5.62 Physical Chemistry II Spring 2008

Transcription

1 MIT OpnCoursWar Physical Chmistry II Spring 2008 For information about citing ths matrials or our Trms of Us, visit:

2 5.62 Lctur #7: Translational Part of Boltzmann Partition Function CNONICL PRTITION FUNCTION FOR INDEPENDENT, INDISTINGUISHBLE MOLECULES Q(N,V,T) = q N /N! approximation valid for q N, not assurd to b always valid corrctd Boltzmann statistics whr q = ε i /kt molcular partition function i sum ovr stats of on of th molculs GOL: to dtrmin for what systms q N is valid. PROCEDURE: 1) dvlop a (simplifid) physical pictur for q 1) PROPERTIES OF q 2) calculat a valu for q q is a masur of th total numbr of molcular stats availabl to on molcul at som tmpratur. ε i /kt proportional to population in stat i at T q total # of stats accssibl at T ( kt population in singl-particl stat i in N atom systm: N ε i whr th trm in parnthss is th probability of finding any singl particl in stat i. Considr molculs and B with nrgy lvls sktchd blow. q)

3 5.62 Spring 2008 Lctur 7, Pag 2 stat of nrgy ε i 1. Molcul has mor stats bcaus thy ar mor closly spacd in nrgy. 2. Th total numbr of thrmally accssibl stats in molcul is largr bcaus thr ar mor stats with εi lss than or comparabl to kt. Contribution of ach stat to th sum in th dfinition of q dpnds on its nrgy rlativ to kt. Thrfor (kt is an nrgy) (k = J/K) q > q B So, q plays an ssntial rol in dtrmining th probability that a molcul is in stat i. Sinc n i = N ε i / kt q n i = = ε i / kt ε i kt = ε m kt N q probability of finding molcul in stat i m dpnds not only on th nrgy of th i-th stat, εi, rlativ to kt, but on q, th total numbr of stats accssibl. ε i kt = = q m ε i kt ε m kt BOLTZMNN DISTRIBUTION FUNCTION Considr molculs and B again. Both and B hav a stat i at nrgy εi. Thrfor rvisd 1/9/08 9:36 M

4 5.62 Spring 2008 Lctur 7, Pag 3 Probability of molcul in stat i Probability of molcul B in stat i = n i N B = = = ε i / kt ε i n / kt i q N q B B It follows that B = q B < 1 q Probability of molcul bing in stat i with nrgy εi is lss than probability of molcul B bing in stat i with nrgy εi bcaus thr ar mor stats in molcul. Considr th sam molcul. Th ratio of th probabilitis of finding in two stats j and k or th ratio of populations in th two stats j and k ar P j P k = n n j k ε j kt / q (ε j ε k )/kt = = ε kt k / q What happns to q as T 0? What happns to all P j? 2. CLCULTION OF q NEED: εi, th nrgis of th stats of a molcul: translation, rotation, vibration, lctronic. STRT: with th nrgis of th translational stats to calculat TRNSLTIONL MOLECULR PRTITION FUNCTION Th εi for translational stats ar solutions to th Schrödingr quation for a particl in a box. Th translational nrgy of a particl of mass m containd in a box of dimnsions a, b, c with quantum numbrs L, M, N is ε(l,m,n) = h2 L 2 M 2 N 2 8m a b 2 c 2 rvisd 1/9/08 9:36 M

5 5.62 Spring 2008 Lctur 7, Pag 4 = ε(l,m,n)/kt L 2 M 2 N 2 q = xp a + trans 8mkT + 2 b 2 c 2 L=1 M=1 N=1 Nd to valuat sums L 2 xp h2 M 2 xp h2 N 2 = xp 8mkTa 2 M =1 8mkTb 2 N=1 8mkTc 2 L=1 L 2 L 2 L 2 xp xp 1 xp = 8mkTa 2 8mkTa 2 8mkTa 2 L=1 L=0 L=0 Now 8mkTa 2 1 Stats ar closly spacd in nrgy. pproximat sum by an intgral. L=0 L 2 L 2 xp dl xp dl xp ( g 2 L 2 ) = π1/2 0 8mkTa = 2 0 8mkTa 2 2g π 1 2 g2 x 2 dx = with g = 0 2g 8mkTa Thrfor L 2 8πa 2 mkt dl xp = 0 8mkTa so πa 2 mkt 2πb 2 mkt 2πc 2 mkt = πmkT 2πmkT = abc = V W hav valuatd qtrans in trms of quantitis w can know!! What idalizations, if any, hav w mad? CHECK VLIDITY CONDITION FOR BOLTZMNN STTISTICS, N. rvisd 1/9/08 9:36 M

6 5.62 Spring 2008 Lctur 7, Pag 5 Calculat for N 2, 1 atm prssur, 1 mol, 273K m = 28g mol 10 3 kg g = 6.0x10 23 mol x kg h = 6.63 x J s k = 1.38 x J K T = 273 K V = 22.4 litrs = 22.4 x 10 m Unit chck: 3 2 3/2 3/2 2πmkT kgjk 1 K kg K 1 K V = m 3 = m 3 J 2 s 2 kg m 2 s 2 s 2 1 m 2 3/2 m 3 UNITLESS Plugging numbrs for N 273K, 1 atm, 1 mol into qtrans yilds: 3 2 2πmkT = V = Chck condition for Boltzmann statistics, q N For 1 mol of N 2 (in our volum of 22.4 litrs), N = N 6x10 23 = = as rquird q 2.8x10 30 So n i = N q ε i /kt < 10 7 bcaus ε i /kt < 1 always on avrag, lss than 10 7 molculs pr stat probability of mor than 1 molcul in any stat is vry small (what is th probability of finding 2 or mor molculs in th ε i lvl?) corrctd Boltzmann statistics OK for molculs at T > 300K lways us simpl short cuts to avoid rptitiv calculations. E.g. dcras T from 273K to 1 K rvisd 1/9/08 9:36 M

7 5.62 Spring 2008 Lctur 7, Pag 6 q ( 1K) = q ( 273K) 1 3/2 273 dcras V from 22.4L = cm 3 to 1 cm 3 q ( 1 cm 3 ) = q ( 22.4L ) Chck condition for corrctd Boltzmann statistics, q > N, for 1 mol of lctrons in V = 22.4 litrs at T = 273K. ll paramtrs ar th sam as in N2 calculation xcpt for mass m = g mol 1 q trans N 2 m N2 Sinc q m 3/2 : m = 3/ /2 q = trans = 28 N2 So N 6x10 23 q = not 1! x10 Can't us corrctd Boltzmann statistics for lctrons at T = 273K. Must us Frmi-Dirac statistics! t what T is Boltzmann statistics OK for an lctron? Sinc corrctd Boltzmann statistics ar valid for atoms and molculs undr th vast majority of conditions, w can now calculat Q, th canonical partition function for indistinguishabl molculs. Frmi-Dirac (frmions) and Bos-Einstin (bosons) statistics nxt lctur. rvisd 1/9/08 9:36 M