14.50. (a)same number of vibrations (b)C 2 H 2 should have
a less number of unique vibrations (c)CH 4 should have fewer
unique vibrations. (d)PF 5 should have fewer unique vibra-
tions.
14.51. (a)proper: E, C 3 ’s; improper: v’s. (b)proper: E, C 3 ’s,
C 2 ’s; improper: S 4 ’s, d’s.
14.53. (a) 6 (c) 3 (d) 6
14.54. (a) 2 (b) 6 (c) 8 (d) 6 (e) 2
14.55.The different symmetries will give a different number
of IR-measurable vibrations: tetrahedral symmetry gives 2 IR-
active vibrations; square planar symmetry (i.e., D4h) yields 3
IR-active vibrations.
14.58.618 cm^1 2 1 , 2337 cm^1 3 2 2 1 ,
and 3715 cm^1 2 1 3 or 2 3.
14.63.r1.00 Å
14.66.There is no such thing as a P(0) line. Pbranches are
J1, and a molecule cannot go from J0 to J1.
Chapter 15
15.1.Possible excited states can only have E1usymmetry.
15.3.R(D) 109,660 cm^1 , R(T) 109,670 cm^1
15.4.15,231 cm^1 , 20,561 cm^1 , 23,029 cm^1 , 24,369 cm^1
15.6. E 0
15.7. (a)L2, 1, 0; ML2, 1, 0, 1, 2 (depending on
the value of L); S1, 0; MS1, 0, 1 (depending on the
value of S); J3, 2, 1, 0 (depending on the values of Land
S); MJ3, 2, 1, 0, 1, 2, 3 (depending on the value
of J).
15.8.MS^12 or ^12 for the H nucleus. If the nucleus and
electron angular momenta are coupled, MScan equal 1, 0,
or 1 for the complete atom.
15.9.Aluminum has a single pelectron in its highest-filled
subshell.
15.10. (a)2, m2, 1, 0, 1, 2, j^52 (mj^52 , ^32 ,
^12 , ^12 , ^32 , or ^52 ) or j^32 (mj^32 , 2 ^1 , or ^12 )
15.11. E 4 10 ^22 J
15.12. (a)^2 S1/2(b)^2 P1/2(c)^2 D3/2
15.14.^3 P 0 is the term symbol of the ground electronic state
of C atoms, and is understood to have an energy of 0 with re-
spect to itself.
15.17.^3 F 2 ,^3 F 3 ,^3 D 1 ,^3 D 2 ,^3 D 3 , and^3 G 3
15.20.Heteronuclear diatomic molecules—and their wave-
functions—do not have a center of inversion symmetry ele-
ment, which is required to use gerade and ungerade labels.
15.21.^3 g,^3 g,^1 g
15.29.There will be no change in the Hückel approximation
of ethylene because the hydrogen atom (any isotope) does
not contribute to the orbitals.
15.30.For cyclobutadiene: 015.31.Cyclopentadiene can accept an electron and have the
maximum possible decrease in energy due to delocalization.
That is, with 6 electrons, Cpis aromatic.
15.34. (a)C 4 n 2 H 4 n 2 , n0, 1, 2,... can be aromatic.
(b)C 4 n 1 H 4 n 1 , n1, 2,... can be aromatic. (c)C 4 nH 4 n^2 ,
n1, 2,... can be aromatic.
15.35.Heating a laser-active material populates energy levels
only thermally, eventually establishing an excited-state popu-
lation dictated by the Boltzmann distribution. Population in-
versions won’t be achieved.
15.39.~1.60 1025 photons per second
15.41.1.20 108 W
15.42.Because the ratio of spontaneous emission to stimu-
lated emission increases proportional to ^3 (see equation
15.33), as the frequency increases, spontaneous emission dom-
inates over stimulated emission. This makes it progressively
more difficult to construct a laser for shorter and shorter wave-
lengths of light.Chapter 16
16.2.0.06 gauss
16.3. (a)Magnetic dipole would be the same. (b)See
Example 16.2b.
16.7. (a) E2.18 10 ^23 J, 0 J, and 2.18 10 ^18 J
16.8.0.357 T
16.9. E(MJ^32 →MJ^32 ) 2.23 10 ^27 J (1.12
10 ^4 cm^1 )
16.11.g1.50115 vs. 1.5
16.12. E3.10 10 ^24 J, 1.03 10 ^24 J, 1.03
10 ^24 J, and 3.10 10 ^24 J
16.15. 5
16.18.0.388 T
16.19.~0.325 cm^1
16.22.7 hyperfine absorptions at 3716, 3638, 3560, 3482,
3404, 3326, and 3248 gauss
16.23.The resonant transitions should occur at the same
wavelength.
16.24.a, d, f, and g have nonzero nuclear spins and so will
show NMR spectra under the right resonant conditions.
16.25. (a)50.5 T (b)8.34 T (c)19.1 T
16.28. E1.432 10 ^26 J, 4.774 10 ^27 J, 4.774
10 ^27 J, and 1.432 10 ^26 J
16.29. E14.41 MHz
0
E0
E
E
0 E
0
Chapter 16 813