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H2O
Mass Spectrometry and Ion Processes
E LS EVI E R
International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
Deuterium exchange in the systems of H20÷/H20 and H30+/H20 Vincent G. Anicich*, Atish D. Sen 1 Jet Propulsion Laboratory, California Institute of Technology, MS 183-301, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Received 26 October 1995; revised 24 June 1997; accepted 28 August 1997
Abstract The reactions of H20 ÷, H3 O÷, D20 ÷, and D3 O+ with neutral H20 and D20 were studied by tandem mass spectrometry. The H20 + and D20 ÷ ion reactions exhibited multiple channels, including charge transfer, proton transfer (or hydrogen atom abstraction), and isotopic exchange. The H3 O+ and DHO+ ion reactions exhibited only isotope exchange. The variation in the abundances of all ions involved in the reactions was measured over a neutral pressure range from 0 to 2 x 10 -5 Torr. A reaction scheme was chosen, which consisted of a sequence of charge transfer, proton transfer, and isotopic exchange reactions. Exact solutions to two groups of simultaneous differential equations were determined; one group started with the reaction of ionized water, and the other group started with the reactions of protonated water. A nonlinear least-squares regression technique was used to determine the rate coefficients of the individual reactions in the schemes from the ion abundance data. Branching ratios and relative rate coefficients were also determined in this manner. A delta chi-squared analysis of the results of the model fitted to the experimental data indicated that the kinetic information about the primary isotopic exchange processes is statistically the most significant. The errors in the derived values of the kinetic information of subsequent channels increased rapidly. Data from previously published selected ion flow tube (SIFT) study were analyzed in the same manner. Rigorous statistical analysis showed that the statistical isotope scrambling model was unable to explain either the SIFT or the tandem mass spectrometry data. This study shows that statistical analysis can be utilized to assess the validity of possible models in explaining experimentally observed kinetic behaviors. Published by Elsevier Science B.V, Keywords: Isotopic exchange; Proton exchange; Charge exchange; Branching ratios; Kinetics
1. Introduction Although deuterium is widely observed in the solar system [1] and interstellar space [2], the deuterium scrambling process has been studied in only a few ion-molecule systems. An understanding of the isotopic fractionation process and the variation in the H-to-D ratio in different molecules is important when interpreting their
* Corresponding author. Fax: 001 818 354 2439. Present address: University of Delaware, Department of Chemistry and Biochemistry, Newark, DE 19. 0168-3659/98/$17.00 Published by Elsevier Science B.V. PII S0168-1 1 7 6 ( 9 7 ) 0 0 2 5 7 - 7
cosmic origin and abundances. Therefore, deuterium scrambling is an important process that needs to be quantified. Ideally, the fundamental process that deserves the most attention is the reaction between an ion and a molecule that have only one deuterium atom between them. The deuterium atom can reside either in the ion or in the neutral molecule. However, in a single-chamber experiment, a series of rapid consecutive reactions occurs and it is very difficult to identify and measure the individual reaction steps. It might seem that partially labeled species can be generated in a
2
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
tandem experiment using a carefully controlled source. However, with larger molecules like water, a network of consecutive reactions allows the unambiguous generation of only the parent and the protonated ions, i.e. H 2 0 +, H30 +, D20 +, and D3O+. All other ions are ambiguous due to mass overlap and isotopic impurities generated from wall reactions. Partially deuterated systems are particularly difficult to study, principally due to the cost of the materials in the quantities needed to passivate the experiments. Deuterium exchange reactions are also particularly difficult to study because of the number of consecutive steps, the extent of branching, and the rapid reaction rates that usually occur in the reaction scheme. In the case of water, where the apparatus retains the isotopic species for a long period of time because of adhesion to exposed surfaces, the purity of the neutral isomer cannot be validated. Even in a tandem experiment where the reactant ion mass is pre-selected, the rate of isotopic scrambling makes the certainty of a 'specific mass' isomeric composition questionable. A case in point is HDO + which cannot be distinguished from the H3O÷ because both ions have a mass of 19 Da. The conclusion of Smith et al. [3] from their selected ion flow tube (SIFT) studies was that the thermal neutral deuterium exchange reactions in the H30+/D20 and D30+/H20 systems exhibited a statistical behavior. They also concluded that the reactions proceeded through an association complex that lived sufficiently long to randomize the hydrogen and the deuterium atoms before unimolecular decomposition to reactants and products. Data were also presented in the same paper for the reaction between D3O÷ and NH3 which exhibited very different results than did the H30+/D20 and D30+/H20 reactions [3]. In this reaction only D ÷ was exchanged, i.e., a proton transfer reaction was observed. The conclusion was that a long-lived intermediate complex was not formed. Similar results were found for CD~/NH3 and other asymmetric exothermic proton transfer reactions [4].
Ion cyclotron resonance (ICR) studies [5] of charge transfer reactions in near-thermoneutral symmetric systems, 15N~]IaN2, 14N~/15N2, 12CO+/13CO, and 13CO÷/12CO measured reaction rate coefficients that were half the collision rates, suggesting the existence of a statistical charge transfer mechanism. The rate coefficients of hydrogen exchange in the reactions H2D÷/HD, HD~/HD, and 14NH~/15NH3 were found to be consistent with the results expected from the statistical model [5,6]. The results obtained from two other systems, CH3D~/CH2D2 and CHzD~/CH2D2, could not be explained by the statistical model [5]. Further studies on isotope exchange reactions have been conducted on the H30÷/D20, D30+/ H20, NH~/ND3, ND~/NH3, CH~/CD4, and CD~/ CH4 systems [7,8]. The reactions were analyzed by considering the formation of an intermediate complex and the average number of protondeuteron jumps, K, occurring during the lifetime of the complex. The water system was found to form a complex with a lifetime longer than the proton-deuteron shuttling time. Therefore, the system exhibited a statistical isotope exchange process, r >-- 20. The ammonia and the methane systems were found to form short-lived complexes with lifetimes shorter than the protondeuteron shuttling time. Therefore, these systems did not exhibit a statistical isotope exchange process, which is consistent with a value of K = 0.25 for the methane reactions. The reactions of H20÷/D20, D20+/H20, H30+/D20, and D30+/H20 reported in the present work were studied by a tandem ICR mass spectrometer. The data from this study and those of a previous SIFT study of H30+/D20 and D30+/H20 reactions were analyzed as a sequence of charge transfer, proton transfer, and isotopic exchange reactions. The overall reaction rate coefficient, the relative rate coefficients, and the branching ratios of these individual steps were determined to improve our understanding of the isotopic exchange process in the water system. Detailed statistical analyses were performed on
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
the tandem ICR and SIFT data to establish confidence levels on the results obtained from the kinetic studies. The statistical analyses had a significant bearing on the conclusions that can be drawn from the experimental results.
2. Experiment A tandem mass spectrometer in a I C R Dempster-ICR configuration [9,10] was used to conduct the present experiments. The instrument was housed within the pole pieces of a 12 inch electromagnet that supplied the magnetic field for the two ICR sections and the Dempster section of the mass spectrometer. The ion source was a two section ICR drift cell. The drift potentials were used to control the reaction time in the source region. A 180 ° magnetic sector was used as a primary mass filter. The volume of the magnetic sector was defined by a metal cage within which all electric fields were uniform. A 3 kV potential provided the acceleration field at the exit of the 'source' ICR cell. The magnetic field was adjusted to give the ions the appropriate radius of curvature needed to enter the 'deceleration-energy selection' region at the front of the 'detector' ICR cell. After the ions were decelerated, a long narrow Wien filter was used to limit the energy of the ions entering the reaction region to less than 30 meV. The first section of the detector ICR cell was the ion-molecule reaction region. The second section of the cell was the ion detection region and was used to measure the ion abundances. A Wronka bridge detector [11] monitored the ion signals in this section. Frequency scans on the detector section were used to obtain a mass spectrum. Vapors from distilled water and deuterated water were the source gases. Both samples were purified by several freeze-pump-thaw cycles. To ensure the predominance of H20 + and D20 + exiting the source region, the pressure of
3
the source gas was limited so that very little of the protonated water was formed. The magnetic field was scanned to obtain the mass spectrum of the ion source output. A biased metal plate connected to an electrometer served as an 'auxiliary' ion detector. The plate was placed behind a slit in the acceleration housing at a smaller radius of curvature than the detector cell. Ions passing through the slit impinged on the plate, and an ion current was observed that was proportional to the number of ions focused on the detector plate. To ensure the predominance of H30 + and D3O+ exiting the source region, the pressure of the source gas was increased so that the protonated water was the dominant ion. Long passivation times were required before the pressure and the isotopic purity became stable in both the source and the detector regions. Pressures above 5 × 10 -6 Torr in the detection cell were measured by a 1 Torr capacitance manometer. Lower pressures were measured by an ionization gauge which was calibrated against the capacitance manometer. The drift times were not measured directly. The rate coefficients of known water reactions were used as a reference [12], and the unknown rate coefficients were determined relative to the known rates.
3. Results Data sets consisting of the variation of fractional abundances of ions with bath gas pressures were obtained for the four pairs of ion-molecule reactants: H20+/D2 O, D20+/H20, H30÷/D20, and D30+/H20, and these are shown in Figs 14. In all cases the primary ion reacted rapidly, forming several isomeric species in intermediate stages. In the end, a single 'terminal' ion, the protonated neutral, dominated the reaction products. Several consecutive reactions are possible at the highest pressures. The terminal ion was always the protonated neutral. The intermediate steps in the reaction scheme were assumed to be a
4
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
1.0 D
tt4 .Q
0.8
0.6
g 0.4
i
--0.2-
0.0
-
-
0
I
I
1
2
Pressure,
3
Torr (xlO "s)
Fig. 1. Ion abundances in the isotope exchange reactions, starting with H3O÷ in a D20 bath. From the tandem ICR-Dempster-ICR spectrometer. The points represent the data, and the line a modeled fit. [~, H30÷; (3, H2DO÷; O, HD20÷; ZX,D3O÷.
c o m b i n a t i o n o f c h a r g e transfer, p r o t o n transfer, and isotope e x c h a n g e reactions.
3.1. Data analyses A n o n l i n e a r least-squares regression t e c h n i q u e
w a s used to a n a l y z e the data. T h e reaction s c h e m e d e v e l o p e d to explain the data is s h o w n below. T h e first g r o u p o f reactions is a list o f the possible p r o d u c t c h a n n e l s f r o m either H2 O+ or H3 O+ as the p r i m a r y ions in a bath o f D20.
1.0
0.8
0.6
i
0.4
0.o,
--0
I
I
I
5
10
15
Pressure,
Torr (xl0 "s)
Fig. 2. Ion abundances in the isoptope exchange reactions, starting with D30 + in an H20 bath. From the tandem ICR-Dempster-ICR spectrometer. The points are the data, and the line a modeled fit. The open and filled symbols represent different data sets of the same experiment. Legend: [], D30+; O, HD20+; <>, H2DO+;A, H30 +.
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
5
1.0
0.8
0.6
|o.4 '~ 0 . 2
0.0
I
0
1
2
3
4
Pressure, Torr ( x l 0 4)
Fig. 3. Ion abundances in the isotope exchange reactions, starting with D20 + in a H20 bath. From the tandem ICR-Dempster-ICR spectrometer. The points represent the data, and the lines a modeled fit. The open and filled symbols represent different data sets of the same experiment. Legend: I-q, (D20 +, H2DO+); A, HD20+; ~, H20+; O, (HDO +, H30+).
1.0
0.8 m
8 c
o.6
c .Q 0.4 o
,,~ 0 . 2
0.0
I I
2
P r e s s u r e , T o r r ( x l 0 "s) Fig. 4. Ion abundances in the isotope exchange reactions, starting with H20 + in a D20 bath. From the tandem ICR-Dempster-ICR spectrometer. The points represent the data, and the lines a modeled fit. The open and filled symbols represent different data sets of the same experiment. Legend: ~ , H20+; - - , HDO+; [:], (D20 +, H2DO+); A, HD20+; C), D3O+.
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V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
D20 reactions with H20 + and H 3 0 + primary ions: H2 O+
+
D20
~ "-* -'-'
HDO ÷ D20 + H2DO + HD20 +
+ + + +
HDO H20 OD OH
(1) (2) (3) (4)
HDO +
+
D20
"-'* ---* ---*
D2 O+ + HD20 + + D3 O+ +
HDO OD OH
(5) (6) (7)
D20 +
+
D20
-"*
D3 O+
OD
(8)
H30 ÷
+
D20
--'* --*
H2DO ÷ + HD2 O+ +
HDO H20
(9) (10)
H2DO + +
D20
~
HD2 O÷ + D3 O+ +
HDO H20
(11) (12)
HD20 + +
D20
"-'*
D30 +
HDO
(13)
+
+
The second group of reactions is the possible product channels from either D20 + or D3 O÷ as the primary ions in a bath of H20.
H20 reactions with D 2 0 + and D3 O+ primary ions: D2 O+
+
H20
~
HDO ÷
HDO
(1)
H20 ÷ + HD2 O+ + H2DO + +
+
D20 OH OD
(2) (3) (4)
HDO +
+
H20
~ --~ ---'
H20 + + H2DO ÷ + H30 ÷ +
HDO OH OD
(5) (6) (7)
H2 O+
+
H20
~
H3 O+
OH
(8)
D3 O+
+
H20
~
HD2 O+ + I-I2DO + +
HDO D20
(9) (10)
HD2 O+ +
H20
-"* --*
H2DO + + H30 + +
HDO D20
(11) (12)
H2DO ÷ +
H20
-"*
H3 O+
HDO
(13)
+
+
Owing to the symmetry of the two reaction schemes, the reaction numbers were duplicated for equivalent reactions. Equivalent reactions can have identical products, but the rate coefficients and the branching ratios may not be equal. Each reaction scheme is further split into two groups of reactions: one with ionized water as the primary ion (Reactions 1-8) and the other with protonated water as the primary ion (Reactions 9-13). The kinetics of the reactions of protonated water, Reactions 9-13, were treated as a set of simultaneous partial differential equations. The equations and their solutions are shown in Appendix A. These solutions to the equations were fitted to the experimental data by a nonlinear least-square regression technique. The 'best fit' solutions then gave values of the branching ratios and rate coefficients for Reactions 9 13. The model fits are shown in Figs 1 and 2 as solid lines. The data sets of the reactions H30÷/ D20 and D30+/H20 were analyzed to determine the branching ratios b9, bl0, b11, and b12, and the rate coefficients k9_10, k11-12, and k13. The subscripts to b refer to the reaction numbers. The subscripts to k indicate the total reaction rate coefficient of an individual reaction, k13 (Reaction 13), or a group of reactions, k9-10 (Reactions 9 and 10). This notation is maintained when evaluating all data sets. Thus, in the protonated water reactions, since b9 + bl0 = 1 and b l l + b 1 2 = 1, there are only five independent parameters that have to be determined. The reaction times of the individual experiments were not measured, but were considered free parameters to be optimized for each reaction set. The value of k9-10 is well established in the literature [12] and was therefore set to the accepted value of 2.2 x 10 -9 cm 3 s -1. The reaction time r was computed relative to the value of k9-10 and then used to calculate the values of k11-12 and k13. The solutions to the reactions H30+/D2 O and D30+/I-I20 are shown in Table 1. The kinetics of the reactions of ionized water, Reactions 1-8 and 11-13, were treated as a different network of simultaneous partial
V.G. Anicich, A.D. Sen~InternationalJournal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
7
Table 1 S u m m a r y o f analysis o f the H 3 0 + / H 2 0 systems D 3° +/I"120, SIFT
H 3 0 +/D 20, SIFT
D 3° +/n 2° (this work)
H 3 0 +/D 2 0 (this work)
Statistical model
b~ b~0
0.71 -+ 0.03 0.29 ± 0.03
0.70 0.30
0.52 - 0.04 0.48 - 0.04
0.42 --- 0.03 0.58 ± 0.03
0.67 0.33
k9b_10
2.20
2.27
2.20 -
2.20 -
bH b Iz
0.72 ± 0.04 0.28 -+ 0.04
-
1.00 + 0 . 0 0 / - 0.30 0.00 + 0 . 3 0 / - 0.00
1.00 + 0 . 0 0 / - 0.46 0.00 + 0 . 4 6 / - 0.00
kH-12
1.95 --+ 0.12
-
3.32 ± 0.42
2.32 + 0 . 4 0 / - 0 . 2 8
k13
0.99 - 0.06
-
2.84 + 0 . 1 2 / - 0.60
0.63 + 0 . 4 6 / - 0.11
0.86 0.14
a The b values are the b r a n c h i n g ratios, w h o s e subscripts reference the equation n u m b e r s used in the text. b The k values are expressed in units o f ( x 10 -9) c m 3 s -1.
differential equations. The equations and their solutions are shown in Appendix B. Reactions 9 and 10 were excluded from the network because the ion H30 ÷ is not a product when the reaction starts with H20 ÷ in a D20 bath. The analysis of the data sets from the reactions H20÷/D20 and D20+/H20 was simplified by using the solutions obtained from the reactions of the protonated species, i.e., the values of b9, blo, bll, b12, k9-10, kll-12, and k13 were considered to be the same as determined from the previous analysis. These values were then used as constants to fit the model to the data from the H20+/D20 and D20+/H20 reactions. The best fit solutions from a non-linear least squares regression analysis then gave values of the branching ratios and rate coefficients for Reactions 1-8. The model fits are shown in Figs 3 and 4 as solid lines. The analysis was simplified by the constraints: and which reduced the number of independent variables to seven (bl, b2, b3, bs, b6, kl-4, and k5_7). The reaction times of the individual experiments were again considered free parameters to be optimized for each reaction set. The value for k8 was taken from the literature [12] and assigned the accepted value of 1.85 × 10 -9 cm 3 s -1. The
bl+b2+b3+b4=l
bs+b6+b7=l,
2 The r o o t - m e a n - s q u a r e d deviation w a s c a l c u l a t e d as the square root of the s u m o f the squares o f the deviation f r o m the m e a n divided b y the m e a n value o f the variable.
solutions to the reactions H20+]D20 and D20+/ H20 are shown in Table 2. A fifth data set was obtained from the work of Smith et al. [7]. This set consisted of the values of the fractional abundances of ion concentrations varying with bath gas pressure for the reactant pair D30+/H20. The experimental data and the model fits are shown in Fig. 5. The actual data of this experiment was taken from the figure showing the data in their paper. The analysis performed was the same used on the ICR data, as described above and below. To assign confidence limits on the values of the rate coefficients and branching ratios determined from the model, a merit function, X2, is defined by Eq. (14) [13]: X2
=
~IYi-Yfit(xi'afit)] 2 i= 1
(1i
(14)
J
(xi,Yi) y~t(xi,ant)
where are a set of N experimental data points, are the values of y modeled using a set of M fitted parameters, ant. For a 'best fit' of the model to the experimental data, the fitted parameters have optimum values. The standard deviation, a, was assumed to be equal for all data points in a set, and was evaluated by the equation
o2=[i=~l[Yi-Yfit(xi,afit)]21](N-M)
(15)
In this study the standard or the root-mean-square
8
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
Table 2 Summary of analysis of the H20+/H20 systems (this work)
b~ b2 b3 b4
D20+/H20
H20+/D20
Statistical model
0.19 0.21 0.59 0.00
0.22 0.00 0.49 0.30
0.26 0.07 0.33 0.33
--- 0.06 --- 0.06 --- 0.11 _ 0.21
_+ 0.03 + 0.11/- 0.00 + 0.03/- 0.14 -+ 0.04
k~_4
1.56 + 0.78/- 0.22
5.22 + 1.3/- 0.7
b5 b6 b7
0.00 + 1.00/- 0.00 1.00 + 0.00/- 1.00 0.00 + 1.00/- 0.00
0.00 + 0.59/- 0.00 1.00 + 0.00/- 0.59 0.00 + 0.46/- 0.00
k5-7
3.0 + 0.00/- 3.0
3.0 + 0.00/- 0.54
k8
1.85 lit.
1.85 lit.
0.23 0.19 0.58
a The b values are the branching ratios, whose subscripts reference the equation numbers used in the text. b The k values are expressed in units of ( × 10 -9) cm 3 s t.
deviation 2 between the experimental data and the model was 1.5% for the H30+/I-I20 system and 1.8% for the H20+/H20 system. In the SIFT study of the H30÷/H20 system, the standard deviation was 0.8%. This indicates that the SIFT data had a signal-to-noise ratio that was about twice that in the ICR data. Confidence limits were set on the estimated model parameters using the chi-squared formalism. A function, AX2 [13], is defined as 2 A X 2 = Xv2 - Xmin
(16)
X2in is the minimum value of X2. The minimum value is obtained from the nonlinear leastsquares regression fit of the reaction model (Appendix 1 and Appendix 2) to the experimental data by adjusting parameter al~t. Any deviation from the optimum value (for the best fit) of a parameter increases the value of X2 to a new value, X. 2. (Then AX2 defines a confidence limit on the parameter.) The confidence limit of a single parameter, ant (equivalent to a _ 1~ limit) is defined by a value of AX2 equal to 1 and encloses a 68.3% confidence interval. In a multi-parameter problem, as when a statistical model is being compared to an experimental data set, the constant AX 2 that confines the 68.3% confidence interval increases from 1.00 for a single fitted parameter to 2.30, 3.53, and where
4.72 for cases with two, three, and four fitted parameters, respectively. The value of the confidence limit of each parameter, afit, was determined by using the selected value of AX2 and searching for values ofafit which have AX2 values of the limit, while continually optimizing all other parameters. The 68.3% confidence limits in the branching ratios and reaction rate coefficients are shown in Tables 1 and 2. The reaction model fits the experimental data very well and has errors between 0.8% and 1.8% for each data point. However, the model, when optimized leads to solutions for the parameters (branching ratios and rate coefficients) that have uncertainties that are larger than 3%. Further, the AX2 was insensitive to the values of some branching ratios. When this occurred it was not possible to determine the values of these branching ratios, since all possible values of b, 0-1, were included in the confidence limit [13]. A test was conducted to ascertain whether the statistical isotope scrambling model of Smith et al. [3] would fit the experimental data (ICR and SIFT) within a 68.3% confidence limit. All the branching ratios and rate coefficients were set at their predicted model values (final columns in Tables 1 and 2) to obtain a value o f x , 2 and therefore AX2. Then the calculated value of AX2 was
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
9
1.0
0.8 m
_8
~
i
0.6
0.4
~0.2 0.0 0
2
4
6
8
0
Pressure, Ton" (xlO "e) Fig. 5. Ion abundances in the isotope exchange reactions, starting with D30 + in a H20 bath. From Ref. [7], a SIFT experiment. The points are the data, and the line a modeled fit. Legend: [~, D30+; O, HD20+; O, H2DO÷; A, H30 +.
compared to the value of AX2 specific to the number of variable parameters in the reaction set. If the value of AX 2 derived from the statistical scrambling model was less than the parameter specific value, then the criterion for the 68.3% confidence limit was satisfied.
4. Discussion
The predictions of the statistical isotope scrambling model of Smith et al. [3] were statistically not supported by the results of the ICR and the SIFT experiments. The branching ratios between channels 9 and 10 predicted by the statistical isotope scrambling model [3] are 0.67 and 0.33 respectively. The test to evaluate the confidence level in the fitted parameters derived from the statistical scrambling model showed that all the expected model values listed in Table 1 were outside the 68.3% interval for both the current ICR data and the re-analyzed SIFT data. The ICR and the SIFT results agree to within three standard deviations ( _+ 3o) of each other for the D30+/H20 data. The measured branching ratios of the charge transfer channels,
Reactions 1 and 2, are not consistent with a statistical distribution of protons and deuterons. We conclude that the proton transfer reactions are near statistical; therefore, they proceed through a long-lived collision complex. The charge transfer reactions are further from statistical and do not always occur through a long-lived collision complex. There appears to be different collision complexes associated with the charge transfer and proton transfer processes. In the ICR and SIFT experiments the lifetime of either complex is not sufficiently long to allow complete statistical scrambling of hydrogens and deuterons. There is a noticeable increase in the uncertainties in the derived values of the branching ratios from the primary reactions, to the secondary reactions, and then to the tertiary reactions. The branching ratios of channels 9 and 10 were determined from the analysis of the data from the H30+/D20 and D30+/H20 reactions (Table 1). The results of the SIFT and the ICR studies have a maximum error of -+ 0.04. The branching ratios of channels 11 and 12 have much larger uncertainties because of propagation of errors. The SIFT experiments were conducted at a
10
V.G. Anicich, A.D. Sen~International Journal o f Mass Spectrometry and Ion Processes 172 (1998) 1-14
higher neutral gas concentration than the ICR experiments. Therefore, the extent of reaction in the SIFT study was larger. The SIFT experiment sampled a different regime of the reaction pressure-time domain, and so a different experimental view of the branching ratios of channels 11 and 12 can be expected. The branching ratios of channels 1 to 4 were determined with errors of _+ 0.21 or less for the HzO+/D20 and DzO+/H20 reactions (Table 2). The branching ratios for channels 5, 6, and 7 could not be determined with any certainty due to large propagation of errors. In the next two paragraphs we will look at the constitution of the overall reaction rate coefficients. There are two possible channels of the reaction H 2 O+ + D 2 0 ; charge transfer with and without isotopic scrambling, and self-protonation of H 2 O ÷ with and without isotopic scrambling. In this discussion no distinction will be made between hydrogen atom abstractions and proton transfer. Since a collision complex is used in our model and direct reactions are not considered, hydrogen atom abstractions and proton transfer are included in what we have called 'self-protonation'. All possible products of the proton-deuteron exchange reactions of H30 + were observed. The literature value of the rate coefficient k8 for the self-protonation reaction with all hydrogens is 1.85 x 10 -9 c m 3 s -1 [12]. This value has a n estimated error of _+ 15%. The collision rate coefficient for this same reaction is 2.97 x 10 -9 c m 3 s -1 [14]. It is our proposition that the difference between the collision rate coefficient and the measured proton transfer rate coefficient is due to a 'symmetric' charge transfer reaction whose reaction rate coefficient cannot be measured directly and whose products are the same as the reactants. In the following arguments we are assuming that: 1. the literature value of the rate coefficient for the self-protonation reaction of D2 O+ + DzO, k8, is the same for the proton transfer portion of the rate coefficients for Reactions 3 and 4
(i.e., k8 = k3-4) and Reactions 6 - 7 (i.e., k8 = k6-7), and
2. the sum of the charge transfer and proton transfer reaction rate coefficients is equal to the collision rate, including the 'symmetric' charge transfer reaction channel, then the values of the rate coefficients k~ 4 and k5 7 should be equal to the difference of the collision rates (3.09 x 10-9cm3s -1 and 2.97 x 10 -9 c m 3 s -1 respectively) and the rate coefficients for 'symmetric' charge transfer. Further, if the symmetric charge transfer channels are assumed to be statistically accessed, then for Reactions 1-4, there are four charge transfer possibilities. There are three forward possibilities: Reaction 1 (counted twice) and Reaction 2 (counted once). The fourth channel is the back charge transfer reaction (the 'symmetric' reaction). Therefore, the rate coefficient for the forward charge transfer channels is kl-2 = 0.75 x (3.09 - 1.85) x 10-9 cm 3 s -1 = 0.93 x 10 -9 cm 3 s -1. Since k3_ 4 = k8, then the value of the rate coefficient kl_ 4 = kl_ 2 + k3-4 = 2.78 x 10 -9 cm 3 s -1. In Reactions 5 - 7 there are four charge transfer possibilities: two forward reactions (Reaction 5) and two back charge transfer reactionss (the symmetric reactions). Then the rate coefficient for the forward charge transfer channels is k5 = 0.5 x (2.97 - 1.85) x 10 -9 c m 3 s -1 = 0.56 x 10 -9 c m 3 s -1. Since k6_ 7 = k8, the value o f the rate coefficient k5-7 = k5 + k6_7 = 2.41 × 1 0 - 9 c m 3 s -1. A chi-squared analysis of the charge transfer rate coefficients gave AX2 values several times larger than the ___ 1o confidence interval, thus not supporting the statistical scrambling model as a viable explanation for the observed charge transfer processes. In a similar manner we make the following observations. Only proton-deuteron exchanges are the possible products of the H3 O ÷ + D 2 0 reaction. The difference between the collision rate coefficient and the measured rate coefficient for proton-deuteron exchange reactions is again the rate coefficient for the 'symmetric' reaction
V.G. A nicich, A.D. Sen~International Journal o f Mass Spectrometry and Ion Processes 172 (1998) 1-14
channel, whose products are the same as the reactants. The values of the rate coefficients k9 10, kll-12, and k13 should be equal to the difference of the collision rates (2.89 x 10 -9 cm 3 s -l, 2.82 x 10 -9 cm 3 s -l, and 2.79 x 10 -9 cm 3 s -1 respectively) and the rate coefficients for symmetric proton exchange reactions. Further, if the symmetric proton exchange channels are assumed to be statistically accessed, then for Reactions 9 and l0 there are ten proton exchange possibilities. There are nine forward possibilities: Reaction 9 (counted six times) and Reaction 10 (counted three times), and the back charge transfer reaction (the symmetric, reaction) (counted once). Therefore, the rate coefficient for the forward proton exchange channels is k9-10 = 0.90 x 2.86 x 10-9cm3s -1 = 2.57 x 10-9cm3s -1. This value is within the errors in the experimental determination of the rate coefficient k9-10 = 2.20 x 10 -9 c m 3 s - l [12]. In Reactions 11 and 12 there are again ten proton exchange possibilities. There are seven forward possibilities: Reaction 11 (counted six times) and Reaction 12 (counted once), and the back charge transfer reaction (the 'symmetric' reaction) (counted three times). Therefore, the rate coefficient for the forward proton exchange channels is kll_12 = 0.70 × 2.82 × 1 0 - 9 c m a s - l = 1.97 x 10 -9 c m 3 s - l . With four deuterons involved in the reaction, channel 13 has four possible forward channels and six backward reaction channels. Thus, the proton exchange rate coefficient is kl3 = 0.40 x 2.79 x 10 -9 c m 3 s -1. A chisquared analysis of the proton exchange rate coefficients gave AX 2 values several times larger than the _+ 1a confidence interval, thus excluding the statistical scrambling model as a viable explanation.
of ion-molecule reactions. This analysis can be utilized to access the validity of possible models used to explain the observed kinetics. The delta chi-square analysis assigns confidence limits to the solutions of a test model with respect to a nonlinear least-squares regression best fit of the experimental data. The predictions of the statistical isotope scrambling model of Smith et al. [3] were not statistically supported by the results of ICR and SIFT experiments for charge transfer and proton transfer reactions. Proton transfer reactions appear to be near statistical and proceed through a complex that is relatively long-lived, yet not sufficiently long-lived to allow for complete statistical scrambling of hydrogens and deuterons. In charge transfer reactions, the complex appears to have an even shorter lifetime.
Acknowledgements The work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. We thank Prof. Murray J. McEwan for his helpful discussions. In addition we thank Prof. Robert Jenerch of UCLA for his advise on the statistical analysis performed in this work. Finally we thank reviewer number 2 for persistent encouragement to analyze our data correctly.
A P P E N D I X 1: P R O B L E M
klo
5. Conclusions The primary and salient feature of this work is the application of rigorous statistical analysis to the experimental results obtained for the kinetics
I1
d[A] _
dt
(k9 + klo)[A][B]
K G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
12
d[C] = k 9 [ A ] [ B ] - (kl~ + k12)[C][B] at
d[E] -= kl 0 [A ] [B] + all [ C] [B] - k 13 [E] [B]
dt
d[G] dt = k12 [C] [B] + k 13[E] [B]
SOL UTION [A] = [A]o exp[ - (k 9 + klo)[B]t]
[C] = [.4]0 k9 + klo ---(-kl 1 + kl2) [exp[
[A]o[ [E]=
- (kll + kl2)[B]t] - exp[ - (k9 + klo)[B]t]]
k9 +~o_k13] [exp[-k13[B]t]-exp[-(k9 + klo)[B]t]]
+[A]o[k9+kaok(kll+kl2)]
k~l +k~2 [kll _kl3]
[exp[-kl3[B]t]-exp[-(kll+kl2)[B]t]]
- [ k9 + ~--ll~_k,3][exp[-kl3[B]t]-exp[ ,(k9 + klo)[B]t]] [ 6 q = 1 - [A] - [ C ] - [ E ]
APPENDIX 2." PROBLEM
V.G. Anicich, A.D. Sen~International Journal o f Mass Spectrometry and Ion Processes 172 (1998) 1- 14
d[X] i dt
(k~ +k2 +k3 + k 4 ) [ x ] [ 8 ]
d[dY] = k I IX] [B] - (k 5 -4-k 6 +
d[Z] = k2 IX] [8] dt
--
d[C] at
=
k7)[Y][B ]
+ k5 [ r] [8] - k8 [z] [8]
k3 IX] [B] - (kll + k 12)[c] [8]
--
d[E] dt = k 4 [ x ] [ 8 ]
+ k6[r'][8] + kl ~[ c ] [ 8 ] - k13[E][8]
d[G] dt = k7[r][8]
+ k8 [ z ] [ 8 ] + k~2 [ c ] [ 8 ] + k~3 [E][8]
SOL UTION [X] = [X]o exp[ -
(kl +k2 + k3 +k4)[B]t
[Y] = [X]o kt +k2 +k3 +k4 -(ks +k6 +kT) [exp[-(ks +k6 +kT)[B]t] -exp[ -(kl +k2 -t- k3 +k4)[B]t]]
[X]o
kl +kz +k3 + k 4 - k 8 [exp[-k8[B]t]-exp[-(kl +k2 +k3 +k4)[B]t]] k5
[ks+k6-~k7-k8] [exp[-k8 [B]t] [Z]
=
- exp[ - (ks + k6 + k7)[B]t]]
k1
+[X]°[kl+k2+k3+k4-(ks+k6+k7)]
_ [kl+k2+_~3+k4_k8][exp[_ks[B]t ] k S -
exp[ - (kl + k2 + k3 + k4)[B]t]]
13
14
14G. Anicich, A.D. Sen~International Journal o f Mass Spectrometry and 1on Processes 172 (1998) 1-14
I
k3
]
[C] = [X]o kl + k2 + k3 + k4 - (kll + k12) [exp[
- (kll + kl2)[B]t] -
exp[
- (kl + k2 + k3 + k4)[B]t]]
k4
[kl + k2+ k-7+k4_k13-] [exp[- kl3[B]t] -exp[-(kl+k2+k3+k4)[B]t]] k6
[ k5 + k6 + k7 _ k13] [eXp[-kl3[B]t] -
+ [X]o
kl +k2 +k3
+k4 - ( k 5 +k6 +k7)
exp[ -
(k 5 + k 6 + k7)[B]t]]
k6 - [ kl +k2 +k3 +k4_k13] [exp[-kl3[B]t]
[E] = -- e x p [ -- ( k 1 + k 2 + k 3 +
k4)[B]t]]
[ kll kll -kl3]
[
k3
]
-
kl +k2 +k3 + k 4 - ( k l l +k12)
- (kll + k12)[B]t]]
exp[ kll
- [ kl +k2 +k3 +k4_kl3] [exp[-k13[B]t] - e x p [ - (kl + k 2 + k 3 +k4)[B]t]]
[G] = 1 - [X] - [Y] - [Z] - [C] - [E]
R e f e r e n c e s
[1] J. S. Lewis, R. G. Prinn, Planets and their Atmospheres, Academic Press, New York, 1984. See page 200 for Earth as an example. [2] W. W. Duley, D. Williams, Interstellar Chemistry, Academic Press; New York, 1984. See section 8.6, Introduction to interstellar isotopes; J. Lequeux, E. Roueff, Physics Reports 200 (1991) 241. [3] D. Smith, N. G. Adams, M. J. Henchman, J. Chem. Phys. 72 (1980) 4951. [4] A. G. Harrison, P-H Lin, C. W. Tsang, Int. J. Mass Spectrom. Ion Phys. 19 (1976) 23. [5] T. B. McMahon, P. G. Miasek, J. L. Beauchamp, Int. J. Mass Spectrom. Ion Phys. 21 (1976) 63. [6] K. Giles, N. G. Adams, D. Smith, J. Phys. Chem. 96 (1992) 7645. [7] N. G. Adams, D. Smith, M. J. Henchman, Int. J. Mass Spectrom. Ion Phys. 42 (11) (1982); M. J. Henchman, D. Smith,
[8]
[9] [10] [11] [12] [13]
[14]
N. G. Adams, J. F. Paulson, Z. Herman, Int. J. Mass Spectrom. Ion Proc. 92 (15) (1989); M. J. Henchman, D. Smith, N. G. Adams, Int. J. Mass Spectrom. Ion Proc. 109 (1991) 105. N. G. Adams, D. Smith, in A. Fontijn, M. A. A. Clyne (Eds.), Reactions of Small Transient Species, Academic Press, New York, 1983, pp. 343-359. V. G. Anicich, A. D. Sen, W. T. Huntress Jr., J. Chem. Phys. 102 (1995) 3256. P. R. Kemper, M. T. Bowers, Int. J. Mass Spectrom. ion Phys. 52 (1983) 1. J. Wronka, D. P. Ridge, Rev. Sci. Instrum. 43 (1982) 49. V. G. Anicich, J. Phys. Chem. Ref. Data 22 (1993) 1469. W. H. Press, S. A. Teukolsky, W. T. Wetterling, B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed., Cambridge University Press, New York, 1992. See under 'Confidence Limits on Estimated Model Parameters, Constant Chi-Squared as confidence Limits', pp. 688690. T. Su, W. J. Chesnavich, J. Chem. Phys. 76 (1982) 5183.
E LS EVI E R
International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
Deuterium exchange in the systems of H20÷/H20 and H30+/H20 Vincent G. Anicich*, Atish D. Sen 1 Jet Propulsion Laboratory, California Institute of Technology, MS 183-301, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Received 26 October 1995; revised 24 June 1997; accepted 28 August 1997
Abstract The reactions of H20 ÷, H3 O÷, D20 ÷, and D3 O+ with neutral H20 and D20 were studied by tandem mass spectrometry. The H20 + and D20 ÷ ion reactions exhibited multiple channels, including charge transfer, proton transfer (or hydrogen atom abstraction), and isotopic exchange. The H3 O+ and DHO+ ion reactions exhibited only isotope exchange. The variation in the abundances of all ions involved in the reactions was measured over a neutral pressure range from 0 to 2 x 10 -5 Torr. A reaction scheme was chosen, which consisted of a sequence of charge transfer, proton transfer, and isotopic exchange reactions. Exact solutions to two groups of simultaneous differential equations were determined; one group started with the reaction of ionized water, and the other group started with the reactions of protonated water. A nonlinear least-squares regression technique was used to determine the rate coefficients of the individual reactions in the schemes from the ion abundance data. Branching ratios and relative rate coefficients were also determined in this manner. A delta chi-squared analysis of the results of the model fitted to the experimental data indicated that the kinetic information about the primary isotopic exchange processes is statistically the most significant. The errors in the derived values of the kinetic information of subsequent channels increased rapidly. Data from previously published selected ion flow tube (SIFT) study were analyzed in the same manner. Rigorous statistical analysis showed that the statistical isotope scrambling model was unable to explain either the SIFT or the tandem mass spectrometry data. This study shows that statistical analysis can be utilized to assess the validity of possible models in explaining experimentally observed kinetic behaviors. Published by Elsevier Science B.V, Keywords: Isotopic exchange; Proton exchange; Charge exchange; Branching ratios; Kinetics
1. Introduction Although deuterium is widely observed in the solar system [1] and interstellar space [2], the deuterium scrambling process has been studied in only a few ion-molecule systems. An understanding of the isotopic fractionation process and the variation in the H-to-D ratio in different molecules is important when interpreting their
* Corresponding author. Fax: 001 818 354 2439. Present address: University of Delaware, Department of Chemistry and Biochemistry, Newark, DE 19. 0168-3659/98/$17.00 Published by Elsevier Science B.V. PII S0168-1 1 7 6 ( 9 7 ) 0 0 2 5 7 - 7
cosmic origin and abundances. Therefore, deuterium scrambling is an important process that needs to be quantified. Ideally, the fundamental process that deserves the most attention is the reaction between an ion and a molecule that have only one deuterium atom between them. The deuterium atom can reside either in the ion or in the neutral molecule. However, in a single-chamber experiment, a series of rapid consecutive reactions occurs and it is very difficult to identify and measure the individual reaction steps. It might seem that partially labeled species can be generated in a
2
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
tandem experiment using a carefully controlled source. However, with larger molecules like water, a network of consecutive reactions allows the unambiguous generation of only the parent and the protonated ions, i.e. H 2 0 +, H30 +, D20 +, and D3O+. All other ions are ambiguous due to mass overlap and isotopic impurities generated from wall reactions. Partially deuterated systems are particularly difficult to study, principally due to the cost of the materials in the quantities needed to passivate the experiments. Deuterium exchange reactions are also particularly difficult to study because of the number of consecutive steps, the extent of branching, and the rapid reaction rates that usually occur in the reaction scheme. In the case of water, where the apparatus retains the isotopic species for a long period of time because of adhesion to exposed surfaces, the purity of the neutral isomer cannot be validated. Even in a tandem experiment where the reactant ion mass is pre-selected, the rate of isotopic scrambling makes the certainty of a 'specific mass' isomeric composition questionable. A case in point is HDO + which cannot be distinguished from the H3O÷ because both ions have a mass of 19 Da. The conclusion of Smith et al. [3] from their selected ion flow tube (SIFT) studies was that the thermal neutral deuterium exchange reactions in the H30+/D20 and D30+/H20 systems exhibited a statistical behavior. They also concluded that the reactions proceeded through an association complex that lived sufficiently long to randomize the hydrogen and the deuterium atoms before unimolecular decomposition to reactants and products. Data were also presented in the same paper for the reaction between D3O÷ and NH3 which exhibited very different results than did the H30+/D20 and D30+/H20 reactions [3]. In this reaction only D ÷ was exchanged, i.e., a proton transfer reaction was observed. The conclusion was that a long-lived intermediate complex was not formed. Similar results were found for CD~/NH3 and other asymmetric exothermic proton transfer reactions [4].
Ion cyclotron resonance (ICR) studies [5] of charge transfer reactions in near-thermoneutral symmetric systems, 15N~]IaN2, 14N~/15N2, 12CO+/13CO, and 13CO÷/12CO measured reaction rate coefficients that were half the collision rates, suggesting the existence of a statistical charge transfer mechanism. The rate coefficients of hydrogen exchange in the reactions H2D÷/HD, HD~/HD, and 14NH~/15NH3 were found to be consistent with the results expected from the statistical model [5,6]. The results obtained from two other systems, CH3D~/CH2D2 and CHzD~/CH2D2, could not be explained by the statistical model [5]. Further studies on isotope exchange reactions have been conducted on the H30÷/D20, D30+/ H20, NH~/ND3, ND~/NH3, CH~/CD4, and CD~/ CH4 systems [7,8]. The reactions were analyzed by considering the formation of an intermediate complex and the average number of protondeuteron jumps, K, occurring during the lifetime of the complex. The water system was found to form a complex with a lifetime longer than the proton-deuteron shuttling time. Therefore, the system exhibited a statistical isotope exchange process, r >-- 20. The ammonia and the methane systems were found to form short-lived complexes with lifetimes shorter than the protondeuteron shuttling time. Therefore, these systems did not exhibit a statistical isotope exchange process, which is consistent with a value of K = 0.25 for the methane reactions. The reactions of H20÷/D20, D20+/H20, H30+/D20, and D30+/H20 reported in the present work were studied by a tandem ICR mass spectrometer. The data from this study and those of a previous SIFT study of H30+/D20 and D30+/H20 reactions were analyzed as a sequence of charge transfer, proton transfer, and isotopic exchange reactions. The overall reaction rate coefficient, the relative rate coefficients, and the branching ratios of these individual steps were determined to improve our understanding of the isotopic exchange process in the water system. Detailed statistical analyses were performed on
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
the tandem ICR and SIFT data to establish confidence levels on the results obtained from the kinetic studies. The statistical analyses had a significant bearing on the conclusions that can be drawn from the experimental results.
2. Experiment A tandem mass spectrometer in a I C R Dempster-ICR configuration [9,10] was used to conduct the present experiments. The instrument was housed within the pole pieces of a 12 inch electromagnet that supplied the magnetic field for the two ICR sections and the Dempster section of the mass spectrometer. The ion source was a two section ICR drift cell. The drift potentials were used to control the reaction time in the source region. A 180 ° magnetic sector was used as a primary mass filter. The volume of the magnetic sector was defined by a metal cage within which all electric fields were uniform. A 3 kV potential provided the acceleration field at the exit of the 'source' ICR cell. The magnetic field was adjusted to give the ions the appropriate radius of curvature needed to enter the 'deceleration-energy selection' region at the front of the 'detector' ICR cell. After the ions were decelerated, a long narrow Wien filter was used to limit the energy of the ions entering the reaction region to less than 30 meV. The first section of the detector ICR cell was the ion-molecule reaction region. The second section of the cell was the ion detection region and was used to measure the ion abundances. A Wronka bridge detector [11] monitored the ion signals in this section. Frequency scans on the detector section were used to obtain a mass spectrum. Vapors from distilled water and deuterated water were the source gases. Both samples were purified by several freeze-pump-thaw cycles. To ensure the predominance of H20 + and D20 + exiting the source region, the pressure of
3
the source gas was limited so that very little of the protonated water was formed. The magnetic field was scanned to obtain the mass spectrum of the ion source output. A biased metal plate connected to an electrometer served as an 'auxiliary' ion detector. The plate was placed behind a slit in the acceleration housing at a smaller radius of curvature than the detector cell. Ions passing through the slit impinged on the plate, and an ion current was observed that was proportional to the number of ions focused on the detector plate. To ensure the predominance of H30 + and D3O+ exiting the source region, the pressure of the source gas was increased so that the protonated water was the dominant ion. Long passivation times were required before the pressure and the isotopic purity became stable in both the source and the detector regions. Pressures above 5 × 10 -6 Torr in the detection cell were measured by a 1 Torr capacitance manometer. Lower pressures were measured by an ionization gauge which was calibrated against the capacitance manometer. The drift times were not measured directly. The rate coefficients of known water reactions were used as a reference [12], and the unknown rate coefficients were determined relative to the known rates.
3. Results Data sets consisting of the variation of fractional abundances of ions with bath gas pressures were obtained for the four pairs of ion-molecule reactants: H20+/D2 O, D20+/H20, H30÷/D20, and D30+/H20, and these are shown in Figs 14. In all cases the primary ion reacted rapidly, forming several isomeric species in intermediate stages. In the end, a single 'terminal' ion, the protonated neutral, dominated the reaction products. Several consecutive reactions are possible at the highest pressures. The terminal ion was always the protonated neutral. The intermediate steps in the reaction scheme were assumed to be a
4
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
1.0 D
tt4 .Q
0.8
0.6
g 0.4
i
--0.2-
0.0
-
-
0
I
I
1
2
Pressure,
3
Torr (xlO "s)
Fig. 1. Ion abundances in the isotope exchange reactions, starting with H3O÷ in a D20 bath. From the tandem ICR-Dempster-ICR spectrometer. The points represent the data, and the line a modeled fit. [~, H30÷; (3, H2DO÷; O, HD20÷; ZX,D3O÷.
c o m b i n a t i o n o f c h a r g e transfer, p r o t o n transfer, and isotope e x c h a n g e reactions.
3.1. Data analyses A n o n l i n e a r least-squares regression t e c h n i q u e
w a s used to a n a l y z e the data. T h e reaction s c h e m e d e v e l o p e d to explain the data is s h o w n below. T h e first g r o u p o f reactions is a list o f the possible p r o d u c t c h a n n e l s f r o m either H2 O+ or H3 O+ as the p r i m a r y ions in a bath o f D20.
1.0
0.8
0.6
i
0.4
0.o,
--0
I
I
I
5
10
15
Pressure,
Torr (xl0 "s)
Fig. 2. Ion abundances in the isoptope exchange reactions, starting with D30 + in an H20 bath. From the tandem ICR-Dempster-ICR spectrometer. The points are the data, and the line a modeled fit. The open and filled symbols represent different data sets of the same experiment. Legend: [], D30+; O, HD20+; <>, H2DO+;A, H30 +.
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
5
1.0
0.8
0.6
|o.4 '~ 0 . 2
0.0
I
0
1
2
3
4
Pressure, Torr ( x l 0 4)
Fig. 3. Ion abundances in the isotope exchange reactions, starting with D20 + in a H20 bath. From the tandem ICR-Dempster-ICR spectrometer. The points represent the data, and the lines a modeled fit. The open and filled symbols represent different data sets of the same experiment. Legend: I-q, (D20 +, H2DO+); A, HD20+; ~, H20+; O, (HDO +, H30+).
1.0
0.8 m
8 c
o.6
c .Q 0.4 o
,,~ 0 . 2
0.0
I I
2
P r e s s u r e , T o r r ( x l 0 "s) Fig. 4. Ion abundances in the isotope exchange reactions, starting with H20 + in a D20 bath. From the tandem ICR-Dempster-ICR spectrometer. The points represent the data, and the lines a modeled fit. The open and filled symbols represent different data sets of the same experiment. Legend: ~ , H20+; - - , HDO+; [:], (D20 +, H2DO+); A, HD20+; C), D3O+.
6
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
D20 reactions with H20 + and H 3 0 + primary ions: H2 O+
+
D20
~ "-* -'-'
HDO ÷ D20 + H2DO + HD20 +
+ + + +
HDO H20 OD OH
(1) (2) (3) (4)
HDO +
+
D20
"-'* ---* ---*
D2 O+ + HD20 + + D3 O+ +
HDO OD OH
(5) (6) (7)
D20 +
+
D20
-"*
D3 O+
OD
(8)
H30 ÷
+
D20
--'* --*
H2DO ÷ + HD2 O+ +
HDO H20
(9) (10)
H2DO + +
D20
~
HD2 O÷ + D3 O+ +
HDO H20
(11) (12)
HD20 + +
D20
"-'*
D30 +
HDO
(13)
+
+
The second group of reactions is the possible product channels from either D20 + or D3 O÷ as the primary ions in a bath of H20.
H20 reactions with D 2 0 + and D3 O+ primary ions: D2 O+
+
H20
~
HDO ÷
HDO
(1)
H20 ÷ + HD2 O+ + H2DO + +
+
D20 OH OD
(2) (3) (4)
HDO +
+
H20
~ --~ ---'
H20 + + H2DO ÷ + H30 ÷ +
HDO OH OD
(5) (6) (7)
H2 O+
+
H20
~
H3 O+
OH
(8)
D3 O+
+
H20
~
HD2 O+ + I-I2DO + +
HDO D20
(9) (10)
HD2 O+ +
H20
-"* --*
H2DO + + H30 + +
HDO D20
(11) (12)
H2DO ÷ +
H20
-"*
H3 O+
HDO
(13)
+
+
Owing to the symmetry of the two reaction schemes, the reaction numbers were duplicated for equivalent reactions. Equivalent reactions can have identical products, but the rate coefficients and the branching ratios may not be equal. Each reaction scheme is further split into two groups of reactions: one with ionized water as the primary ion (Reactions 1-8) and the other with protonated water as the primary ion (Reactions 9-13). The kinetics of the reactions of protonated water, Reactions 9-13, were treated as a set of simultaneous partial differential equations. The equations and their solutions are shown in Appendix A. These solutions to the equations were fitted to the experimental data by a nonlinear least-square regression technique. The 'best fit' solutions then gave values of the branching ratios and rate coefficients for Reactions 9 13. The model fits are shown in Figs 1 and 2 as solid lines. The data sets of the reactions H30÷/ D20 and D30+/H20 were analyzed to determine the branching ratios b9, bl0, b11, and b12, and the rate coefficients k9_10, k11-12, and k13. The subscripts to b refer to the reaction numbers. The subscripts to k indicate the total reaction rate coefficient of an individual reaction, k13 (Reaction 13), or a group of reactions, k9-10 (Reactions 9 and 10). This notation is maintained when evaluating all data sets. Thus, in the protonated water reactions, since b9 + bl0 = 1 and b l l + b 1 2 = 1, there are only five independent parameters that have to be determined. The reaction times of the individual experiments were not measured, but were considered free parameters to be optimized for each reaction set. The value of k9-10 is well established in the literature [12] and was therefore set to the accepted value of 2.2 x 10 -9 cm 3 s -1. The reaction time r was computed relative to the value of k9-10 and then used to calculate the values of k11-12 and k13. The solutions to the reactions H30+/D2 O and D30+/I-I20 are shown in Table 1. The kinetics of the reactions of ionized water, Reactions 1-8 and 11-13, were treated as a different network of simultaneous partial
V.G. Anicich, A.D. Sen~InternationalJournal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
7
Table 1 S u m m a r y o f analysis o f the H 3 0 + / H 2 0 systems D 3° +/I"120, SIFT
H 3 0 +/D 20, SIFT
D 3° +/n 2° (this work)
H 3 0 +/D 2 0 (this work)
Statistical model
b~ b~0
0.71 -+ 0.03 0.29 ± 0.03
0.70 0.30
0.52 - 0.04 0.48 - 0.04
0.42 --- 0.03 0.58 ± 0.03
0.67 0.33
k9b_10
2.20
2.27
2.20 -
2.20 -
bH b Iz
0.72 ± 0.04 0.28 -+ 0.04
-
1.00 + 0 . 0 0 / - 0.30 0.00 + 0 . 3 0 / - 0.00
1.00 + 0 . 0 0 / - 0.46 0.00 + 0 . 4 6 / - 0.00
kH-12
1.95 --+ 0.12
-
3.32 ± 0.42
2.32 + 0 . 4 0 / - 0 . 2 8
k13
0.99 - 0.06
-
2.84 + 0 . 1 2 / - 0.60
0.63 + 0 . 4 6 / - 0.11
0.86 0.14
a The b values are the b r a n c h i n g ratios, w h o s e subscripts reference the equation n u m b e r s used in the text. b The k values are expressed in units o f ( x 10 -9) c m 3 s -1.
differential equations. The equations and their solutions are shown in Appendix B. Reactions 9 and 10 were excluded from the network because the ion H30 ÷ is not a product when the reaction starts with H20 ÷ in a D20 bath. The analysis of the data sets from the reactions H20÷/D20 and D20+/H20 was simplified by using the solutions obtained from the reactions of the protonated species, i.e., the values of b9, blo, bll, b12, k9-10, kll-12, and k13 were considered to be the same as determined from the previous analysis. These values were then used as constants to fit the model to the data from the H20+/D20 and D20+/H20 reactions. The best fit solutions from a non-linear least squares regression analysis then gave values of the branching ratios and rate coefficients for Reactions 1-8. The model fits are shown in Figs 3 and 4 as solid lines. The analysis was simplified by the constraints: and which reduced the number of independent variables to seven (bl, b2, b3, bs, b6, kl-4, and k5_7). The reaction times of the individual experiments were again considered free parameters to be optimized for each reaction set. The value for k8 was taken from the literature [12] and assigned the accepted value of 1.85 × 10 -9 cm 3 s -1. The
bl+b2+b3+b4=l
bs+b6+b7=l,
2 The r o o t - m e a n - s q u a r e d deviation w a s c a l c u l a t e d as the square root of the s u m o f the squares o f the deviation f r o m the m e a n divided b y the m e a n value o f the variable.
solutions to the reactions H20+]D20 and D20+/ H20 are shown in Table 2. A fifth data set was obtained from the work of Smith et al. [7]. This set consisted of the values of the fractional abundances of ion concentrations varying with bath gas pressure for the reactant pair D30+/H20. The experimental data and the model fits are shown in Fig. 5. The actual data of this experiment was taken from the figure showing the data in their paper. The analysis performed was the same used on the ICR data, as described above and below. To assign confidence limits on the values of the rate coefficients and branching ratios determined from the model, a merit function, X2, is defined by Eq. (14) [13]: X2
=
~IYi-Yfit(xi'afit)] 2 i= 1
(1i
(14)
J
(xi,Yi) y~t(xi,ant)
where are a set of N experimental data points, are the values of y modeled using a set of M fitted parameters, ant. For a 'best fit' of the model to the experimental data, the fitted parameters have optimum values. The standard deviation, a, was assumed to be equal for all data points in a set, and was evaluated by the equation
o2=[i=~l[Yi-Yfit(xi,afit)]21](N-M)
(15)
In this study the standard or the root-mean-square
8
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
Table 2 Summary of analysis of the H20+/H20 systems (this work)
b~ b2 b3 b4
D20+/H20
H20+/D20
Statistical model
0.19 0.21 0.59 0.00
0.22 0.00 0.49 0.30
0.26 0.07 0.33 0.33
--- 0.06 --- 0.06 --- 0.11 _ 0.21
_+ 0.03 + 0.11/- 0.00 + 0.03/- 0.14 -+ 0.04
k~_4
1.56 + 0.78/- 0.22
5.22 + 1.3/- 0.7
b5 b6 b7
0.00 + 1.00/- 0.00 1.00 + 0.00/- 1.00 0.00 + 1.00/- 0.00
0.00 + 0.59/- 0.00 1.00 + 0.00/- 0.59 0.00 + 0.46/- 0.00
k5-7
3.0 + 0.00/- 3.0
3.0 + 0.00/- 0.54
k8
1.85 lit.
1.85 lit.
0.23 0.19 0.58
a The b values are the branching ratios, whose subscripts reference the equation numbers used in the text. b The k values are expressed in units of ( × 10 -9) cm 3 s t.
deviation 2 between the experimental data and the model was 1.5% for the H30+/I-I20 system and 1.8% for the H20+/H20 system. In the SIFT study of the H30÷/H20 system, the standard deviation was 0.8%. This indicates that the SIFT data had a signal-to-noise ratio that was about twice that in the ICR data. Confidence limits were set on the estimated model parameters using the chi-squared formalism. A function, AX2 [13], is defined as 2 A X 2 = Xv2 - Xmin
(16)
X2in is the minimum value of X2. The minimum value is obtained from the nonlinear leastsquares regression fit of the reaction model (Appendix 1 and Appendix 2) to the experimental data by adjusting parameter al~t. Any deviation from the optimum value (for the best fit) of a parameter increases the value of X2 to a new value, X. 2. (Then AX2 defines a confidence limit on the parameter.) The confidence limit of a single parameter, ant (equivalent to a _ 1~ limit) is defined by a value of AX2 equal to 1 and encloses a 68.3% confidence interval. In a multi-parameter problem, as when a statistical model is being compared to an experimental data set, the constant AX 2 that confines the 68.3% confidence interval increases from 1.00 for a single fitted parameter to 2.30, 3.53, and where
4.72 for cases with two, three, and four fitted parameters, respectively. The value of the confidence limit of each parameter, afit, was determined by using the selected value of AX2 and searching for values ofafit which have AX2 values of the limit, while continually optimizing all other parameters. The 68.3% confidence limits in the branching ratios and reaction rate coefficients are shown in Tables 1 and 2. The reaction model fits the experimental data very well and has errors between 0.8% and 1.8% for each data point. However, the model, when optimized leads to solutions for the parameters (branching ratios and rate coefficients) that have uncertainties that are larger than 3%. Further, the AX2 was insensitive to the values of some branching ratios. When this occurred it was not possible to determine the values of these branching ratios, since all possible values of b, 0-1, were included in the confidence limit [13]. A test was conducted to ascertain whether the statistical isotope scrambling model of Smith et al. [3] would fit the experimental data (ICR and SIFT) within a 68.3% confidence limit. All the branching ratios and rate coefficients were set at their predicted model values (final columns in Tables 1 and 2) to obtain a value o f x , 2 and therefore AX2. Then the calculated value of AX2 was
V.G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
9
1.0
0.8 m
_8
~
i
0.6
0.4
~0.2 0.0 0
2
4
6
8
0
Pressure, Ton" (xlO "e) Fig. 5. Ion abundances in the isotope exchange reactions, starting with D30 + in a H20 bath. From Ref. [7], a SIFT experiment. The points are the data, and the line a modeled fit. Legend: [~, D30+; O, HD20+; O, H2DO÷; A, H30 +.
compared to the value of AX2 specific to the number of variable parameters in the reaction set. If the value of AX 2 derived from the statistical scrambling model was less than the parameter specific value, then the criterion for the 68.3% confidence limit was satisfied.
4. Discussion
The predictions of the statistical isotope scrambling model of Smith et al. [3] were statistically not supported by the results of the ICR and the SIFT experiments. The branching ratios between channels 9 and 10 predicted by the statistical isotope scrambling model [3] are 0.67 and 0.33 respectively. The test to evaluate the confidence level in the fitted parameters derived from the statistical scrambling model showed that all the expected model values listed in Table 1 were outside the 68.3% interval for both the current ICR data and the re-analyzed SIFT data. The ICR and the SIFT results agree to within three standard deviations ( _+ 3o) of each other for the D30+/H20 data. The measured branching ratios of the charge transfer channels,
Reactions 1 and 2, are not consistent with a statistical distribution of protons and deuterons. We conclude that the proton transfer reactions are near statistical; therefore, they proceed through a long-lived collision complex. The charge transfer reactions are further from statistical and do not always occur through a long-lived collision complex. There appears to be different collision complexes associated with the charge transfer and proton transfer processes. In the ICR and SIFT experiments the lifetime of either complex is not sufficiently long to allow complete statistical scrambling of hydrogens and deuterons. There is a noticeable increase in the uncertainties in the derived values of the branching ratios from the primary reactions, to the secondary reactions, and then to the tertiary reactions. The branching ratios of channels 9 and 10 were determined from the analysis of the data from the H30+/D20 and D30+/H20 reactions (Table 1). The results of the SIFT and the ICR studies have a maximum error of -+ 0.04. The branching ratios of channels 11 and 12 have much larger uncertainties because of propagation of errors. The SIFT experiments were conducted at a
10
V.G. Anicich, A.D. Sen~International Journal o f Mass Spectrometry and Ion Processes 172 (1998) 1-14
higher neutral gas concentration than the ICR experiments. Therefore, the extent of reaction in the SIFT study was larger. The SIFT experiment sampled a different regime of the reaction pressure-time domain, and so a different experimental view of the branching ratios of channels 11 and 12 can be expected. The branching ratios of channels 1 to 4 were determined with errors of _+ 0.21 or less for the HzO+/D20 and DzO+/H20 reactions (Table 2). The branching ratios for channels 5, 6, and 7 could not be determined with any certainty due to large propagation of errors. In the next two paragraphs we will look at the constitution of the overall reaction rate coefficients. There are two possible channels of the reaction H 2 O+ + D 2 0 ; charge transfer with and without isotopic scrambling, and self-protonation of H 2 O ÷ with and without isotopic scrambling. In this discussion no distinction will be made between hydrogen atom abstractions and proton transfer. Since a collision complex is used in our model and direct reactions are not considered, hydrogen atom abstractions and proton transfer are included in what we have called 'self-protonation'. All possible products of the proton-deuteron exchange reactions of H30 + were observed. The literature value of the rate coefficient k8 for the self-protonation reaction with all hydrogens is 1.85 x 10 -9 c m 3 s -1 [12]. This value has a n estimated error of _+ 15%. The collision rate coefficient for this same reaction is 2.97 x 10 -9 c m 3 s -1 [14]. It is our proposition that the difference between the collision rate coefficient and the measured proton transfer rate coefficient is due to a 'symmetric' charge transfer reaction whose reaction rate coefficient cannot be measured directly and whose products are the same as the reactants. In the following arguments we are assuming that: 1. the literature value of the rate coefficient for the self-protonation reaction of D2 O+ + DzO, k8, is the same for the proton transfer portion of the rate coefficients for Reactions 3 and 4
(i.e., k8 = k3-4) and Reactions 6 - 7 (i.e., k8 = k6-7), and
2. the sum of the charge transfer and proton transfer reaction rate coefficients is equal to the collision rate, including the 'symmetric' charge transfer reaction channel, then the values of the rate coefficients k~ 4 and k5 7 should be equal to the difference of the collision rates (3.09 x 10-9cm3s -1 and 2.97 x 10 -9 c m 3 s -1 respectively) and the rate coefficients for 'symmetric' charge transfer. Further, if the symmetric charge transfer channels are assumed to be statistically accessed, then for Reactions 1-4, there are four charge transfer possibilities. There are three forward possibilities: Reaction 1 (counted twice) and Reaction 2 (counted once). The fourth channel is the back charge transfer reaction (the 'symmetric' reaction). Therefore, the rate coefficient for the forward charge transfer channels is kl-2 = 0.75 x (3.09 - 1.85) x 10-9 cm 3 s -1 = 0.93 x 10 -9 cm 3 s -1. Since k3_ 4 = k8, then the value of the rate coefficient kl_ 4 = kl_ 2 + k3-4 = 2.78 x 10 -9 cm 3 s -1. In Reactions 5 - 7 there are four charge transfer possibilities: two forward reactions (Reaction 5) and two back charge transfer reactionss (the symmetric reactions). Then the rate coefficient for the forward charge transfer channels is k5 = 0.5 x (2.97 - 1.85) x 10 -9 c m 3 s -1 = 0.56 x 10 -9 c m 3 s -1. Since k6_ 7 = k8, the value o f the rate coefficient k5-7 = k5 + k6_7 = 2.41 × 1 0 - 9 c m 3 s -1. A chi-squared analysis of the charge transfer rate coefficients gave AX2 values several times larger than the ___ 1o confidence interval, thus not supporting the statistical scrambling model as a viable explanation for the observed charge transfer processes. In a similar manner we make the following observations. Only proton-deuteron exchanges are the possible products of the H3 O ÷ + D 2 0 reaction. The difference between the collision rate coefficient and the measured rate coefficient for proton-deuteron exchange reactions is again the rate coefficient for the 'symmetric' reaction
V.G. A nicich, A.D. Sen~International Journal o f Mass Spectrometry and Ion Processes 172 (1998) 1-14
channel, whose products are the same as the reactants. The values of the rate coefficients k9 10, kll-12, and k13 should be equal to the difference of the collision rates (2.89 x 10 -9 cm 3 s -l, 2.82 x 10 -9 cm 3 s -l, and 2.79 x 10 -9 cm 3 s -1 respectively) and the rate coefficients for symmetric proton exchange reactions. Further, if the symmetric proton exchange channels are assumed to be statistically accessed, then for Reactions 9 and l0 there are ten proton exchange possibilities. There are nine forward possibilities: Reaction 9 (counted six times) and Reaction 10 (counted three times), and the back charge transfer reaction (the symmetric, reaction) (counted once). Therefore, the rate coefficient for the forward proton exchange channels is k9-10 = 0.90 x 2.86 x 10-9cm3s -1 = 2.57 x 10-9cm3s -1. This value is within the errors in the experimental determination of the rate coefficient k9-10 = 2.20 x 10 -9 c m 3 s - l [12]. In Reactions 11 and 12 there are again ten proton exchange possibilities. There are seven forward possibilities: Reaction 11 (counted six times) and Reaction 12 (counted once), and the back charge transfer reaction (the 'symmetric' reaction) (counted three times). Therefore, the rate coefficient for the forward proton exchange channels is kll_12 = 0.70 × 2.82 × 1 0 - 9 c m a s - l = 1.97 x 10 -9 c m 3 s - l . With four deuterons involved in the reaction, channel 13 has four possible forward channels and six backward reaction channels. Thus, the proton exchange rate coefficient is kl3 = 0.40 x 2.79 x 10 -9 c m 3 s -1. A chisquared analysis of the proton exchange rate coefficients gave AX 2 values several times larger than the _+ 1a confidence interval, thus excluding the statistical scrambling model as a viable explanation.
of ion-molecule reactions. This analysis can be utilized to access the validity of possible models used to explain the observed kinetics. The delta chi-square analysis assigns confidence limits to the solutions of a test model with respect to a nonlinear least-squares regression best fit of the experimental data. The predictions of the statistical isotope scrambling model of Smith et al. [3] were not statistically supported by the results of ICR and SIFT experiments for charge transfer and proton transfer reactions. Proton transfer reactions appear to be near statistical and proceed through a complex that is relatively long-lived, yet not sufficiently long-lived to allow for complete statistical scrambling of hydrogens and deuterons. In charge transfer reactions, the complex appears to have an even shorter lifetime.
Acknowledgements The work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. We thank Prof. Murray J. McEwan for his helpful discussions. In addition we thank Prof. Robert Jenerch of UCLA for his advise on the statistical analysis performed in this work. Finally we thank reviewer number 2 for persistent encouragement to analyze our data correctly.
A P P E N D I X 1: P R O B L E M
klo
5. Conclusions The primary and salient feature of this work is the application of rigorous statistical analysis to the experimental results obtained for the kinetics
I1
d[A] _
dt
(k9 + klo)[A][B]
K G. Anicich, A.D. Sen~International Journal of Mass Spectrometry and Ion Processes 172 (1998) 1-14
12
d[C] = k 9 [ A ] [ B ] - (kl~ + k12)[C][B] at
d[E] -= kl 0 [A ] [B] + all [ C] [B] - k 13 [E] [B]
dt
d[G] dt = k12 [C] [B] + k 13[E] [B]
SOL UTION [A] = [A]o exp[ - (k 9 + klo)[B]t]
[C] = [.4]0 k9 + klo ---(-kl 1 + kl2) [exp[
[A]o[ [E]=
- (kll + kl2)[B]t] - exp[ - (k9 + klo)[B]t]]
k9 +~o_k13] [exp[-k13[B]t]-exp[-(k9 + klo)[B]t]]
+[A]o[k9+kaok(kll+kl2)]
k~l +k~2 [kll _kl3]
[exp[-kl3[B]t]-exp[-(kll+kl2)[B]t]]
- [ k9 + ~--ll~_k,3][exp[-kl3[B]t]-exp[ ,(k9 + klo)[B]t]] [ 6 q = 1 - [A] - [ C ] - [ E ]
APPENDIX 2." PROBLEM
V.G. Anicich, A.D. Sen~International Journal o f Mass Spectrometry and Ion Processes 172 (1998) 1- 14
d[X] i dt
(k~ +k2 +k3 + k 4 ) [ x ] [ 8 ]
d[dY] = k I IX] [B] - (k 5 -4-k 6 +
d[Z] = k2 IX] [8] dt
--
d[C] at
=
k7)[Y][B ]
+ k5 [ r] [8] - k8 [z] [8]
k3 IX] [B] - (kll + k 12)[c] [8]
--
d[E] dt = k 4 [ x ] [ 8 ]
+ k6[r'][8] + kl ~[ c ] [ 8 ] - k13[E][8]
d[G] dt = k7[r][8]
+ k8 [ z ] [ 8 ] + k~2 [ c ] [ 8 ] + k~3 [E][8]
SOL UTION [X] = [X]o exp[ -
(kl +k2 + k3 +k4)[B]t
[Y] = [X]o kt +k2 +k3 +k4 -(ks +k6 +kT) [exp[-(ks +k6 +kT)[B]t] -exp[ -(kl +k2 -t- k3 +k4)[B]t]]
[X]o
kl +kz +k3 + k 4 - k 8 [exp[-k8[B]t]-exp[-(kl +k2 +k3 +k4)[B]t]] k5
[ks+k6-~k7-k8] [exp[-k8 [B]t] [Z]
=
- exp[ - (ks + k6 + k7)[B]t]]
k1
+[X]°[kl+k2+k3+k4-(ks+k6+k7)]
_ [kl+k2+_~3+k4_k8][exp[_ks[B]t ] k S -
exp[ - (kl + k2 + k3 + k4)[B]t]]
13
14
14G. Anicich, A.D. Sen~International Journal o f Mass Spectrometry and 1on Processes 172 (1998) 1-14
I
k3
]
[C] = [X]o kl + k2 + k3 + k4 - (kll + k12) [exp[
- (kll + kl2)[B]t] -
exp[
- (kl + k2 + k3 + k4)[B]t]]
k4
[kl + k2+ k-7+k4_k13-] [exp[- kl3[B]t] -exp[-(kl+k2+k3+k4)[B]t]] k6
[ k5 + k6 + k7 _ k13] [eXp[-kl3[B]t] -
+ [X]o
kl +k2 +k3
+k4 - ( k 5 +k6 +k7)
exp[ -
(k 5 + k 6 + k7)[B]t]]
k6 - [ kl +k2 +k3 +k4_k13] [exp[-kl3[B]t]
[E] = -- e x p [ -- ( k 1 + k 2 + k 3 +
k4)[B]t]]
[ kll kll -kl3]
[
k3
]
-
kl +k2 +k3 + k 4 - ( k l l +k12)
- (kll + k12)[B]t]]
exp[ kll
- [ kl +k2 +k3 +k4_kl3] [exp[-k13[B]t] - e x p [ - (kl + k 2 + k 3 +k4)[B]t]]
[G] = 1 - [X] - [Y] - [Z] - [C] - [E]
R e f e r e n c e s
[1] J. S. Lewis, R. G. Prinn, Planets and their Atmospheres, Academic Press, New York, 1984. See page 200 for Earth as an example. [2] W. W. Duley, D. Williams, Interstellar Chemistry, Academic Press; New York, 1984. See section 8.6, Introduction to interstellar isotopes; J. Lequeux, E. Roueff, Physics Reports 200 (1991) 241. [3] D. Smith, N. G. Adams, M. J. Henchman, J. Chem. Phys. 72 (1980) 4951. [4] A. G. Harrison, P-H Lin, C. W. Tsang, Int. J. Mass Spectrom. Ion Phys. 19 (1976) 23. [5] T. B. McMahon, P. G. Miasek, J. L. Beauchamp, Int. J. Mass Spectrom. Ion Phys. 21 (1976) 63. [6] K. Giles, N. G. Adams, D. Smith, J. Phys. Chem. 96 (1992) 7645. [7] N. G. Adams, D. Smith, M. J. Henchman, Int. J. Mass Spectrom. Ion Phys. 42 (11) (1982); M. J. Henchman, D. Smith,
[8]
[9] [10] [11] [12] [13]
[14]
N. G. Adams, J. F. Paulson, Z. Herman, Int. J. Mass Spectrom. Ion Proc. 92 (15) (1989); M. J. Henchman, D. Smith, N. G. Adams, Int. J. Mass Spectrom. Ion Proc. 109 (1991) 105. N. G. Adams, D. Smith, in A. Fontijn, M. A. A. Clyne (Eds.), Reactions of Small Transient Species, Academic Press, New York, 1983, pp. 343-359. V. G. Anicich, A. D. Sen, W. T. Huntress Jr., J. Chem. Phys. 102 (1995) 3256. P. R. Kemper, M. T. Bowers, Int. J. Mass Spectrom. ion Phys. 52 (1983) 1. J. Wronka, D. P. Ridge, Rev. Sci. Instrum. 43 (1982) 49. V. G. Anicich, J. Phys. Chem. Ref. Data 22 (1993) 1469. W. H. Press, S. A. Teukolsky, W. T. Wetterling, B. P. Flannery, Numerical Recipes in FORTRAN, The Art of Scientific Computing, 2nd ed., Cambridge University Press, New York, 1992. See under 'Confidence Limits on Estimated Model Parameters, Constant Chi-Squared as confidence Limits', pp. 688690. T. Su, W. J. Chesnavich, J. Chem. Phys. 76 (1982) 5183.