, whc fcltte thermodynmc clcultons on open systems wth choce of ntensve stte functons held constnt ) The Helmholtz free energy H = U[S] (The squre brcket ndctes trnsform S T) U[S] = A(T,V,n ) = U U S S V,n From ths follows = U TS A da = SdT dv + µ dn nd T = S. V,n When s A mnmzed nsted of U? The nswer s, t constnt T, nsted of constnt S. To see ths, consder system open such tht ts T s constnt (het flow n or out s llowed s needed qussttclly). Mnmzng U of the totl closed system (open system of nterest nd reservor t constnt T ), we hve 0 = du tot = du + du r (by 1) = du + TdS r (V r nd n r re constnt) = du TdS ds r = ds sys = dq r T = dq T = du TdS SdT (dt=0 ssumed) = d(u TS) = da In ths equton, the subscrpt r ndctes the reservor, no subscrpt ndctes the open system of nterest. Thus, A s ndeed the functon mnmzed when the totl system reches equlbrum, whch mens tht the open system of nterest reches equlbrum t constnt T. Note tht da = du tot, but cn be clculted usng only vrbles of the open system, U, T, nd S. In ddton, d 2 A = d 2 (U + U r ) = d 2 U tot > 0 t constnt T,V,nd n. Thus, n nlogy to the energy of the closed system, A s mnmzed n system t constnt temperture (n contct wth het reservor). Consder smple pplcton of the Helmholtz free energy, provng tht pressure s equlzed between two subsystems seprted by fleble wll. For two systems coupled by movble mpermeble membrne t constnt T t equlbrum: da = 0 = S 1 dt 1 dv 1 + µ 1 dn 1 S 2 dt 2 dv 2 + µ 2 dn 2 Becuse T s constnt nd the wll s mpermeble, only the pressure terms re nonzero, nd snce da = 0 t equlbrum, 0 = ( 1 2 )dv 2 (becuse dv 1 = dv 2 ) 1 = 2 ressure s the quntty equlzed between two systems seprted by moveble membrne. We lredy proved ths erler n the entropy representton, but the proof s lttle shorter n the Helmholtz representton becuse temperture s lredy tken cre of.

4 b) The Enthlpy: we trnsform V U[V ] = H (S,,n ) = U U V V S,n = U + V From ths follows, usng the Gbbs-Duhem relton, dh = TdS + Vd + µ dn H = V. S,n When system s t constnt, then 0 = du tot = dh. Agn, unlke du tot, dh cn be clculted usng only system prmeters. As n emple, consder smple system t constnt nd vnshng µ dn (no mss flow). It then follows tht dh = TdS = dq f the recton s done qussttclly. Thus the enthlpy s the het relese of qussttc recton t constnt pressure, just lke the energy s the het relese when the volume s held constnt. Becuse mny rectons run t constnt pressure nd do not echnge prtcles wth the envronment, ths s very useful for clcultng het echnge. c) The Gbbs free energy: we trnsform twce, S T, V U[S,V ] = H[S] = A[V ] = G(T,,n ) = H TS = A + V = U TS V From ths nd the Gbbs-Duhem relton follows dg = SdT + Vd + µ dn s well s G T = S,n G = V nd G = µ n T,n for smple mult-component systems. Agn, dervton nlogous to ) shows tht dg = du tot = 0 s mnmzed n system n contct wth het nd pressure reservors,.e. T, constnt. Emple: For one-component system, G = µn, so µ (the chemcl potentl) s Emple: pressure nd concentrton dependence of the chemcl potentl, lso clled the molr free energy. Usng the Gbbs-Duhem equton, ndµ = DdT + Vd = Vd t constnt T. b b b nrt n dµ = Vd = d = nrt ln b = n(µ b µ ) for n del gs µ = µ (0) + RT ln f we mke the reference presure 0 =1 (usully tm) Becuse c = n V = RT for n del gs µ = µ (0) c + RT lnc f c 0 =1 (usully moles L ) Ths knd of equton for the chemcl potentl cn lso hold ppromtely n dlute solutons. If c n moles/lter s used for the concentrton of non-nterctng component. The reson s tht dlute solutes re screened from one nother by the solvent, nd effectvely move lke rndom gs prtcles. For n moles of solute, we cn wrte

5 n µ = n µ 0 + n RT lnc = n µ 0 n + RT ln{ c } In relty, the chemcl potentl does not scle ectly s ln becuse of nterctons mong prtcles, especlly n soluton, where screenng s not perfect, or solvent molecules cn nterct strongly wth solutes (e.g. H-bondng n n queous sugr soluton). In tht cse we cn correct the chemcl potentl: µ = µ 0 + RT lnc + Δµ corr. Lettng γ ( c Δµ corr RT,T) e be the ctvty coeffcent. ( ) µ = µ 0 + RT ln γ c γ s often used n tbultons nd mesures the devton from the del non-nterctng prtcle model. Fnlly we note the relton between the temperture dervtves of chemcl potentl nd free energy, whch yeld the concentrton dependence of the entropy: µ = 1 G = S µ(0) = S S = + Rlnc T n T n T (del soluton or gs, nd smlrly for severl nonnterctng components) As concentrton (or pressure) ncreses, nd prtcles become more confned, entropy decreses. d) Msseu functons: Legendre trnsforms of the entropy The homework wll ntroduce Msseu functons: these re the Legendre trnsforms of the entropy. For emple, S[U,V.n] s functon of T, nd n just lke the Gbbs free energy. The Msseu functons re closely relted to the prtton functons of sttstcl mechncs. For emple, S yelds the mcrocnoncl prtton functon (when U s constnt). Trnsformng S[U] we cn obtn the cnoncl prtton functon, whch s functon of T. Let us consder these two n some detl. From S tself, whch s mmzed t constnt energy U, we obtn the number of mcrosttes populted t constnt energy, W U (U) = e S /k B. Ths s lso known s the mcrocnoncl prtton functon. Now consder the Legendre trnsform S[U] = f (T ) = S 1 T U snce S = (1/T) U + (/T) V (µ/t)n for smple system. The number of mcrosttes t energy U populted t constnt temperture s W T = e (S U /T )/k B = es /k B e U /k BT = W (U)e U /k BT. The probblty of the system hvng energy E = U out of ll possble energes s gven by

6 p = W (E )e E /k B T j W (E )e E /k B T = 1 Q W (E )e E /k B T. Q s the cnoncl prtton functon, or the effectve number of mcrosttes populted t temperture T. For sttes of very hgh energy E, the Boltzmnn fctor e -E/kT s very smll, nd ther mcrosttes counted by W(E ) do not contrbute sgnfcntly to the sum of sttes.