Assignment #1 Solutions

Transcription

1 Chm 453/544 Fall 003 9/0/03 Assignmnt # Solutions. Thrmodynamics is all about nrgis. Th standard unit of nrgy in chmical problms is kj/mol (not kcal/mol!). It is usful to b abl to quickly convrt this basic unit into othr commonly usd nrgy masurs. Exprss kj/mol in trms of th othr nrgy units: (a) V, (b) kcal/mol, (c) cm -, and (d) K. I usd th Mathcad worksht shown blow for this problm. : C : mol kj : 0 3 J V V cal 4.87 J kcal : 0 3 cal kcal mol 4.87 kcal mol h : J s c : m s cm : 0 m hc cm hc cm k B : J K k B K k B K 0.7 Not that with th xcption of kcal/mol, th rmaining convrsions rquird th us of fundamntal constants. Th unit V is a tru nrgy masur, but it is on a pr molcul rathr than a pr mol basis, so that th convrsion rquirs us of Avogadro s numbr. Th units cm - and K ar not nrgy units at all. Us of such quantitis as masurs of nrgy rquirs that on apply th basic rlationships ε h ν hcν and ε k B T. For purposs of mmory kj/mol is roughly quivalnt to: 0.0 V, ¼ kcal/mol, 80 cm -, and 00 K

2 Chm 453/544 Fall 003 9/0/03. Us th formulas and data in Chaptr of M&S to calculat th nrgy diffrncs btwn th two lowst translational, rotational, rational, and lctronic stats of Cl and compar ths nrgis to th thrmal nrgy k B T at 98 K. (Th st lctronically xcitd stat of Cl occurs at 7,60 cm -.) I carrid out th calculations using th Mathcad worksht appndd blow. Th final rsults ar: Mod / J / kj mol - / k B T translation rotation ration lctronic In this xampl, on finds that nrgy spacings ar much smallr than thrmal nrgis at room tmpratur in th cas of translational and rotational mods but gratr than thrmal nrgis in th cas of rational and lctronic stats. (This obsrvation is gnrally corrct and nabls us of classical approximations for translational and rotational nrgis in most thrmodynamic applications as w will s in Chaptr 3.)

3 Chm 453/544 Fall 003 9/0/03 3. (a) Using th data in Tabl 7. of th Brry, Ric, & Ross rading, dtrmin th rational quantum numbrs at which th actual rational nrgy lvl spacings (i.. thos computd using Eq. 7.0 of BR&R) of F and Cl dviat by 0% from th harmonic spacings basd on ν alon. (b) Compar th nrgis at which ths dviations occur to th dissociation nrgis (D ) of th molculs. (c) What tmpraturs do ths nrgis corrspond to? Vibrational nrgy lvls ar dscribd by: 3 ε h {( n + ) x ( n + ) + y ( n + ) +...} (3-) ν 3

4 Chm 453/544 Fall 003 9/0/03 Th spacing btwn adjacnt nrgy lvls (i.. th nrgy of th transition btwn stats n n+) is givn by: ε ( n + ) ε hν { x ( n + ) +...} and th quilibrium dissociation nrgy is: D D ε n 0) D hν { x + y...} (3-3) 0 ( Thr ar a numbr of ways that on could solv this problm. I chos to simply calculat nrgis for th first svral rational stats using th Excl spradsht shown blow. I chos to work in units of th rational quantum hν and to throw in a calculation for H for good masur. Th columns in th lowr part of th tabl ar th rational quantum numbr n, th anharmonic nrgy ( E ), th anharmonic nrgy spacings ( E ) and th anharmonic nrgy xprssd in units of th dissociation nrgy D and as a tmpratur. In units of hν th harmonic spacing is unity, so that finding th rational quantum numbr whr th anharmonic nrgy spacing dviats by 0% from th harmonic valu amounts to looking for th lowst valu of n for which ε / hν Ths valus ar highlightd in th tabl. F F F F Cl Cl Cl Cl H H H H v 99 D0/V.604 v D0/V.484 v D0/V x 3.6 cm/v vx.7 vx 0.85 x D0/hv 4.08 x D0/hv x D0/hv 8. y 0 D/hv 3.58 y 0 D/hv y D/hv 7.7 kb/hv 0.00 kbt/hv 0.00 kbt/hv v E E E E /kb v E E E E /kb v E E E E /kb /hv /hv /D / K /hv /hv /D / K /hv /hv /D / K (3-) Ths calculations show that 0% dviation from harmonic frquncis (highlightd) occurs nar v3 for F, v9 in Cl and v in H. Th nrgis at ths lvls corrspond to about 0-5% of th dissociation nrgy in all thr cass. (This commonality rflcts th similar shap of th rational potntial surfac for ths and many othr closd shll diatomic molculs.) Equating th nrgis of ths lvls to tmpraturs, 0% dviations ar quivalnt to tmpraturs in th thousands of dgrs K. Altrnativly, on could hav chosn to solv th problm algbraically. Ltting f dnot th fraction of th harmonic spacing w r looking for v such that: 4

5 Chm 453/544 Fall 003 9/0/03 harm anh harm f Using Eq. 3- for ε anh and ε harm hν this bcoms, x ( n + ) f or n f / x. Solving this quation for v I find n.4 in th cas of F and n 9.4 in th cas of Cl. 4. M&S -47: Exprss th nrgy both in units of cm - and kj/mol. Th zro point nrgy of a molcul (in th harmonic approximation) is: ε h ν. In th cas of CH 4 f , with many of mods bing dgnrat du to th molcul s high symmtry. Dgnracis ar indicatd by th valus in parnthss in Tabl.6. Using th information listd thr: ε zp - or ε.6 kj mol. zp - / hc / cm { } M&S -44: Dtrmin th numbr of dgrs of frdom of N, C H, C H 4,C H 6, and C 6 H zp f i i (3-4) Molcul N f tot typ f tr f rot f N 6 L 3 C H 4 L 3 7 C H NL 3 3 C H NL C 6 H 6 36 NL whr N is th numbr of atoms in th molcul, f tot is th total numbr of dgrs of frdom ( f tot 3N ), f tr is th numbr of translational (3), f rot th numbr of rotational ( for a linar molcul or 3 for a nonlinar molcul), and f th numbr of rational dgrs of frdom ( f f tot ftr f rot ). 5