Ερωτήσεις και ασκησεις Κεφ. 10 (για μόρια) ΠΑΡΑΔΟΣΗ 29/11/2016. (d)

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1 Ερωτήσεις και ασκησεις Κεφ 0 (για μόρια ΠΑΡΑΔΟΣΗ 9//06 Th coffcnt A of th van r Waals ntracton s: (a A r r / ( r r ( (c a a a a A r r / ( r r ( a a a a A r r / ( r r a a a a A r r / ( r r 4 a a a a 0 Th on lngth of s aout (n unts of 0 m (a ( 06 (c 57 ( Th on lngth of N s aout (n unts of 0 m (a 05 ( (c 6 ( 0 4 Th on lngth of O s aout (n unts of 0 m (a ( 45 (c 8 ( 0 5 Th on lngth of Na s aout (n unts of 0 m (a 0 ( 5 (c 08 ( 49 6 Th ssocaton nrgy of Na s aout (n unts of V (a 5 ( 4 (c 7 ( 08 7 Th ssocaton nrgy of N s aout (n unts of V (a ( (c 0 ( 8 Th van r Waals ntracton twn two nutral atoms at a stanc A 0A form (a A/ 4 ( A/ 5 (c A/ 6 ( A/ 9 Th vratonal quantum nrgy of N n mv s aout (a 0 ( 00 (c 800 ( 00 0 Th vratonal quantum of Na n mv s aout (a 0 ( 00 (c 800 ( 00 Th rotatonal quantum / J of Na n mv s aout (a ( 8 (c 004 ( 000 Th rotatonal quantum / J of n mv s aout (a,5 ( 5 (c 50 ( 0,5 a s fn as follows ( V / : Th olarty n (a (c a V / V V ( a V / V V a V V / V V ( V V / V s of th 4 W assum that n th formaton of a atomc molcul ach atom s mloyng only on atomc ortal of gnnrgy, (, V ˆ, In trms of th quantts ( /, an V / th groun stat nrgy of th molcul s gvn y th formula (omt U (a V, ( V, (c V V ( V V

2 5 W assum that n th formaton of a atomc molcul ach atom s mloyng only on atomc ortal of gnnrgy, (, In trms of th quantts a V / V V, V ˆ, an V / 0 th groun stat of th molcul s gvn y th formula (a a a ( a a (c ( 6 In th fgur low th two s hyrs, an, ar shown ontng along th -as Whch on of th followng formula s th rght on? (a ( ( s ( s y ( s ( s (c (, 7 In th fgur low th thr rght on? s hyrs ar shown Whch on of th followng formula s th (a (, ( s ( s y ( y (c s, ( ( s y 8 In th fgur low th four on? (a ( (c ( s hyrs ar shown Whch on of th followng formula s th rght ( y ( y ( y ( y, 4 s s, s s 9 Th matr lmnt Vh of th amltonan twn two s hyrs longng to nghorng atoms, lyng on th sam ln, an ontng n oost rctons s gvn y ˆ Takng nto account that ˆ ( /, s s j 4 for j ss s w fn that j j

3 (a V h = 77 ( V h = 4 (c V h = 9 ( V h = 0 Th matr lmnt Vh of th amltonan twn two s hyrs longng to nghorng atoms, lyng on th sam ln, an ontng n oost rctons s gvn y ˆ Takng nto account that ˆ ( /, s s j 4 for j ss s w fn that (a V h = 6 ( V h = 9 (c V h = 48 ( V h = 444 Th on of th atomc molcul O s (a sngl twn,, ( oul, a strong on twn, (c oul, a strong on twn, an a wak on twn y y an a wak on twn y, j (, (, / ( trl, a strong on twn, an two wak ons twn, an, In th fgur low th four normal vratonal mos of th O molcul ar shown (mo (c s ouly gnrat Whch on(s ar caal of asorng or mttng lctromagntc (EM raaton? y y j (a a, (,c (ca,c ( all of thm In th fgur low th four normal vratonal mos of th O molcul ar shown (Mo (c s ouly gnrat an s rsonsl for th grnhous ffct Th gnfrquncy of ths vraton s aromatly (n mv: (a 00 ( 56 (c 60 ( 80 4 Th nrgy of th systm of two nutral hyrogn atoms as a functon of th stanc twn thr rotons s gvn aromatly y th rlaton, E (04/ (049 / n atomc unts an for aroun th on lngth Th on lngth (n Angstroms of th molcul s accorng to ths rlaton:

4 (a 074 ( 05 (c 06 ( 4 5 Th nrgy of th systm of two nutral hyrogn atoms as a functon of th stanc twn thr rotons s gvn aromatly y th rlaton, E (04/ (049 / n atomc unts an for aroun th on lngth Th vratonal frquncy of th molcul n mv s accorng to ths rlaton: (a 56 mv ( 58 mv (c 80 mv ( 760 mv 8 Solv rolms Th nrgy of th systm of two nutral hyrogn atoms as a functon of th stanc twn thr rotons s gvn aromatly y th rlaton, E (04/ (049/ n atomc unts an for aroun th on lngth Otan th ssocaton nrgy of th molcul Soluton: Th ssocaton nrgy D s qual to E0, / mr 04/ an E/ (, an mr m / 98m E = /, whr E 0 s th mnmum of E, Thus th stanc at mnmum E s ; (/ 049 / 0786; 0786 / Thus, fnally, D V Show that th matr lmnt twn two s hyrs longng to two nghorng atoms, lyng on th sam ln, an havng oost rctons s gvn y th formula,( / Soluton: W shall choos th ln connctng th two atoms as th -as Thn th V = h s hyr lyng along th -as an longng to th atom on th lft has th form l s, whl th hyr lyng along th -as an longng to th atom on th rght has th form r s Thus l r s s s

5 onsr th nn molcul shown low an ts s unhyr atomc ortals Show that th nrgy gan woul qual to 6 V, f thr ons wr for twn thr ars of nghorng ortals, whl th gan s 8 V, f local molcular ortals accorng to Eq (6 ar for(oulom rulson ffcts hav n omtt Soluton: Accorng to Eq (0 y lacng two lctrons at th onng lvl for ach of th thr ons twn nghorng atomc ortals th lowrng of th nrgy s qual to V, snc n th rsnt cas V 0 Thus th total lowrng of th nrgy for th thr on cas s n V 6 V On th othr han for th actual cas whr th local molcular ortals ar gvn y (6 w shall occuy y two lctrons th lowst nrgy gvn y V V so that th contruton to th lowrng of th nrgy wll 4 V ; th nt ouly gnrat lvl at V cos( / V wll occu y th rmanng four lctrons contrutng thus to th lowrng of th nrgy y anothr 4 V to a total ructon y 8 V W s that th local confguraton accorng to Eq (6 roucs lowr total nrgy comar to th local thr on confguraton whch s fntly not th groun stat of th s ortals n nn 4 In th fgur low th four normal vratonal mos of th O molcul ar shown (th mo (c s ouly gnrat Prov that th four vratonal gnfrquncs of ths molcul satsfy th followng oul nqualty: c a an stmat thr valus Soluton: W assum that th strchng/comrsson srng constant of th /O on s κ, whl th nng on s Snc nng s much asr than strchng/comrssng w ct that For th mo (a th caron atom rmans unmov so that / m / (6 u For th mo (, as for vry ur vratonal mo, th cntr of mass of th systm s not movng so that m m 0 m / m ; th quaton of moton s O O O O O O O O O O O m k( m k m / m Thus w hav a O

6 ( m mo / (46 u, hnc a For th mo (c w hav as α rsult of th mmol cntr of mass 0 / m y k(y y m y k m / m y c O O O c O O O O m y m y y m m y an O O O O from th quaton of moton Thus c ( m mo / (46 u an th orrng of th gnfrquncs s c a To stmat th valus of ths gnfrquncs w shall start wth a an w shall us Eq (5 wth th rmntal valu of 9 m a c v 58 mv m m A 9 6 r Wr whl th rmntal valu s 65 mv From th aov analyss w hav that / 6 / 46 9 so that mv vs 9mV for th rmntal a valu Fnally, to stmat w shall us th nformaton that mo c s rsonsl for th grnhous c ffct; ths mls that th gnfrquncy of ths mo s clos to th frquncy m of mamum msson from th groun of Earth Assumng an avrag tmratur of th groun qual to 90K (s scton, 96 an a lack oy msson w hav m 8kT B 705 mv Actually th gnfrquncy s qual to 87 mv whch mans that 008 c 9 Unsolv rolms Th ntracton nrgy twn two ntcal nol atoms as a functon of th stanc twn thr nucl s gvn y th formula 6 j ( 4, / 0 For th Ar atoms 0 mv an 4 ( 0 m Dos th atomc argon molcul st at asolut ro tmratur? If ys, what s th on lngth? What s ts vraton frquncy? Wll ths molcul survv at room tmratur? onsr th O molcul Whch atomc ortals ar nvolv n ts on? Is th lattr sngl, oul, or trl? Any smlarty wth th ntrogn molcul? Fn th nrgy ga ( th nrgy ffrnc twn th lowst unoccu molcular ortal (LUMO an th hghst occu molcular ortal (OMO of th molcul 6 6