c:374-7-ivap-statmch.docx mar7 Statistical Thrmodynamics: Sublimation of Solid Iodin Chm 374 For March 3, 7 Prof. Patrik Callis Purpos:. To rviw basic fundamntals idas of Statistical Mchanics as applid to a pur solid and pur diatomic gas. Apply common quations for translational, rotational, and vibrational partition functions to comput th quilibrium constant for I(s) ---> I(g), i.., th vapor prssur, and H, both as a function of tmpratur.. Compar th calculations with what you masurd arlir in th smstr. Introduction: Hr w giv som nots to clarify th svral pags of Statistical-Mchanical background and application to th vapor prssur of I solid rproducd blow from a txtbook. a. chmical potntial Th Van t Hoff Equation tlls how G and chang with tmpratur, thrfor how K changs with tmpratur. This includs th quilibrium constants for phas changs. is calld th chmical potntial. At constant T and P, it is dfind as: G dg T dg S dt P, n, n G dt P G whr G is th partial molar Gibbs fr nrgy, n T, P, n V T, n, n dp G dp n T, P, n dn but is most commonly calld th"chmicalpotntial"of componnt. G dn n T, P, n dn dn But our xprimnt is don at constant T and V. Now thr is no non-pv work to worry about. It is nrgy instad of nthalpy that w ar concrnd with. Th Hlmholz fr nrgy, A, is what dtrmins quilibrium constant, usful work, and spontanity at constant volum. Pag of 4
A = E TS by dfinition, whr E in this documnt is nrgy. In trms of A, th chmical potntial bcoms: A da T da V, n, n S dt A dt V A whr A n T, V, n T, n, n P dv G dv n T, V, n dn This is th chmical potntial whn V is constant. G dn n T, V, n dn is th partialmolar Hlmholtz fr nrgy for componnt Partial molar quantitis ar ncssary to talk about proprtis of mixturs in which th composition changs. W usd a mixtur of substancs for illustration, but th sam quations apply to any numbr of componnts. For a pur substanc, lik pur I = molar G with units of J/mol at constant tmpratur and prssur. = molar A at constant tmpratur and volum. Whn or mor phass ar in quilibrium, th chmical potntial is th sam in ach phas for vry componnt. This is anothr way of saying th obvious: G = for transfrring any componnt btwn any two phass at constant T and P. No mattr what th conditions, whn or mor phass ar in quilibrium, th chmical potntial is th sam in ach phas for vry componnt. A = for transfrring any componnt btwn any two phass at constant T and V. By th way, th Van t Hoff Equation for th constant V cas is: dn ln K A A E S Divid by - : ln K R Subtract for twodiffrnt valus of T, assuming constant H K( T) r E ln K( T ) R E T TS T and S This mans that in th vapor prssur xprimnt, th plot of lnp vs /T has a slop of -E /R, not -H /R. It is asy to show that H = E + from th dfinition H = E + PV. Pag of 4
b. Statistical Thrmodynamics Statistical mchanics was invntd by Boltzmann. It is concptually quit simpl, but is unfortunatly prsntd in txtbooks in such a way as to appar frightning and impossibl to larn. You alrady know th basic ida: G = -lnk or th quivalnt statmnt: G G.3 G 57 K at constant T=98 K and P. At constant T and V, this bcoms K A A.3 A 57 at constant T =98. This mans if A = -57 J, K= x ; if A = +57 J, K= x - Th abov xprssion for K is th wll-known Boltzmann distribution, which w hav bn constantly applying this smstr in lctur and lab. This is bst mmorizd as th simpl ratio of probabilitis to b in nrgy lvls and at quilibrium: K/ = P P N N g g E A.3 A 57, whr N and N ar th numbrs of molculs in stats and, and g and g ar th dgnracis of th stats and. g is th numbr of diffrnt stats with nrgy = E. Th dgnracy is what Boltzmann calld th numbr of availabl stats in his rmarkabl molcular statmnt of ntropy: S = kbln(g). (usually writtn as S = kblnw). Boltzmann s constant kb =.38x -3 J K - molcul -. Whn multiplid by Avogadro s Numbr, Boltzmann s constant bcoms R = 8.345 J K - mol -. (Thus whn on ss th xprssion xp(-e/kbt), you immdiatly know that th units of E ar J/molcul, instad of J/mol) Thrfor S = S S= Rln (g) - Rln (g) = Rln(g/ g). And, g g S R giving: K g g E S R E E TS E TS ( E TS ) A Pag 3 of 4
c. Partition functions At th outst, lt s b clar that this trribl thing (partition function) as usd hr is nothing mor than th numbr of availabl stats in a constant tmpratur systm. Q Ei kt stats, i, a wightd sum of stats wightd by Boltzmann factors, which is what is mant by availabl. As th stat nrgy incrass, it is lss availabl at a givn tmpratur. (Th most vidnt display of this is atmosphric prssur as a function of altitud!) Now, if w sum ovr E lvls, multiplying ach Boltzmann factor by th dgnracy of th nrgy lvl, w gt th quivalnt statmnt: Q g i lvls, i Ei kt lvls, i TSi k Ei kt lvls, i Ai kt A kt, or A ln Q, or A ktln Q kt This is coincidntly closly rlatd to th Q in G = G +lnq. d. molcular partition functions In th gasous stat, a molcul s translational, rotational, vibrational, and lctronic dgrs of frdom bhav indpndntly. Th total numbr of availabl stats, qg, is just th product of th individual partition functions for th various dgrs of frdom: qg= qtrans qrot qvib ql. Each of ths is th sum ovr all quantum stats, ach wightd by its dgnracy and Boltzmann factor. Ths ar wll approximatd by simpl intgrals for translational and rotational, bcaus th nrgy lvls ar quit clos togthr. For vibrational and lctronic w must sum. Vibrations ar assumd to b harmonic oscillators, for which th sum of Boltzmann factors is a simpl powr sris that can asily b shown to b q vib ( h kt ). Bcaus in most molculs all lctronic xcitd stats ar so high abov th ground stat, qlctronic=. This is tru for I bcaus th ground stat of I is non-dgnrat.. Vapor Prssur = Kq = xp(-a /) Finally, instad of quation (34), which has bn mad compltly baffling by simplifying it to dath, w will us A= Agas -Asolid = -lnqgas + Qsolid) and vary th concntration (which appars in qtrans disguisd as th volum = n/prssur) in th attachd spradsht, until w find th prssur that maks A =. That will b quilibrium, and that will b th vapor prssur Pag 4 of 4
A ln[( ) ( h vib ) ] ln( q mkt 3/ kt kt kt h p hcb solid ), whr qsolid is givn in quations 3 and 33, and on th sprad sht. ------------------------------------------- A fw mor hlpful dtails will b mntiond during our lab mting, during which w will work on stting up your sprad sht. Th spradsht is complt xcpt for th formulas for th gas partition functions, which hav all bn st =. You should mak a start on filling in ths formulas bfor coming to class if possibl. Th tabl of xprimntal valus is from a prvious yar. You should ntr th data you took this yar in plac of that data. Pag 5 of 4
Blow is from: Exprimnts in Physical Chmistry, 5 th Ed., D.P. Shomakr, C.W. Garland, and J. W. Niblr, 998 Pag 6 of 4
Pag 7 of 4
Pag 8 of 4
Pag 9 of 4
Pag of 4
Pag of 4
Pag of 4
Pag 3 of 4
Pag 4 of 4