APPLICATIONS: CHEMICAL AND PHASE EQUILIBRIA

Transcription

1 5.60 Sprn 2007 Lecture #28 pae PPLICTIOS: CHMICL D PHS QUILIBRI pply tattcal mechanc to develop mcrocopc model for problem you ve treated o far wth macrocopc thermodynamc 0 Product Reactant Separated atom Product & reactant enery level ε r = -D 0,r ε p = -D 0,p Chemcal equlbra Ga phae: Calculate Kp from mcrocopc properte a + bb Φ cc + dd Δ G = RT K = cg + dg ag + bg p C D B eed G 0 for each pece G = + pv = Q + pv = Q + ( tran!) Q = q q nt ( tran nt ) ( tran nt ) ( tran nt ) Q = q q! = q q + G = Q + = q q For molecule n the a phae, nternal deree of freedom are rotaton and vbraton and electronc tate.

2 5.60 Sprn 2007 Lecture #28 pae 2 qnt = qrotqvbqelec qelec D0 = e Docaton enery D 0 from round electronc level Uually no other electronc level matter qvb = e = + e + e + e + = e n nε0 ε0 2ε0 3ε 0 ε0 ote we ve et the zero of vbratonal enery a the lowet vbratonal level. The zero-pont vbratonal enery ha been ncluded n q elec by un the docaton enery rather than the bottom of the electronc potental enery. (300 K) 200 cm - Mot molecular vbratonal frequence > 500 cm - q vb - eed to calculate t, but t not lare We have not treated q rot. Level are OT evenly paced: ε rot = J(J + )ε 0,rot where J = 0,, 2,, and ε 0,rot cm -. Hh-T lmt: q rot = /ε 0,rot We ve een that q tran 0 30, q tran / 0 6. For multple pece n chemcal equlbrum, need to ue for each pece and ue partal preure value n pv term. Uually enery effect domnate n chemcal equlbra. Relatvely mple cae: Sold-tate chemcal reacton a Φ bb e.. omerzaton n an oranc molecular crytal molecule of, B molecule of B, + B = o tranlaton or rotaton, no chane n pv all the chane n the electronc and vbratonal enery If vbratonal frequence are the ame, then only the dfference n docaton enere needed.

3 5.60 Sprn 2007 Lecture #28 pae 3 Q e = = Ω e mcrotate enere Recall the entropy due to mxn of two pece!! D+ BDB Q = Ω ( ) e = e = e e!!!! ( D ) ( D ) = 0 B = 0 B enere! B B = q qb = ( q + qb) = q ( + ) where = qb q = e = e!! = 0 B ( D D ) ΔD0 The bnomal theorem ve a mple cloed form for the um. It convenent here to redefne the zero of enery a D o q = and Q = + = + a + a + a + + a B B The round tate of the ytem ha all molecule Sytem enery ( B ) = B ΔD 0. Probablty that the ytem ha th enery p = a Q The averae number of B molecule 2 3 a + 2a 2 + 3a a Q Q = p( ) = = = Q Q B B B B and = U = B Δ D0 If vbratonal enere dffer then q vb alo need to be ncluded. In eneral, low vbratonal frequency favor a pece nce the level are cloely paced, o there are many low-enery tate of that pece. For chemcal reacton nvolvn covalent bond, the bond enere domnate over entropy. But the treatment we ve ven could alo be ued for much more ubtle chane lke rotaton of CO molecule n a CO crytal.

4 5.60 Sprn 2007 Lecture #28 pae 4 ote that th treatment aume no nteracton between dfferent unt cell. Th ve re to a farly entle T-dependence of the equlbrum. Sold-old phae equlbra Phae tranton between dfferent crytale phae may be eaer to treat then chemcal reacton, becaue there jut one tate at any temperature: the crytal n phae α or the crytal n phae β. Problem et problem nten model for crytale phae α and β, frequence ν α, ν β Bndn enery per atom: ε α, ε β 0 Phae α Phae β ε β ε α Bndn enery lke docaton enery. It nclude zero-pont vbratonal enery, o no need to account for th eparately. You can predct the phae tranton temperature baed on a mple frt prncple model. Phae tranton only occur f the crytal wth troner bndn enery, e.. α phae, alo ha hher vbratonal frequency. ote that th treatment aume complete cooperatvty: the crytal ether all one or all the other phae. Th ve re to an abrupt T-dependence of the equlbrum.

5 5.60 Sprn 2007 Lecture #28 pae 5 Many ntermedate cae occur. xample: helx-col tranton n bopolymer. Interacton make few lon ement of helx or col more table than many hort ement. (Smlar for manetc and nonmanetc doman n a ferromanetc crytal, and n all ort of nteractn ytem). In th cae the T-dependence not completely abrupt, but not nearly a radual a n the non-nteractn lmt. Can alo calculate old-a phae equlbra,.e. vapor preure over the crytal. ume monatomc deal a only tranlatonal partton functon. Famlar reult: 2π m = 2! = μ = h TV, Q V Q 2π m Q = V 2 + h TV, Q 2πm 2πm V = V + = 2 2 h h μ = 2π m 2 h p Condton for phae equlbrum: ( 2πm ) 2 ( ) (deal a) μ = μ 2π m μ = ε 3 = μ = 2 hν h 52 3 ( ) hν ( 2 m π ) p ( ) 3 m ν 2 2 2π ε = = h p p = 3 ν e ε ve vapor preure p(t) over the crytal! Stron bndn enery or low T ve low preure a expected. Low vbratonal frequency alo ve low p, nce th allow hher entropy n the crytal. So you can calculate the p-t phae daram that you decrbed before n macrocopc term only. p 52