Lasers, Laser Spectroscopy, and Photochemistry
SO
2.3038Cl = a N1
2 303 N N
. = a
8 NA
2.303; _
N _
U
A
For the benzene solution in Problem 1539, 8 = 629 mz-mol", so
2.303 629 2- l"
a = (27% = 2.41
MATHCHAPTER G
Numerical Methods
PROBLEMS AND SOLUTIONS
Excel was used to create spreadsheets for the approximations in the problems below. Any spreadsheet
program can be used, but programs such as E
Lasers, Laser Spectroscopy, and Photochemistry 45 7
1510. Use the equation given in Problem 159 and Equation 15.13 to derive the quantum mechanical
expression for the Einstein B coefcient. Consider th
Nuclear Magnetic Resonance Spectroscopy
Use the energies given by Equations 14.65, recalling that E = hv:
hJ h a +0 hJ
EFEI=_T*["3(Un02)2+12]l/2+h"0(1_ 2 2)_T
h] h hv
_2 E [305 _ 72)2 + J2]l/2 + (20
474
MathChapter G
for the value of . This formula was discovered by a Babylonian mathematician more than 2000
years ago. Use this formula to evaluate J2 to ve signicant gures.
x2 = A, so x2 A = 0 = f(
470 Chapter 15
The following four problems examine how the intensity of absorption lines are quantied.
1538. The decadic absorbance A of a sample is dened by A = log(IO/1), where 10 is the light
int
Lasers, Laser Spectroscopy, and Photochemistry 467
This means that we need
(1.00 X 1022 molecule-s)
071 = 1.41 x 1022photon-s'
or an output radiant power of
nhc _ (1.41 x 1022 s)(6.626 x1034J-s)(2.998
The functions we will use in the spreadsheet created for Problem Gl are then
f(x) = 0 = 27x4 26x2 + 52x 26
f'(x) : 108x3 52x + 52
We know that x must be between 0 and 1, so we can take x0 to be 0.5. T
Lasers, Laser Spectroscopy, and Photochemistry 463
1524. A CO2 laser operating at 9.6 12m uses an electrical power of 5.00 kW. If this laser produces
100-ns pulses at a repetition rate of 10 Hz and ha
476 MathChapter G
Setting f(x) equal to zero, we nd that the inection points of the equation x3 3x + 1 are 1
and 1. We can therefore set our xos to 0, 1.5, and 1.5.
n It,l f(x,) f(xn)
0 0.00000 1 .000
Nuclear Magnetic Resonance Spectroscopy 449
where 1:1) is the Hamiltonian operator that causes the transitions from one state to another. In
NMR spectroscopy, there are two magnetic elds to consider.
472 Chapter 15
where a is a constant and 17m is the maximum frequency of absorption. Plot K07). How is oz related
to Al/Z, the width of the absorption line at half of its maximum intensity? Now show t
Lasers, Laser Spectroscopy, and Photochemistry 469
Because the sample absorbs only 10% of these photons, 1.31 x 10H photons are absorbed by the
sample, and because the quantum yield is one, 1.31 x 10'
460 Chapter 15
1517. Using the method explained in Section 89, show that the states associated with a 2p5ns
electron conguration are 3P2, 3Pl, 3P0, and lPl.
There are six distinct ways of assigning th
Nuclear Magnetic Resonance Spectroscopy
Transitions are possible only if PM]. # 0, so the transitions 1 > 2, 1 ~> 3, 2 > 4, and 3 > 4 are
possible - the others are forbidden.
1439. Using the spin func
Lasers, Laser Spectroscopy, and Photochemistry 459
and a radiative lifetime of 1/ 2?: AM. The radiative lifetime of neon would be
1
Tm" _ (0.48 + 0.60 + 0.70 + 6.56 +1.35 + 1.28 + 0.68 + 0.56) x 106 s
Lasers, Laser Spectroscopy, and Photochemistry 46]
Because each pulse is 25 fs long, the radiant energy of each laser pulse (which is measured in watts)
is
1.4 X 10'8 J-pulse"
_15 = 560 kW
25 x 10 s
T
45 8 Chapter 15
1512. Consider the nondegenerate three-level system shown in Figure 15.8. Suppose that an incident
light beam of energy hv 2 E3 El is turned on for a while and then turned off. Show th
454
Chapter 15
The spectral radiant energy density was dened as pu(v) = dp/dv. Replacing v with i; or A gives
units of
~ dp J-m3
_ : :1 = J _2
pu(v) di rnl m
or
dp J-m3
A, = = = J. _4
10A ) dk m m
468
Chapter 15
v = O, J = 3 of the X22)+ ground state to the v = 0, J = 3 level of the B22+ excited state?
Use the following spectroscopic data.
" 1 ~ 1 ~ 1 ~
State TP/cm vP/cm vyxg/cm Be/cm ate/cm
B
464 Chapter 1 5
excited state and the v = 12 level of the ground state. Use the following spectroscopic data to
determine the wavelength of the laser light from the H2(g) laser.
1 l
State 7~;/Cm_l 17,
450 Chapter 14
with a similar equation for Py. Using the notation given by Equations 14.30, show that the only
allowed transitions are for l '> 2, 1 > 3, 2 > 4, and 3 > 4.
The functions given by Equat
Lasers, Laser Spectroscopy, and Photochemistry 455
mm+mm=N
mm, so we can write Equation 15.15 as
d N20)
dt
Taking the derivative of Equation 15.16 gives
= BOu(ul2) [Ntotal(t) 2N2(l)] _ AN2(t) (1)
_
462 Chapter 15
1521. Which laser pulse contains more photons, a lO-ns, 1.60-mJ pulse at 760 nm or a 500-ms,
1.60-mJ pulse at 532 nm?
The pulse duration does not affect the number of photons a laser pu
Lasers, Laser Spectroscopy, and Photochemistry 465
Some of the spectroscopic constants for the X ground state and the A excited state of ICl(g) are
tabulated below.
State i/cm-1 Df/cml fair/cm" Evie/c
466 Chapter 15
First, determine the number of molecules that decompose:
[ 0.153 x103 gHI
-j (6.022 X 1023 moll) = 7.203 X 10'7 molecules
(126.904 + 1.008) g-mol
Now determine the number of photons abs
456
Chapter 15
157. Prove that Equation 15.17 implies that Nz/Nt is less than 1/2 because A > 0.
01211
If A = 0, the ratio is
N2(t > oo) _ Bpu(v]2) l
N _ A +ZBpU(vn) = 2
total
Because A > O, A + 23pv(
CHAPTER 1 5
Lasers, Laser Spectroscopy, and Photochemistry
PROBLEMS AND SOLUTIONS
151 . The ground-state term symbol for O; is 2Hg. The rst electronic excited state has an energy
of 38795 cm*1 above