Statistical Mechanics

Transcription

1 Statistical Mchanics h analysis of th physical and chmical natur of systms from a microscopic prspctiv is only usful if this analysis can b connctd to th macroscopic world. Onc having such a connction, thn w can not only undrstand th dtails of th micro world, but w can also prdict chmical bhavior from lmntary analysis. his is th job of statistical mchanics. Our approach will b to dtrmin what th microscopic stats ar and thn to calculat th probability of having particls distributd among thos stats in a givn configuration. Molcular Quantum Stat Distributions: W know now, from our studis in th quantum mchanics of culs, that thr xist many quantum stats that vary with nrgy. As is oftn th cas in natur, thr is a balanc btwn th tndncy of a cul to distribut itslf among th quantum stats (ntropy) and th nrgy that is availabl in which to do that. his balanc rsults in a distribution of culs among th quantum stats that is a function of th availabl nrgy. As w shall s, that availabl nrgy is usually thrmal in natur. W can altr th distribution by putting nrgy into th systm such as subjcting th systm to E-M radiation, magntic filds, DC voltag sourcs, tc. For now, w will considr only thrmal nrgis. Considr, for th momnt, a cular st of quantum stats dfind as follows: Lt a i # of systms in stat, i, having nrgy, E i. W can spcify th stat by writing... Stat # L Enrgy E E 2 E E 4 E 5 E L # of systms a a 2 a a 4 a 5 a L in stat (occupation #) thus i i a i A otal # of systms (culs for xampl) (constraints) a i E i E otal Enrgy h lattr st of summations ar not only dfind from th systm dfinitions, but ar also limitations on th systm. hr is only so much nrgy availabl and so many culs. Both of ths two limitations, calld constraints, must b accountd for whn dtrmining th statistical distribution of th cul among ths quantum stats.

2 2 Now, considr that thr is som distribution of culs among ths stats. If th particls ar distinguishabl, th following holds: h numbr of ways, W, that any particular distribution of th a j 's can b ralizd is th numbr of ways that A distinguishabl objcts can b arrangd into groups such that a ar in group, a 2 ar in group 2, tc., and is: Wa ( ) A a a a... 2 A a k k his xprssion givs th total numbr of ways that a particular distribution of a's can b arrangd. h distributions ar subjct to th constraints givn abov, namly thr ar a fixd numbr of particls and a fixd or limitd amount of nrgy. hr ar many arrangmnts that fit th constraints, howvr, w will wish to find th distribution of a's in which thr ar a maximum numbr of arrangmnts. h rason for this will bcom apparnt in th following xampl, howvr, th tndncy towards disordr or ntropy is th driving forc hr. his particular distribution will b th maximum of W undr th givn constraints and is found using th mthod of Lagrang Multiplirs. his is discussd in Raff, Chaptr 7. and will not b rviwd hr. I will only prsnt som argumnts hr and rsults. Sinc som calculations will rquir calculating larg factorials, it is convnint to us an approximation schm known as "Stirlings" approximation. For factorials in th rang of 80! to 2000!, you may us ln( N) 2 ln( 2π) N 2 ln( N) N 2N and for factorials > 2000! ln( N) Nln( N) N 2

3 Considr 5 particls in 2 E stats such that E tot E E 2 5 using th xprssion givn abov, w hav th following numbr of distributions and arrangmnts.. i 0 5 E i E 2i W i W 5 E i E 2i Now, rpat th calculation for 500 particls i 499 E i i E 2i 500 i W i i E E2 W f( N) 2 ln( 2π) N 2 ln( N) N 2N ( f ( 500) f ( 260) f ( 240) ) p f ( 500) f E i 2i fe i p i Notic how sharp th distribution has bcom. his is only for an incras in th numbr of particls from 2 to 500. What if w us Avogadro's numbr of particls. W find that in this typ of analysis, th distribution bcoms xtrmly sharp i

4 4 Now, considr Avogadro's numbr of particl. Using Stirlings approximation, w can mak som rudimntary calculations. E E2 W(a) x x 0 22 xp( x 0 22 ) 5.0 x x 0 22 xp( x 0 22 ) xp(2 x 0 7 factor diffrnc!) Clarly, as th numbr of particls incrass, th distribution bcoms mor pakd and for Avogadro's numbr, typical of a lab condition, it bcoms VERY pakd! (his distribution maximizing is fundamntal to th concpt of ntropy) Sinc w know that thr is virtually only ralizd distribution, (all othrs ar statistically improbabl) w will ultimatly b abl to us this distribution and it's charactristics in ordr to prdict global variabls (, P, E, S, tc.) W must now dtrmin undr what conditions W is maximizd subjct to th constraints of fixd total N and total E. h rsult w obtain will rprsnt HE distribution w will s almost xclusivly. h Molcular Partition Function In ordr for a cul to b in a quantum stat that is at a highr nrgy, th nrgy must b availabl thrmally to xcit th cul. ypical nrgy transitions for translation, rotation, vibration and atomic-lctron ar givn blow and compard to th avrag thrmal nrgy ransition ypical nrgy of transition ranslation vry small Rotation kj/ Vibration 25 kj/ Elctronic 00 kj/ Avrag hrmal Enrgy at 00 K.7 kj/ As can b sn, for th most part th intrnal motions and transitions ar wll suitd to a larg N/s ratio as thr ar far mor culs than th quantum stats availabl thrmally. hat mans that lowr quantum stats ar th only ons populatd. 4

5 5 h mthod usd to find th maximum (most likly distribution) undr th givn constraints (N and E fixd) is that of Lagrang Multiplirs.(Rviw Calculus). Although w will not rviw it hr. I will only prsnt th rsults of such a maximization. h rsult of such an analysis is givn by.. P j E j k b his xprssion givs th probability that a givn systm of E j particls will b in nrgy stat, E j, whr k b is Boltzmann's k b constant j his probability xprssion will b usd to connct all ths thrmal particls to macroscopic laboratory masurmnts. Of particular not is th sum in th dnominator. his is a vry spcial sum ovr all stats and it's valuation will play a major part of th analysis of our quantum systms. his sum is known as th "partition function" and is frquntly givn th symbol Z or Q, or Physical Intrprtation of z: k b Z VERY IMPORAN!!! j E j h cular partition function rprsnts th numbr of quantum stats thrmally availabl to th cul at th givn. A qualification nds to b mad for vibrations as quantum stat n 0 has an nrgy calld th "zro point nrgy" hus th cul has n 0 most availabl as w will s. It is important to not that th sum is ovr quantum stats. W can asily mak this a sum ovr nrgy stats if w rcogniz that thr ar oftn multipl quantum stats with th sam nrgy. his w hav sn bfor and is th dgnracy. If w dnot th dgnracy as g j for a givn quantum stat, thn th summations abov can b writtn as: P j E j k b g j Z his xprssion is Boltzmann in it's form. hrfor, w s that th Boltzmann Distribution rprsnts th most likly distribution of particls among stats! his distribution is so ovrwhlmingly likly that all othr distributions ar statistically insignificant. 5

6 6 Exampl: A systm of nrgy stats and dgnracis is givn blow. Calculat th probability of a particl bing in ach stat at 500 K. Additionally, for a systm of 500 particls, calculat th numbr of particls in ach stat. i 0 Ek i 0K 250K 750K 75K g i z ( ) 500K for convninc N 500 z( ) xp i g xp i Ek 0 Ek i 5xp Ek z( 500K) 8.7 7xp Ek 2 or xpanding... 9xp z( 500K) 8.7 Ek P i Ek i g xp i z ( ) P Num NP 0.9 i i 0.07 Num Additional Quantitis: Rcall from arlir discussions that in ordr to calculat an avrag of a valu, on multiplis th valu by fraction of particls having that valu and adding all th rsults, or: X avg P X i i i Lts us this to calculat th avrag nrgy and th avrag squar nrgy. E avg E P i i i i Ek i g xp i k b E i z Ek i E g xp i i k b i k b J z K Excuting.. E avg ( ) Ek Ek Ek 0 Ek xp 0 Ek 5xp 2 Ek 7xp 2 Ek 9xp z ( ) Ek k b E avg ( 500K) J 6

7 7 Exampl: h diffrnc in nrgy lvls btwn th chair and boat configurations of cyclohxan is 5.5 kcal/. What fraction of cyclohxan culs ar in th boat configuration rlativ to th chair configuration at 00 K and at 800 K. cal R K 5500 cal/ E chair 0 cal E boat 5500 cal z cyclohxan ( ) 0 E boat R h probability of ach configuration is xprssd as.. P chair ( ) E chair R z cyclohxan ( ) E boat R P boat ( ) ΔE 5500 cal z cyclohxan ( ) Fraction( ) P boat ( ) P chair ( ) Eboat R z cyclohxan ( ) Echair R z cyclohxan ( ) E boat E boat E chair ΔE R R R Fraction( ) E chair R Fraction( 29K) Fraction( 800K) 0.0 Important Not: h Fraction givn in th abov xprssion is xactly corrct providd thr is no dgnracy prsnt. For systm with dgnracy, th propr xprssion bcoms. Fraction( ) E j E i g j R g i 7

8 8 Partition Functions for Individual Quantum Systms: Finally, th nrgy of a particl can b distributd in a varity of placs including translation, vibration, rotation, lctronic lvls, nuclar and othrs. h partition function for a cul can b sparatd into partition function for individual nrgy systms IF ach systm is indpndnt of th othrs. his is not always tru, but will provid an xcllnt first assumption. hus.. Z β j E j j k b E trans E vib E rot... j E trans k b E vib k b E rot k b... k b k Z b k b... z trans z vib j E trans j E vib j E rot z rot z lc z nucl... h objct is thn to xamin ach systm and build a cular partition function. Application: -D particl in a box systm. W found in th -D PIB systm that thr wr O(0 6 ) availabl quantum stats at room tmpratur. Furthr a typical nrgy diffrnc btwn 2 stats is much small than k (a rough masur of thrmal accssability). hus th nrgy stats can b considrd to b continuous and Boltzmann-lik statistics can b applid. W bgin with th partition function.. Z trans E j k b h whr E n j 8M a 2 x n y n z for a cubical box!! Sinc th avrag n is vary larg, w can to an xcllnt approximation assum that.. n x n y n z n so th partition function bcoms... substituting... 8

9 9 Z trans h 2 n 2 8M a 2 k b Aftr appropriat summation ovr all possibl stats, th rsult is givn as: n Z trans Z trans 2 V 2πMk b If rcogniz that V k b h p o ( 2πM) kb whr p h o bar p o hrmodynamic Connction Sinc w now know th Z for translational motion, w also know th probability of bing in ach quantum stat. If w multiply th fraction of culs in ach quantum stat by th nrgy of that stat, w ar, in fact, calculating an avrag nrgy! his nrgy, whn scald up to Avogadro's numbr is what w masur as th avrag kintic nrgy of a systm. If th systm is an idal monatomic gas, thn w call this th Intrnal Enrgy. (wll..almost as w'll s.) hus, from fundamntal quantum mchanics and avraging ovr thrmally availabl nrgy stats, w hav com full circl to classical thrmodynamics! Now, to valuat th avrag thrmodynamic valus from a statistical viw, th avrag is calculatd as follows... M X avg P a X a whr M is th total numbr of availabl stats. a P a in this xprssion is th probability of a systm bing in a stat having th valu X a. W ar now in a position to dtrmin som avrag valus for a systm of culs.hs avrag valus ar thos obsrvd macroscopically!! Onc w connct th quantum stats to th avrag valus, thn w hav succssfully tid th microscopic quantum world to th macroscopic world, as it should b! 9

10 0 E avg Exprssing th avrag nrgy as a function of th partition function. U P j E j j E j E j k b j h avrag masurd nrgy would b th intrnal E j nrgy. j k b Using som mor advancd mathmatical tchniqus, w can show that E avg k b 2 d( lnz) d his is a CALCULUS xprssion and th dtails of solving nd not concrn you for this cours. Without proof (providd upon rqust) w find whn w prdict th avrag nrgy for a translational systm, w gt E avg 2 k b From kintic thory whr w connctd lab to motion, w found that th avrag translational nrgy for an nsmbl of particls to b: E Kitic_Avg 2 k b which fit xprimnt prfctly Comparing.. W now hav th thrmal connction and it is th sam as w found bfor. h Boltzmann constant conncts nrgis to th masurmnt of tmpratur, which is a laboratory dvic Lt's now tst this furthr... 0

11 Exprssing th cular prssur as a function of th partition function. Rcall from thrmodynamics that for an isolatd systm, thr is no nrgy bing introducd into th systm, that is q 0. Undr such conditions.. de de pdv or p but p avg dv j P j p j Not that a lowr cas p is dlinatd as th prssur to distinguish it from probability. W s that, aftr furthr tratmnt that.. p avg p k b dlnz dv his is anothr CALCULUS rsult for which I'll simply provid th rsults of application. Prssur for a ranslational Systm: Whn applid to th translational partition function, w arriv at PV k b or th idal gas law!!!! Do not forgt that w ar still at th cular lvl. Whn scald up for Avogadro's numbr of atoms w gt PV N a k b R Rcall that for translational motion, th idal gas law is.. PV R for on.

12 2 h Avrag Enrgy E avg k b 2 and th prssur p k b Othr hrmodynamic Functions dln( Z) dv dln( Z) d h Enthalpy - his on is fairly dirct and asy from th prvious dfinitions.. H U PV and insrting from abov... H k b 2 dln( Z) d k b V dln( Z) dv Entropy - his on involvs a littl work, howvr, following... du ds PdV w find that.. S k b dln( Z) d k b ln( Z) hs xprssions ar advancd. h point to tak away from this and what you should undrstand is that it is possibl to add up all th quantum stats in a spcial way to provid a constant at a givn tmpratur calld th "Partition function" his spcial sum can b rlatd dirctly to quantitis w masur in th laboratory using only th Boltzmann constant. hus w will hav a fundamntal quantum dscription of th macroscopic valus w s in th physical world. All through th partition function Intrnal Motions Having xamind translational motion, w can now turn our attntion to finding th partition function for rotation, vibrational, lctronic, nuclar, tc... As bfor, w can find ach individually and multiply th rsults togthr. his assums that th motions ar indpndnt of ach othr. his is an approximation that is found to hav minimum rror at lowr nrgis. 2

13 Rotational Partition Function: hbar 2 From th nrgy xprssion of rotation, w now hav E J J( J ) 2I At this point, w nd only to xcut th summation rquird by th partition function. Excuting... z rot J 0 h 2 J( J) 8π 2 Ik b ( 2J ) 8π 2 Ik b h 2 W s that w hav a larg collction of constants. Collcting th constants producs a valu known as th "charactristic rotational tmpratur" h nam is givn bcaus th valu has units of tmpratur. So w writ our rsult as.. z rot h 2 whr: Θ rot Θ rot 8π 2 Ik b For this drivation, w must includ on othr trm. his ariss from th rquirmnts on a quantum mchanical wav function with rspct to symmtry. Physically, thr is an intrsting xplanation. If th cul of intrst posssss an inhrnt symmtry, such as H 2, thn it can b said that rotating th cul 80 dgrs producs an idntical cul. In producing th quantum stats for culs, this xchang symmtry is ovr countd whn nuclar spin stats ar ignord. Whn w procd to count stats, w must b carful not to ovr count ths idntical stats. hus, to proprly account for th stats, liminating th symmtry forbiddn stats, w must divid by th symmtry numbr for th cul. For a homonuclar diatomic, this is 2. For a cul such as BH, th numbr would b thr. tc. W includ this symmtry numbr by th symbol,. Doing so producs our final rsult. z rot h 2 whr: Θ rot Θ rot σ 8π 2 Ik b

14 4 Exampl: What is th rotational distribution of HCl culs tratd as a rigid rotor? First collcting all ndd constants... N a k b J h Js K Now, th cular valus.. hbar h 2π m.0079 gm m gm R m m m 2 μ μ kg m m 2 N a Notic this is in kilograms, not kg/! I μr 2 I kgm 2 h rotational tmpratur and partition function is computd. θ r h 2 θ r K 8π 2 Ik b z rot ( ) θ r Now th probability of bing in any givn quantum stat is P E L k b g L Z Now th quantity in th argumnt of th xponnt is... E L k b h 2 θ r L( L ) L( L ) 8π 2 Ik b 4

15 5 Putting it all togthr, w hav Prob( L) θ r ( 2L ) xp L( L ) z rot ( ) Now, plotting th rsult... L 0 0 z rot ( 00K) Prob( L00K) Prob( L00K) L Notic that most culs ar in a quantum stat highr than th ground stat. In this cas L is most probabl. As th tmpratur is raisd, most probably L stat gos to highr valus as xpctd. 5

16 6 Non-Linar Polyatomic Molculs Sinc such culs possss diffrnt rotational momnts of Inrtia, contributions from ach nd to b addrssd. Without th xtnsiv drivation ncssary, th rsult is simply givn hr as: z rot π σ 8π 2 I a k b h 2 2 8π 2 I b k b h 2 2 8π 2 I c k b h 2 2 π σ Θ rot_a Θ rot_b Θ rot_c Vibrational Partition Function As bfor, w must bgin with an xamination of th nrgis and dgnracis of this particular systm. hs wr xamind in th study on th harmonic oscillator and anharmonic corrctions. h rsult was: E vib n 2 h ν o Now, procding as bfor... z vib n hν o hν o nh 2 k b 2k b k b n n ν o From Calculus study of sris, w know that th sris convrgs to a givn rsult. Without proof (availabl by rqust), th rsult bcoms.. z vib hυ o 2k hυ o k Θ vib 2 hν o or commonly z vib whr Θ vib Θ vib k b whr w hav, again, combind all of th constants in ordr to produc a "charactristic vibrational tmpratur" Again, this nam is giv only bcaus it has units of tmpratur and is not to b construd as having any connction any othr tmpratur rlationship. 6

17 7 Exampl: Calculat th probability of an HF cul bing in quantum stat, n0 and n as a function of tmpratur. From our discussion of statistical probabilitis... PE ( ) E k g z For this systm.. υ o 48.5cm hυ o c so θ v θ v K k b For vibration, from abov.. z vib ( ) xp xp θ v 2 θ v Combining... Pn ( ) n 2 z vib ( ) θ v 0K20K 6000K. 0.9 P0 ( )

18 8 0. P ( ) hs figurs confirm our analysis arlir, that lowr quantum stats ar populatd at typical lab tmpraturs. Not that it isn't until w rach tmps in xcss of 500 K that any significant population of th nxt quantum stat is ralizd! Poly-vibrational culs: For culs that hav multipl vibrational mods, th vibrational partition function is simply a product of vibrational mods, or z vib m Θ vib_m 2 Θ vib_m whr m gos ovr all vibrational mods 8

19 9 Elctronic contribution For tmpraturs that go to vry high valus, it is asily shown that lctronic contributions du to xcitd stats ar virtually nonxistnt. ak, for xampl, th xcitation of I 2. h principl lctron xcitation occurs at about 5790 cm -. h ground stat is 4-fold dgnrat. hus, th nrgy associatd with this transition is h Js c m k b J s K E lc_trans 5790cm hc E lc_trans J Calculating th first two trms of th partition function E lc_trans z lc 4 0 k b 500K z lc or z is 4 to 8 digits past th dcimal. hus, usually w hav only to worry about dgnracis at th ground lvl. Summary so far... What w now hav is a st of partition functions for th thr motions and for lctronic nrgy. hs ar calld Molcular Partition Functions. Physically, ths rprsnt th numbr of thrmally availabl quantum stats for culs. hy giv th probability distribution of culs among th quantum stats. Now, w nd to considr an "nsmbl" of such culs..or for our purposs a of culs. Whn w scal up ths cular partition functions to that of a of particls, w will b abl to calculat vrything w nd from our quantum systm. 9

20 20 Canonical Systms and th Canonical Partition Function Now, w ar in a position to scal this analysis to includ a complt systm of particls, not just of quantum stats as has bn don. o do this, w nd to xpand out dfinitions of systms and stats. Systm dfinitions: Considr, now, an nsmbl mad up of a systm of particls subjct to th constraint that: a.) h # of particls in th systm, dnotd by N, will rmain fixd. b.) h tmpratur of th systm,, is also fixd. c.) h Volum of th systm, V, is fixd. A systm constraind in such a fashion is calld a "canonical nsmbl". his is th systm that w will study. Othr systms of intrst ar: W now wish to carry out a statistical counting of distributions (also calld microstats) as w did in individual culs. An important obsrvation and similar mathmatical formalism will provid us with an asy rout to this nd. If on maks th obsrvation that, for this dfind systm, all th particls ar in thrmal contact with ach othr, thn, th distribution of stats of an nsmbl of particl systms will b th sam as a singl systm. As a rsult, th partition function for th canonical systm will b idntical to that of a cular systm, only augmntd by N, th total numbr of particls in th systm. hrfor, w may writ as a start.. E nsmbl NE particls and bcaus th nrgy is in th xponnt of th partition function ie nsmbl β E particl E particl E i 2 particl... β i i z canonical or i i z canonical NE particlβ N N Z Molcular or simply z canonical Z Molcular i Z Molcular rprsnts th distribution of a cul among quantum stats th cular partition function. z canonical rprsnts th distribution an nsmbl of culs among all possibl stats th canonical partition function. Now, w must tak car! Particls, ar inhrntly indistinguishabl. It is not possibl to distinguish a systm in which Nitrogn cul A is in quantum stat and Nitrogn cul B is in quantum stat 5 from th condition in which cul A is in quantum stat 5 and cul B is in quantum stat. Propr counting bcoms important. Considr th following simplistic xampl: 20

21 2 Exampl : Considr 5 particls in 2 quantum stats such that all but ar in th ground stat. Lt us count th stats for distinguishabl and indistinguishabl particls. Counting th possibilitis... distinguishabl indistinguishabl W 5 W distin or W indis N Now considr 4 particls in 4 stats. Counting th possibilitis... distinguishabl indistinguishabl W 24 W distin or W indis N Clarly, how w count stats dpnds on how th particls ar distributd. Of th possibl prmutations of ths typs of counting gams, for ralistic systm involving larg numbrs of particls, w will find that trating systms as pr th scond xampl will provid an xcllnt, narly xact approximation to th problm of counting stats. Raff, Physical Chmistry pp provid mor quantitativ argumnts should you dsir mor information. hus, for our systms, w may apply th following.. for indistinguishabl systms. Z z N N W can now complt our thrmodynamic connctions to th partition function. Bginning with th intrnal nrgy. Excuting propr counting and applying all w know, w can construct th following sts of statistical thrmodynamic functions: 2

22 22 Molcular Partition functions Summary abls z trans 2 ( 2πm) kb ph 5 2 Θ vib _j N 5 2 hυ o z vib or N - 6 for non-linar culs Θ vib Θ vib _j k b j π h 2 z rot z rot Θ rot σθ rot σ Θ rot_a Θ rot_b Θ rot_c 8π 2 Ik b Diatomic systm Polyatomic systms Elctronic z lc g j Complt Molcular systms Intrnal Enrgy U 5 N5 2 R R Θ vib _j R Θ vib _j U R 2 Θ vib _j j N6 j R Θ vib _j 2 R Θ vib _j Θ vib _j For Linar Systms For Non-Linar Systms 22

23 2 Hat Capacity C v ( ) Θ vib _j N 5 R Θ 2 vib _j 5 2 R C v ( ) R 2 j Θ vib _j N 6 j R Θ vib _j Θ vib _j 2 Θ vib _j 2 For Linar Systms For Non-Linar Systms Entropy S trans Rln ( 2πm) 2 2 ( k ) ph R S rot Rln R S rot Rln σθ rot π σ Θ rot_a Θ rot_b Θ rot_c 2 R For Linar systms For Non-Linar Systms S vib ( ) N 6 j Rln Θ vib _j R Θ vib _j Θ vib _j or N - 5 for Linar systms Gibbs Fr Enrgy G tr R ln z tr G int R N lnz int 2

24 24 Putting it all togthr: Rprsntativ Exampls Calculat and plot th partition function, th hat capacity and th ntropy for CO gas as a function of mpratur. o do ths problms, I rcommnd collcting all th constants and cular paramtrs togthr and thn applying th rquird xprssions. Establishing constants... pm 0 2 m N a k b J h Js K J R 8.45 p 0 5 Pa K Molcular paramtrs kg kg kg ( ) M CO M C M O N a N a N a Vibrational wavnumbr ν o 270.2cm Equilibrium Bond Distanc R 2.8pm Symmtry Numbr σ Calculating th charactristic rotational tmpratur μ CO M C M O μ CO kg I CO μ CO R M CO hbar 2 Θ rot Θ rot K 2I CO k b Calculating th charactristic vibrational tmpratur Θ vib hc ν o Θ vib 22 K k b 24

25 25 Calculating th partition function From abov... 5 Θ vib z trans ( ) 2 2 2πM CO kb ph z vib ( ) 2 Θ vib z rot ( ) σθ rot z ( ) z trans ( ) z rot ( ) z vib ( ) z ( 00K) K60K 000K z ( ) Notic how rapidly th numbr of stats incras with tmpratur. h largst contributor is th translational partition function. Compar th plot of Z trans vs tmpratur with that of Z rot vs tmpratur for xampl.50 8 ranslational Partition Function vs. mpratur Z(trans) mpratur (K) 25

26 Rotational Partition Function vs. mpratur 00 Z(rot) mpratur (K) Hat Capacity C v ( ) 5 2 R R Θ vib Θ vib 2 Θ vib 2 J C v ( 298.5K) K C v_lit J K C v_lit C v ( 298.5K) %rror %rror 0.08 % C v_lit!!!!! 50K60K 000K Hat Capacity (J/-K) Hat Capacity vs. mpratur mpratur (K) Not that this shap is govrnd by th vibrational motion. ranslation and rotation provid a constant valu of 5/2 R. 26

27 27 Entropy 5 S CO_trans ( ) R ln 2π M 2 2 CO kb ph S CO_rot ( ) Rln σθ rot R 5 2 R S CO_vib ( ) Rln Θ vib R Θ vib Θ vib S CO ( ) S CO_trans ( ) S CO_rot ( ) S CO_vib ( ) J J S CO ( 298.5K) S CO_lit K K S CO_lit S CO ( 298K) %rror %rror 0.02 % S CO_lit 5 Entropy of Carbon Monoxid vs mpratur Entropy (J/-K) mpratur (K) 27

28 28 A Mor complx xampl: CO 2 Calculat and plot th ntropy as a function of tmpratur. In addition, calculat and plot th hat capacity as a function of tmpratur. Compar th calculatd valus to xprimntally fittd data. For th ntropy, th only significant diffrnc is in th vibrational contributions. hr ar four vibrational mods with thr vibrational wavnumbrs. hrfor, prcding as bfor.. bar 0 5 Pa p.025bar kg M CO2 σ 2 I kgm 2 N a Θ rot h 2 Θ rot 0.56 K 8π 2 Ik b ν 26cm ν 66.2cm ν 66.2cm ν.9cm S trans ( ) R ln 2π M 2 2 CO2 kb ph 5 2 R Θ vibi hc ν i k b Θ vib K S rot ( ) Rln σθ rot R (Not: CO 2 is linar) S vib ( ) j 0 Rln Θ vibj R Θ vibj Θ vibj S CO2 ( ) S trans ( ) S rot ( ) S vib ( ) J S CO2 ( 298.5K) K 28

29 29 Entropy (J/-K) Entropy of Carbon Dioxid vs. mpratur mpratur (K) S CO2_lit 2.8 J K S CO2_lit S CO2 ( 298.5K) %rror %rror % S CO2_lit Hat Capacity C v ( ) 5 2 R j 0 R Θ vibj Θ vibj 2 Θ vibj 2 C p ( ) C v ( ) R Lt's compar this calculation to an xprimntally fittd hat capacity polynomial. J J C p_lit ( ) J JK 2 K K 2 K J C p ( 298K) K C p_co2_ J K %rror C p ( 298K) C p_co2_298 C p_co2_298 %rror 0.62 % 29

30 0 0K0K 2500K 65 Carbon Dioxid Hat Capacity vs mpratur Hat Capacity (J/-K) Quantum-Stat Lit Polynomial Fit mpratur (K) 0

31 Equilibrium Constants W saw in our dvlopmnt of classical thrmodynamics, that th Gibbs Fr nrgy provids for us a mans by which to calculat th quilibrium of a systm through th quilibrium constant. Sinc w now hav a statistical Gibbs Fr nrgy, w can now us fundamntal quantum rsults in ordr to produc an quilibrium constant using partition functions. h following xampl dmonstrats th procss. In this xampl, I us all th functions w hav dvlopd abov. From G Rln z w can writ for an quation N aa + bb cd + dd ΔG cg C dg D ag A bb B Substituting th statistical G abov and using similar algbra as was usd in our prvious dvlopmnt of K, w arriv at K P c d z C zd a b z A zb Considr, for xampl, th raction.. I 2 ( g) 2I( g) Lt us calculat th partition functions for I 2 and I, and thn combin. Important Not: In ordr to compar ach ths itms with ach othr, w must tak into account th non-zro ground lctronic stat. his is information obtaind by solving th lctronic systms quantum mchanically. Although w hav bn abl to produc accurat thrmodynamic quantitis for individual itms, whn comparing 2 or mor, ths diffrncs must b accountd for.

32 2 Iodin Atoms For iodin atoms, thr is translational motion and no vibration or rotation. hrfor th partition function is without th lctronic ground stat.. M I 26.9gm N a z I_tr ( ) πM I kb ph From lctronic tabls, w find that Iodin posssss a 4-fold ground lctronic stat and a 2-fold first xcitd lctronic stat. h diffrnc in nrgy in ths two stats is 7580 cm -. Lt us calculat this partition function z I_lc ( ) cm ch k b z I_lc ( 00K) 4 Now, dfin constants and calculating... z I ( ) z I_tr ( ) z I_lc ( ) Iodin Molculs Hr, w build a partition function as abov. P o 0 5 Pa σ 2 Θ vib 08.64K Θ rot 0.057K M I2 2M I 5 z trans ( ) 2 2 2πM I2 kb P o h z rot ( ) σθ rot 2

33 Finally, thr is on mor thing to considr. As mntiond abov, ach of ths itms must b considrd rlativ to th sam nrgy rfrnc. o account for this, all w nd do is to calculat th diffrnc btwn th nrgy stats. his is rlativly simpl as th diffrnc btwn two iodin atoms and th iodin cul is th Dissociation nrgy, D o. I 2 has a ground lctronic nrgy stat, D o 48,800 J/ and a dgnracy of. h bst and asist way to account for this is to produc an xprssion for th diffrncs in th ground stat nrgis, ΔD o and thn multiply th K p by this valu as.. ΔD o R z lc ( ) whr ΔD o cd o_c dd o_d ad o_a bd o_b For rasons that ar byond our discussion, this practic causs th changs in th vibrational partition function. Without furthr discussion, w procd as follows: Stp : h z vib now bcoms... g z vib ( ) g Θ vib whr g is th dgnracy of th lctronic ground stat. Stp 2: For this Iodin systm, w hav th uniqu condition that th individual atoms hav D o 0. hus, for this systm For Iodin, w hav ΔD o 2D o_iodin_atoms D o_iodin_molculs D o_iodin_molculs kj 48.8 D o_i2 D o_i 0 N a kj 000J ΔD o ΔD o 2D o_i D o_i2 z lc_systm ( ) k b Stp : Now, finishing... z I2 ( ) z trans ( ) z rot ( ) z vib ( ) K p ( ) z I ( ) 2 z I2 ( ) z lc_systm( )

34 4 Bfor calculating and plotting, lt m includ som xprimntal data and comparison... i 0 4 K xpi xpi 800K 900K 000K 00K 200K K p xpi prc_rror i K p xpi K xpi K xpi prc_rror i % From our prvious xamination of K's, w found that lnk p ΔH rxn R Constant hrfor, as bfor, a plot of ln(k p ) vs. / will giv a straight lin whos slop is -ΔH rxn /R Now, plotting both... 4

35 5 800K80K 200K ln(k) ln(k) vs / Statistical Litratur / (x 000) Now, calculating th raction nthalpy from th slop... kj 000J ΔH rxn_stat Rslop lnk xp ΔH rxn_stat 5.8 kj xp 5

36 6 Comparing to xprimnt, w nd to calculat th raction nthalpy at 298 K and thn us th hat capacitis in ordr to calculat a rsult at th highr tmps. I'll slct th middl tmpratur valu as my stimat, 000K. Using tabls and thn Kirchoff's formula... From thrmodynamic abls ΔH f_i kj ΔH f_i kj ΔH rxm_298 2ΔH f_i ΔH f_i2 ΔH rxm_ kj Raction nthalpy at K. Now, calculating th lvatd tmp... J J C p_i C p_i ΔC p_rxm 2C p_i K K C p_i2 Δ 000K 298.5K ΔH 000 ΔH rxm_298 ΔC p_rxm Δ ΔH kj ΔH 000 ΔH rxn_stat prc_rror prc_rror % ΔH 000 6

37 7 Exampl 2 CO 2 CO 2 O 2 Molcular Data: kj 596 CO 2 D o_co2 Θ rot_co2 0.56K σ CO2 2 i 0 N a Θ vib_co2i 60K 954K 954K 890K M CO2 44.0gm N a CO kj 070 D o_co Θ vib_co 2K Θ rot_co 2.778K N a σ CO M CO 28.0gm N a 49.6 kj O 2 D o_o2 Θ vib_o2 227K Θ rot_o2 2.07K N a gm 2.00 g O2 σ O2 2 M O2 N a Ground stat Elctronic dgnracy 7

38 8 Calculating partition functions.. 5 O 2 z trans_o2 ( ) 2 2 2πM O2 kb ph z rot_o2 ( ) Θ rot_o2 σ O2 z vib_o2 ( ) Θ vib_o2 z O2 ( ) z trans_o2 ( ) z rot_o2 ( ) z vib_o2 ( ) CO 5 z trans_co ( ) 2 2 2πM CO kb ph z rot_co ( ) Θ rot_co σ CO z vib_co ( ) Θ vib_co z CO ( ) z trans_co ( ) z rot_co ( ) z vib_co ( ) 8

39 9 CO 2 z trans_co2 ( ) πM CO2 kb ph z rot_co2 ( ) Θ rot_co2 σ CO2 z vib_co2 ( ) Θ vib_co20 Θ vib_co2 Θ vib_co22 Θ vib_co2 z CO2 ( ) z trans_co2 ( ) z rot_co2 ( ) z vib_co2 ( ) Now, having th partition functions, w can calculat K p... D o_co 2 D o_o2d o_co2 k b z lc_systm ( ) K p ( ) z CO ( ) z O2 ( ) z lc_systm ( ) z CO2 ( ) K p ( 2000K) K p_lit_ K p ( 2000K) K p_lit_ % K p_lit_2000 kj kj G o_co 7.7 G o_co o 298.5K ΔG calc G o_co G o_co2 K p_calc_298 ΔG calc R o K p_calc_ K p ( 298.5K)

40 40 K p_calc_298 K p ( 298.5K) Error Error 5.65 % K p_calc_298 his rror is du mostly to th assumption of rotational-vibrational indpndnc. k K 0.kK lnk ln K k k p k ln(kq) ln(kq) vs / / (/K) x 000 SLP slop lnk ΔH rxn RSLP From hrmo abls... ΔH rxn kj kj kj H o_co 0.5 H o_co2 9.5 ΔH calc H o_co H o_co2 ΔH calc kj ΔH calc ΔH rxn Error Error % ΔH calc 40