Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz # Nam KEY Problm (3 points). Ammonia gas is vry hygroscopic (asily racts with watr), so it is packagd for shipping in a small dry gas cylindr of 50 ml volum. Standard shipping conditions ar room tmpratur and th prssur of 0 atmosphrs insid th cylindr. For this sampl in transit, dtrmin th ratio of rms and most probabl spd basd on kintic thory of gass. Answr: This is a vry simpl problm sinc for a givn mass, this ratio is indpndnt of any of th paramtrs listd abov: u % 3kT M v # % kt 3. M Of cours, as xpctd, th most probabl spd is lowr than th rms spd. Problm (4 points). Estimat th man fr path for mthan molculs at atmosphr prssur and room tmpratur basd on kintic thory of gass. Answr: λ πd n 456 RT πd PN ; λ atm 035Pa atm 8.444 mol K 98.5K 6.0 0 mol π (0.3783 0MN m) 6.45 0 MP m Problm 3 (3 points). Th Cs film was vaporatd in vacuum onto on sid of a cubic sampl of th carbon matrial. At room tmpratur in 3 hours, on avrag th pntration dpth of Cs into this matrial was stimatd to b 0 µm basd on microscopic analysis. Estimat a diffusion cofficint for csium through msoporous carbon basd on ths xprimntal data. Answr: This is a D procss: x Dt; D TU WXY XYZ[ U \] c[dd fg V ^ `ab h ij 4.63 0 MXk \U b
Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz # Nam KEY Problm (8 points). Exprimntal studis of a dcomposition raction dtrmind that it is formally a scond ordr procss. Th following st of data was obtaind xprimntally: T/K 556 69 700 78 k/(l mol - s - ) 3.5 x 0-7 3.0 x 0-5.6 x 0-3 3.95 x 0 - Dtrmin th Arrhnius paramtrs for this raction. On th graph blow, dmonstrat your rsults clarly with an appropriat plot (vn if you us your calculator for linar rgrssion). Indicat th units of th obtaind paramtrs. Answr: First, w nd to plot ln(k) vs. invrs tmpratur, so lt s convrt th abov tabl into th appropriat axs: 0 3 K/T.799.590.49.80 Ln(k/L mol - s - ) -4.860-0.408-6.759-3.3-4 -6-8 ln(k) -0 - -4.3.4.5.6 /T(K) Nd th slop and th intrcpt. E R4386.35 k/mol; A xp (5.35).03 0 litr/mol/sc Problm ( points). Driv th xprssion for th half-lif of th first ordr procss starting with th tim dpndnc of th ractant concntration for th first-ordr kintics. Answr: c(t) c(0)xp(k, t); c(t,/) c(0).7.8x0-3 xp3k,t,/ 4; t,/ ln k,
Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz #3 Nam KEY Problm (6 points). Lindmann s mchanism for a unimolcular raction A Product involvs a mdiator, M, that crats an xcitd-stat spcis, A. Th mchanism has th following lmntary stps. k A M ¾¾ A M A M k ¾¾ A M k3 A ¾¾ Product Driv th rat law for th apparanc of product at stady stat. Show all work clarly. This is a simpl straightforward, but vry important mchanism that ultimatly is found in th RRKM thory. Product is producd only in th last stp, so d[ P] k3[ A ] ss dt Howvr, th xcitd-stat spcis is an intrmdiat. So, on assums a stady-stat concntration of ths intrmdiat spcis. d[ A ] 0 k[ A][ M ] - k[ A ][ M ] - k3[ A ] dt Rarranging this to solv for th stady-stat concntration of th intrmdiat givs: k[ A][ M ] [ A ] ss k[ M ] k3 Substitution of this into th quation abov givs th final rsult: d[ P] k3k[ A][ M ] k3[ A ] ss dt k[ M ] k3 Problm (3 points). From th statmnts in column B, slct th bst match for ach phras in column A and put its lttr in th adjacnt blank. Thr is only on bst match for ach phras. Column A Column B. In Eyring thory, d rprsnts th systm at th transition stat. A raction mchanism that consists of a singl lmntary stp is known as a. 3. Th dscription of a raction mchanism whr quilibrium btwn th ractants and th intrmdiat is maintaind during th cours of th raction and only th intrmdiat undrgos dcay to form a product is rfrrd to as c. a) stady-stat approximation b) diffusion-controlld limit c) prquilibrium approximation d) activatd complx ) simpl raction f) stoichiomtric numbr g) funky approximation h) activation nrgy i) complx raction j) activatd ractant approximation
Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz #4 Nam KEY Problm (9 points). Find th commutators for th following pairs of oprators: a)! "!"!#" and!( " )!" ;!"!# "!( " f(x. y)!" 3!"!# "!(!" f4 3!" f4 0, so th oprators commut "!( "!#" Or you could simply stat that th two oprators work in diffrnt indpndnt dimnsions, so thy must commut and th commutator is zro. b) y 7 and!!( 8y 7 ; d dy 9 f(y) y7 : d dy f; d dy (y7 f) y 7 : d dy f; <yf y7 : d f;? yf dy so th commutator is qual to -y c) sinθ!!c and! "!C " Dsinθ d dφ ; d7 d dφ7f f(θ, φ) sinθ dφ H d7 d7 df fi :sinθ dφ7 dφ7 dφ ; sinθ d f dφ inθ d f dφ 0 so th commutator is qual to zro and th oprators commut Problm ( point). In a discrt nrgy spctrum two lins ar sparatd by.6 k/mol. Calculat th frquncy corrsponding to th transition btwn th two corrsponding nrgy lvls in cm -. From Tabl 3.5: cm - 0.09666 k/mol Thn Dn cm - x.6 k/mol/0.09666 k/mol 33.75 cm -
Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz #5 Nam KEY Problm (7 points). Compos a corrct ignquation. Indicat th ignfunction and th ignvalu of th oprator. Any quation that fits th dfinition is appropriat. Any quation that uss an appropriat function with a corrsponding oprator, such that th oprator usd on this function rsults in a product of a constant and a function itslf is corrct. For xampl: d (cos(x)) cos (x) dd Hr cos(x) is an ignfunction of th oprator d, th ignvalu is -. dd Problm (3 points). From th statmnts in column B, slct th bst match for ach phras in column A and put its lttr in th adjacnt blank. Thr is only on bst match for ach phras. Column A Column B. Bcaus all valus of nrgy ar allowd for a classical systm, it is said to hav j nrgy spctrum.. R H in th quation ν R H n n is calld f constant. 3. Th proportionality constant btwn nrgy radiatd by black body and frquncy was proposd by k. a) undtrmind b) work function c) Ruthrford d) photolctric ) rsonanc f) Rydbrg g) Ritz h) discrt i) undfind j) continuous k) Plank l) Rachl m) dbrogli
Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz #6 Nam KEY Problm (8 points). Normaliz th wavfunction corrsponding to th scond lowst nrgy lvl of a quantum mchanical harmonic oscillator. Th answr should xplicitly prsnt th normalization constant for this function, but you may lav constant α in that formula. Th gnral formula for constructing wavfunctions of QM harmonic oscillator is providd on pag - of th Handbook. ψ n A n H n x x α α A x x n α α ψ n dd 4(A n ) x α x α dd 4α(A n ) x α x α d x α 8α(A n) x α x α d x α 0 and 8α(A n ) 4 π π 4 A n α π απ /4 oo ψ n x α x απ/4 α x xxxx α 3 π/4 α Not, that th dfinition of α is diffrnt in th Handbook and in th txt. Th abov formulas rfr to α ħ kk Problm ( points). What is th dgnracy of th scond lowst lvl of nrgy for a particl in a cubic 3D box (abc) Answr: Th scond lowst nrgy lvl for this systm would hav on of th thr quantum numbrs n qual and th othr two qual, maning that thr diffrnt solutions ((n x, n y, n z ), (,,), and (,,) would yild xactly th sam nrgy. In othr words, th dgnracy of this lvl is 3. /4
Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz #7 Nam KEY Problm (8 points). Fill in th tabl blow. All atoms ar in thir ground stat configurations. In th trm symbols, you do not hav to provid th indication of th total angular momntum () Atom Configuration S L List all trm symbols. Circl th lowst nrgy trm for multipl trms. H s ½ 0 S B s s 0 0 S O s s p 4 0 or 0,, or S, 3 P, D Problm ( points). What is th lowst nrgy trm for d 5 configuration and is thr spin-orbit coupling contribution to this trm (dos th nrgy lvl corrsponding to this trm split in multipl lvls bcaus of spin-orbit coupling)? According to Tabl.6 in th Handbook, 6 S trm for d 5 configuration has th highst spin multiplicity, maning that it corrsponds to th lowst nrgy trm. Thr is only on sxtuplt trm corrsponding to this configuration. Sinc th spin stat for this trm is S0, thr is no spin-orbit coupling contribution to th nrgy of this trm.
Physical Chmistry 444-0 Nam KEY Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz #8 Problm (7 points). Us data in Tabl. to calculat th frquncy (you can lav th answr in trms of commonly usd cm - units) of th transition from n0, 0 to n, for H 35 Cl in cm - as accuratly as you can. You can disrgard th cntrifugal distortion trm. ) ( ) ( hd hb x n n D E c n n ( ) n B h x n B h x E α α 3 ) ( 4 4 9 ( ) 905.89 0.309 3 0.5909 5.05 989.74 cm cm cm cm cm E n As you can s, for this molcul, th corrctions to th fundamntal harmonic trm ar vry substantial! Problm (3 points). As was discussd in class, X-ray photolctron spctroscopy is a vry powrful surfac analytical tchniqu, and quantum mchanics hlps us idntify th spin-orbit coupling contributions to th xprimntal spctra. Blow is an XPS spctrum of gold foil with a sris of obsrvd faturs. For th paks corrsponding to 4d and 4p lctrons, indicat th corrsponding spinorbit coupling by placing th valu of th total angular momnta in th boxs providd.
Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz #9 Nam KEY Problm (0 points). Fill in th tabl blow. All molculs and ions ar in thir ground stat configurations. In th trm symbols, you must indicat th symmtry undr rflction in th σ plan only for Σ trms and symmtry undr invrsion for ALL trms Molcul Configuration S Λ List all trm symbols. Circl th lowst nrgy trm for multipl trms. H σ g 0 0 Σ g B σ g σ u σ g σ u 0 0 Σ g B σ g σ u σ g σ u / 0 Σ u N σ g σ u σ g σ u π u 4 3σ g 0 0 Σ g N σ g σ u σ g σ u π u 4 3σ g / 0 Σ g Problm ( point, bonus). In class w covrd spin-orbit coupling and hyprfin coupling (intraction btwn nuclar spin and lctron spin). In both cass, th total Hamiltonian of th systm was simply corrctd by prturbation, adding anothr trm to th Hamiltonian. As I also mntiond in class, fundamntally, coupling of any angular momnta is possibl. For xampl, thr is a vry small coupling btwn nuclar spin and lctron orbital angular momntum. Writ a vry gnral Hamiltonian dscribing this intraction. You can assum that similarly to spin-orbit coupling constant and to hyprfin coupling constant, th intraction btwn nuclar spin and lctron orbital angular momntum is dscribd by a constant A n,spin,,ang. For hyprfin coupling, H hf ha I. S; whr A is th hyprfin coupling constant, I is th oprator of nuclar spin angular momntum, and S is th oprator of lctron spin angular momntum. Similarly, for coupling btwn nuclar spin and lctron orbital angular momntum: H n,spin,,ang ha n,spin,,ang I. L, whr I is th oprator of nuclar spin angular momntum, and L is th oprator of lctron spin angular momntum.
Physical Chmistry 444-0 Spring 08 TR 5:00-6:5 pm, 07 BrL Quiz #0 Nam KEY Problm (7 points). For ach molcul blow, giv th point group in th box to its right. Molcul HI Point group C v O D h,,3,4,5,6-hxachlorobnzn D 6h,,-trichlorothyln C s Bromomthan C 3v dibromomthan C v mthan T d Problm (3 points). Tru of fals (mark appropriat column) Statmnt Tru Fals Th diffrnc btwn fluorscnc and phosphorscnc is th wavlngth of radiation X In HF molcul, th σ orbital is primarily s orbital of a fluorin atom. Th valnc shll lctron pair rpulsion modl provids a qualitativ rationalization of molcular structurs using Lwis concpts of localizd bonds and bond pairs X X