NS1
NO, Ct?~
A KINETIC STUDY OF THE RECOMBINATION
REACTION Na + SO2 + Ar
THESIS
Presented to the Graduate Council of the
University of North Texas in Partial
Fulfillment of the Requirements
For the Degree of
MASTER OF SCIENCE
by
Youchun Shi, B.S., M.A.
Denton, Texas
December, 1990
Shi, Youchun, A Kinetic Study of the Recombination
Reaction Na + S0 + Ar. Master of Science (Chemistry) ,
December, 1990, 88 pp., 15 figures, bibliography, 85 titles.
The recombination reaction Na + S02 + Ar was
investigated at 787 16 K and at pressures from 1.7 to 80
kPa. NaI vapor was photolyzed by an excimer laser at 308 nm
to create Na atoms, whose concentration was monitored by
time-resolved resonance absorption at 589 nm.
The rate constant at the low pressure limit is ko =
(2.7 0.2) x 10-21 cm6 molecule-2 s~1. The Na-SO 2 dissociation
energy E0 = 170 35 kJ mol1 was calculated with RRKM
theory. The equilibrium constant gave a lower limit E0 > 172
kJ mol~1. By combination of these two results, E0 = 190 15
kJ mol~ 1 is obtained.
The high pressure limit is k, = (1 - 3) x 10-10 cm3
molecule 1 s~1, depending on the extrapolation method used.
Two versions of collision theory were employed to estimate
k,.. The 'harpoon' model shows the best agreement with
experiment.
ACKNOWLEDGEMENTS
I would like to thank my Major Professor, Dr. Paul
Marshall. His encouragement and support are very important
factors which helped me finish the present thesis
successfully. Dr. Marshall used his enthusiasm and patience
to lead me into the field of gas kinetics. He interested me
very much in this research field and made work in his
research group a pleasure.
I am grateful D.M. Baker, A.L. Cook, M. Cordonnier,
C.E. Pittman, R. Ramirez and S.W. Timmons for their help in
constructing the apparatus, to L. Ding for assistance with
some of the experiments and to Prof. J.A. Roberts for
providing samples of S02.
Finally, I thank the Robert A. Welch Foundation (Grant
B-1174) for their support.
iii
TABLE OF CONTENTS
Page
LIST OF FIGURES v
Chapter
1. THEORIES
Early Work on Chemical KineticsThe Arrhenius Law - Temperature Effect and
Activation EnergyCollision TheoryTransition - State Theory
Unimolecular ReactionsRecombination Reactions
Pseudo-order Techniques
2. EXPERIMENT AND DATA ANALYSIS 35
Experimental Technique ReviewApparatus and Experimental ProcedureData Analysis
3. RESULTS AND DISCUSSIONS 57
Results
Low Pressure Limit
High Pressure Limit
APPENDIX 67
REFERENCES 82
iv
LIST OF FIGURES
Page
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
The reaction barriers, E0 and E0 ', the energy
difference between the reactants and products and
the transition state. 5
A collision between particles with the potential
V(r) = -C6/r6. 9
Theoretical and experimental behavior when 1/kPS1is plotted against 1/[M]. 20
Reduced 'fall-off' curves of a unimolecular
reaction. 21
Relationship between the various energy quantities
that enter into the formulation of RRKM theory. 26
Schematic diagram of a discharge-flow apparatus.
36
Schematic diagram of a flash photolysis apparatus
with detection by resonance fluorescence. 40
Schematic diagram of a shock tube experiment
employing optical monitoring.
Schematic diagram of the high-temperature reactor.45
The linear least-square calibration plot for the
0-2000 sccm mass-flow controller. 49
(A) Trace of transmitted light intensity in
arbitrary units vs time.
(B) Plot of transmittance vs time after
photolysis.
V
55
42
Figure 12 Plot of the pseudo first-order decay constant for
Na as a function of [SO2] in Ar bath gas. 56
Figure 13 Plot of pseudo second-order rate constant for Na
+S02 against [Ar]. 59
Figure 14 Lindemann plot of reciprocal pseudo second order
rate constant for Na + S02 against reciprocal
[Ar]. 61
Figure 15 Extrapolation of measured kps2 for Na + S02 (solid
circles) to higher densities, showing low and high
pressure limits from the NASA RRKM expression. 63
vi
CHAPTER 1
THEORIES
1.1 Early Work on Chemical Kinetics
Research on reaction rates began in the mid-1800's. In
1850, Ludwig WilhelmyI studied the rate of acidic hydrolysis
of sucrose, which changes the rotation from dextro to levo
following the hydrolysis, by a polarimeter. He found that
the rate was proportional to the concentration of sucrose
and that -(dC/dt) = kC, where C and t represent
concentration and time respectively, and k is a constant.
Integration gave kt = 2.37og(CO/C), where C represents the
initial concentration. First order kinetics were
demonstrated.
In 1863, Guldberg and Waage2 published an important
paper which emphasized the dynamic nature of chemical
equilibrium. Their basic idea on the 'law of mass action'
was apparent on their ttudes sur les Affinites Chimiques3, in
1867. Ten years later, van't Hoff set the equilibrium
constant equal to the ratio of rate constants for forward
and reverse reactions: K = kf/kb4
In 1884, van't Hoff4 clearly showed the influence of
1
2
temperature on equilibrium, while a few years later Svante
Arrhenius 5 formulated the effect of temperature on reaction
rates and introduced the concept of 'activated molecules'.
Chemical kinetics steadily declined from 1900. Perhaps
the early emphasis on solution kinetics retarded the
development of the theory because of the great difficulty in
treating liquids on a molecular basis. After the kinetic
molecular theory of gases, kinetics made great progress.
During the period around 1920, Trautz6, Lewis7 , and others
developed a quantitative treatment of gas-phase reactions on
the basis that only colliding molecules which possessed or
exceeded a critical energy E could react. Beginning in the
mid-1920's, Rodebush'9, Hinshelwood , Lindemann , Rice,
Kassel13 , and other investigated the collision hypothesis
exhaustively. The 'collision theory' soon was established.
For calculating the frequency of collisions, the collision
theory assumes that molecules are hard spheres. It
undoubtedly is very crude and only species that do behave
approximately as hard spheres can agree with the theory.
However, certain reactions were found to proceed more slowly
than the calculated. In order account for deviations from
the simply collision theory, it has been postulated that the
number of effective collisions may also be less than that
given by kinetic theory, since for reaction to take place a
critical orientation of the molecules on collision may be
3
14necessary . The rate constant, therefore, was written as k =
PZeE/RT , where P is referred to as a probability, or steric
factor.
In order to overcome these weaknesses, a more accurate
and detailed treatment of reaction rates was developed. It
is known as transition-state theory, or, sometimes, as
activated complex theory or the theory of absolute reaction
rates. This theory was first proposed by Pelzer and Wigner15
in 1932, who calculated the rate of reaction between
hydrogen atoms and molecules. A particularly clear
formulation of the problem was made by Eyring and his
collaborators16,17, who have applied his method with
considerable success to a large number of physical and
chemical processes. A somewhat similar formulation of the
18,19problem was made by Evans and Polanyil
1.2 The Arrhenius Law - Temperature Effect and Activation
Energy
In 1889, an empirical equation proposed by Arrhenius5
k = A exp(-Ea/RT) (1.2-1)
where k is the rate constant, or rate coefficient, of a
chemical reaction, A is the pre-exponential factor, R is the
gas constant, T is the absolute temperature and Ea is called
'activation energy' which is the difference between the
average energies of 'activated' molecules and the reactant
4
20molecules. This equation describes the dependence of
reaction rates on the temperature. Equation (1.2-1) may be
rewritten in a logarithmic form
log k = -Ea/2.303RT + log A (1.2-2)
According to this equation, a straight line should be
obtained when the logarithm of the rate constant k is
plotted against the reciprocal of the absolute temperature
T. Differentiating equation (1.2-2) with respect to
temperature and then integrating between limits gives
dlnk/dT = Ea/RT 2 (1.2-3)
and
log(k2 /k 1 ) = (Ea/2.303R) (T 2-T 1 )/TT2 (1.2-4)
If k, and k2 , the rate constants at different temperatures T,
and T2 respectively, are known, the activation energy Ea can
be calculated by equation (1.2-4). A better method is to use
a plot of log k vs. T, for which the slope is -Ea/2.303RT and
the intercept is log A.
In a simple reaction, the activation energy Ea can be
visualized approximately as an energy barrier between
reactants and products, E0 , which is represented
diagrammatically in Figure 1. E0 and E0 ' are the reaction
barriers corresponding to the forward and the reverse
reactions. AH is the heat of the reaction, or the energy
difference of the reactants and products.
Employing the van't Hoff equation
5
E,'
C 2 EO
AH
Reaction coordinate
Figure 1. The reaction barriers, E0 and E0 ', and the energy
difference between the reactants and products,
i.e. heat of the reaction, AH. The transition
state is labeled +.
6
dlnKc/dt = AH/RT 2 (1.2-5)
where KC is the reaction equilibrium constant, and applying
the 'principle of detailed balancing' to the Arrhenius
equation (1.2-3)
KC = kf/kr (1.2-6)
where kf and kr are the rate constants of the forward
reaction and the reverse reaction, leads to the conclusion
that
AH = EO - EO' (1.2-7)
which is shown in Figure 1.
1.3 Collision Theory
This theory is based on the kinetic theory of gases. It
suggests that, for a reaction to occur, reactants must
collide and the energy of the colliding molecules must be at
least equal to a critical energy, E0. The rate of reaction
is then given by the number of effective collisions per unit
time in unit volume of gas. The pre-exponential factor A in
the Arrhenius equation (2.1-1) is directly proportional to
the number of collisions, while the quantity exp(EO/RT)
represents the fraction of number of collisions having the
necessary energy E0. Using the concentration in molecules
cm-3 to express the reaction rate and nA and nB as the
numbers of molecules per volume of reactants A and B
respectively, the collision theory gives
7
-dnA/dt = -dn/dt = Z exp (-E 0 /RT) (1.3-1)
where Z is the number of collisions. According to the
calculation of the kinetic theory of gases 21, the number of
collisions Z of non-identical spherical molecules per unit
volume per unit time is expressed as
Z = ngnera2 (8RT/rINA) h (1.3-2)
where a is the collision diameter of the spherical
molecules, and A the reduced mass of particle A and particle
B. A = mAmB/(mA+MB) , where mA and mB are masses of particle A
and particle B, respectively. Thus, the number of collisions
is proportional to the product of the concentrations of
reactant A and reactant B.
The rate law of a second-order reaction is
-dcA/dt = -dc/dt = k cAcB (1.3-3)
The concentration c1 in moles liter' is related to the
concentration in molecules cm-3 by ci = 103ni/NA, where NA is
the Avogadro constant. Therefore,
-dnA/dt = -10-3NAdcA/dt = k[10-3NAnAn(103/NA) 2 ]
(1.3-4)
Comparing equation (1.3-1) and (1.3-4), the second-order
rate constant can be derived
k = (l0-3 NAZ/nAnB)exp(-E/RT) (1.3-5)
Substituting equation (1.3-2) to equation (1.3-5) gives
k = (a2 7rNA/1000) (8RT/7rNA)exp (-EO/RT) (1.3-6)
From this equation, the frequency factor is obtained
8
A = (a2lrNA/1000) (8RT/rMNA) (1.3-7)
The typical 'gas kinetic' collision frequency factor is
about 101Q cm3molecule~is-i. Benson gave some data of atomic
reactions22 which show agreement between experimental and
calculated values. In many cases, however, the calculated
rates are too large, especially for molecular reactions .
To account for such deviations, equation (1.2-1) was
modified by addition of an empirical factor P called the
'steric factor', where P = Aobserved/AcalculatedI
k = P A exp (-E0 /RT) (1.3-8)
The factor P is justified qualitatively on the grounds that
colliding molecules may not be suitably oriented for
reaction and it represents the fraction of energetically
suitable collisions for which the orientation is also
favorable. For atomic reactions, P z 1.
Now, more complicated collision theories which include
long-range potentials will be discussed24 ,25. Figure 2 shows a
collision between structureless molecules. There are several
features:
(1) the potential V(r) is derived from a spherically
symmetric force field;
(2) the relative-velocity vectors are not used; instead, one
molecule is held fixed at origin and the other is given
aninitial relative velocity, with the result that the
trajectory lies in a plane and corresponds to the
9
Figure 2 A collision between particles with the potential
V(r) = -C6/r6. b is the impact parameter.
description of collision in terms of center-of-mass rather
than laboratory coordinates;
(3) the impact parameter is shown as b, and the
instantaneous separation between the two molecules as r.
The impact parameter is the perpendicular distance
between the center of one of the molecules which is held
fixed and the straight line which the center of the moving
molecule would follow if the fixed molecule was not there to
deflect it.
The collision cross section is given by S = rb2max,
where b is the largest value of impact parameter for
which a collision can occur.
The total collision energy can be calculated from the
collision dynamics:
ET = j((d2 r/dt2 ) + [r(d/dt)] 2} + V(r) (1.3-9)
where A is the reduced mass of the two colliding particles,
and r and e are the radical and angular coordinates,
respectively. The magnitude of the angular momentum L is
L = pr2 (d9/dt) (1.3-10)
The initial energy and angular momentum when the particles
are infinitely far apart are pv2 and Avb and can be equated
to equations (1.3-9) and (1.3-10), respectively, where v is
the relative velocity of the two particles. Thus
ET2= yv2 = 1-g[ (d2 r/dt 2 ) + [r(de/dt) ]2 + V(r)
(1.3-11)
11
and
Mvb = pr2 (de/dt) (1.3-12)
Elimination of d8/dt yields
ET = 11-A(d 2 r/dt) + ETb 2 /r 2 + V (r) (1.3-13)
This equation represents the one-dimensional motion of a
single particle of mass A and total energy ET in an
'effective potential'
Veff (r) = ETb 2 /r 2 + V (r) (1.3-14)
The term ETb2 /r 2 is frequently called the 'centrifugal
potential'. Taking account of long-range attractive
potentials of the form
V(r) = -C6/r6 (1.3-15)
one obtains
Veff (r)2= Eyb2 /r2 - C6 /6 (1.3-16)
The value r at the centrifugal maximum, rmax, is determined
by setting aVeff (r)/ar equal to zero:
0 = -2Eyb 2 /rmax3 + 6C6rm ax7 (1.3-17)
so
r = (3C6/Eb2)1/4 (1.3-18)
Thus, the value of Veff (r) at rmax is found
Veff (rmax) = (2/3) ETb 2 (ETb2 /C 6 )h (1. 3-19)
This expression for Veff(rmax) is equated to ET to discover
the maximum impact parameter for which collisions of this
energy will surmount the centrifugal barrier, which yields
12
bmax2 (3/2) 1 (3C3/E) 1/3 (1.3-20)
Plane and Saltzman26 made the approximation of setting
collision diameter a equal to bmax, the square of which is
only weakly dependence on ET, at the mean collision energy,
ET = (3/2)RT. Integration of equation (1.3-20) over a
thermal energy distribution yields the simple collision
theory rate constant as
k(T) = 7ra 2 (8kBT/r)exp(-EO/RT) (1.3-21)
The 'harpoon model'2 is an alternative way to estimate
the collision diameter a. When two atoms, or molecules,
collide to form an ionic compound, the formation energy for
the ion-pair is AE = IP - EA, where IP is the ionization
potential of the atom which tends to form a cation and EA is
the electronic affinity of the atom which tends to form an
anion. The 'harpoon model' employs the electronic potential
energy, which equals -Qq/4reoa, as a critical energy, where
Q and q are electric charges of the cation and anion, co is
the vacuum dielectric constant and the a is the distance
between the two ions: if the formation energy is smaller
than electronic potential energy, the formation of the ionic
compound will occur. The electronic potential energy depends
upon the distance between two ions with the opposite
charges. At the critical distance, the formation energy of
the ionic compound is equal to the electronic potential
energy
13
-Qq/47rE 0ac = IP - EA (1.3-22)
The IP and EA values can be obtained from various handbooks
or manuals so that the critical distance ac can be
calculated. Employing ac in simple collision theory, the
rate constant can thus be evaluated.
1.4 Transition - State Theory25 ,28
This theory postulates that in order for any chemical
change to take place, it is necessary for atoms or molecules
involved to come together to form an 'activated complex',
which is regarded as being situated at the top of an energy
barrier lying between the reactants and pruducts (see Figure
1). The rate of reaction is the number of activated
complexes passing per second over the top of the potential-
energy barrier. For a general reaction
A + B = AB+ -+ product (1.4-1)
the rate is equal to the concentration of the activated
complexes, [AB+], multiplied by the average frequency, v+,
with which a complex moves across the product side, or
Rate = v+[AB+] (1.4-2)
If the activated complexes AB+ are in equilibrium with
reactants, the equilibrium constant for the formation of
complexes is
K+ = [AB+]/[A][B] (1.4-3)
Thus, the concentration of the complexes may be expressed as
14
[AB+] = K+[A][B] (1.4-4)
When K+ is written in terms of the molecule partition
functions per unit volume23 , one obtains
[AB+] = [A] [B] (QAB/QAQB)exp(-Eo/RT) (1.4-5)
where Q1 represent partition functions of appropriate
species involving in the reaction, except that one of their
vibrational degrees of freedom of AB+ corresponds to passage
over the barrier, i.e. to translation along the reaction
coordinate, the reaction trajectory with the lowest energy
on potential energy surface, and E0 is the height of the
lowest energy level of the complex above the sum of the
lowest energy levels of the reactants A + B. Substituting
equation (1.4-5) to the (1.4-2) yields
Rate = [A] [B]v+(QABW/QAQB)exp(-E/RT) (1.4-6)
The rate law for the reaction (1.4-1) is
Rate = -d[A]/dt = k[A][B] (1.4-7)
From equations (1.4-6) and (1.4-7), one can therefore derive
the rate constant,
k = v+(QA/QAQB)exp(-Eo/RT) (1.4-8)
This expression for k should be multiplied by a factor x,
the 'transmission coefficient' which is the probability that
the complex will dissociate into products instead of back
into reactants, so that
k = Kv+(QAW+/QAQB)exp(-Eo/RT) (1.4-9)
The vibration of the activated complexes, which becomes
15
a translation in crossing the barrier, is assumed to have
its classical energy RT/NA. The vibration energy is
therefore given by
hv+ = RT/NA (1.4-10)
where h is the Planck's constant. Hence,
Y+ = RT/NAh (1.4-11)
Now, the rate constant can be rewritten as
k = (KRT/NAh) (QABV/QAQB)exp(-EO/RT) (1.4-12)
This is the theoretical expression given by transition-state
theory for a bimolecular rate constant in terms of partition
functions. Sometimes, the formalism of the transition-state
theory is expressed in terms of thermodynamic functions,
i.e. free energy G, enthalpy H and entropy S. Since
-RT InK+ = AG+ = AH+ - TAS+ (1.4-13)
equation (1.4-4) becomes
[AB+] = [A] [B) (AS+/R - AH+/RT) (1.4-14)
When the equation (1.4-14) is substituted into (1.4-2), the
result is
-d[A]/dt = [A] [B]v+exp(A S+/R - AH+/RT) (1.4-15)
Thus by comparing equation (1.4-5) to the rate law (1.4-7),
the expression of the rate constant is given by
k = v+exp(AS+/R - AH+/RT)
= (RT/NAh) exp (AS+/R) exp (-AH+/RT) (1.4-16)
In most cases, the activation enthalpy AH+ approaches
critical energy E0, so that
16
k = (RT/NAh) exp (AS+/R) exp (-Eo/RT) (1.4-17)
If it is compared with equation (1.3-8), the collision
theory, one obtains
P A = (RT/NAh) exp (AS+/RT) (1.4-18)
This expression indicates that the factor P A includes the
change of entropy. The entropy decreases with the formation
of activated complexes, which implies a loss of degrees of
freedom. The more complex the reactants, the more negative
AS+. Thus, transition-state theory explains the steric
factor in a reasonable manner.
The transition-state theory has brought the properties
of the molecules into the picture in a realistic way and has
been used to estimate quantitatively the rate coefficients
of many reactions29 ,20 ,31. Difficulties, however, still exist,
such as the energy calculation of complicated systems, and
determination of the properties of activated complexes.
1.5 Unimolecular Reactions
I Lindemann - Hinshelwood Mechanism10,11,32,33
In 1922, Lindemann11 proposed a collision mechanism for
unimolecular decomposition reactions. He supposed that a
molecule may be energized by a bimolecular collision and
that there may be a time lag before decomposition, and
during this time lag, the energized molecule may lose its
extra energy in a second bimolecular collision. Since such
17
reactions are usually studied by diluting the reaction gas A
with an excess of inert gas M, the energization and
deenergization collisions are mostly collisions of A with M.
The reaction steps are indicated in the following mechanism
A + M A+M (1.5-1)
A - products (1.5-2)
where A is energized molecule of A. Using k2 and k-2 to
represent the forward rate constant and reverse rate
constant of reaction (1.5-1) and kI to represent rate
constant of reaction (1.5-2), the energisation rate is equal
to k2 [A][M] and the deenergisation rate is k-2 [A ][M]. The
overall rate of reaction (1.5-2) is the rate of product
formation, k[A*].
Since A* is very labile, the rate law for this
mechanism may be derived by assuming that [A*] is in the
steady state
0 = -d[A*]/dt = k2[A][M] - (k2 [M] + k1) [A]
(1.5-3)
Hence
[A] = k2 [A][M]/(k-2 [M]+k1 ) (1.5-4)
The total decomposition rate is therefore
Rate = k1[A*] = kk2 [A][M]/(k-2 [ M]+k1 ) (1.5-5)
At very low pressures, k-2 [M]J<<k so that
Rate = k2 [A][M] (1.5-6)
*
Thus, the reaction becomes second order, and all A
18
molecules decompose to products before they can be
deactivated in a collision with a second M. At sufficiently
high pressures, k-2 [M]>>k and
Rate = k k/k-2[A] = k[A] (1.5-7)
Now, the reaction is first order in [A]. The concentration
of A is at its equilibrium value, [A*]=k2 [A]/k-2, at the high
pressure limit so that the rate of the reaction is
determined by the equilibrium constant for the production of
A* and by k1 .
As the pressure decreases in the reacting system, the
rate changes from first order to second order, which is
observed experimentally. If a pseudo first-order rate
coefficient kpsi (See section 1.7) is defined as
kpsI = kk2 [M]/(k-2[M]+kl) (1.5-8)
the equation (1.5-5) can then be expressed as
Rate = k ps[A] (1.5-9)
As in the equation (1.5-7), the limiting first order rate
constant at high pressure is
k, = kIk2/k-2 (1.5-10)
One can rewrite the equation (1.5-8) in an alternative way
i/kPSI = 1/k, + l/k2 [M] (1.5-11)
Hence a graph of 1/kPS 1 against the reciprocal of the
pressure should be linear. However, deviations from
linearity have been found and the Figure 3 shows the
19
theoretically predicted and experimental curves.
Kassel13,34,Rice and Ramsperger12 explained this behavior.
Rewriting equation (1.5-8) gives
ks S-k/ (1+kl/k-2 M]) (1.5-12)
A plot of kpsi against [M] gives a curve shown in Figure 4.
The kPS becomes a constant, k, in the high pressure range
but falls to zero at lower pressure. The transition from the
high-pressure rate constant,kkps1 to the low-pressure
linear decrease in kpsi is called the 'fall-off region'. The
pressure at which kps1 k = 1/2 is denoted by p .
It may easily be seen from equation (1.5-12) that when
k- 2 = kkPpsI becomes equal to (1/2)k. Therefore, defining
[M] = kg/k-2 (1.5-13)
gives
k2[M] = k1k2 /k-2 = k (1.5-14)
Thus, one obtains
[M] = k,/k2 (1.5-15)
The value of k. can be obtained from experiments, and,
according to the simple collision theory, k2 should equal
Zexp (-E /RT) , where E is the energized energy. For several
reactions, however, this procedure gives rise to the result
that the kPs should fall off at much higher pressure that
actually observed. Since there can be no doubt about ku,, the
error must be in the estimation of k2. It is therefore
20
theoretical
experimental
1/[MI
Figure 3 Predicted, by equation (1.5-11) and experimental
behavior when 1/kPSj is ploted against 1/[M].
21
'kok/
[M]
Figure 4 Reduced ' fall-of f' curves of a unimolecular
reaction. The dashed line is given by equation
(1.5-12) and the solid line is the observed
behavior.
22
necessary for the collision theory to be modified in such a
manner as to give large value for k2 . Hinshelwood5 proposed
a method to give a much larger value for k2 and the
theoretical fall-off curves that are in much better
agreement with experiments. Hinshelwood noticed that the
exp (-E*/RT) term used to calculate the kPS is based on the
condition that the critical energy is acquired in two
translational degrees of freedom only. He proposed that
internal degrees of freedom also can contribute to the
energization energy E . Therefore, a molecule has a greater
probability of acquiring the necessary energy E . As the
result, the energization rate constant k2 with s degrees of
vibrational freedom becomes
k2 = [Z/ (s-1) ! ] (E/RT) s-lexp (-E*/RT) (1.5-15)
In practice, s is usually found by a method of trial and
error, and it is usually possible to explain the results by
using a value of s that is usually smaller than the total
number of vibrational normal modes in the molecule.
The treatment of Hinshelwood is best considered in
terms of the following mechanism:
A + M = A + M (1.5-16)
A* + A(1.5-17)
A+ -+ products (1.5-18)
k2 and k-2 represent the rate constants of forward and
reverse reaction of equation (1.5-16) and ki and k*
23
represent rate constants of equations (1.5-17) and (1.5-18),
respectively. In above mechanism, a distinction has been
made between an activated molecule, represented by A+, and
an energized molecule, represented by A*. An activated
molecule A+ is one that is passing smoothly into the final
state, while an energized molecule is one that has
sufficient energy and can become an activated molecule
without acquiring further energy. An energized molecule may
undergo extensive vibration before its internal energy can
become localized in the particular bond or bonds that are to
be broken during the course of reaction. The essence of
these modifications to the Lindemann theory is that
molecules may become energized much more readily than had
been considered possible on the basis of simple collision
theory, but that a long period of time may elapse before an
energized molecule can become an activated molecule.
Hinshelwood's treatment has been seen to predict an
abnormally large value for k2, and k, must be
correspondingly low. The theories discussed so far still
postulate a large value for k2 . Rice, Ramsperger12 and
Kassel 13 made a modification, RRK theory35, and later
Marcus36,37 derived the most successful theory, RRKM
theory38 ,39 ,40, in both classical and quantum forms.
II RRKM Theory
Marcus extended the RRK theory by combination with the
24
activated-complex model, to overcome the shortcomings of
basic RRK theory.
RRKM theory contains several assumptions:
(1) Free exchange of energy between oscillators
The theory classified energy of molecules as fixed and non-
fixed energies. The non-fixed energy can be redistributed
between the various degrees of freedom of a molecule, and
the fixed energy cannot. The theory assumes that the non-
fixed energy of the active vibrations and rotations is
subject to rapid statistical redistribution, which means
that every sufficiently energetic molecule will eventually
be converted into products unless deactivated by collision.
(2) Strong collisions
This assumption means that relatively large amounts of
energy are transferred in molecular collisions. Thus the
basic RRKM model treats the processes of activation and
deactivation as essentially single-step processes.
(3) The equilibrium hypothesis
In RRKM theory, as in transition-state theory, the
concentration of forward-crossing complexes is treated as
the same in the steady state as it would be at total
equilibrium where no net reaction was occurring.
(4) Random lifetimes
This assumption indicates that the energized molecules A
have random lifetimes before their dissociation. This
25
inplies that the process A - A+ is governed by purely
statistical considerations, and there is no particular
tendency for all A* to decompose soon after formation, or to
exist for some particular length of time before decomposing.
Figure 5 shows the fixed energies (Ej and Ej+),, non-fixed
energies (E and E+) and the zero point energies of the
reagent and transition state. The energy difference between
the reagent and transition state zero levels, including zero
point energies, is AEQ+. The following equation gives the
relationships between these energy quantities
E + Ej = E+ + E + AE0+ (1.5-19)
RRKM theory considers that the fixed energy arises from
'adiabatic' rotation modes, which are related to J. Based on
such assumptions the simple Lindemann mechanism can be
modified to include rate constants which depend upon
specific E and J
A + M = A+(E,J) + M (1.5-20)
A+(E,J) - products (1.5-21)
Now the rate constants are represented by k2(E,J), k-2 (EJ)
and k1(E,J) respectively. The expression for the reaction is
Rate(E,J) = k1 (E,J)k2 (E,J)[M]/{k-2(E,J)[M]+k2 (E,J)}[A]
(1.5-22)
and the rate constant is
Skuni = k1(EJ)(k2 (EJ)/k-2(EJ)]/{1+k1(E,J)/k-2(E,J)[M]}
(1.5-23)
26
E +Ej
4- E+
E P+CuJ
E
Reaction coordinate
Figure 5 Relationship between the various energy quantities
that enter into the formulation of RRKM theory.
27
The rate and rate constant of the unimolecular reaction
given above only take account of the molecules at the energy
level E and J. Taking a properly weighted sum of Sk uni(EJ)
over J (from 0 to oo) and E (from critical energy E0 to o)
gives the total kuni'k k2 (EJ)/k-2(E,J) can be defined as a
function P(E,J) which describes the Boltzmann distribution
of molecules over internal energy and angular momentum
states.
k = I SkunidE/NA (1.5-24)
(1) Low pressure limit41,42
As [M] -+ 0, the equation becomes
kuni,o-(T)=2(E J) P (EJ)[M]dE/NA (1.5-25)
It is assumed that k-2 = Z under the strong collision
assumption, where Z is the collision frequency at unit
pressure for all bimolecular collisions between A+ and M and
can be calculated by simple collision theory. Often, better
agreement with experiment is obtained with a weak collision
assumption. Here, k_2 = a Z, where P < 1, to account for the
possibility that several deenergizing collisions may be
needed to remove enough energy from A+ to bring its internal
energy below the critical energy E0. Using this assumption
for equation (1.5-25), one obtains00 CO
kuni,o(T) = PZ[M] S P(EJ)dE/NA (1.5-26)
Eo ,=
where, strictly, E0 depends upon J. Troe43 has evaluated this
equation and has developed approximate equations to avoid
28
the need for summation and integration. Making the usual
assumption that the vibrational levels comprise a closely
packed manifold at internal energies at and above the
critical energy, a first-order approximation for kuni,o(T) is
given by
k'uni,o(T) = 3Z[M] [Pvib,h(Eo) /vib]exp(-E/RT)dE/NAEc
= fZ[M] [pvib,h(E)kBT/Qvib]exp(-Eo/RT)
(1.5-27)
where Pvib,h(Eo) is the density of harmonic vibration energy
levels of the reagent molecule at an internal energy
corresponding to the critical energy for reaction, E0 , while
Qvib is the vibrational partition function of the reagent.
Sb-1Qvib=,H [1-exp (-hv j/kBT)](1.5-28)
I=/
The factor Pvib,h(Eo) is usually evaluated with Whitten
and Rabinovitch's equation"4,
Pvib,h(E) = [E+a(E)Ez]~S1/[(s-l)!NAsl'l hi] (1.5-29)
where Ez= NA E h is the zero-point energy in a molecule
with s harmonic vibrations of frequency vi and a (E) is a
constant which can be approximated by the following
equations
a(E) = 1 - bW (1.5-30)
logW = -1.0506(Eo/Ez) at EO>Ez (1.5-31)
W= 5(E0/Ez)+2.73(Eo/Ez) +3.51 at E0<Ez (1.5-32)
S 2 S 2b = (s-1) ( F V)/( V)2 (1.5-33)
Normally, a(E) ~ 1. Many studies have been made of the
29
collision factor p45~48. This quantity can be related to the
average energy transferred per collision <AE>. The usually
accepted calculation is given by Troe49
P/(1-p ) = -<AE>/FERT (1.5-34)
The factor FE accounts for the energy dependence of the
density of states. It is given byS-1
FE E [(s-l)!/(s-i-l)!]{RT/[E+a(E)Ez])} (1.5-35)
The first-order estimate, k'uni,o(T), is multiplied by
factors to correct for the neglect effects of, for example,
J. For a reaction not involving internal rotation
kuni,o(T) = k'uniOFanhFEFrot (1.5-36)
The factor Fanh makes a correction for neglect of vibrational
anharmonicity. The largest anharmonic effects are associated
with those vibrational modes which are 'lost' as the
molecule dissociates. Troe proposed the formula
Fanh = [(s-1)/(s-3/2)]m (1.5-37)
where m denotes the number of oscillators which disappear
during the reaction.
The factor Frot accounts for the effect of molecular
rotation. It is approximated by
Frot Frotmax[(I+/I)/(I+/I - 1 + Frotmax)] (1.5-38)
where I+ is the moment of inertia of the dissociating
molecule at the centrifugal barrier (see section 1.3), and I
is the moment of inertia of the molecule. For a linear
molecule
30
Frotmx [E0 + a(E 0)Ez]/sRT (1.5-39)
and for a non-linear molecule
Frotmax [E0 + a (EO) Ez/RT ] (s-1)!/(s+-)! (1.5-40)
for a simple bond fission reaction. It/I is obtained to a
first approximation by using a van der Waals potential
I+/I = 2.15(E,/RT)1/ 3 (1.5-41)
(2) High pressure limit
The function P(E,J), mentioned above, can be derived
from a Boltzmann distribution
P(EJ) = [p(E)exp(-E/RT)/Qvib)/[(2J+1)exp(-Ej/RT)Qj]
(1.5-43)
Combining equations (1.5-19) and (1.5-42) drives
k,1 (EIJ) P (ErJ) = (1/hQvibQJ) exp (-Eo+/RT) (2J+l)
-exp(-E+/RT)N(E)exp(-E+/RT) (1.5-44)
and
k 1(EJ) = N*(E+) /hp (E++AEO++E+-Ei) (1.5-45)
where p(E) is density of rovibrational energy states and
N*(E+) is the number of internal states at the transition
state which have energy less than or equal to E+. RRKM
theory treats N*(E+) in terms of a continuously variable
39energy rather than of quantized energy
Taking mean values of El and E defined as <E.+> = 7RT/2
and <E,> = (I+/I)JIRT/2, where 7 is the number of adiabatic
rotations, and then substituting equation (1.5-44)into the
kun expression, equation (1.5-26) and taking the sum over J
31
gives
kuni (T) = (kBT/h) (Q+/QQib) exp (-AEo+/RT)
- N(E+) exp (-E+/RT)/( (1+k1 (E, J) /3Z [M] }(dE+/RT)
0(1.5-46)
In the limit [M) - co, this equation reduces to the usual
expression of transition-state theory
kuni,(T) = (k8T/h) (Qj+ QVb+/QJQvib)exp (-AEo+/RT) (1.5-47)
(3) Fall-off region
For calculation of the fall-off region, the first step
is to choose the 'reasonable' structure and frequencies of
the transition state, which sometimes can be aided by using
quantum calculations, so that Q, +Qvib, AEO+ and N*(E+) can be
estimated. Taking these estimates into equation (1.5-46),
one obtains kunico(T). If the derived kunio(T) agree with the
values from experiments, the transition state chosen can be
used for the calculation of fall-off curves.
1.6 Recombination Reactions42
The mechanism of a unimolecular dissociation reaction
may be written as
AB + M -+ AB + M ( k2
AB + M -AB + M ( k 2 )
AB -+ A + B (k1 )
Recombination is the reverse reaction and then can be
written as
32
A + B -+ AB (k 1 )
AB -+ A + B (k_ 1 )
AB* + M -+ AB + M ( k2 )
The reaction rates for dissociation reaction and
recombination reaction are
Rate(diss) -d[AB]/dt = kdiss[AB] (1.6-1)
Rate(rec) +d[AB]/dt a krec[A][B] (1.6-2)
so that
kdiss - (1/[AB])d[AB]/dt (1.6-3)
krec +(l/[A][B])d[AB]/dt (1.6-4)
where kdiss is defined as an [M]-dependent first-order rate
constant and krec an [M]-dependent second-order rate
constant. Applying the steady-state assumption d[AB ]/dt=O
to the reaction mechanisms yields
kdiss = k2 [M] {k1l/(k+k-2 [ M] ) ) (1.6-5)
krec = k_{k-2 [M]/(k+k-2 [M]) (1.6-6)
It can easily be found that the ratio kdiss rec, which
follows the principle of detailed balancing, is an
equilibrium constant
kdiss/krec = k1k2 /klk-2 = ([A] [B]/[AB])eq = Kc (1.6-7)
The limiting rate coefficients can be derived from
equations (1.6-5) and (1.6-6). The low pressure limits ([M]
-+ 0) are
kdiss,o = lidississ([M]-+0) = k1.6-8 (1.6-B)
33
krec,O lim krec([M]-O) = (k/k)k-2[M] (1.6-9)
and the high pressure limits ([M] - co) are
kdiss,O kdiss([M]*o) = (k2 /k-2)k, (1.6-10)
krec, lim krec([M]+) = (1.61)
Interpretation of equations (1.6-8) to (1.6-11) may
give some useful information. For the dissociation, equation
(1.6-8) means that at low pressures the rate is equal to the
rate of collision activation, k2 [M); equation (1.6-10)
points out that at high pressures, the collisional
activation/ deactivation processes establish an equilibrium
ratio of AB and AB*, described by the rate-coefficient ratio
k2/k -2,and the unimolecular dissociation process of AB
becomes rate determining. In recombination, equation (1.6-9)
indicates that at low pressure, formation and redissociation
of AB are much more frequent than collisional
stabilization, so that an equilibrium between AB and AB is
established, as described by k_1 /k1 , and collisional
stabilization of AB* is then rate determining; and equation
(1.6-11) tells us that at high pressures collisional
stabilization is so frequent that the rate of association of
A and B to form an initially excited adduct determines the
recombination rate.
1.7 Pseudo-Order Techniquess5
Imagine a reaction having a rate which depends upon the
34
concentrations of several substances. For example, consider
the following elementary reaction
aA + bB + cC -+ products (1.7-1)
The rate for this reaction is given by
-d[A]/dt = k[A]a[B] b [Cc (1.7-2)
Also, imagine that conditions for a kinetic run are so
selected that all but one of concentrations are high,
compared to the one reagent, say A, present at lower
concentration. While the concentration of A changes
appreciably during the course of the reaction run, the
others effectively remain constant. If the order of [A] is
unity, the reaction is said to follow pseudo first-order
kinetics in that particular run. Hence, one can define a
pseudo first-order rate constant
kpsI = k[B]b[C]c (1.7-3)
and rewrite the rate law as
-d[A]/dt = kps1[A] (1.7-4)
The methods used for data analysis for a real first-
order reaction can also be employed for a pseudo first-order
reaction to acquire the rate coefficient. The pseudo second-
order method may also be applied. Such an experimental
technique is referred to as 'flooding'.
CHAPTER 2
EXPERIMENT AND DATA ANALYSIS
2.1 Experimental Technique Review5 1,52 ,53
Several experimental techniques are used for kinetic
investigations of gas-phase elementary reactions of
radicals. When choosing a specific method, many factors
should be considered. These include, for instance, the range
of the reaction rate, the type of the reaction, the nature
and the extent of the surface of containing vessel, the
effect of the products on the reaction, the sensitivity of
the reaction rate to the presence of inert gases, etc. In
this section, a brief description of some experimental
techniques popularly used today and some of their advantages
and disadvantages is given.
Discharge-flow Systems,54 55 The main features of a
discharge-flow system apparatus suitable for studying
elementary gas-phase reactions are shown in Figure 6.
Radicals are created by partial dissociation of a flowing
gas, usually in a microwave discharge. In this way H, N, 0
and halogen atoms can be produced in yields of 1 - 10% from
the parent diatomic molecules, which are usually diluted in
He or Ar. These atoms may be converted into different
35
36
~~ DC
- --Dpum p
d - - --
Figure 6 Schematic diagram of a discharge-flow apparatus.
DC is a discharge cavity to create atoms which may
be converted into other free radicals by titration
at inlet Il. The molecular reagent enters through
a movable injector 12. D is the fixed detector.
37
radicals by adding, at an inlet downstream of the discharge,
a species which rapidly reacts with the atoms converting
then into the radical that is required.
After their production, the radicals in their diluent
enter the main flow tube, which is typically 1 m long with a
25 mm internal diameter. Their (relative) concentration is
measured at a point near the downstream end of the flow
tube. The method depends upon being able to alter the
distance between this observation point and the inlet
upstream through which the molecular reagent is admitted.
This is usually achieved by keeping the detector fixed and
using a movable injector.
Many methods5-60 are used to determine the
concentrations of radicals in experiments of this type,
e.g., chemiluminescence, resonance absorption/fluorescence,
electron spin resonance and mass spectroscopy.
The major strength of the discharge-flow method is that
it can be applied to reactions of a wide variety of atoms.
It has also been used to study the reactions of several
diatomic radicals, but only rarely those of larger radicals.
A second major advantage is that the flow system is in a
steady-state so that, however the radicals are observed, the
signal can, if necessary, be averaged over several minutes
to improve the signal-to-noise ratio. The flow technique
also has been adapted to study of ion-molecule reactions,
38
for which these experiments are the main source of thermal
rate constants.
Some weaknesses of the flow tube experiments are caused
by the 'wall effect', such as the collision of reactant
molecules with the wall of the tube, and the consequent
surface-catalyzed combination of the radicals. Also, with
this method it is difficult to measure second-order rate
constants below 10-16 cm3molecule~ s1 without taking special
precautions.
Flash-photolysis6 l The second major technique is based
on the production of the free radical reactant in a short,
intense flash of radiation. Although only flash photolysis
is considered here, the related techniques of pulse
radiolysis62, where reaction is initiated by a burst of X-
rays or high-energy electrons, and modulated photolysis63
are now also being successfully applied to gas-phase
reactions. The application of laser flash photolysis to the
study of elementary gas-phase reactions is likely to
increase, as powerful lasers for the far and vacuum
ultraviolet regions of the spectrum become increasingly
available.
In experiments where the radicals are created by flash
photolysis,. a static gas mixture is prepared which contains
the photochemical precursor of the radicals, the molecular
reagent, and sufficient chemically inert gas to ensure that
39
the temperature does not rise significantly as heat is
released in photochemical and chemical processes.
Now, resonance absorption and resonance fluorescence
64-67are the popular methods for detection. Figure 7 provides
an illustration of a typical design. Radicals are produced
by photodissociation of one of the components of the mixture
in the reaction vessel (RV) using radiation from the flash
lamp (FL). The fluorescence is excited by radiation from the
resonance lamp (RL) and observed with a photon detector
(PD). The reaction vessel is enclosed in a vacuum housing
allowing both photolysis and detection in the vacuum
ultraviolet region. Light from the pulsed photolysis lamp
and radiation from the resonance lamp illuminate the same
portion of the gas mixture and the fluorescence is observed
in a direction perpendicular to the radiation from both the
photolyzing flash and the resonance lamp.
One important advantage of flash photolysis over
discharge-flow technique is that heterogeneous reactions can
be eliminated. The second is that measurements can be over a
wide of total pressures, which is particularly useful in the
investigation of association reactions. The main problems
with quantitative kinetic studies are those associated with
finding suitable photochemical source for the radicals of
interest and with minimizing the photolytic production of
unwanted reactive species in high concentration. Not all
40
RL
PMT
;ii
1- FL
Figure 7 Schematic diagram of a flash photolysis apparatus
with detection by resonance fluorescence. Radicals
are produced in the reaction vessel (RV) by UV
radiation from the flash lamp (FL). Their
fluorescence is excited by radiation from the
resonance lamp (RL) and observed with a
photomultiplier tube (PMT).
I
RV
41
species have a discrete spectrum. There can be a low
sensitivity of detection, because the absorption strength
may be dispersed over a wide range of wavelengths, or become
an unstable upper electronic state will not fluoresce.
Shock Tube6'9 The two methods described above are
employed for reactions at relatively low temperatures, up to
1000 K. The shock tube technique is used typically for
reactions in the range of 800 - 2500 K.
The basic design of a conventional shock tube apparatus
is illustrated in Figure 8. A thin diaphragm initially
separates a short region of the tube containing inert
'driver' gas at high pressure, typically at several
atmospheres, from a section several meters long, which
contains the potential reactants usually diluted in an inert
gas and at much lower pressure. When the diaphragm is
punctured, pressure waves passes into the low-pressure
section of the tube. Because the sound velocity in a gas
increases with temperature, the later waves catch up the
ones at the front so that they soon unite to form a single
sharply defined shock front passing down the shock tube.
This shock front compresses the gas containing reactants
adiabatically and raises the translational temperature at
any point in the shock tube in less than 1 microsecond.
The wide range of temperature, as well as of pressure,
42
MC
PMT
delay _-+
Figure 8 Schematic diagram of a shock tube experiment
employing optical monitoring. The experimental
section of the tube is separated from the driver
section (DS) by a diaphragm (D). Once the
diaphragm is ruptured, the shock wave propagates
along the tube passing a trigger (T) and velocity
gauge (VG). L is the analyzing light source, MC is
a monochromator and PMT is a photomultiplier tube.
43
that can be covered in shock tube experiments is one of its
major attractions as an experimental technique.
More experimental methods and details are discussed in
Gas Kinetics51 by Mulcahy, Kinetics and Dynamics of Elementary Gas
Reactions52 by Smith and Chemical Kinetics and Dynamics by
Steinfeld et a153.
2.2 Apparatus and Experimental Procedure
This thesis concerns the recombination reaction of
sodium atom and sulfur dioxide
Na + S02 (+ Ar) -+ NaSO2 + (+ Ar) (2.2-1)
This reaction was investigated in a high temperature gas-
flow reactor at 787 16 K.
A slow flow of argon bath gas containing S02 passed
through a heated reactor and entrained NaI vapor. This vapor
was photolyzed by a XeCl excimer UV laser at 308 nm yielding
Na atoms which reacted with the S02' The rate of Na-atom
disappearance following the laser pulse was monitored by
resonance absorption of the Na D-lines, 32 1/2 -+ 3 2P1/2,3/2, in
real time. The pseudo second-order rate coefficient, kps2'
at a given pressure was obtained from 6 measurements of
kps1, the pseudo first-order rate coefficient, as a function
of [S02]. To investigate the pressure dependence of the
reaction rate coefficient, the total reactor pressure was
changed from 1.7 to 80 kPa.
44
Reactor Figure 9 is a schematic diagram of the high
temperature reactor. The reactor consisted of three
stainless steel tubes (each 25 cm long with 2.2 cm i.d.),
crossed at right angles to each other. The intersection
region defined the reaction zone. Standard NW 25 ISO-KF
fittings connected all six ports of the reactor. The reactor
was housed in a thermally insulating box (20cm X 20cm X 20cm
outside dimensions with 2.5 cm thick walls) made of ZAL-5
alumina insulation boards (Zircar Products). The insulating
box was supported on an aluminum plate, through which three
screws were threaded. Each screw stood on a lab jack. Both
the screws and the lab jacks could adjust the height and the
angle of the reactor.
Temperature Control and Measurement Nichrome resistance
heating wire (maximum power 650 W at 115 V), electrically
insulated with ceramic beads, was wrapped around the outside
of each sidearm of the reactor for a length of 6.5 cm inside
the thermal insulation. Cooling water travelled through two
loops of 1/8" copper tubing around each end of the sidearms
for 1 cm outside the insulation. Two sheathed thermocouples
(Omega, type K, chromel(+) vs alumel(-)) were used. A
temperature controller (Omega CN 3910 KC/S) monitored the
reactor temperature with a thermocouple put outside of the
reactor in the insulating box. A solid-state relay (Omega
SSR240DC25, maximum current load 25 A) was connected in
45
EXCIMER LASER
EXHAUST
MONO-CHROMATOR
PMT
AMPLIFIER
DIGITALOSCILLOSCOPE J
THEFT
COMPUTER
'I
RMOCC
GAS INLET
BOAT
HOLLOWCATHODE
OVEN LAMP
)UPLE
Figure 9 Schematic diagram of the high-temperature reactor.
46
series to the laboratory 115 V AC supply and powered the
resistance heater (21 a resistance). The temperature
controller operated the relay to control the current through
the heating wire. In this way, the temperature could be
maintained constant, within about 1 K, from room
temperature up to about 900 K. The other thermocouple could
slide to the center of the reaction zone, to observe the gas
temperature at a thermocouple readout (Omega DP 285 K).
Optical An XeCl excimer laser (Questek 2110) photolyzed
the NaI vapor. The UV laser pulse at 308 nm passed through
the reaction zone of the reactor. A black-painted glass
Rayleigh horn, attached to the far-side port of the reactor,
terminated the laser beam after passing through the reaction
zone. Filters, 25% and 50% neutral density, were used in
some situations to cut down the overall energy of the laser
pulse to check the effect of the pulse energy. Resonance
radiation at 589 nm from a sodium hollow cathode lamp
(Fisher Scientific) was collimated by a focus lens and
entered a quartz window of the reactor, at right angle to
the photolysis laser beam. The Na resonance light passed
through the reaction zone and was isolated spectrally at the
exit of the reactor, first with an interference filter
(Oriel, centered at 590 nm, FWHM 10 nm), and then by
focusing onto the entrance slit of a monochromator (Oriel
77250), employed with a resolution of 2 nm. To detect the
47
concentration change of Na atoms during reactions, a
photomultiplier tube (PMT, Hamamatsu 1P28) monitored the
transmitted light intensity through the reaction zone. The
operating voltage of the PMT was about 700 - 800 V supplied
by a high-voltage power supply (Thorn EMI, PM28RA).
Electronics A four-channel digital delay/pulse generator
(Stanford Research Systems, DG 535) controlled the timing
for the experiments. It provided trigger pulses at a fixed
delay (25 Ms) to the photolysis laser and at a variable
delay to a computer-controlled digital oscilloscope (Rapid
Systems, R402, with 2048 time channels) for collecting the
probe resonance absorption signal from the PMT via an
amplifier (Thorn EMI, C632-Al). The repetition frequency was
set at 1 Hz. Up to 100 decays of sodium concentration were
accumulated for averaging, and kps, was fitted to the
averaged decay. All the data captured were stored and
analyzed by a microcomputer (CompuAdd, Turbo-10, IBM-PC XT
compatible).
Gas Handling The glass gas handling system includes four
gas reservoir bulbs, three of two liters and one of five
liters, and several cooling traps for purifying gases and
attaining vacuum.
Gas flow rates were controlled via a 4-channel readout
(MKS, 247C) which operated two mass-flow controllers, one
(MKS, 1159B-00050SV) for the buffer mixture of S02 and
48
another (MKS, 1159A-02000SV) for pure argon flow directly
from an argon cylinder. The mass-flow controllers were
calibrated with a Hastings Mini-Flo Calibrator (HBM-lA). The
time taken for movement of liquid soap film through a
graduated tube was measured after a flow rate of a flow-mass
controller was set. Three measurements were made at each set
flow rate. Corrections for water vapor pressure, gas
temperature and atmospheric pressure could be made by using
the appropriate tables, and the volume that the film
travelled through could then be converted to the standard
volume, at 760 torr and 0 *C. A standardized mass-flow was
calculated as STANDARD VOLUME (cc)/TIME (min.),, "sccm", as
the true mass-flow of the controller. Appendix B gives an
example of the correction. After calibrations of five
different flows were finished for a mass-flow controller,
the measured flow rate was fitted to a linear function of
the set flow. Figure 10 illustrates a typical linear least-
square fitting correction curve.
During the experiments, fresh gas flowed through the
reaction zone and the photolyzed mixture with reaction
products was pumped away via the exhaust system, so that any
interference caused by the last photolyzed mixture and
accumulated reaction products was avoided. Compared to the
reaction timescale (typically 1 ms), the time taken to sweep
reacted gas out of the reaction zone was long enough to make
49
duou
02L L..LIgl i C n tm
Figure 10 The linear least-square calibration plot for the
0-2000 sccm mass-flow controller.
50
the reactor kinetically equivalent to a static system. The
average residence time of the gas before photolysis is rresi
which was varied from 0.3 to 7.0 seconds.
A two-stage mechanical pump (Edwards, E2M8) pumped
reacted gas mixture out of the reactor. Two pressure gauges
(MKS, 122AA-01000AB and 122AA-00010AB) with a power supply
digital readout (MKS, PDR-C-2C) measured gas pressure over
the range 10-3 Torr to 103 Torr (1 Torr = 133 Pa). Another
gauge (Edwards, Penning 505) measured lower pressures, from
10-2 Torr to 10~7 Torr.
Experimental procedure The gases used were Linde 99.997%
Ar and J.T.Baker 99.96% SO2. Before making mixtures of S02
with pure argon, the S02 was purified. In the gas handling
system, S02 gas from the cylinder was condensed in a liquid
nitrogen trap and impurity gases which had higher vapor
pressures were pumped away. Then the purified S02 was warmed
and stored in a reservoir bulb. For further purification,
the above 'freeze-pump-thaw' procedure was repeated two more
times before the gas was used for making reaction mixtures.
After the gas handling system was pumped down to 104 Torr,
the S02 bath gas mixture, typically from 0.01 to 0.6%, was
made by diluting the S02 with pure argon and then stored in
the 5-liter bulb for about 1 hour to allow time for mixing.
To start the kinetic measurements, the 4-channel mass-
flow controller readout set the flows of SO2 mixture and the
51
Ar bath. The final mixture of So2 and Ar entered the heated
reactor through 1/4" stainless tubing. A stopcock, between
the gas exit of the reactor and the exhaust system, adjusted
the reaction pressure. A ceramic boat (about 75mm X 8mm X
8mm) contained a solid sample of NaI in the heated sidearm
of the reactor upstream of the reaction zone. The gas
mixture entrained vapor from the solid sample of NaI.
It was noticed that [S02 ] in the reactor took 30 to 60
minutes to stabilize at the start of an experiment, which
was attributed to adsorption onto the walls of the reactor
and the connecting line, similar to that observed by Plane
and Saltzman in the case of HC1 27. After the reactor
temperature had stabilized at about 787 K and gases mixed
homogeneously, the four-channel digital delay/pulse
generator started to trigger the excimer laser to generate 1
Hz pulses at 308 nm and to trigger the computer-controlled
digital oscilloscope. The triggering time of the excimer
laser was fixed at 25 ms after the digital delay generator
starting a trigger. The triggering time and time scale per
channel for the computer-controlled digital oscilloscope
were regulated according to the reaction speed to obtain a
suitable decay curve for reliable analysis. Sometimes, 25%
and/or 50% neutral density filters were used to reduce the
intensity of excimer laser photolysis beam. The averaged
curve of one hundred decays was stored in the computer for
52
analysis.
So far one experiment cycle under certain
concentrations of S02 and Ar (i.e. reaction pressure) has
been described. Six cycles for different concentration of
S02 under same reaction pressure were done in order to
obtain a pseudo second-order rate coefficient, kps 2 (see
next section). When the flow of S02 mixture was changed, in
the range of 0 to 50 sccm, the flow of Ar bath was changed,
in the range of 50 to 1000 sccm according to the reaction
pressure and reaction rate, to keep the total flow same, and
so to keep the reaction pressure constant.
2.3 Data analysis
The rate for the reaction (2.2-1) can be expressed as
follow
-d[Na]/dt = k[Na] [S2][Ar]n (2.3-1)
where k is the rate constant and the order of [Ar] may not
simple. Considering the rate of Na atom loss caused by
processes other than the reaction (2.2-1), mainly diffusion
to the wall of the reactor, another term is added to
equation (2.3-1)
-d[Na]/dt = k[Na][So2][Ar]" + kdiff[Na] (2.3-2)
where kdiff accounts for the any other processes except the
reaction (2.2-1). For any set of experiments at a constant
[Ar], equation (2.3-2) can be rewritten as
53
-d[Na]/dt = kps 2 [Na][SO2 ] + kdiff[Na] (2.3-3)
where kps 2 =k [Ar]" is the pseudo second-order rate constant.
Compared to the concentration of Na, the concentration of
S02 is much greater (see Appendix D), so that a pseudo
first-order method can be used here (see section 1.7).
Appendix D shows [S02] ~ 100[Na] which satisfies the
condition [SO2] >> [Na]. When the pseudo first-order method
is applied, equation (2.3-3) becomes
-d[Na]/dt = ks Na] (2.3-4)
where
kps1 = kps2 [ 0 2] + kdiff (2.3-5)
which is the pseudo fist-order rate constant. The integrated
form of equation (2.3-4) is the general first-order reaction
result
[Na] = [Na]oexp(-kps1t) (2.3-6)
where t is the time after the photolysis pulse and [Na]0 is
the concentration of Na atom at t=O in the experiment.
Applying the Beer-Lambert law to equation (2.3-6) yields the
expression for the time-resolved transmitted light intensity
I = Igexp{-E7[Na]0exP(-kPS1 t)) (2.3-7)
where e is the absorption coefficient of atomic Na and 7 is
the absorption path length. It has been estimated 70 that e =
1.9 X 10-12 cm2molecule&1, and 7 = 1.9 cm for the reactor used
here.
A computer program using the non-linear least-square
54
method, in the BASIC language7"772, fit each decay curve of
the concentration of Na with 2000 I(t) points, and gave kPSI
and the quantity A = el[Na]0 , and thus [Na]0 , see Figure 11.
Before obtaining the relationship between kps1 and the
concentration [SO2], several corrections and error estimates
were required, and performed by a computer program which
employed the pressure, the temperature and the gas flow
rates. (See Appendix C). After typically six measurements of
kps1 , corresponding to six different concentrations of S02 at
a constant total pressure, together with their uncertainties
akpsi and a[S02], which are described in Appendix C, had been
obtained, kPSI was plotted against [02]* These plots were
linear with intercept kdiff and slope kps2 ks2 together with
its uncertainty kps2 at a constant reaction pressure, i.e.
a constant concentration of Ar, were obtained with a
computer program using linear least-square data fitting.73
Figure 12 shows a typical plot of kPSI against [S2]
(A)1180 88
97.58
85.88
72.58
68.888.888 ms
1.18
8.95
8.88
8.65
Figure 11
(B)
8.268 as 8.255 ias 16.258 ras
(A) Trace of transmitted light intensity in
arbitrary units vs time, showing decrease after
photolysis of Na at t a 4.3 ins.
(B) Plot of transmittance vs time after
photolysis, showing the fit to a pseudo first-
order decay of [Na).
55
T
10.235 s 20.470 ms
0 + + 40 0 *.1* 4111.6 4 4-101111111114411019 +46W
is
'low
+ + 46 040080 0 900440 *W - -# + 40004" 4." * * + ### . +
+ 41110 0 6
40 40*0
46
40WIM"D #*It
ip "Ism" I&
4pow
C)
2400
2000
1600
1200
800
400
0 4 8 12
[SO2 ] / 101 3 cm-3
16 20
Figure 12 Plot of the pseudo first-order decay coefficient
for Na as a function of [SO2] in Ar bath gas.
T = 787 K and p = 6.0 kPa.
56
Na + SO 2
-I
mmm
CHAPTER 3
RESULTS AND DISCUSSION74
3.1 Results
Twenty two measurements for the recombination reaction
(2.2-1) Na + S02 + Ar - NaSO2 + Ar have been made as
described in Chapter 2. Appendix D summarizes the
experimental results, at a temperature T = 787 16 K, with
the total pressure varying from 1.7 to 80 kPa.
The sources of error in this experiment include both
the statistical scatter in the pseudo first-order decay
plots as well as uncertainties in the mass-flow controller
readings, pressure readings and temperature fluctuations.
These independent errors were added in quadrature to provide
the overall error for each rate constant.
The initial concentration of sodium, [Na]0 , was varied
from 1.0 to 3.8 x 1011 cm-3. There was no significant
influence of [Na]0 on the kps 2 values which demonstrates that
neither photolysis nor reaction products affected the
observed kinetics. Reaction (2.2-1) was therefore isolated
from any interfering processes.
The measurements show that thermal decomposition of S02
was unimportant, because when the residence time was
57
58
changed, which should have affected the degree of thermal
decomposition of SO2 , the rate of reaction (2.2-1) did not
vary.
It was noticed that for a given actinic intensity,
[Na]O increased with rres, which indicates that the heated
reactive gas mixture did not reach equilibrium with the
solid sample NaI. According to the work of Cogin and
Kimball 75 , the equilibrium vapor pressure of NaI at 787 K is
1.2 x 1012 cm , so that the actually employed concentration
of NaI should not be over this value. Davidovits and
Brodhead 7 have reported that NaBr has a similar absorption
coefficient to NaI at 308 nm. However, the present work did
not succeed in obtaining an absorption signal by Na when
NaBr was used as a photolysis precursor, even at
temperatures over 850 K.
Preliminary experiments showed that even at very large
[Na]O, not used for kinetic measurement, about 30% of the
resonance light was still transmitted. Presumably it passed
around the photolysis region in the reactor. A correction
was therefore subtracted from I to 10 before analysis, the
effect of which was increase kps, slightly (by less than
5%).
Figure 13 is a plot of kps2 against the concentration
of Ar, or the reaction pressure, which demonstrates that in
the low pressure region, the first 8 points, a linear
59
10
S8
a) 0e
S0 agis0 0.Tesrigtln orepnst0 4 e 4a6
[Ar]/1 018 cm-3
Figure 13 Plot of pseudo second-order rate constant for Na +
SO2 against [Ar). The straight line corresponds to
a linear least-squares fit to the first 8 points.
60
relationship exists between kps2 and [Ar), and that above
20.0 kPa ([Ar] = 1.8 x 1018 cm-3 ) the curve has entered the
fall-off region. The intercept is insignificantly different
from zero, which demonstrates that the reaction
Na + SO2 -+ NaO + SO (3.1-1)
is negligible. This is in accord with its large
endothermicity, of 273 42 kJ mol~1 at 0 K77 .
3.2 Low Pressure Limit
A good linear relationship between kps2 and [Ar] at
[Ar] < 7 x 10"cm-3 is illustrated in Figure 13. This
relationship implies a third-order reaction in the low
pressure region, first-order in every reagent species, or
Na, S0 and Ar.
The Lindemann mechanism, which is discussed in section
1.5, can interpret such a result. For this specific
reaction, the Lindemann mechanism steps are
Na + S02 -+ NaSO2 (3.2-1)
NaSO2 -+ Na + S02 (3.2-2)
NaSO2 + M -+ NaSO2 + M (3.2-3)
Using the result discussed in section 1.6 for recombination
reactions at the low pressure limit, third-order kinetics
are expected
kps2,0-= k0[M] = kk2 [MJ/k-2 (3.2-4)
Figure 14 is a plot of l/kps2 against 1/[M]. The slope
61
4(1)0
E
ECu)
'0
ClJ0.
3
0
0 1 2 3 4 5 6 7
[Ar]V1/10-1 cm3
Figure 14 Lindemann plot of reciprocal pseudo second order
rate constant for Na + SO2 against reciprocal rAr]
-mom-
62
is (2.4 0.2) x 10-" cmmolecule-2s~1 corresponding to k0 in
the equation (3.2-4). Quoted uncertainties are la. The
defects of Lindemann mechanism have been discussed in
Chapter 2, and a more realistic empirical expression for
kps2 is used in NASA rate constant compilations 7
log kPs2 (NASA) = [l+(log k0 [M]/k,) 2'-'7og(0.6)
+ log kps2 (Lindemann)
(3.2-5)
Using the experimental data to fit to this equation yields
ko = (2.7 0.2) x lO-29 cm6molecule-2 s-1. This fit is shown in
Figure 15, where it is extrapolated to high [M]. Both the
Lindemann and NASA parameterizations describe the
observations equally closely, with root-mean-square
deviations of 14% from the experiment data. Allowing for
potential systematic errors, 2a confidence limits of 20%
for these fits were estimated.
RRKM theory is employed to analyze the experiments.
Troe's method42,45,46, which is presented in section 1.5, was
used for the present experiment
k 0= PZ [ p (EO)RT/Qvib (NaSO2 ) ]FEFanhFrot
-Q (NaS 2) /[Q (Na) Q (S0 2 )1] (3.2-6)
The detailed calculation is given in Appendix A. In equation
(3.2-6) p(E0), the vibrational density of states of NaSO2 at
the critical energy E0 for NaSO2 dissociation to Na + SO2
and FE both depend on E0 . Other terms of equation (3.2-6)
63
10"--
k
k0
1 0 10---0
75EE
10
10-1217 1018 1019 1020
[Ar]/cm'3
Figure 15 Extrapolation of measured kps2 for Na + SO2 (solid
circles) to higher densities, showing low and high
pressure limits from the NASA RRKM expression. The
solid curve corresponds to the NASA fit. The
dashed curve corresponds to the Lindemann fit.
64
are calculated. Information about the structure and
vibrational frequencies of NaSO2 comes from the recent ab
initlo calculation by Ramondo and Bencivenni79 . When E0 = 170
kJ mol1 is selected, the calculated k0 agrees with the
experimental value. Allowance for an uncertainty of a factor
of 2 in all other non-E0 -dependent terms yields an
uncertainty of 35 kJ mol'in E0 .
An alternative way to make an estimation of the
critical energy E0 for the reaction (2.2-1) is to use the
equilibrium constant Kc, a method which was employed by
70Marshall et al to investigate the Na-02 bond energy . No
evidence was seen for the reverse reaction NaSO2 -+ Na + S02
in the present experiment, i.e. all the Na atoms were
apparently consumed and [Na] ~ 0 at equilibrium. As a limit
assumption, it is supposed that at least 50% of Na was
removed by reaction with S02 at the lowest non-zero
concentration employed, which was 6 x 1012 cm-3 . This implies
that KC is at least 1.7 x 1013 cm3. Using statistical
mechanics with partition functions calculated to fit Kc, E0
= 172 kJ molV' is obtained. This is a lower limit.
Combination of these two results for E0 obtained in
different ways gives an estimate of E0 a 190 15 kJ mol1.
This result agrees with the recent work of Steinberg and
Schofield0 very well. Their flame modeling gave an estimate
of EO = 197 20 kJ mol~1.
65
The ab initio calculation by Ramondo and Bencivenni for
the dissociation energy of NaSO2 to ions, Na+ and SO2-, is D
= 643 kJ mol~1. Employing experiment values for the
ionization potential of Na77 , 495.8 kJ mol1 and the electron
affinity of SO281, 106.8 0.8 kJ mol10, gives E0 250 kJ
mol~1 . This gives a k much larger than the present
observation. Such a difference is caused by an overestimate
of the density of the states at the critical dissociation
with the ab initio E0 value.
The pseudo second-order rate constant derived by Bawn
and Evans8 at 49 kPa at 511 K was about 2.4 x 1011
cm3molecule-1, which is larger by a factor of 10 compared to
the present results. Much of the difference may come from
the experimental technique which they used, the diffusion
flame method with a circulating system83. They investigated
some other similar reactions with a modified technique
besides this circulation method. They found the circulation
method always gave a larger rate constant, and they
discouraged use of the circulation method to obtain
quantitative kinetic data.
3.3 High Pressure Limit
The present experiments were not performed at the high
pressure limit. However, the high pressure limit rate
constant can still be estimated fome the data in the fall-
66
off region. Fitting the data to a simple Lindemann
mechanism, equation (1.6-6), and the modified NASA formula,
equation (3.2-5), gives k,(Lindemann) = (1.2 0.2) x 10~
10cm3molecule~1 s1 and k,(NASA) = (2.8 0.5) x 10~
1 0cm 3molecule 1 s~1.
Ham and Kinsey4 have investigated the reactions of K +
S02 and Cs + S02 by crossed molecular beams. They found that
complexes were formed with lifetimes long compared to
rotation times of these complexes. The large value of k,
close to 'gas kinetic', indicates a low reaction barrier for
these recombination reactions. Therefore, it is difficult to
determine a transition state unambiguously. RRKM theory,
which is identical to transition-state theory at the high
pressure limit, is hard to apply to the calculation of k..
The simple harpoon model27, one of the versions of collision
theory, is applied here. The calculation for electron
transfer from Na to S02 shows that the Na+-SO2~ configuration
is favored for separations of up to 0.36 nm. Using this
estimate to calculate the rate constant for ion-pair
formation, one derives 4.0 x 101 cm3molecule-1 s~1 . It can be
seen that this value approaches the experimental k, obtained
by the Lindemann mechanism and the NASA model.
Collision theory with a long-range attractive potential
form V(r) = -Clr' has been used to investigate the reaction
Na + 0224,85 Polarization data for Na and S02 are used to
67
estimate the long-range attractive potential. Using the
method described in section 1.3, b for reaction (2.2-1)
is 0.52 nm which leads to a rate constant of 8.5 x 10-10
cm3molecule1s-1, which is larger than observed. The results
may indicate the harpoon model is a better one for reaction
(2.2-1).
3.4 Conclusions
The flash photolysis/resonance absorption method has
been employed for the investigation of the recombination
reaction Na + S02. Treating experimental data by a simple
Lindemann mechanism and the modified NASA expression shows
agreement with RRKM theory. The Na-SO 2 bond energy, E0, has
been measured. Compared with the present results, the
critical energy E0 calculated via ab initio methods by
Ramondo and Bencivenni seems too large.
In the high pressure region, treatment of RRKM theory
for this reaction is difficult, because the determination of
transition state for this low barrier reaction is difficult.
Two treatments based on collision theory are employed, and
the harpoon model gives the closest accord with experiment.
APPENDIX A
RRKM CALCULATION OF THE LOW PRESSURE LIMIT OF THE REACTION
Na + SO2 (+ Ar) - NaSO 2 (+ Ar)
68
APPENDIX A
RRKM CALCULATION OF THE LOW PRESSURE LIMIT OF THE REACTION
Na + SO2 (+ Ar) - NaSO2 (+ Ar)
The basic idea and general calculation of RRKM theory
have been discussed in section 1.5. Here a rate constant
calculation at the low pressure limit for the recombination
reaction (2.2-1) is presented.
First, the partition functions of Na, SO2 and NaSO2
must be calculated. Corresponding to experiment, all
following calculations are set at a temperature of 787 K.
The translational, vibrational and rotational partition
functions are given by
Qtr = (2rmkBT/h2 )3/2 (A.A-1)
where m is molecular weight, kB Boltzmann's constant, T
absolute temperature, and h Planck's constant.
S
Qv =1 [1 - exp (-hyv /kBT)]' (A.A-2)
where s is the number of vibrational degrees of freedom with
frequencies V1 and
Qrot = (IxY Iz)4-87r2 (2rk8T)3/2/h 3a (A.A-3)
where Ii are the principal moments of inertia and a is the
rotational symmetry factor.
69
70
Substituting the relative molecular masses of Na, S02
and NaSO2 (23, 64 and 87 respectively), one obtains
Qtr(Na) = 4.58 X 10 m-3 (A.A-4)
Qtr(S02) = 2.12 X 103 m-3 (A.A-5)
Qtr(NaSO2) = 3.37 X 10 m-3 (A.A-6)
The sodium atom has no vibrational and rotational
degrees of freedom. Vibrational frequencies of So2 are
measured to be 1151, 518 and 1362 cm1 and those of NaSO2 are
calculated to be 1040, 983, 500, 337, 258, 117 cm-. it
should be noted that if the frequencies are given in cm1 ,
they need to be multiplied by the light speed constant when
they are substituted into equation (A.A-2). The vibrational
partition functions are
Qvib(SO2) = 2.03 (A.A-7)
Qvib(NaSO2 ) = 70.6 (A.A-8)
The principal moments of inertia are 1.380 X 10-46, 8.131 X
10-46 , 9.534 X 10-46 kg-m2 for S02 and 7.836 X 10-46, 1.6653 X
10-45 and 2.4489 X 1045 kg-m2 for NaSO2. Both of the
molecules have a symmetry factor of 2. Using equation (A.A-
3),
Qrot = 2.50 X 10 4(A.A-9)
Qrot = 1.37 X 10 4 (A.A-10)
Taking account of only electronic ground states results
in the electronic partition functions of Qe(Na) = 2, Qe(S02)
= 1 and Qe(NaSO2 ) = 2.
71
The total partition function is the product of
translational, vibrational, rotational and electronic
partition functions, or Q = QtrQvibQrotQe, and thus the total
partition functions are
Q(Na) = 9.15 X 1032 m-3 (A.A-ll)
Q (S02) = 1.08 X 1038 m-3 (A.A-12)
Q(NaSO 2 ) = 6.50 X 1040 m-3 (A.A-13)
The zero point energy of NaSO 2 is calculated by
= 1-2NA E hv1iEz ='=NA
= x 6.022 x 1023 x 6.626 x 10-34 x 2.9979 x 1010
x (1040 + 983 + 500 + 337 + 258 + 117)
= 1.934 x 104 Jmol 1 (A.A-14)
Selecting E0 = 170 kJ moli1 as the critical energy, E0 > E7 s
that
logW = -1.0506 (Eo/Ez ' -1.0506 (170/19.34)4
= -1.8090 (A.A-15)
W = 0.01552 (A.A-16)
Now, b and then a(E0) can be calculated
SZ Vi 2 = (2.9979 x 1010)2 (10402 + 9832 + 5QQ2 +
3372
+ 2582 + 1172 ) = 2.239 x 10 2
S 10)( Z v)2 = (2.9979 x 101)2 (1040 + 983 +
+ 258 + 117)2 = 9.406 x 10 27
b = (6 - 1) x 2.239 x 102/9.406 x 1027
= 1.190
(A.A-17)
500 + 337
(A.A-18)
(A.A-19)
%0 6-
'I1
72
a(EO) = 1 - 1.190 x 0.01552 = 0.9815 (A.A-20)
Thus, the harmonic oscillator density of states at the
critical energy is given by
Pvib,h(Eo) = (170000 + 0.9815 x 19340)5
/[5!(6.626 x 10-34 x 2.9979 x 1010 x 6.022 x
1023 6 x (1040 x 938 x 500 x 337 x 258 x 117)]
= 138.2
(A.A-21)
Each correction factor is calculated as follows
RT = 8.314 x 787 = 6543 J mol~1 (A.A-22)
RT/(EQ + a(EO)Ez) = 6543/(170000 + 0.9815 x 19340)
= 0.03462 (A.A-23)s-I
FE =, (6 - 1) !/(6 - 1 - i)! x (0.03462)1=0
= 1 + 5 x 0.03462 + 20 x 0.034622 + 60 x 0.034623
+ 120 x 0.034624 + 120 x 0.03462 x 0.034625
= 1.200 (A.A-24)
The number of oscillators lost in the reaction is 3. Hence,
Fanh [(6 - 1)/(6 - 3/2) ]3 = 1.372 (A.A-25)
1+/I 2.15 x (170000/6543)1/3 = 6.368 (A.A-26)
Frot max = (1/0.03462)3/2(6 - 1)!/(6 + ) !
= 9.955 (A.A-27)
Frot = 9.955 x [6.368/[6.368 - 1 + 9.955)]
= 4.137 (A.A-28)
Substituting all these factors into equation (1.5-36), one
can derive the rate constant for the reverse of reaction
73
(2.2-1). The unimolecular rate constants kuni ,(T) and
k'uni,o(T) are converted to recombination rate constants
krec,0 and k'rec,0(T) via the equilibrium constant
[Q (NaSO2) /Q (Na) Q (S0 2) ] exp (EO/RT) . Thus,
krec, (T) = k'rec,OFanhFEFrot
= $Z[Q(NASO2) /Q (Na) Q (S0 2) ] [p(EO)RT/Qv ib (NaSO2)
XFanhFEFrot (A.A-29)
Using a reasonable estimate of PZ = 5 x 10-10 cm3molecule&1s~ 1
results in
krec, 5 x 10-10 x[6. 50 x 104 0/ (1. 08 x 10 38 x 9.15 x
1032
x(138.2 x 6543/70.6) x 1.372 x 1.200 x 4.137 x
106
= 2.87 x 10-29 cm6molecule- 2s~1 (A.A-30)
In the above expression, the last term 106 converts m3
implicit in the partition functions to cm3.
APPENDIX B
CALIBRATION OF MASS-FLOW CONTROLLERS
74
APPENDIX B
CALIBRATION OF MASS-FLOW CONTROLLERS
The method for calibration of the mass-flow controllers
is described in section 2.2. An example set of data which
was obtained with a mass-flow controller for the range 0 -
2000 sccm is analyzed here. The data in group S are the
standard flow rate, set via a 4-channel readout, in a units
of sccm (standard cubic centimeter per minute), and the data
in group T are the measured average times, in minutes, which
it took for 1000 cm3 of gas to flow.
S (sccm) 1899 1600 1200 800 400
T (min.) 0.4897 0.5797 0.7717 1.153 2.279
While the calibration was being done, the atmospheric
pressure was read as 756.7 Torr from a mercury barometer and
the temperature was 24.9 *C. In order to convert the data to
sccm, several corrections are necessary.
First, the reading of the barometer is reduced to the
standard atmospheric pressure. This correction include two
parts: the temperature correction at 24.9 *C and 756.7 Torr
is -3.1 Torr, and the gravity correction in Denton is about
75
76
-0.8 Torr. The calibration was done above the water, so that
a second correction is for vapor pressure of water. The
vapor pressure of water at 24.9 0C is 23.6 Torr.
Therefore, the real pressure of the gas used for calibration
at 24.9 0C is
Pgas = 756.7 -23.6 - 3.1 - 0.8 = 729.2 Torr
The third correction is to convert the gas volume at 24.9 *C
and 729.2 Torr to that under the standard state
corresponding to 0 *C and 760 Torr. Using the ideal gas
equation gives
Vgas = 1000 x (729.2/760) x (273/298) = 879.0 cm3
Now, using the times listed in group T to divide 879.0 cm3
the measured flow rates are obtained, which are listed in
the following table. The group S is the same as before and
the group T' is now the measured flow rates in units of
SCCm.
S (sccm) 1899 1600 1200 800 400 0
T'(sccm) 1795.0 1516.3 1139.0 762.4 385.7 0
The fitted result is
T' = 4.55 + 0.944 S
Figure 10 shows the calibration curve.
APPENDIX C
CALCULATION OF THE UNCERTAINTY OF [SO2]
77
APPENDIX C
CALCULATION OF THE UNCERTAINTY OF [SO2 ]
The concentration of S02 is calculated as
[S02] = Ps02/RT = (fraction of S02 in gas mixture/RT)
x (Fso2/F) p (A.C-1)
where T is the reaction temperature, PSO2 is the partial
pressure of SO2, p is the total pressure in the reactor, the
Fso2 is the flow rate of the S02 mixture and F is the total
flow rate.
Only the uncertainties of the measurements of p, Fs02
and F are considered. The measurement of F comes from two
individual measurements, the measurement of Fs02 and the
measurement of FAr (the flow rate of bath gas Ar), so that
F = Fso2 + FAr (A.C-2)
The scatters of the calibrations of mass-flow
controllers are used as the uncertainties of Fs02 and FArn
CFSO2 and UFAr, respectively. The relative uncertainty of
pressure ar/p = 0.005 + 0.4/p is estimated according to the
manual of the pressure gauge.
If y = a-b/c, the propagation of deviation rule is
ay = y [(aa/a) 2 + (ab/b) 2 + (ac/c) 2])h (A.C-3)
and if y = a + b, then the rule is
78
79
CY = (Ca2 + ab2 (A.C-4)
Using equation (A.C-4) for the uncertainty of total flow rate,
one derives
CF = (aFS02 2 + UFAr 2 )h (A.C.-5)
Substituting the pressure and the flow rates and their
uncertainties to the equation (A.C-3) results in
'[S 02] = [[S 2C (aplp) 2 + (UFS02/FSO2) 2
+ (aFS02 2 + UFAr2 )/(FS02 + FAr) 2 ] (A.C-6)
If FSo2 >> FAr and UFSO2 >aFAr, the (A.C-6) can be approximated
as
"[S02] = [S02] 1 (ap/p) 2 + (aFS02/FSO2) 2 + (aFAr/FAr) 2
(A.C-7)
APPENDIX D
SUMMARY OF RATE CONSTANT MEASUREMENTS
FOR Na + SO2 + Ar
80
APPENDIX D
SUMMARY OF RATE CONSTANT MEASUREMENTS FOR Na + S02 + Ar
P, [M]~[Ar], Tres, [Na]0 , [SO2mx, k +aps2 kps2,10-12
kPa 1017cm-3 s 1011cm-3 10 13cm-3 cm3molecule-Is~1
1.67 1.53 0.5 2.2 14 3.32 0.10
2.33 2.15 0.3 1.4 13 4.24 0.17
2.36 2.17 0.6 1.7 25 5.31 0.22
3.60 3.31 2.0 2.2 22 7.85 0.40
6.00 5.52 1.6 1.9 17 12.2 0.7
9.84 9.05 0.7 1.5 8.8 20.5 0.5
9.84 9.05 0.7 3.0 8.8 18.8 0.7
12.3 11.3 1.7 2.2 4.8 20.3 0.8
20.0 18.4 1.8 1.6 20 33.3 1.4
25.7 23.7 1.8 1.9 10 37.3 1.2
25.9 23.8 1.8 1.7 6.3 32.9 0.5
25.9 23.8 7.0 3.8 15 30.5 1.3
32.7 30.1 1.8 1.5 7.5 47.5 1.8
33.3 30.7 0.9 1.9 3.7 54.6 2.6
33.9 31.2 0.9 1.0 3.9 61.4 2.6
40.0 36.8 1.8 1.6 6.5 51.8 3.5
46.7 42.9 1.6 1.0 1.4 39.4 2.4
46.7 42.9 1.6 1.7 3.0 53.0 3.8
47.3 43.6 6.4 1.8 7.2 48.2 2.1
53.6 49.3 1.8 1.1 6.6 63.0 5.8
66.7 61.3 2.3 1.6 2.9 69.5 6.0
80.0 73.6 2.2 1.1 2.7 84.1 8.5
81
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