T H E
J O U R N A L
O F
PHYSICAL CHEMISTRY Registered i n U S . Patent. Office 0 Copyright, 1980, by the American Chemical Society
VOLUME 84, NUMBER 2
JANUARY 24,1980
The Onset of Diffusion Control for Ion-Ion Neutralization in Ammonia Vapor E. S. Sennhauser,’ C). A. Armsfrcmg,* and F. Wllkinson* Department of Chemistry, University of Colgary, Calgary, Alberta, Canada T2N 1N4 (Recelved August 16, 1979) Publication costs assisted by the University of Calgary
The overall ion-ion neutralization coefficient, a, has been measured for ions recombining in pulse-irradiated ammonia vapor at 296 K with 1torr of either CC14or O2added as electron scavenger. Over the range of pressures from 10 to 1200 torr the results with the two scavengers were identical. The mobilities of the positive and negative ions were determined in the same pressure range, and exhibited a negative pressure exponent close to uinity. The reduced mobilities were po+ = 0.67 f 0.04; po- with C C 4 = 0.71 f 0.05; po- with 02 = 0.72 f 0.05 cm? V-l s-l. They were used to calculate diffusion coefficients and mean free paths for the ions. With the aid of these parameters the pressure dependence of a was analyzed in terms of Bates and Flannery’s modification of Natanson’s equation. This equation was shown to have a common origin with Noyes’ equation for diffusion-controlled reactions in liquilds. In gases the decrease in diffusion coefficient with pressure causes a to pass through a maximum, which for the present ions in ammonia occurred near 500 torr. This is at a lower pressure than that in oxygen because of the more polar gas. The latter effect also increased a in the low pressure region, where three-lbody neutralization dominates.
Introduction Diffusion control in fast liquid phase chemical reactions is a well-recognized phenomenon. ‘The theoretical treatment of these reactions has been reviewed and discussed in detail by no ye^,^ and a great deal of experimental work has been done on reactions of both neutral and charged reactant^.^ By contrast relatively little is known about diffusion-controlled reactions in gaties. Although neutral reactants should not be affected until high pressures are reached, for ionic species the onset of diffusion control is expected in the regEon of 1 atm.5 An equation describing the density or pressure dependence of gas phase ion neutralization reactiona was put forward by Natanson.6 Like many of the equations of Noyes, it has its origin in the Debye-Smolouchowski equation. Hecent comprehensive solution^^^^ to that equation give the following expression for the “long time” or steady-state rate constant for a reaction between ions of unit positive and negative charge: 0022-36~4/80/2084-0’123$01 .OO/O
This is equivalent to eq 6.5 of no ye^,^ in whose terminology E is the rate constant that would apply if the reaction were slow enough that the concentration of ions a t the reaction radius R were as predicted by equilibrium statistics. The quantity e is the electron charge, while D is the sum of the ionic diffusion coefficients D+and D-.For ions in thermal equilibrium with the molecules of a surrounding fluid it can normally be replaced by ~ ( k T / e(=) ~(p+ + p J ( k T / e ) ) . Natanson’s equation can be derived from (1) by substituting his values of R and k,viz. RN and b N . Natanson’s 0 1980 Amerlcan Chemical Society
124
The Journal of Physical Chemistry, Vol. 84, No. 2, 1980
Sennhauser, Armstrong, and Wilkinson
EN included only three-body ion neutralization as conceived by Thomson;lo this rate coefficient is given by
e-
+
Ir
1-1
or, in the formalism of Bates and Flannery13
+ (a’TN)-l)-l
(4) While a’TN increases with pressure, CY’L“ decreases, and therefore a eventually passes through a maximum. However, up to the present time the pressure dependence of a, has not been examined from the extreme low-pressure three-body region to above the maximum in one laboratory. For example, theoretical neutralization rates for oxygen ions13had to be compared to separate sets of data of McGowan14and Ma~h1er.l~ Neither of these actually spanned the pressure range where a reached its maximum, and in fact Machler’s results were for air while McGowan’s were for oxygen. The present investigation is concerned with the turnover region for ions recombining in pulseirradiated ammonia, which was chosen for the following reasons: (a) the radiation chemistry of the system has been extensively investigated, and the products Nz and H2 are nonpolar and nonreactive;’, (b) the ammonium ion is highly stable17 and dominates the positive ion spectra at long time^;^^^^^ (c) the equilibrium constants for clustering of NH4+,viz. NH,’.(n - l)NH3 + NH3 + NH4+.nNH3 (5) are known. l7-I9 Separate sets of experiments were performed with small amounts of carbon tetrachloride or oxygen added. These scavengers suppress electron-ion recombination by converting electrons to C1- or 02-: anm
= {(a’L”)-’
C1-
+ CC13
k, = 3.6 X lo-’ cm3 s-l
eThe parameter rN is a “critical radius”, within which any collision of either ion with a neutral causes stabilization of the ion pair and eventual neutralization; X and v are the mean free path and mean relative velocity of the ions; w is the probability of collision with a neutral while within rN,and it is used as derived by ThomsonlO and Loeb.ll The term in backets is a correction for the curvature of the ion paths in the Coulombic fieldagOn substitution of the value it simof rN,which is (X/2){(1 + 5e2/(3mockTX))1/2- 1)6, plifies to 17/5. The magnitude of RN is taken as rN PA, with generally assumed to be unity. As it stands eq 2 is only valid for ions of mass equal to the neutral^.^^^ However, Bates and Flannery12,13have modified the equations of Natanson to take account of different masses and mean free paths. In so doing they also normalized the equation to the quasi-equilibrium theoryL3of ion neutralization in the low-pressure threebody-dominated region. Thus some of the arbitrariness in the evaluation of rN, which has been criticized as a source of weakness in Natanson’s theory,13was overcome. The details of their approach are described in ref 13. Writing R’and EN’ for the revised Natanson parameters, one obtains, for anm,the steady-state ion neutralization coefficient for ions of species n and m
-
+ CCll
+ O2 + NH, k7 = 7.5
-
Oz-
cm6
X
(6)
2o
+ NH3
(7)
21
Both reactions are faster than electron capture in N20;3124 which was used as the electron scavenger in earlier Because of this feature and improvements in the apparatus, it was possible to obtain data at lower pressures. The mobilities of the positive and negative ions in the NH3systems were also determined. They are CCl, and “3-02 used for finding diffusion coefficients and mean free paths of the ions in order to calculate values of anmwith eq 4. For comparison with the experimental recombination coefficient a, the values of a , must be weighted in accord with the expression a = Canmfnfm
(8)
n,m
where f, and fm are the fractional abundances of clusters of different size.25This equation assumes that equilibrium cluster distributions are maintained during ion decay, which is a good approximation for the relatively long ion lifetimes ( 1 2 ms) used. Finally EN’ must be modified by adding a term p to take account of the two-body ion neutralization process, which is the only process at ultralow p r e ~ s u r e . ~The J ~ proportion of ions undergoing this process at ambient pressure is equal to the fraction which penetrate from rN to the reaction radius for the two-body neutralization, rz, without suffering a collision.26 Using a factor of the same form as Natanson, viz. (1 + e 2 / (4acockT)(l/r2- l/R’)), to account for the effect of the Coulombic field on the ion trajectories and the standard reduction in the intensity of a particle one finds, therefore
Writing (rN/X) = x and realizing that R ’ > > r2 in the pressure regime of interest here, one obtains
Further, if the small change in u (-20%) with changes in mean cluster size as pressure rises (see below) is neglected, this simplifies to
p
=
(11)
where Po can be identified as the intercept of an experimental a vs. pressure plot. The incorporation of this into eq 3 is deferred until later.
Experimental Section Materials. All gases were handled in a mercury-free vacuum line. Electronic grade ammonia (99.999% minimum purity) and research grade oxygen (99.99% minimum purity) were obtained from the Matheson Co. The ammonia was sublimed several times to remove noncondensibles and then stored at liquid nitrogen temperature in a bulb attached to the vacuum line. The oxygen cylinder was connected directly to the line with metal tubing and that gas was used as supplied. Carbon tetrachloride (spectrophotometric grade) from Baker Chemicals was degassed by freeze-pump-thaw cycles and kept in a vial. Apparatus and Procedures. Ion Recombination. The design of the cylindrical ionization chamber and bridge
Ion-Ion Neutralization in Ammonia Vapor
The Journal of Physical Chemistry, Vol. 84, No. 2, 1980 625
CELL
: : F j ‘0
H.V.-
”
1
O00
5
1 10
I 15
1 20
TIME (ms)
Flgure 1.
Bridge circuit and cell containing ionization chamber:
(A)
gold coating; (S) quartz.
circuit is shown in Figure 1. From a chemical point of view the use of a gold coating for the electrodes is an improvement over the graphite used p r e v i o u ~ l y .The ~ ~ calibrated collecting volume was 46.6 f 1.0 mL. Pracedures for baking and filling the chamber, which was fitted with a grease-free stopcock, were the same as those used before.24i27 At pressures of 25 torr and above ions were produced by exposing the ionization chamber to repetitive pulses of X rays generated by stopping electrons from a 1.5-MeV high-voltage Engineering Corporation van de Graaff accelerator in a 0.3-cm thick water-cooled gold target. The pulse frequency, v, and the pulse width were varied over the ranges 15-25 s-l and 50-250 ps, respectively. At a pressure of 10 torr and also at 25 torr, 120-kV X rays were produced by the Muller MG 150 generator and rotating sector previously described.24 These lower energ:{ X rays yielded higher dose rates below 30 torr. However, they were not used at higher pressures, where the smaller ranges of the photoelectrons and Compton scattered electrons lead to inhomogeneous doses. The diose distribulion was investigated with microelectrodes placed inside the ionization chamber. At a selected time, t , after each ionizing pulse, high voltage was applied across the bridge (see Figure 1) for a period of almost 4 ms. The charge due to ions remaining in the collection volume was swept onto the inner electrode and registered by the Victoreen Model 475 A electrometer as a time-averaged current. This is referred to as I+ or I-, depending on the polarity of the aplplied voltage. The high-voltage switch and timer have been described elsewhere.24 This timer was driven either by the van de Graaff pulser or by a light pulse from the rotating sector used with the Muller X-ray tube.24 The average concentration of positive or negative ions in the cell a t time t (n+or n-) is given by ni = l*/evV, where e is the electron charge, v the frequency of the radiation pulses, and V the collection volume. When necessary corrections were applied to allow for the small number of ions that recombined after application of the high voltage.z4
Figure 2. Plots of reciprocal of ion current vs. time: (a) 1000 torr of NH, with CCI4 (lower line) and 0, (upper line); (b) 100 torr of NH3 with CCI4 (upper line) and O2 (lower line); (c) 10 torr of NH3 with CCI,. Symbols are as follows: (A)positive ions; (0)negative ions; (0)
computer calculations.
Mobility. The drift cell and pulsed ion collection system used for mobility determinations has been described elsewhere.= Ions were detected with the amplifier system described in ref 29. The cell was evacuated to torr on the mercury-free vacuum line before being filled to the described pressures of ammonia and CC14 or O2 for each experiment. Periodically it was also heated with a hot air gun during evacuation.
Results and Calculations Ion Recombination Coefficients. When the end of pulse positive and negative ion concentrations are equal and only homogeneous second-order recombination occurs, the ion concentrations should follow the relation (n)-l = (n),,-I + at. Here (n)and (nIorepresent n, or n- at time t and zero, respectively, and a is the recombination coefficient. Writing I , = evVni one has
Examples of data plotted in accord with this expression are given in Figure 2. For both the NH3-02 and NH3-CC1, systems eq 12 was obeyed over two to three half-lives at high pressures (see Figure 2a,b), while at very low pressures there was an obvious upward curvature at long times (Figure 2c). The latter effect is due to the occurrence of wall diffusion, which actually also increases the slope a t short t i r n e ~ , ~ , ~ O Values of a for such conditions are therefore referred to as aaPFThey were corrected by the following procedures. For our geometry the contribution of wall diffusion to the rate of ion decay is to a good approximation a linear function of (n)o-l. This has been demonstrated with the computer model described in ref 31. Values of a obtained from the NH3-N20 systemz4 and similar to those for corresponding pressures in the present system were used along with diffusion coefficients derived from mobility datazs to compute the buildup and decay of ion concentrations. Values of aiapp were then found by plotting the inverse of the average ion concentrations against time as
128
The Journal of Physical Chemistry, Vol. 84, No. 2, 1980
io7 0,/(nI0
Sennhauser, Arrnstrong, and Wiikinson PRESSURE l t x r l
A* (cm3 < I )
0
1
2
3
I
I
I
1
I
i 3C
400
203
600
800
1000
PRESSURE ( t o r r )
Figure 4. Dependence of a on pressure at 296 K. ( - - - - ) best line through results with N,0:24 present data: (V)with CCI4; (0) with 02; ref 27 with O2(m). Figure 3. Dependence of aaW on reciprocal of initial ion concentration. (a) Computed aaPp/a vs. D,l(n)o.~2for 100 (a)and 150 (0) torr of NH3. (b) aapp at 10 torr NH,: computed with a = 3.7 X (0) and 3.0 X IO- [m);observed with 1 torr of 0,(V),1 torr of CCI, (0).
in Figure 2a-c. Typical examples of computed values of aapp are plotted in Figure 3a. The values of (n)o-lfor the two different pressures have been normalized by multiplying by @,/Az), where D, is the ambipolar diffusion coefficient and A the characteristic diffusion length of the ionization ~ h a m b e r .The ~ extrapolation to within 10% of a is clearly seen. This uncertainty is comparable to the present experimental error below 150 torr, but the apparent tendency for the intercept to overestimate a must be borne in mind. For every ammonia pressure with each additive (Le,, CC14 or 0,) a was determined at several different values of (nIo. Significant contributions of wall diffusion were only apparent below 250 torr. Between this pressure and 100 torr corrected a values were obtained from extrapolations of plots of aaPpvs. l / ( r ~to) l~/ ( r ~=)0.~ For pressures of 10,25, and 50 torr “preliminary values” of a were found by the above procedure. These were then used in the computer program to obtain decay plots and calculated values of aappat various ( r ~ ) ~Comparison . of these with the experimental values of aaPp allows one to set more precise limits on a. The approach is illustrated by the two lines which bracket the experimental points in Figure 3b. They were computed with a = 3.7 and 3.0 X lo4 cm3 s-’. We estimate a value of a = 3.1 f 0.3 X cm3 s-l at 10 torr for the NH3-CC14 system. The value for NH3-02 is, within experimental error, the same. The closeness of the fit between the computed and experimental LY values is also illustrated in Figure 3, where some of the computed values of l / ( n ) were plotted against time. The overall dependence of the corrected values of a on pressure of NH3 is displayed in Figure 4a,b. One should recall that each mixture also contained 1 torr of CC14 or 02.The “zero pressure” intercept corresponds to the value of Po, the two-body ion recombination coefficient. From Figure 4a Po is the same for both the NH3-02 and NH3C C 4 systems. Bates and M e n d a have ~ ~ ~pointed out that /3 may be strongly pressure dependent in this region, causing a likelihood of overestimating Po. From Figure 4a Po cannot exceed 2 X cm3 ion-I s-l and the most probable value is 1.0 f 0.5 X cm3 ion-l s-l. The lower limit was found by using Po = rrz2u(l + e2/(4mockTrz)) from above, and setting r2 equal to the hard-sphere collision diameters of the ions (0.8 nm for n = 4 and m = 3, see below).
100
-
c
C;m
>
N
10
E
u
t.
t
-1 m
I
0.1
40
100
400
1000
P R E S S U R E (torr)
Flgure 5. Mobility vs. pressure: positive ions (0, 0); negative ions with NH,-CCi, ( 0 ) ,NH3-0, (A)displaced by one decade upward and downward, respectively
The dashed line in Figure 4b is the best line through the earlier results with NzO as electron scavenger.24 The difference is not due to the change of ion ionization chamber, since the chamber used in the earlier work gave the same a values for NH3-02 as the newer chamber in a few check runs. The single point for “3-02 at 500 torr is taken from earlier work in this laboratory by Wilson and McCrackenUz7 Mobility Measurements. The mobilities of the ions are plotted against pressure in Figure 5. The positive ion mobility was, within experimental error, the same for the NH3-CC14 and NH3-02 systems. The best lines through the data points lead to the following empirical relations: p’ = (0.67 f 0.04)[NL/~1.02, p- with CC14 = (0.71 f 0.O5)[NL/Nl1.O3,and p- with O2 = (0.72 f 0.05)[NL/N]1.02, molecule ~ m - N~ is , the molecule where NL = 2.69 X concentration a t any given pressure and the numbers in parentheses are “reduced mobilities”. The positive ion mobilities are slightly smaller than those reported in ref 28. The close to inverse dependence on pressure (or on
The Journal
Ion-Ion NeutralLtation In Ammonia Vapor
TABLE I: Fractional Distributions of Cluster Sizes for NH,'.nNH, IO^
NH, press., torr 10 100 400
a
f3'
f,'
0.19
0.79
0.04
0.80
0.39 0.19
800 1200 0.11 Based on data in ref 17-19.
f,'
of Physical
Chemistry, Vol. 84, No. 2, 1980 127
Here P is given by a form of (11)in which the single x is replaced
f6'
0.02 0.20 0.40 0.40
0.21 0.42
01.35
0.54
0.03
N) for both positive and negative species is in accord with
the earlier results.28 A simplified mobility formula p = 0.72,(N~/N)!was used for all ion mobilities in the compuitations below. This agrees with all three empirical formulae to within 8% over the pressure range 50-1200 torr (i.e., N = 0.16 X :1019-3.89 x 1019). Calculations. Values of cy were calculated with eq 8 from a,,, f,, and f,, where the subscrilpts now deinote the numbers of NH3 molecules clustering positive and negative ions, respectively. Examples off, a t a few presriures are given in Table I. The negative ion cores were assumed to have a mass of 34 amu (see Discussion), and, in view of the generally weaker clustering of negative ions (cf. AH3 for H30+and C1- or C), in Table 8 of ref 17), the ,f values a t any given pressure were taken as f2- = f3+, f3-= f4+, ,.., etc. Abundances below f3+were negligible and we have not considered ions of n > 6 because of lack of data on dissociation constanr,~.'~This did not seriously affect a, but one may note that a7,, would be slightly smaller than %,m, etc. As in ref 25 hard-sphere radii of the clustered ions were calculated from a hexagonal closest packing model, using r",t = rNH8 0.22, rcl- = 0.18, and roz-= 0.21 nm1.33As an example, they ranged from r = 0.38-0.46 nm for NH4+.nNH3when n was increased from 3 to 6. These radii were used to estimate ion-neutral hard-sphere collision diameters, which are needed along with the individual ion masses in order to find Bates and Flannery's parameter ~ i 3 . l Here ~ and in other parameters below i is 1 for a positive and 2 for a negative ion, while 3 denotes the neutral. Mean free paths and rN1 values at each pressure were calculated from the simplified mobility formula above and equations given in ref 13. While referring the reader to ref 13 for full definitions of Bates-Flannery parameters, we note that &' for species of particular n and m is given in their symbolism by y-
RI, = ( ~ ~ / ~ ) ~ P ? ~ N , ~ W ( +P P~ ~XN N, ~ JU J ( P ~ X N Z-) P < ' ~ N ~ ' ' W( P ~ Y N , )w (P
