chem03 - (5 Problem#1[Nuclear Chemistry(worth 15 of the...

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Unformatted text preview: ' . [(5% Problem #1 [Nuclear Chemistry] (worth 15 % of the grade on this exam) 3 P; (a) Complete (balance) the following nuclear equations: 9L (i) “N + ? --> 17O + 1p EH6 an. 20L Cm I (ii) ‘2 + '03 --> 249Bk + _ [5- %, 3 '19; (b) In the nuclear process known as "electron capture", an electron in which type of orbital is most likely to be captured, 3, p, d, or f? Explain. AN 8 mfg/7794 We amid art/Ms We 22w flow/52527? x97” 77%. NuCLew, V 3% (c) What is the thermodynamic criterion for why some nuclei are radioactive, i.e., decay spontaneously? Express this in terms of both AG and AE. awe, £E<G 3 ft (d) What is the Einstein mass-energy relationship? . r 2..., 5," ”— ‘ 5?. K ! M” «- JIR (A ‘5 fa ... 1% G va-£>~ fl : e a J fl”? k 3 I?" (6) Together, parts (0) and (d) above lead to what additional cn'terion for radioactive decay? } mu“ {(5% (a) Consider a particle in a one-dimensional box. If its width, L, is such that its n=2 f-> n=1 Problem #2 [Particle in a Box] (worth 20 % ) transition emits radiation having a frequency of 1015 sec1 (that's visible light), how much bigger (that is, wider) will this box (with the same particle in it) have to be expanded in order for the same transition to emit radiation at the frequency of an AM radio (106 $604)? ’3 ”72/52 Age/5,2“ fi‘ I; M be 3x N EN» N Zr» x N “m. “z \}i «Ema rs ii“ l s *' ' '* " 4/ ' Lfi:g,3)<,y fiAf ,. x [U m M 23 we W gm #3; Ma! (b) Consider a two-dimensional particle in a box with sides of length, L1 and L2. Let L1 = 2L2. For this problem, there are two quantum numbers, II] and n2. What is the energy for the lowest energy level in this system? Give your answer in terms of L2, h, and m. , r: 2;. gr a it? a, f ('1 . ,.. _-" .. f t; g i a , / “a m“ f 4" [f , £2; 2., \) if: ‘ 1‘ ’57 ,. j t W2 a 5 W77" ,3 M {a}, :s , e: 9.1:— , f r :5»!- fig (“‘1 g ( f 3 m f K m . 5: («we -: t A.» m (if! g r , fl __ I; 5”?!) I? ”‘°‘ 6, 5" I .3 {L {3‘ ti: f f; t; E" 9' third-Mi, ,z’avaz be? i "t 5"“ {’kfflifilg r» f I ’ I j? M M » f x71 g [f E I i f' x. M 1,—— m 5 J W ,,,,, .. t “ , ’ “° WW”; 1““ g F: W“ “‘ a. » . a , f )2, /;| 1 I m éegw I,» at.» .- ‘ ’ » KEY Problem #3 [Hydrogen Atom] (worth 20 points) /’_____\‘ /, Drag! (quantitaively to scale) energy level diagram for the hydrogen atom showing the energies (not the wavefunctions) of all allowed orbitals (states) from n=1 through n=3. Label all of them with their quantum numbers (n, 1, m, but not ms) as well as their orbital (eg., 23) designations. 3 (p j Z X ”:50 24: 2'0 2/ £-/ ,(~/ 18:2 ,sz =2 - =1 £5$2b0 3’3 gaze? 312s 59/; 3a.; 43% 590’. baa/i— Q/ugsO) CMfi'QOJ‘z’) (CM/g: 2J5 03—6—2) R20 fih/ J22! ,6—1/ flc'L20 ZS 26X 2%,! .25; I: l) ’/ (M’Q’O) {M} : I) O) _/) - #20) ‘ 0:0 ‘%QF% 3 mw.M—MM _ u )I . ‘53 fit {Mam/ Problem #4 [Quantum Mechanical Principles] (worth 30 points) 3?; (a) Write the Schrodinger equation (in either of the two forms that we discussed). H l” i E l” ’— 2 ' i” , _ 51/ - 0)? “—L' 0Q %r) 7L V60 09 _ I 00 flay; (b) What 13 a Hamiltonian? It IS the sum of what two things? r P Welt/3W Ext/em o AWE )f 4&5,» 0F 1:5 + V at 5”,:sz 7" V60 4% (c) If m represents a normalized wavefunction, evaluate the integral, wawd‘c, where (11: is a volume element, H is the Hamiltonian, and the integration is canied out over all space. We»: 77/47 fy/VOM: ,/ W W—zf’ MAI/rm :f/wzmyn: [SIM =5 2% (d) What is the equation which defines the de Broglie wavelength? )1 : h/fi [9 : M, V .2 3% (e) Write the equation for the Heisenberg Uncertainty Principle in terms of position and linear momentum. I!) > M AX9 / E‘Lfr ark (f) How did Born interpret the probability of finding a particle in terms of 11:3 wavefunction, ‘1’? EPA/hf; _ . . . . 2, 7/092 (3) State the Panel: glow M , 73% M Q‘ , . .4 a a i W 6"“ “W c11e I, 1 1 3”“ W MM W M of 3W 1.1% _ K a -» 2 r a . ,1? 1' ~ ,.- ' W fififw (h) State Hund’s rules. . W W /“’4€’Q5“ CM Magi“ 5’ NW? g __ , LIL/{r VJ; mv‘ l [1X ,IAICL, f, r i A ~ f 1 :- /‘ 7'5 n _ 0 q ‘(j _ "W“ ’ C7(\- PM : G. g I)! a 5'33, /% '1}; ,- ma 1 . w\ ‘. " 3 as: I): > w {\ fl. , . u M‘Qfi UM t 3% (i) Beyond the first level, how are the energy levels spaced 1n the one- -dimensional harmonic oscillator problem? KEY Problem #5 [The Periodic Table] (worth 15 points) (a) Write the electronic configurations of the first 21 elements in the periodic table. 2 / O ‘ H ,[3’ M [We] 352 3/) I Va; He [32 «21' [N9] 35‘ 3/7: L; [3222}: P (”£392 3/3 Be 132 3 me] 335% 9, Z / 5W 3 W4 N #1322793 A?” Md $73 2.2/6 0 Maw/7% K [Ar] 46’ F 3223223" Ce» CM #3 J’c MU 31/53 I {if (b) Kr" should be chemically similar to what Group of neutral atoms? Why? . Hawk/w 7:7 32:33 4 {film/9 Cami? @333!” a,» [K (c) Which has the larger ionization energy, Na or Ar? Why? fl?” fit” 5: .35 :35, arfl J/é/e f /i fife: [:9 waft [ fi’ (d) Which is physically larger, I or F? w l H ’ (e) What is the electron affinity of Xe? IF (f) What is the ionization potential of H? L . j end of exam Problem #1 (The Hydrogen Atom) (25 points) All parts of this problem pertain to the hydrogen atom. l 0 (a) Draw (quantitaively to scale) an energy level diagram for the hydrogen atom showing the energies (not the wavefunctions) of all allowed orbitals (states) from n=1 through n=4. Label all of them with their quantum numbers (n, 1, m1, but not ms) as well as their orbital (eg., 28, 2px, etc.) designations. 1 2‘0 W 11’ ’50 l3 w- . W W (b) For n = 12, there are a total of how many orbitals (allowed sets of quantum numbers)? 9— .. n — /¥f‘ r ‘gifi (c) For] 2 5, there are how many m1 values? (1 is the orbital angular momentum quantum number). i - Le+/ : // 39g ((1) The 3s orbitalhas how many radial nodes in its wavefunction? H “l = 2“ fifl; (e) In order to have a finite probability of being found right at the nucleus, an orbital (Wavefunction) must have what value for its quantum number, 1? 1 3 0 Problem #2 [Compare & Contrast Simple Quantum Mechanical Systems] (25 pts.) Consider the following quantum mechanical systems: (i) a one-dimensional particle in a box with infinitely high walls, (ii) a one—dimensional harmonic oscillator, and (iii) the hydrogen atom. (a) Sketch their potential energy diagrams, side by side, below. (a KEY (b) Within the confines of each potential energy diagram you have drawn above, sketch the first four energy levels for each, being careful to sketch them to scale (within that system). Note: The energy level spacings may increase, decrease, or remain constant with increasing energy. It is important to show this clearly for each system. )1)- M (c) Label each energy level with its quantum number, starting om its ground state. iu (d) For (i) and (ii), sketch the wavefunctions on top of each of the energy levels you have drawn above.‘ 291 GM (6) For (i) and (ii), what is the behavior of their wavefunctions at the edges of their potential wells? flan (’3) corresponding force, F(x or r). F; "VV Problem #3 [Quantum Mechanical Concepts] (20 points) (a) What is a Hamiltonian? It is the sum of what two kinds of things? Kfi/ (b) The expectation value of energy, <E>, is Swde‘c, where (it: is a volume element, H is the Hamiltonian, w is a normalized wavefunction, and the integration is carried out over all space. Show that <E> = E. ””55” <E> “fr/m Wyn 5 Mr E b (c) What is the equation which defines the de Broglie wavelength? h = _..___. P=MV ’ P (d) If 1111 represents the wavefunction of a px orbital 1n the hydrogen atom, and In represents the wavefunction Of a py orbital in the same hydrogen atom, what 18 the value of the integral, j‘l’l‘l’zd'c, where d1 18 a volume element, ‘1’] and T2 are both normalized wavefunctions, and the integration is carried out over all space. What is the value of the integral,5‘l’1‘l'1df? jflgj}:o' 55%Qh= (e) Show that sin (Zinc/7t) 1s an eigenfunction of the operator, dZIdxz, and find the corresponding eigenvalue. .. a owe/TX) _ fl; (Lani £77)! 2 h algae/11??) # Wfl’fl: a ‘KE/ (a) Write the ground state, electronic configurations for the f0110wing atoms in the periodic table. . . 2’ (i) Mg ‘ If fi‘i/‘fi (ii) AP“ [,9 2,1 "279‘35’ 3/5 l (m) Sc .rf/l JJJ 6‘3 (iv) 1- fi/(yj 5‘ ”5312576 on [X93 ’ (V)Na 0/633 M- Mil-#177533 (vi) Zn 047,] 30/”)6‘5'1' (b) Are the electron affinities of the following atoms likely to be positive (they want an electron) or negative? Why, 1n each case. . 2. (1) C1 fay/7 / Y6 Md 3,? 3f flT/ayeecfll’f/Z/frgl;%fl/II <fi>Mg Nam/177m Mejis'z 17,3, ,4 Problem #4 [The Periodic Table] ' (30 points) 2 MM - 7 MW? (iii) Xe ”9 game Io 05192.0 .9ch [fix/0186 deem/«J ‘ WW3z9€_ ime [W MIN/god (c) Chemical similarities (i) The atomic anions of the alklai metals should be chemically ysEm milar to which Group of (neutral) atoms? 7,1212 M Q (ii) The atomic cations of the rare (noble) gases should be chemically similar to which Group of (neutral) atoms? . W 432959; (iii) The Group containing Zn should most closelyfiw‘fié share chemical similarity with which other Group 1n the periodic table? m 92 a (d) Size, ionization energies, and magnetic properties. State your reasons. (i) Which IS larger, Cs or K? (:9 9 W W 0060M CL Gran—m (ii) Which has the higher ionization energy, Kr or K? K7) J’ a M M (iii) Which 1s paramagnetic, B or Ar? 3) J?— 4 MW m end of exam ...
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