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Unformatted text preview: CHM4411—02 [PhysChem. II], Spring 2012
Instructor: H. Mattoussi
(6 pages total)
Exam 1 (Monday Jan. 30, 2012) _____ “Ki You name; Exercise 1 (6 points) Prior to Planck’s derivation of the energy distribution law for black—body radiation, Wien found
empirically a closely related distribution function that is very nearly but not exactly in agreement with the experimental results, namely: pOtJT) = (ea/ks) >< exp(—b/7thT). This formula shows
small deviations from Planck’s expression at long wavelengths. 8nhc hc
Afem ~l] (a) By ﬁtting (comparing) Wien’s empirical formula to Planck’s at short wavelengths determine
the constants a and Z7. ' (b) Demonstrate that Wien’s formula is consistent with Wien’s law (XmaxXT = constant). Planck’s distribution function is given by: ppmnck (AT) : Answers:
(a) We first need to simplify the Planck expression at small X. For 7t small (hc/kkBT) >>l. This he hc also implies that we can use the transformation: (emT — l) = e’V‘BT hc
8nhc MET
A~5 This expression is identical to that of Wien with a = 87thc and b = hc The Planck formula can then be written as: ppmck (LT) = e (b) Wiens’ formula is consistent with Wiens’ law, because:
At kmax the distribution function must satisfy the differential equation: [98] :0
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“AV/waif A A Zesl Q d {shah Q Q/_ r N 33 a A >3 *‘L W “JawMN; X a ( {45% 4; 652 (lava) Exercise 2 (4 paints) The work function for a metal Mediated by an incident light signal is 2.09 eV. Calculate the kinetic energy and speed of the electrons ejected by an incident light of wavelength equal to:
(a) 1000 nm (b) 300 nm
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{L M 3*}; an” {:23} iii 5%) LPV‘Qf/“J ‘ k ‘ i Exercise 3 (5 points) An un—normalized Wavefunction ‘P for an electron in a carbon nanotube of length L is:
(IJ(X) : sin(2xn/L). Normalize this function. This implies ﬁnding the corresponding normalized function ‘PN(X). Hint: sin2(y) =(1~cos(2y))/2 7f? 5/ ,3 L We .94 ea, :1 ’"f‘ fig. 71%” 4‘91) ; oer/“crew ﬂ» V“? 1 x 3 L g" 2“ 27; x, 2 [K73] it. C» K k? ,. ig’i “Tag J i»; X, (ix/,9
p « Exercise 4 (5 points) To provide a physical sense to the idea proposed by de Broglie, that to a moving particle we can
also associate a wave with a wavelength A = h/p = h/mv (as done by Einstein and Planck for
electromagnetic radiation), Schrodinger introduced the quantum mechanical wavefunetion w. 2 2
In one dimension, this equation is given by: ~3~d Mx) + V(X)i,b(x): E§U(X)
1 What is the Born interpretation of that quantum mechanical wavefunction u}? 2m dx2
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Respond to the statement by TRUE or FALSE. Justify your respor/se in one of two sentences. 2— As a consequence of the Born interpretation and normalization of the particle wavefunction u},
u;(x) can be inﬁnite. Viiﬁrgia’; “x ’ ~31 m If?) {gagiﬂé’
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V I 3 Because the Schrodinger equation imposes a second derivative on \J/(X), the wavefunction
does not have to be continuous. i ’ 3,55,» i,/15J£ZWQ~ [x3 ,3“ a, M}; 172:; 3 an A. is 1
’T‘éil/ég, . “1’”: W I I 54?) Li“ was; the. Gift? i7 ﬂue“ ‘ 4 The wavefunction cannot be zero everywhere. 3
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"‘ > t” ‘ ~ a win m f“? 6/3/12 Uth t . v («be m); e y 1 max . .L. W i J {:5}? 1/7 1/? is?!“ W £52,? V3 Zig’fér ‘ l 5 w(x) must be squareintegrable. ‘ {1/ X. J5 I ’ (ft 16/ i Extra {Points gamed here will be added 2‘0 your total grade) Exercise 2 (4 points) We assume that the state of a vibrating atom is described (in l dimension, x) by the 2
wavefunction 41(x) = A x x x exp[— x ] 2a2 The probability density associated with this function has a maximum value at a unique value of x
: XM. 1) Write the differential equation that must be satisﬁed at x 2 XM.
2) Show that the most probable location of the particle is at x = XM = a. A 3" 1 3 7' v; w‘ ~ n “" a a “ ’Z’“ 1"” ” “
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 Spring '12
 Dr.Matoussi
 Physical chemistry, pH, Distribution function, un—normalized Wavefunction ‘P, quantum mechanical wavefunction, related distribution function, energy distribution law

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