1. Solutions to Homwork 8 chm 44 Sp 14
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4. All diffrnt orbitals mans thy could all b paralll spins 5. Sinc lctrons ar in diffrnt orbitals any combination is possibl paird or unpaird spins 6. Equivalnt to min J lowst for lss than half-filld a. Allowd, b. Allowd,S = & J = 1, c. Forbiddn by spin,s = 1 7.
8. 9.
1. P15.6) Show that th total nrgy ignfunctions 1 (r,,) and 11 (r,,) ar orthogonal. Do you hav to intgrat ovr all thr variabls to show that th functions ar orthogonal? This intgral is zro bcaus sin cos sin d. It is sufficint to valuat th intgral ovr 11. P15.17) Ions with a singl lctron such as H +, Li +, and B + ar dscribd by th H atom wav functions with Z/a substitutd for 1/a, whr Z is th nuclar charg. Th 1s wav function bcoms r 1 Z a Zr a. Using this rsult, compar th man valu of th radius r at which you would find th 1s lctron in H, H +, Li +, and B +. Using th standard intgral: r 4 4Z 6a a 16Z Z a 4 1 r a ; r a ; r a ; r a 4 8 H H Li B 1. P15.) Cor lctrons shild valnc lctrons so that thy xprinc an ffctiv nuclar charg Z ff rathr than th full nuclar charg. Givn that th first ionization nrgy of Li is 5.9 V, us th formula in Problm P15.18 to stimat th ffctiv nuclar charg xprincd by th s lctron in Li. From th prvious problm, Z I 1.6 V ; Z n ff ni 45.9V 1.6 1.6 V 1.6 V
1. P15.4) Th forc acting btwn th lctron and th proton in th H atom is givn by F = /4 r. Calculat th avrag valu F for th 1s and p z stats of th H atom in trms of,, and a. F 1s 4 1 r * d 1 r / a 1 r / a sin a r F d d r d r 1s 4 4 r / a 4 a r / a a a a F dr 1s 4 4 F pz 4 1 r * d 1 r r/ a 1 cos sin a a r F d d pz r d r 4 1 cos r/ a F r dr pz 5 4 16 a 1 r/ a F r dr pz 4 4 a Using th standard intgral F pz 1 4 4 48 n r r n! n1 a 5 a a 14. P15.) You hav commissiond a masurmnt of th scond ionization nrgy from two indpndnt rsarch tams. You find that thy do not agr and dcid to plot th data togthr with known valus of th first ionization nrgy. Th rsults ar shown hr: Th lowst curv is for th first ionization nrgy and th uppr two curvs ar th rsults for th scond ionization nrgy from th two rsarch tams. Th upprmost curv has bn shiftd vrtically to avoid an ovrlap with th othr nw data st. On th basis of your knowldg of th priodic tabl, you suddnly know which of th two sts of data is corrct and th rror that on of th tams of rsarchrs mad. Which data st is corrct? Explain your rasoning. Th data st shown by th dashd (purpl) lin is corrct, th rd on is incorrct. Although th alkali atoms hav
th lowst ionization nrgis, thy must hav th highst ionization nrgy for th scond ionization potntial bcaus th singly chargd positiv ions hav th rar gas filld shll configuration. Th xprimntrs that producd th data st shown by th gray (rd) lin had assignd atomic numbrs that wr too low by on. 15. P15.) An approximat formula for th nrgy lvls in a multilctron atom is E n Z ff 8 a n, n 1,,,., whr Z ff is th ffctiv nuclar charg flt by an lctron in a givn orbital. Calculat valus for Z ff from th first ionization nrgis for th lmnts Li through N (SEE www.wblmnts.com). Compar ths valus for Z ff with thos listd in TABLE 15.1. How wll do thy compar? Using: Z ff 8 E ion n a N A 8 E ion 1 (8.85418781-19 (1.61771-1 -1 C J m ) (5.91771-1 C) (6.1 mol ) th ffctiv chargs basd on th first ionization nrgis for th lmnts Li to N ar calculatd as: Elmnt Li B B C N O F N E ionization in kj mol -1 5. 899.5 8.6 186.5 14. 11.9 1681. 8.7 Z ff 1.6 1.66 1.56 1.8.7..6.5 A comparison with th data in Tabl 15.1 shows that th approximation rproducs th ffctiv chargs rasonably wll for th scond-row lmnts with only s lctrons, howvr, fails to prdict th chargs for lmnts with p lctrons. Extra Problms 1. -1-11 m),
.. 4. 5. 6. a. Allowd, b. Allowd, c. Forbiddn
7. P15.8) How many radial and angular nods ar thr in th following H orbitals? a. px r,, radial nod and 1 angular nods b. s (r) 1 radial nod and angular nod c. dxz r,, d. dx y radial nod and angular nods r,, radial nod and angular nods 8. P15.16) Calculat th man valu of th radius r at which you would find th lctron if th H atom wav function is 1 (r,,). r 5 a cos sin 5 1 r d d r dr a r r r 5 a 5 a 5 a 5 5 5 cos 1 r r dr r dr r dr a a 4 a Using th standard intgral: 1 6 r 5 5! a 5 a 4a 9. P15.18) Th nrgy lvls for ions with a singl lctron such as H +, Li +, and B + ar givn by E n = Z /8 a n, n = 1,,,... Calculat th ionization nrgis of H, H +, Li +, and B + in thir ground stats in units of lctron-volts (V). Th ionization potntial is th ngativ of th orbital nrgy. Z 19 1.61 C Z 1V I n 1 1 1 11 19 8 a n 8 8.8541 J C m 5.91 m 1.61 J Z I 1.6 V n I 1.6V; I 54.4 V; I 1.4 V; I 17.7 V H H Li B