Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY

Transcription

1 Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl many quantum lvls availabl, but only on is occupid at any point in tim Thrmal mods thos quantum intractions with nrgy gaps small nough that changs in tmpratur can affct a chang in population of stats Non-thrmal mods thos quantum intractions whos nrgy gaps btwn adjacnt quantum stats ar too larg for population to b affctd by tmpratur Statistical Mchanics statistical analysis of nsmbl of atoms, molculs & stats xpctation valu <E i > avrag nrgy for a singl molcul avragd across all possibl quantum stats and wightd by probability of ach stat Total nrgy N <E i >, whr N total numbr of particls Thrmodynamics study of macroscopic proprtis of a systm Focus is on mol of molculs, not individual molculs th capacity to do work Intrnal nrgy of a systm (U) combind nrgy of all th molcular stats Hat (q) thrmal transfr of nrgy to/from th systm to th surroundings. Occurs through random collisions of nighboring molculs. Tmpratur (T) paramtr that dscribs th nrgy distribution across th quantum stats availabl to th systm Thrmal quilibrium whn thrmal nrgy distribution of a systm matchs th nrgy of th narby surroundings i.. thr is no mor combind transfr of hat nrgy. Thrmal nrgy avrag Boltzmann nrgy lvl of molculs in surroundings i β Ei Boltzmann population factor, gnral β / Population ratio btwn two stats p p ( E ) 0 th Law of Thrmodynamics: If systms A&B ar in thrmal quilibrium with ach othr, and if systms B&C ar in thrmal quilibrium with ach othr, thn A&C ar also in thrmal quilibrium with ach othr S Gntry, 04

2 Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY (takn from Fundamntals Chaptr and Chaptr 5: Statistical Thrmodynamics) Lctur Outlin Not: Som of this matrial coms from topics that will b xplord in much gratr dtail latr in th yar whn w gt to QM. Th sction is mant to introduc you to som of th concpts includd in statistical mchanics as thy rlat to hat and nrgy, not provid a comprhnsiv discussion of th topic. W ar covring th information now, howvr, so as to undrstand th fundamntal natur of th intrnal nrgy of a systm as it rlats to thrmodynamics, and how that nrgy is transfrrd to anothr body. Thrmodynamics vs. Quantum Mchanics and what about Statistical Mchanics Quantum Mchanics QM dscribs microscopic proprtis of individual atoms and molculs Thrmodynamics & Kintics Thrmodynamics and kintics study macroscopic proprtis that ar obsrvd in th lab. Thy look at th proprtis of laboratory sampls avragd ovr ~0 3 molculs Thrmodynamics study of nrgtics and stability of physical systms with particular mphasis on dscribing systms that hav rachd quilibrium Kintics study of th rat at which systms ract Thrmodynamics may prdict a raction will vntually mov to a mor stabl stat, But actual raction procss dpnds on () rat of collision of molculs and () probability of raction whn thy do collid May tak sconds, minuts, hours, or vn yars to rach quilibrium stat. Statistical Mchanics Bridgs th gap btwn Quantum Mchanic s individual atoms and molculs and th larg laboratory sampls usd in Thrmo. and Kintics. Statistical mchanics provids th framwork to calculat avrag proprtis basd on a larg sampling of individual molculs. Individual Atoms & Molculs Quantum Mchanics - Translations - Rotations - Vibrations - Elctron Lvls - Bond Enrgis Statistical Mchanics - Avrags Larg Lab Sampls (~0 3 molculs) Thrmodynamics - Intrnal - Entropy - Equilibrium Stats Kintics - Rats of Raction - -

3 MICROSCOPIC ENERGY LEVELS Total nrgy is summation of nrgis for all th individual mods - Translational nrgy: motion of atoms or molculs through spac Motion dfind by chang in th location of cntr-of-mass for particl - Rotational nrgy: Changs in orintation of molcul in spac I.., dos molcul point up or sidways, and dos orintation rotat with tim - Vibrational nrgy: Strtching of bonds and bond angls - Elctronic nrgy: Includs both lctronic atomic orbitals and lctronic molcularbond nrgis Can hav simultanous multipl mods for xampl rotat in 3 dirctions in spac and hav multipl vibrations all occurring at sam tim Translational Rotational Vibrational Molcular Bond Atomic Orbital Intr-molcular p x p Z p Y Quantum Mchanical Calculation of Microscopic Lvls Enrgis occur in discrt lvls (quantum stats) rathr than continuous rang This is tru vn for translations (movmnt through spac) MACROSCOPIC ENERGY Thrmodynamics focuss on th total Intrnal, U, of th systm Thrmodynamics dos not particularly concrn itslf with th spcifics of th individual nrgy stats or typs of mods, instad looks at total systm U total combind nrgy of all th molculs in th systm I.., translations + rotations + vibrations + lctronic lvls for ach molcul thn summd ovr all of th molculs in whatvr w dfin as th systm - 3 -

4 STATISTICAL MECHANICS Qustion #: How dos th population of an nrgy stat dpnd on th surrounding thrmal nrgy? Common-sns xprinc: if put a cold soda glass in a warm room, thn th soda will vntually warm up (absorb nrgy) until it rachs thrmal quilibrium with th surroundings. But what is happning at th quantum mchanical lvl for individual molculs? Answr: th systm and surroundings xchang nrgy until thy ar at thrmal quilibrium with on anothr. Initial Distribution of Final Distribution of E 5 Thrmal E 4 E 3 E Thrmal Equilibration Thrmal E DEFINITIONS: of a singl molcul dpnds on which quantum stat is occupid. A singl molcul will only xist in on stat at any givn tim vn though a numbr of possibl quantum stats ar availabl to it. Intrnal nrgy of a systm (U) combind nrgy of all molculs in th systm Tmpratur (T) paramtr that dscribs th nrgy distribution across th quantum stats availabl to th systm Hat (q) thrmal transfr of nrgy to/from th systm to th surroundings. Occurs through random inlastic collisions of nighboring molculs. Thrmal quilibrium whn thrmal nrgy distribution of a systm matchs th nrgy of th narby surroundings i.. thr is no mor combind transfr of hat nrgy. Is a rsult of probabilitis of hat bing xchangd ovr countlss individual collisions of molculs. Equilibrium rachd whn qual chanc that collisions rsult in nrgy bing transfrrd in on dirction as thr is in othr dirction

5 Qustion #: What controls distribution of nrgy across individual molculs in th systm? Boltzmann Distribution Can b shown that population of nrgis in th systm dpnds on th availabl thrmal nrgy in th surroundings, But this wighting factor is xponntial, not a stp function. Boltzmann factor i (population distribution of nrgy stats as function of tmp) Population ratio btwn two stats β Ei β /, T masurd in Klvin (-73 C) p p E i nrgy of stat i ( E ) Stp Function Distribution (wrong) Boltzmann Factor (corrct) Stp Function E/ E/ Thrmal nrgy availabl from surroundings Thrmal k Boltzmann constant J/K cm - /K At 300K (room tmpratur), th thrmal nrgy is J 07 cm - Assumptions with Boltzmann Distribution All stats hav qual starting probability (i.. only nrgy affcts probability of populating on stat ovr anothr, stats of qual nrgy ar qually probabl) is allowd to rach quilibrium with surrounding nvironmnt is transfrrd back and forth as particls randomly collid. act as vry larg nrgy rsrvoir so larg that small amount of nrgy transfrrd to or from systm is not nough to significantly chang th nrgy of th surrounding nvironmnt. Lik taking a cup of watr out of th ocan th ocan lvl dosn t chang

6 Qustion #3: How can molculs in th systm hav highr nrgy than th surrounding thrmal nrgy? Th prcding discussion was a littl mislading. In actual fact, thr is no singl wll-dfind surrounding thrmal nrgy lvl. Th surroundings (b it air, watr, mtal containr, tc.) ar thmslvs mad up of atoms and molculs that abid by th sam Boltzmann distribution. Thr ar just mor surrounding molculs than thr ar systm molculs, hnc thy can giv up som nrgy with only vry small impact on thir ovrall combind nrgy (i.. taking a drop of watr from th sa dosn t chang th lvl of th sa) Random collisions caus constant intrchang of nrgy from on molcul to anothr (for both systm molculs and surrounding molculs) Corollary: Tmpratur by dfinition dscribs th rlativ sprad of populatd nrgis. It is not som singl discrt phnomnon that xists in isolation. Instad two systms continually xchang nrgy with on anothr through molcular collisions until th two ar at thrmal quilibrium with ach othr. 0 th Law of Thrmodynamics: If systms A&B ar in thrmal quilibrium with ach othr, and if systms B&C ar in thrmal quilibrium with ach othr, thn A&C ar also in thrmal quilibrium with ach othr - 6 -

7 Qustion #4: Which typs of nrgis ar most influncd by Boltzmann distribution? All of th microscopic nrgy lvls on pag ar subjct to th Boltzmann quation Howvr, thy can b dividd into two groups basd on whthr th nrgy gap btwn lvls is smallr than or largr than th xtrnal thrmal nrgy. Thrmal motion thos nrgy mods that hav transition nrgis substantially lss than th surrounding thrmal nrgy. Exampls ar translations, rotations, and som vibrations - which all dscrib motions. Sinc th gap btwn succssiv nrgy lvls is small, thrmal nrgy is abl to populat multipl nrgy lvls. Incrasing th surrounding tmpratur causs th population of th systm to shift to highr nrgy lvls. Th rsult is that incrasing th tmpratur incrass th molcular motion of th systm. Non-thrmal mods lctronic orbitals, molcular bonding nrgis, and most vibrations hav transition nrgis that ar much largr than th surrounding thrmal nrgy. Raising th tmpratur (.g. from 0ºC to 00ºC) is not nough to provid th nrgy ncssary to populat th highr-nrgy lvls. Ths transitions ar typically only accssibl by adding larg incrmnts of othr typs of nrgy such as optical radiation. Thrmal Motion (translations, rotations, vibrations) Non-Thrmal Mods E Small nrgy gap Incrasing Thrmal En. Larg nrgy gap E Transition Enrgis btwn Quantum Stats for Molcular Oxygn (O ) in a Litr Box Typical Transition Thrmal at 98K 08 cm - Translational.6 * 0-7 cm - Elctro-magntic Spctrum Rotational 3 cm - Microwav Vibrational 580 cm - Infrard Molcular Orbital 0,000 cm - UV / Vis Atomic Orbital 400,000 cm - UV - 7 -

8 Qustion 5: How should w dscrib th nrgy of a systm that is rally mad up of a rang of Boltzmann distribution populations? Answr: calculat th avrag nrgy for th molculs i.. avrag th nrgis of ach individual atom or molcul to gt an nsmbl avrag. Chaptr 6 gos into th mthodologis usd by th fild of Statistical Mchanics to calculat ths nsmbl avrags. Basd on th wightd avrag i.. find avrag nrgy for all th stats, wightd by thir population. Wightd Avrag E i all stats ( of i i ) ( population of ( population of stat i ) stat i ) EXAMPLE: If popl wigh 00 pounds, 5 popl wigh 50 pounds, and prson wighs 00 pounds, thn th avrag wight of th group is: wight i wight numbr of popl i numbr of popl (00 ) + (50 5) + (00 ) lbs. Sam basic probability quation works with molcular systms and nrgy lvls Th challng that will b lft for Chaptr 6 is how to do this for a larg numbr of atoms or molculs in th systm (.g molculs). Solv by using th Boltzmann quation, and th libral application of calculus. BOTTOM LINE Whnvr w talk about th nrgy of a collction of molculs, what w rally ar saying is that w ar looking at th avrag nrgy summd ovr a larg st of molculs. Each molcul will hav its own uniqu nrgy dfind by discrt occupid quantum stats. But if w hav a 0 3 molculs prsnt, thos individual diffrncs avrag out to a common valu. Total nrgy of systm N ε i N # of molculs, <ɛ i > avrag nrgy of on molcul n E n # of mols, <E i > avrag nrgy of on mol i - 8 -

9 HEAT CAPACITIES (s also sction A.4) Hat capacity amount of nrgy (in th form of hat, q) ndd to rais tmpratur of a sampl by an incrmntal amount C q T Or stating it anothr way, hat capacity addrsss th qustion of how much nrgy (in th form of hat) is rquird to chang th Boltzmann distribution to a nw tmpratur. This changs from on matrial to th nxt dpnding on th numbr of diffrnt nrgy mods that ar prsnt and th spacing btwn nrgy lvls for ach mod. T Translations Rotations Vibrations Elctrons Atoms Molculs Intrmolcular Forcs gaps not drawn to scal Hat capacity is an important paramtr bcaus what w oftn masur in th lab is th chang in tmpratur of a sampl, and what w rally want to know is th amount of nrgy that has bn xchangd with th surroundings. Molar Hat Capacity, C m Hat capacity, lik nrgy and hat, dpnds on th numbr of particls prsnt. Th biggr th sampl th mor th nrgy that is rquird to rais th tmpratur. It is thrfor usful to dfin th hat capacity pr mol of matrial C C m, whr n numbr of mols of matrial (C v,m and C p,m ) n Th total amount of hat absorbd (+ hat) or rlasd (- hat) whn th sampl changs tmpratur is givn by th intgral of th abov dfinition T q C(T )dt whr C(T) hat capacity as a function of tmpratur (Klvin) T T q CdT C ( T T) or q Cm n T for C indpndnt of tmpratur T - 9 -

10 SI UNITS Unlss othrwis spcifid, our standard units of masurmnts for th class will b SI units. Distanc mtr (m) Mass kilogram (kg) Volum mtr 3 (m 3 ) Joul (J) Tmpratur Klvin (K) Prssur Pascal (Pa) NOTE: Volum is m 3, not litrs - m L. Mass is kg, not grams - Molcular wight of O is 0.03 kg/mol Whn in doubt whn using a mathmatical formula, convrt all valus to SI units. This will insur that units all cancl proprly. Do not attmpt to mix and match units. For xampl: whn solving for Boltzmann distribution, -E/ do not us Boltzmann constant (J/K) with nrgy lvls givn in caloris Must first convrt nrgis to Jouls, thn can solv Boltzmann distribution. Mor silly rrors on homwork and xams ar mad by not convrting to SI units than by all othr silly mistaks combind! CONVERSIONS (SI unit J) V.60 * 0 9 J cm * 0 3 J cal 4.84 J L*atm 0.3 J L*bar 00 J Prssur (SI unit Pa) bar 0 5 Pa atm.03 bar atm.03 * 0 5 Pa torr 33.3 Pa Exprssd in Wavnumbrs (cm - ) On xcption to th rul that w will us SI units is our intrmittnt us of cm - for nrgy Wavnumbrs ar oftn usd for nrgy whn doing optical spctroscopy. Wavlngth (λ) and frquncy (υ): c λ υ, c spd of light (3.00*0 8 m/s) of light wavs: E h υ, h Plank s constant (6.63*0 34 J s) Algbraic manipulation givs: E h c /λ Th wavnumbr (cm - ) is dfind as /λ in units of /cm. Not that strictly spaking, cm - dos not hav units of nrgy, it has units of invrs distanc By convntion w rally man th nrgy of light corrsponding to th invrs wavlngth Easist to simply us th dfind nrgy convrsion: cm * 0 3 J Exampl: 400nm light has wavnumbr /(400.*0-7 cm).5*0 4 cm - and its nrgy is E [.5*0 4 cm - *.986*0-3 J/cm - ) 4.96*0-9 J - 0 -