Unimolecular Dissociation of Ions. Effect on Mass ... - ACS Publications

Effect on Mass-Spectrometric Measurements of. Ion-Molecule ... high-pressure mass spectrometer or a flowing afterglow apparatus. This dissociation can...
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J. Phys. Chem. 1981, 85,327-335

327

Unimolecular Dissociation of Ions. Effect on Mass-Spectrometric Measurements of Ion-Molecule Association Equilibria J. Sunner and P. Kebarle' Chemistry Department, University of Alberta, Edmonton, Canada T8G 2G2 (Received: August 4, 1980)

Ion clusters like K+(H20), can dissociate unimolecularly in the vacuum of the mass analysis system of a high-pressuremass spectrometer or a flowing afterglow apparatus. This dissociation can affect the measurement of the equilibrium constants Kn-l,n. A theoretical analysis of the kinetics for the unimolecular decomposition is presented. The binding energies of the clusters K+(H20), for n = 1-6 are calculated with an improved electrostatic technique. The technique gives binding energies in good agreement with those obtained by ab initio calculations and experiment. The vibrational frequencies of the clusters are then obtained from electrostatically calculated potential energy surfaces for the internal motions of the ligands. The calculated Al-Zan-l,n and Asan+, are found in agreement with those obtained by other methods and experiment. The rate constants for the unimolecular decomposition of the clusters are obtained from RRKM calculations using the above frequencies. The rate constants are then used to calculate the extent of unimolecular dissociation of clusters in the vacuum region of the mass spectrometer. It is found that unimolecular decomposition occurs for clusters Measurement with n > 4. It has the effect of increasing somewhat the experimental -AHan-l,nand of the metastables for dissociationK+(HzO), K+(HzO)n-Iare in fair agreement with theoretical predictions. Experimental conditions which eliminate or at least minimize the problem are described. Equilibrium constants Kn-l,nmust be measured at the lowest possible temperatures. Their values should be no lower than 10 torr-'. Similar analysis for H+(H20)zand (C6H&+shows that no unimolecular decomposition occurs for these systems over the entire experimental range. Diagnostic tests for the occurrence or nonoccurrence of unimolecular dissociation are described.

-

Introduction The measurement of gas-phase ion-molecule reaction equilibria has generated a vast quantity of thermochemical on organic and inorganic ions which are of great value not only to fields dealing with gaseous ions but also to the much wider area of ions in condensed media. For the purposes of the present discussion, the measured ion-molecule equilibria can be divided into two classes: exchange or transfer equlibria (eq 1-3) and association equilibria (eq 4-7). Reaction 2 gives an example of the A+ + B = C+ + D (1) (2) H30+ + CH30H = H 2 0 CH30Hz+

protonated acetic acid, from the association of the acetyl cation with water (eq 7). From an instrumental standpoint there are important differences between exchange-transfer reactions and association reactions. Since a single entity is formed in association reactions, this class of reactions is dependent on the presence of third bodies M which act as deactivators in the forward direction, which is exothermic, and activators in the reverse direction, which is endothermic. Reaction 4 is therefore better represented by eq 8. The

A+ + B =AB+ kf

+

H602+ + CH30H = (CH30H)H+(Hz0)+ HzO (3) f

A+ + B F'c AB+

(4)

Na+ + OHz = NaOHz+

(5)

NHd+(NH3),-1

+ NH3 = NHd+(NH,),

A+ + B

(6)

CH,CO+ + HzO = CH,C(OH)Z+ (7) extremely important proton-transfer eq~ilibria,l-~ which provide quantitative information of Bronsted basicities and acidities, while reaction 3 gives an example of an exchange equilibrium in which a water molecule is replaced by a methanol molecule. Reactions 5-7 give examples of association equilibria, addition of solvent m o l e ~ u l e sto l~~~ a given ion (eq 5-6) and formation of a new m o l e ~ u l e , ~ (1) P. Kebarle, Annu. Reu. Phys. Chem., 28, 445 (1977). (2) R. W. Taft and E. M. Arnett in "Proton Transfer Reactions", E. F. Caldin and V. Gold, Eds., Chapman and Hall, London, 1975. (3) J. F. Wolf, R. H. Staley, I. Koppel, M. Taagepera, R. T. McIver, J. L. Beauchamp, and R. W. Taft, J . Am. Chem. Soc., 99, 5419 (1977). (4) S. K. Searles and P. Kebarle, Can. J. Chem., 47, 2620 (1969). ( 5 ) I. Dzidic and P. Kebarle, J. Phys. Chem., 74, 1466 (1970). (6) W. R. Davidson and P. Kebarle, J. Am. Chem. SOC.,98,6125,6133 (1976). nnnn n e e

4/81/2085-0327$01.00/0

kc

(8)

kr

(ABf)*

kAM1

AB4

rate of reaction 8 in the forward direction becomes third order and in the reverse direction becomes second order at low pressures (generally below -104.1 torr). For this reason association equilibria become extremely slow at lo+ torr, the pressure range used in equilibria measurements with the ICR te~hnique.~ Therefore, association equilibria cannot be measured by ICR, yet ICR is the only technique in which ion concentrations are determined in situ. Association equilibria have been measured only at higher pressures (0.1-10 torr) by several groups of workers8'12with techniques (high-pressure mass spectrometry,8-11flowing (7) W. R. Davidson, S. Meza-Hojer, and P. Kebarle, Can. J. Chem., 57, 3205 (1979). (8) A. J. Cunningham, J. D. Payzant, and P. Kebarle, J.Am. Chem. Soc., 94, 7627 (1972). (9) I. N. Tang and A. W. Castleman, J. Chem. Phys., 67, 3636 (1972). (10) D. P. Beggs and F. H. Field, J.Am. Chem. Soc., 93,1567 (1971); J. J. Solomon, M. Meot-Ner, and F. H. Field, J. am. Chem. SOC.,96,3727 (1974). (11) M. R. Arshadi and J. H. Futrell, J. Phys. Chem., 78, 1482 (1974). (12) F. C. Fehsenfeld and E. E. Ferguson, J. Phys. Chem., 61, 3181 (1974).

0 1981 American Chemical Society

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The Journal of Physical Chemistry, Vol. 85, No. 4, 1981

afterglow12)in which the ions are not detected in situ, but where a small fraction of the gas mixture is bled through a leak into an evacuated region. Mass analysis and detection is then executed under vacuum by essentially conventionalmass-spectrometricmethods. It was realized from the very beginning that this method of sampling can lead to err0r5.l~ Three problems were recogni~ed'~ as potentially serious: (a) cooling of the sampling gas due to adiabatic expansion in vacuum, (b) collisional stripping of the ions in the region of high gas density immediately outside the leak where electric fields are present, and (c) unimolecular dissociation of the ions by the reaction AB+ = A+ + B, in the vacuum region outside the reaction chamber (ion source). Problems a and b were tackled by narrowing the leak to dimensions where molecular flow prevails14and reducing the magnitude of the electric fields immediately outside the leak.8 Problem c, unimolecular dissociation, was assumed not to be serious on the basis of the following consideration. The ions AB+ on entering the vacuum are assumed to have a thermal energy distribution corresponding to the temperature of the ion source. Only a fraction of the ions equal to f,,will have energy higher than the threshold energy Eo required for the dissociation: AB+ A+ B (Eox -AH4). Since the thermal energy distribution is not maintained in vacuum, only the fraction f,,will be able to dissociate in vacuum. Probably f,,is small. Actually the situation is more complicated, as is shown below. The fraction f,,will be the smaller the lower the temperature at which the equilibrium is measured. In practice there is a limiting temperature below which the equilibria cannot be measured. Since ion diffusion to the wall of the reaction chamber is a competitive reaction, the forward and reverse rates of reaction 4 should be kept faster than the diffusion rates. The presence of a major gas M reduces the diffusion of the ions to the wall; i.e., M acts not only as a third body in reaction 8 but also as an ion-trapping gas. It was found that the pressure of M cannot be raised above 10 torr without adverse instrumental effects. Since the ion diffusion half-lives at this pressure are in the 100-ps range, one needs to have halflives for reaction 4 which are shorter than 100 ps. In order to keep f,,as low as possible, Le., keep the temperature as low as possible, one should not let the equilibrium half-lives become very much faster than the diffusion half-lives. This is achieved by keeping the pressure of B low (0.001-0.1 torr). With a low pressure of B, the forward rate is relatively slow, and at equilibrium the reverse rate must be slow also. Since the reverse rate does not depend on B, to slow it down one must select a low temperature, which is what is desired. If the neat gas B at some few torr pressure had been used instead, the forward rate would have been 100-1000 times faster, and, if one is to observe a reasonable ratio of [AB+]/[B+](Le., near unity), one has to raise the temperature. Conditions of high pressure of M and low pressure of B were used whenever possible in measurements undertaken in our laboratory in the last 10 yr or s0.l However, the potential seriousness of the problem with unimolecular dissociation was underestimated, and in some experiments the neat gas B was used at ca. 0.3-3-torr pressure. This was the case in the measurements involving the alkali positive ions.P6 The detection sensitivities of these ions, which are produced by thermionic emission, are somewhat low and deteriorate further in the presence of a large ratio -+

+

(13) A. M. Hogg and P. Kebarle, J. Chem. Phys., 43, 449 (1965). (14) A. M. Hogg, R. N. Haynes, and P. Kebarle, J. Am. Chem. SOC., 88, 28 (1966).

Sunner and Kebarle

of third gas M to solvent gas B. Recently experiments involving the equilibria of the potassium ion with dimethylacetamide (DMA), represented in eq 9, were found to give anomalous results. The K+(DMA),-l + DMA = K+(DMA), (9) equilibrium constants K9 were not independent of the pressure of DMA, and the van't Hoff plots of K9showed curvature. These deviations became noticeable for n = 2 and increased with n. The fact that the problem was observed with DMA, a relatively large molecule with many internal degrees of freedom, and increased with n indicated that unimolecular dissociation of the clusters with high n may be occurring in the vacuum of the instrument. One can expect that clusters having a large number of internal degrees of freedom will be more likely to dissociate in the mass analysis system since such clusters will have relatively large internal energies. Therefore we decided to reexamine the whole problem of unimolecular decomposition. The present paper deals with the systems H+(H20)2,(C&6)2+, and K+(H20),for which the earlier experimental data did not give indication of problems. The K+(DMA), results will be discussed in a separate paper.15 It is important to note that no problem with unimolecular decomposition is to be expected for exchangetransfer reactions (eq 1)measured with high-pressure mass spectrometers. For this type of reaction, the temperature is increased to the point where the endothermic direction of reaction 1is sufficiently fast, but this temperature, in general, is much lower than the temperature required for A+ (or C+) to dissociate into two fragments. The calculations required for the unimolecular dissociation analysis provide some results of interest in their own right. Thus the binding energies, heat capacities, and entropies of the clusters are obtained as well as data on the lifetimes of the thermal clusters at different temperatures.

Experimental Section A 60°, 6411. radius magnetic sector mass spectrometer was used for the metastable measurements of K+(H20),. The accelerating potential was 1800 V. Except for the ion source reaction chamber, which was a somewhat improved version over that used by Davidson? the instrument and experimental conditions were the same as those used in the earlier work.4 The ion flight times were estimated from the potential distribution in the ion acceleration system and the dimensions of the instrument. The following relationships tl = 1.17m1/2, were obtained t , = 0.265m1/2,t, = 0.75m1/', t b = 1 . 4 n ~ l / Ions ~ . K+(H20),dissociating from 0 to t, are detected as n - 1; t , is the time when the ion enters the field free region before the magnetic sector; tl is the time that the ion leaves the field free region; t b is the time when the ion enters the field free region past the magnetic sector. On substituting the mass of the ion in atomic units, one obtains the time (microseconds)from the above equations. Results and Discussion General. Central to the present work are calculations of the normalized thermal internal energy distribution P(E) of AB+,at different temperatures. The temperatures are so selected as to cover the range of the experimental van't Hoff plots. To determine P(E) one needs the frequencies for the normal vibrations and the reduced moments of inertia for the internal rotations. For K+(H20), (15) J. Sunner and P. Kebarle, to be submitted for publication in J. Phys. Chem.

Unimolecuiar Dissociation of Ions

The Journal of Physical Chemistry, Vol. 85, No. 4, 1981 329

TABLE I : Comparison of Stabilization Energies (AE), Enthalpies ( A H ) , and Entropies ( A S ) for the Reaction K+(H,O),-, + H,O = K+(H,O)na -AEo, n

-AHon-i, n

RK-o e

1

elsb 16.6

HFC 17.5

PPd 17.5

elsb 2.62

PPd 2.67

2 3 4 5 6

31.7 44.9 56.2 64.4 71.3

33.5 47.5

34.1 48.9 62.0 72.6 82.4

2.65 2.68 2.71 2.76 2.81

2.69 2.72 2.74 2.69 2.84

elsf 17.5

Monteg exptlh exptlc' 17.5 17.9 17.9 16.9 16.9 14.4 16.2 16.1 16.0 13.5 14.5 13.2 12.9 11.0 12.6 11.8 11.1 8.1 11.2 10.7 9.6 10.0 8.4 6.9 10.5

-ASon-l, n

els Monteg exptlh exptlci temph 18.6 19.0 21.6 21.6 550 19.9 19.9 20.2 20.9 24.2 24.1 460 20.2 21.4 23.0 22.1 390 23.0 20.9 24.7 22.2 330 22.6 20.4 25.2 21.8 300 23.7 21.3 25.7 19.2 270

All energies in kcal mol-' ;entropy in cal deg-' mol-' ;standard state, 1 atm. Electrostatic calculations, present work. Hartree-Fock calculations by Clementi et a1.I9 Clementi et al. pair potential function^.'^ e Distance in angstroms between potassium and oxygen nucleus. The PP distances are for the symmetric clusters. f Based on moments of inertia and vibrational frequencies predicted by electrostatic calculations and evaluation of heat capacities and entropies with use of statistical mechanics. g Monte Carlo calculations by Mruzik e t aLZobased on energies provided by potential functions of Experimental results of Searles and Kebarle;4 lower number given for R = 1 from Davidson Clementi; temperature, 298 K. and KebarleS6 Temperature given in table corresponds to average temperature at which the equilibria measurements were done. I Experimental results as above but approximately corrected for unimolecular dissociation, with use of eq 15.

and K+(DMA), no suitable data were available. The required data were obtained from electrostatic calculations. These are described in Appendix I. The assumption was made that the internal motions of the ligand molecules remain the same as in the isolated ligands. Additional frequencies for the clusters were then obtained by considering the various motions of the ligand molecules in the field of the ion and the field resulting from ligand-ligand interactions. Electrostatic calculations for the alkali positive and halide negative ions and single solvent molecules have been reported before.6 These calculations predicted binding energies in fairly good agreement with the experimental results. For the present purpose the calculations were extended to handle clusters containing more than one ligand (see Appendix I). Once the normalized thermal distribution functions P(E) are available, it is easy to evaluate (numerically) the fraction of AB+ molecules f,, which is excited above the threshold energy Eo (see eq 10 and 11). Obviously iff,, EO = m 4 r (10)

is very much smaller than unity, e.g., f,, < 0.01, then the equilibrium measurements will not be affected by unimolecular decomposition during mass analysis. However, iff,, is close to unity, there might be a serious problem provided that the rate constants for unimolecular dissociation k,,(E) = k(E) are of such magnitude that dissociation occurs within the flight time of the ions to the ion detector, i.e., within 0-20 ps. The rate constants k(E) were obtained from RRKM calculations.16 Details about the RRKM calculations are given in Appendix 11. The availability of k ( E ) and P(E) permits the numerical evaluation of It, the time dependence of the AB+ ion beam intensity due to unimolecular dissociation. This is given in eq 12. Applying eq 12 to two different times tl and t2,

I ( t ) = j-P(E)e-k(E)t 0 dE

(12)

one is able to evaluate the fraction that dissociates in the time interval tz - tl = At (see eq 13). Of course, since k(E) f A t = Wl) - m 2 ) (13) = 0 for E < Eo, the numerical summation equivalent to (16)W. Forst, "Theoryof UnimolecularCalculations"Academic Press, New York, 1973.

eq 12 has Eo as lower limit. The numerical results providing I ( t )show how many ions decompose in what region of the instrument, including also the ions that will be observed as metastables in a single focusing magnetic sector instrument. It is well known that RRKM calculations often do not have quantitative predictive value. On the other hand Brauman"J* has used recently with good success such calculations for an analysis of the rates of third-order ion-molecule association reactions. In the present work there is also the additional uncertainty introduced by the electrostatic calculations of the required frequencies. Obviously the results can be expected to be only of orientational value. However, maybe surprisingly, results of considerable usefulness are obtained, as the reader will see. Results for K+(HzO),. The frequencies, moments of inertia, etc., obtained for the K+(HzO),and the transition states for the dissociation of K3(H20),to K'(H201n-1 + HzO are given in Appendixes I and 11. From these data one can obtain AEosl, and AS,-l,n predicted by the electrostatic calculations (els). hEo,, values are the stabilization energies for the reaction K+ + nH20 = K+(H20),. The electrostatic stabilization energies are shown in Table I together with theoretical results obtained by Clementi and co-workers.le Clementi obtained, with large basis set Hartree-Fock (HF), stabilization energies Ai?,,, for n = 1-3. In addition to these energies Clementi et al.19obtained AE0,, from pair potential functions (PP) for ion-water molecule and water-water molecule.1g These do of course neglect multibody effects. Comparing the AEor obtained with electrostatic calculations, HF, and PP, one finds fair agreement. The electrostatic-calculation results predict consistently somewhat lower (absolute) values. From A E O , , one can calculate U n - 1 , n = A E o n hEo,,-l (not given in table). The availability of the electrostatic-calculation vibrational frequencies and the electrostatic-calculation data on the moments of inertia allows one to calculate predicted by electrostatic calculations. The change from hE,-l,, to AHn+ includes consideration of the zero-point energies and the thermal energies due to vibration and rotation. AEn-lp and M n - i , n are generally quite close because of cancellations. For example, hE3,4(els)= -11.3 kcal/mol while AH3,Jels) = (17)W. N. Olmstead, M. L. On, D. M. Golden, and J. I. Brauman, J. Am. Chem. SOC., 99, 992 (1977). (18)J. M. Jasinski, R. N. Rosenfeld, D. M. Golden,and J. I. Brauman, J. Am. Chem. SOC.,101, 2259 (1979). (19)H. Kistenmacher, H.Popkie, and E. Clementi,J. Chem. Phys., 61, 799 (1974).

330

Sunner and Kebarle

The Journal of Physical Chemistry, Vol. 85,No. 4, 1981 8

10

O . 1 0.4

I

f l

'o-:oo

300

400

To K Figure 1. RRKM calculated results for unimolecular decay of K+(H20)" clusters taken with thermal energies distribution at temperature T

(shown on ordinate) and put under vacuum as isolated molecules. Abscissa gives fractions where (B) f,, is fraction of ion cluster n which dissociates from time 0 to m , (0)f, is fraction which dissociates from time 0 to the time required to pass the magnetic mass analysis region, and (0)fa is the fraction of ions n that dissociates from 0 to 2 s. This fraction is detected as n - 1 ions. Temperature range covered for each n corresponds to temperature at which experimental measurements are generally made. Values for equilibrium constants given in figure correspond to Kn-l,n experimental values (Searles and Kebarle4) observed at given temperature.

-11.0 kcal/mol. The AHon-l,n(els)can be compared with results by Mruzik et aL20obtained on the basis of the pair potential functions of Clementilgand a Monte Carlo calculation of a chain of configurations from which ensemble averages are obtained. These results are shown in Table I in the column labeled Monte. It may be noted that AHn-l,n electrostatic-calculation, Monte Carlo, and experimental (exptl) data, are in fair agreement although the experimental values (Searles and Kebarle4)are generally numerically somewhat larger for large n. Third-law entropies for the clusters K+(H20),can be readily obtained from electrostatic-calculationfrequencies, etc., and standard statistical-mechanics expressions. The ASon-l,n values obtained in this manner are shown as ASon-l,n, els. These data are in substantial agreement with Mruzik's Monte Carlo entropies (obtained via the free energy change).20 The experimental ASon-l,n values are fairly close too, but the experimental -ASon-l,n values are somewhat higher, particularly for n = 5 and 6. As will be seen later, there is reason to believe that the experimental -AS04,5 and values are too high. The general agreement between the electrostatic-calculation, Monte Carlo, and experimental ASon-l,n values lends support for the electrostatic calculation predicted frequencies and rotational parameters. However, one must point out that (20) M. R. Mruzik, F.F.Abraham, D. E.Schreiber, and G. M. Pound, J. Chem. Phys., 64,481 (1976).

I

-.

The Journal of Physical Chemistry, Vol. 85, No, 4, 198 1 33 1

Unimolecular Dissociation of Ions

lead to Kn+" = 103-10, i.e., equilibrium constants in the range where no significant dissociation occurs for n up to 5. Working with n / ( n - 1) >> 1 would seem to be even more favorable; however, this condition may not be advisable if collision-induced dissociation21is suspected to occur. (c) Increase of the size of the cluster leads to an increase of the difference between f,,, f,, and f b . Thus for n = 2, fa a f , a fb; however, for n = 6, f,, >> f, > f b This behavior is understood easily. The small clusters have relatively few internal degrees of freedom and dissociate rapidly (on the microsecond scale) when E > Eo. It is also observed in Figure 1 that for a cluster with a given n the difference between f,,, fa, and f b increases as the temperature is decreased. This behavior reflects the different energy distribution in the cluster for different temperatures. At low temperatures the energy distribution above Eofalls off rapidly with E; therefore, it takes a longer average time for this energy-poor population of excited clusters to decompose. The time dependence of two potassium hydrates, evaluated with eq 12, is given in Figure 2 as an illustration of two cases in which severe dissociation occurs (Kn-l = 0.1 torr-l and Kn+ = 0.01 torr). The figure shows that, for these larger n clusters and low values of Kn-l,n,most of the dissociation occurs within the first 2-3 ps after exposure to vacuum. Very little dissociation occurs in the metastable region, which involves a time interval of -5 ps occurring some -13 ps (depending on mass of ions) after exposure to vacuum. Thus little experimental indication of trouble is available from the observation of metastables. A more detailed discussion of the metastables is given later on in the text. Considering that the predicted decomposition (see Figures 1 and 2) can be so severe for the higher n, the following question immediately arises: why is there no indication in the experimental results of something being wrong? As will be shown below, a cancellation occurs which reduces the error. The relationship between the true equilibrium constant Pb= and the observed equilibrium constant Know for the situation where the ion n dissociates such that a fraction f," is detected as ion n - 1 and a fraction 1 - fbn is detected as ions n is given by eq 14. The treatment used

Knobsd/Kntr= (1 - fbn)/(l + Kn,rfanP) (14) to obtain eq 14 neglects the fact that there might be also n + 1 ions which dissociate within 2 ps and are detected as ions n. An expression taking this into account as well as the dissociation of n - 1 to n - 2 is shown in eq 15.

Shown in Figure 3 are van't Hoff plots in which eq 14 and 15 were used to calculate K n o w The fractions fb" and fan were taken from Figure 1. The calculations with eq 14 and 15 are for a water pressure p = 1 torr. For K",, the experimental (Searless and Kebarle4) Kn+, values were taken. As is evident in Figure 3, the Pow values obtained by eq 14 show larger curvature than the Knobad from the more accurate eq 15. Actually, K40w(eq15) and Ksow(eq 15) show so little curvature that they could be mistaken for straight lines. The curvature will be even less noticeable when a narrow temperature range is covered. For example, the range of temperature covered in the experto imental determination4 was only from 3.4 x (21) K. Hiraoka and P. Kebarle, J. Am. Chem. Soc., 97,4179 (1975).

Flgure 3. Effect of unlmdecular dissociation of KC(H20), to K+(H20),-, In vacuum on van't Hoff plots: (-) K,,-,,, = K", = assumed true equilibrium constants; (-- -) observed equilibrium Constants flObsd as as predicted by eq 14; (-. -) observed equilibrium constants K", predicted by the more accurate eq 1 4 which takes into account dissociations n 1 n, n n - 1, and n - 1 n - 2. Calculations are for p(H,O) = 1 torr.

+

- -

-

4X K-l, which represents only one-third of the range shown in Figure 3. The Knobsd(eq 15) values (see Figure 3) are generally steeper than Knt,, This means that the experimental error will be in the direction of larger -AH,+, and values. The values designated exptlc in Table I were obtained by calculating from eq 15 (p = 1 torr) what change in slope would be expected if the experimental4 K,-,, values were affected by unimolecular decomposition. p = 1 torr represents an average pressure for the K4,6 and K5,6determination^.^ These and AS,+, exptlc values are probably better than the exptl results. It is interesting to note that the exptlc results for n = 5 and 6 are in better agreement with the calculated values. This is particularly the case for the AS results from the Monte Carlo calculation (see Table I). Equation 15 predicts Powvalues which are dependent on the pressure p . The experimental determinations4for the K+(H20),equilibria were taken over a limited water pressure range so that eq 15 cannot be tested; however, the new experiments on the K+(DMA), equilibria gave results16 supporting eq 15. Metastables in K+(H20),.Unimolecular dissociation of K+(H20),with mass m, in the vaccum of the mass analysis system should give rise to metastable ions, i.e., ions which dissociate in the field free region after acceleration and before magnetic separation in the sector field mass spectrometer. These ions appear at nominal mass given by eq 16. (16)

The fraction of ions n that dissociate in the metastable region f," can be calculated by substituting in eq 13 the times tenand tIncorresponding to times of entrance and leaving of the metastable region (eq 17). The evaluation

me")

fmn = - I(t1") (17) of ten and tln is discussed in the Experimental Section. Unfortunately f," cannot be determined directly by experiment. What can be observed is the ratio of the metastable ions to the ions detected as ions n. The ions detected as n are ions n which do not dissociate before having passed the magnetic sector, Le., before time tb, and ions n 1 which dissociate to n within t , = 2 ps. If one calls

+

332

The Journal of Physical Chemistry, Vol. 85, No. 4, 7981

Rn

10-2

I

Sunner and Kebarle

20

t

'

'

.4

IO

r

.

1

/

300

200

,.,a00

Figure 4. Comparison of R predicted from RRKM calculations (eq 18) (open symbols) and R measured experimentally (solid symbols). R is the ratio of metastable ions (K+(H20), K+(H20),-,) over ions detected as K+(HzO),. The different shapes of R,,, and R,, are believed to be due to the contribution of collision-induced decompositlon K+(H,O), M = K+(H20),-, M in the metastable region of the mass spectrometer. This collision-induced decomposition dominates #leXptl in the region where 6''RRKM is small, Le., at low temperatures. Ion source pressure, 1 torr of H20. Ion flight tube pressure, -3 X lo-' torr. Abscissa gives ion source temperature. -+

+

+

R" the ratio of metastables n over the ions detected as n, it is easy to show that eq 18 holds. The first term in the r n

denominator gives the fraction of ions n + 1 (per ion n) that dissociate within 2 k s and are detected as n, while the second term gives the fraction of ions n that do not dissociate. Shown in Figure 4 is a comparison of Rn predicted by the RRKM calculations and eq 18 (1torr of HzO) and Rnexptlwhich was obtained from measurement. The Rnexpd was obtained with the assumption that the detection sensitivity for the metastable ions is the same as that for the ions n. The pressure was 1 torr of H20. The figure shows that Rnexptlis of the same order of magnitude as RnRRKMfor the temperatures where R" is large. However, the shapes of the two curves are different, and Rnexptlbecomes larger in the region of small R". An approximate analysis given below indicates that this discrepancy is due to metastables induced by collisions of the ions n with background gas present in the mass analysis region. The Rnexptl was measured for n = 3 and 4 (Pionsource = 1torr of HzO, T = 310 K) in separate experiments in which air was admitted into the mass analysis region. The results are shown in Figure 5. Rnexptlis observed to increase linearly with pressure. This increase must be due to collision-induced dissociation (CID).The results in Figure 5 obey eq 19, which expresses Rexptlas the sum of unimolecular deRnerpd =

Rnuni+ RncD = Rnuni + PA"

(19)

composition R d which is independent of pressure and RcD which is proportional to pressure. The proportionality RcD = PA" is expected under the thin target conditions prevailing in the metastable region. The intercept with the ordinate at p = 0 in Figure 5 gives the values for Rnmi, while the slope gives A". One finds the following: Rluni = 1.6 X RsUni= 0, A4 = 3.4 X lo3 torr-', A3 = 1.4 X lo3torr-'. The results for Rd are in qualitative agreement with the values for Runi predicted by RRKM given in and Figure 4, which are R 4 = 3 x~ ~ R ~ 3e ~ 1 x~ lo4 (1torr of H20, T = 310 K). The pressure in the flight

Figure 5. Plot of Re, (see Figure 4) as a functlon of ion flight tube pressure (air). Ion source pressure, 1 torr of H20; temperature, 210 K. Pressure dependence of Re, shows that collision-induced decomposition produces metastables. Extrapolation of R4expd and R3,* to p = 0 leads to Intercepts whlch correspond to metastable ratlos due to unimolecular decomposition. These are found In approximate agreement with RRKM calculated ratios.

tube for the experiments in Figure 4 was p c 3 X lo-' torr. Using the experimentalvalues for A3 and A4 quoted above one obtains R3crr,= 0.4 X and R4cD = 1X for the experiments in Figure 4. At 310 K Rluniis large but Rsmi is small. Therefore R 4 dominates ~ over R4CIDand R4,, = R 4 On~ the other hand R 3 c >> ~ R3,i so that R3exptl R3CID. The observation in Figure 4 that Rnexptlis much larger than Rnmiin the region of low temperatures where R,i is small is thus explained by the dominant contribution of RcrDin that range. The values A4 = 3.4 X lo3torr-' and A3 = 1.4 X lo3torr1 can be converted into CID cross sections u by means of eq 20, where L is the length of the metastable region. A" = anL (20) Using L = 30 cm one obtains a4 = 34 A2 and u3 = 15 A2. These cross sections are of the same order of magnitude as the expected hard-sphere collision cross sections for the corresponding neutral species. The observed difference between A4 and A3 is larger than what could be expected on basis of the radii of the two clusters. Therefore it seems that the dissociation energy A",,, and the internal energy of the clusters are also influencing the value of the cross section. The observation of metastables could be a valuable aid to the experimentalist engaged in the measurement of clustering equilibria. Unfortunately, the above discussion shows that this test can be applied only if rather low ion tube pressures are available. For example, the 3 X lo-' torr pressure prevailing in our instrument (l-torr ion source pressure) can be considered low for high-pressure mass spectrometers, yet even at this pressure CID produced metastables are observed in abundance and only an analysis, such as that shown in Figure 5, permits one to separate the CID from the unimolecular metastables. operating at higher ion tube ~Clearly~ for instruments ~ pressures, such an analysis will be impossible. It is

The Journal of Physical Chemistty, Vol. 85, No. 4, 1981 333

Unimolecular Dissociation of Ions

TABLE 11: Data for H,O+.H,O and (C6H6),+ temp,a K ~ , tom-' a fex b fex(eq 26)c 10z'ktoc

617 103 4.2 X 4.0 X 2

temp, K

301 3 x 10-3

lo*

H30+*Hz0 7 50 10' 1.9 x 10-4 1.4 x 10-4 0.7

331 1.5 x

368 5.5 x 10'2

677.7

lov6

2.9 x 10-5 2.4 x 10-5 1.2

(C6H6

fex

843 100 1.1 x 10-3 1.1 x 10-3 0.5

96 0.3 10-1 6.3 x 10-3 8~ 0.5

1115.4

414 0.17

474 0.42

5 54 0.74

2 . 9 ~105.6 X 100.3

)l.+

a Temperatures and corresponding equilibrium constants from ref 8. f e x calculated from frequencies of Brauman.' '9" f e x calculated from eq 26 and experimentally measured8 third-order rate constant k t , given in table. Units for kt, are cm6

molecules-2 s-'.

therefore desirable to have available other tests. One such test is described in the next section. Unimolecular Decomposition in H30t-Hz0 and (c&,&+.Experimental Tests for Occurrence or Nonoccurrence of Unimolecular Decomposition. Calculations for the thermal energy distribution P(E) for the systems H30+.Hz0and (C&6)zt were made by using the parameters for internal motion given by Braurnan.'* f e x , the calculated fraction of complexes excited over the threshold energy (see eq ll),is given in Table 11. The experimental measurements for the equilibrium constant Kzl for reaction 21 were mades with Kzl ranging H30+ + HzO = Ht(HZO), (21) from 100 to 10 (t0rr-l). Examination of Table I1 shows that f,, even for Kzl = 1 is only equal to 1.1 X This means that no significant error due to unimolecular dissociation was present in the Kzl measurements described in ref 8. Unfortunately the same cannot be said for the earlier measurements22which were made with neat water. By analogy with the results for Kt(HzO),, one may suspect that the datazzfor Ht(H20), will be affected in the region of n = 6-8. We suspect that the true and -ASo,..l,n in that range are somewhat smaller than the experimental results of ref 22. Results for f e x based on frequencies provided by Brauman17p18 were obtained also for (C6H6)2 The temperature range shown in Table I1 covers the range 330-475 K used in the experimental measurements of Fieldz3which were made at 0.5-torr constant pressure of benzene and no third gas. It will be noted that the upper experimental temperature falls in the region where appreciable mimolecular dissociation of the dimers will be occurring. However, most of the range of the van't Hoff plot falls in a safe region where f e x is small. The experimental AHo and ASovalues are probably affected only little by unimolecular dissociation. The RRKM calculation of k(E) for the rate constant for unimolecular decomposition of H30-HzOtand (C6&)2+ predicts that essentially all of the molecules which are excited above the threshold energy decompose within microseconds, i.e. f,, = fa = f b As mentioned earlier, this is a typical result for clusters with only a small number of internal degrees of freedom (see Figure 1, n = 1 and 2, K+(H20)n) Since the calculations off and k ( E ) require knowledge of the vibrational frequencies and internal rotations of the ion and activated complex, and these can only be estimated, it is clearly very desirable to have some experimental method available which will allow one to judge (22) P. Kebarle, S. K. Searles, A. Zolla, J. Scarborough, and M. Arshadi, J.Am. Chem. SOC.,89, 6393 (1967). (23)F. H. Field, P. Hamlet, and W. F. Libby, J. Am. Chem. SOC.,91,

2839 (1969). (24)K. Hiraoka and P. Kebarle,J. Chem. Phys., 63, 746 (1975).

TABLE 111 Qoxy e n = - 0.86 elementary charges @hydro e n =

+ 0.38 elementary charges orr(0-H) = 0.72 A 3 ~q(0-H= ) 0.72 A 3 IP(H,O) = 12.4 eV J(H,O)= 2.5 IP(K+)= 31.7 eV J ( K + )= 2.25

orion = 0.89 A 3 aion = 3.62 A -' aOxyg= 3.6 56 A-' ahydr= 3.01 a-' cion= 220000 kcal/mol-' coXyg=18700 kcal/mol-' Chydr= 696 kcal/mol-'

whether danger from unimolecular decomposition is present. Such a test is presented below. The general association reaction with third-body dependence was shown in eq 8. The Lindeman treatment of this reaction leads to eq 22-24. Under conditions where

[ABt*]/[AB+] = k,/k, Keq=

kcks/kdka

(23) (24)

the forward reaction is third order Itd

>> k s [ M l

such that

k f = kcks[Ml/kd = k,[M1 (25) where k , = k,k,/kd is the third-order rate constant for the forward direction. Substituting eq 24 and 25 into eq 23, one obtains eq 26. The numerical factor on the right side [ABt*] f e x = -[AB1 3 X 10z5k,,(molecules-2cm6 s-l) k, -=5 (26) ksKeq Keq(torr-l) of eq 26 was obtained by assuming that k, = cm3 molecules-' s-l, which is the Langevin rate constant for orbiting collisions,1s and introducing the proper conversion factors. f,, values calculated from eq 26 for reaction 21 are given in Table 11. The k , and Keqvalues required for eq 26 are experimental values taken from our earlier publication.s The agreement between f,, (eq 26) and f,, obtained from the statistical-mechanics equations is extremely close. The close agreement is obviously fortuitous; however, the result supports both eq 26 and the frequencies used by Braumans which enter the calculation of t,,. Equation 26 can be used as follows when one wishes to examine whether unimolecular decomposition is occurring. Having measured Kes at a given Teq,one substitutes Ke and k,(T,) into eq 26. If f e x C 0.05, the experiment shoul2

The Journal of Physical Chemistry, Vol. 85, No. 4, 1981

334

Sunner and Kebarle

TABLE IV: Summary of Calculated Frequencies in K+(H,O),a internal vibration of water asymmetric stretch symmetric stretch bending ion-ligand stretch ligand rock

3700 (1) 3600 (1) 1600 ( 1 ) 147 (1) 287 (1) 231 (1)

ligand bend a

Frequencies in cm".

3700 (2) 3600 (2) 1600 (2) 114 ( 1 ) 1 5 9 (1) 270 ( 2 ) 217 (2) 2 1 (2)

k,[M]

= [AB+*]/[AB1 = kc/(kdKeq)

(27)

< kd, one obtains eq 28. The approximate ex-

pression on the right is obtained with the assumption kc = k, (both probably Langevin rate constants). Equation 28 shows that, iff,, is to be small, e.g., then, for a pressure of M in the 1-10-torr range, measured Keqshould be in the 100-10 (torrV1)range. It should be noted that eq 28 is valid also for reactions in which the forward rate is second order but the unequal sign must be reversed. Evidently an even larger Kq should be selected in this case iffexis to be kept small. The problem arising from unimolecular decomposition could be solved by instrumental changes. One method eliminating ion sampling errors is the ion T jump technique which is being developed in our labor at or^.^^ However, this technique will eliminate the problems only if all ions detected as n originate from ions n in the ion source. Another approach is to use controlled adiabatic expansion in the ion exit leak to cool the clusters down before their passage through vacuum. Experiments along these lines are planned. Conclusions. Unimolecular decomposition of ion clusters n can occur in the vacuum region of the mass analysis system. The effect of the decomposition is to increase somewhat -aHon-l,nand -ASo,-l,n. The problem occurs for n > 4 and is aggravated by ligand molecules which have a large number of internal degrees of freedom. The problem can be avoided by proper choice of experimental conditions. One must measure equilibrium constants K,,+ at the lowest experimentally accessible temperature; i.e., the values of the constants should be in the range K,-',, = 103-10 (torr-'). Such values can be measured with present apparatus by using pressures of the clustering gas in the 1-100-mtorr range and a third gas in the torr range. Extension of the measurements to even lower clustering gas pressure and higher third-gas pressure, i.e., to Kn+ i= lo4torr-' is possible. Corrections for the (25) T. Magnera, J. Sunner, and P. Kebarle, to be submitted for

publication.

3700 (4) 3600 (4) 1600 (4) 1 0 8 (4)

3700 (5) 3600 (5) 1 6 0 0 (5) 9 5 (5)

3700 (6) 3600 (6) 1600 (6) 85 ( 6 )

250 (3) 202 (3) 3 2 (3)

241 (4) 194 (4) 4 1 (5)

217 (5) 1 7 5 (5) 5 1 (7)

199 (6) 160 (6) 63 (9)

Multiplicities are given in parentheses.

not be significantly affected by sampling error due to unimolecular decomposition. The kto will have to be determined in separate kinetic measurements at temperatures below Teq. Plots of log k,, vs. log T are common practice in ion-molecule kinetic measurements and almost always lead to straight lines. Such plots should be used to obtain an extrapolated kto at Teq. An even simpler condition not requiring the knowledge of kt,is obtained from eq 22-25 for conditions where the forward reaction is third order. From eq 23 and 24 one obtains eq 27. Substituting the third-order condition fex

3700 (3) 3600 ( 3 ) 1600 (3) 116 (3)

diffusion rates of the ions n and n - 1 might have to be applied for such conditions if the clustering rates are relatively slow compared to the diffusion rates. Extension of the equilibrium measurements to high n (n > 6) might be possible with new instrumental techniques. Acknowledgment. We thank Dr. I. Safarik for help with the RRKM computer programs.

Appendix I Electrostatic Calculations of the Potential Energies and Vibrational Frequencies of K+(H,O),. The potential energies were calculated by using the equations given by Davidson.e However, since the Davidson calculations involved only the ion and one ligand, the equations were modified to cover interactions of the ion with several ligands. The total energy, Et, was made up of the terms in eq 29. I and L refer to the ion and the ligands, respecTt = Edip(I,L) + Edip(L,L) + E r e p U & ) + Erep(L,L) + Eind(L) + Eiml(I) + Edi,(I,L) (29) tively. Below, Lk (and L1) are often used, and the energy terms then refer to the interaction for a particular ligand. The energy for all ligands, as used in eq 29, is then obtained by summation (for all ligands). Edip(I,Lk)is the energy of ligand Lk in the field of the ion. The water ligand is treated with a point-charge model; Le., each atom is assigned a net charge (Table 111) such that the correct dipole moment of water is reproduced. Edip(I,Lk)is given by eq 30, where i is summed over all atoms in ligand Lk, (30) QI is the charge of the ion, Qi is the net atomic charge on atom i in the ligand, and Ri is the distance between that atom and the ion. The electrostatic interaction between two ligands was obtained from eq 31, where i and j are

summed over all atoms in ligands Lk and L1, respectively. The total ligand-ligand electrostatic interaction was obtained by summing over all pairs of ligands (eq 32). The Edip(L,L) =

E . Edip(Lk,Ll)

(k,l)pslrs

k

f

1

(32)

closed shell repulsion energy was evaluated by assuming an exponential dependence on (interatomic) distance (eq 33). cij = (cicj)'I2and aij = l/z(ai aj) where ci, cj, ai, and Erep= cij exp(-aijRij) (33)

+

aj are characteristic constants of the participating atoms

or ion. The values used are given in Table 111. Rij is the distance between the atoms. The repulsion energy between the ion and a ligand was obtained from eq 34. The re-

The Journal of Physical Chemistry, Vol. 85, No. 4, 1981 335

Unimolecular Dissociation of Ions

Erep(I,Lk) =

CCi,I i

exp(-ai,&i,I)

(34)

pulsion energy between a pair of ligands was calculated from eq 35. To obtain the total ligand-ligand repulsion Erep(Lk,LJ = CCcij ed-aijRij) I

(35)

J

enerky, EreP(L,L),eq 35 was summed over all ligand pairs. For the calculations of the energy due to polarization of a ligand in the electric field set up by the ion and the other ligands, each bond in the ligand was assigned a lateral and a transverse polarizability, such that the overall polarizability of the molecule was reproduced. The energy was obtained from eq 36, where b is summed over all bonds

in Lk. “l,b and (Yt,b are the lateral and transverse polarizabilities, respectively, of bond b. flb is the angle between the bond and the electric field vector. The latter was calculated from eq 37, where i is summed over all charges (37) outside ligand Lk. Rib is the vector from charge Qi to the center of bond b. The energy associated with polarization of the ion in the field of the ligands was obtained from eq 38. The dis(38) persion energy between an ion and a ligand was obtained from the modified London equation, eq 39. II and I L are Eind(1)

= -%“d~aI2

v2

= f/(47&)

mass for the motion. The calculated frequencies are given in Table IV. The ion-ligand stretch vibrations for K+(H20),were calculated for both the symmetrical and asymmetrical mode. For the higher clusters, however, the frequency for the “single ligand stretch”, Le., where only one ligand changes its bond distance, was used for all n ion-ligand stretch vibrations. The same approach was taken for the ligand rocking vibrations; i.e., the frequencies for the vibrations where only one water ligand is allowed ita rocking motions were calculated and later used for all rocking motions. For the higher clusters, the bending motion for a ligand is close to harmonic. However, for the lower clusters, in particular for n = 2 and 3, the bending motions are far from harmonic and have very large amplitudes. As our RRKM program requires harmonic frequencies, the problem was to find frequencies that gave a good estimate of the density of states as thermal energies. Since the “bending” motion is more nearly a free motion on a twodimensional surface, the bending frequencies v were obtained by equating the density of states for a two-dimensional box with area equal to that available to the ligand with the density of states for harmonic oscillators with frequency v. These bending frequencies are given in Table IV. For n = 6 the bending frequency obtained by the particle in the box model agreed closely with a calculated single ligand bend frequency.

Appendix I1 RRKM Calculations. The unimolecular rate constant at an energy E*, for K+(H20), K+(H20),-I + H20, was obtained from eq 4LZ6 N*(E*) is the density of vibra-

-

E+

T i

the ionization potentials of the ion and the ligand, respectively. J I and J L are the correction factors suggested by Pitzer. ob is the angle between the vector from the ion to the center of the bond and the bond vector. The total dispersion energy, E&,(I,L), was obtained by summing eq 39 over all ligands. In order to use eq 29-39 to calculate vibrational frequencies, it was necessary first to obtain minimum energy geometries of K+(H20). The structures were assumed to be linear for n = 2, planar trigonal for n = 3, tetrahedral for n = 4, trigonal bipyramidal for n = 5, and octahedral for n = 6. All ion-oxygen distances were identical. The water ligands were oriented so that a line from the ion through the oxygen bisected the H-O-H angle. Rotation of the ligands around the ion-oxygen bond axis changed the energy with only negligible amounts. The calculated energies for these geometries are given in the first column in Table I. As for the present purpose, a full vibrational analysis is neither needed nor justified; a simpler approach was taken. The internal vibrational frequencies of the ligands were considered to be unchanged. The cluster vibrations, Le., all vibrations except those internal to the ligands, were divided into ion-ligand stretch and rocking vibrations and ligand-ion-ligand bending vibrations. The potential well associated with each motion was calculated from the electrostatic model. The force constant, f , was obtained by graphical fitting. The frequency, v, was then calculated from eq 40, where /I is the reduced

(40)

tional-rotational states of K+(H20),;CP(E,+) is the total number of (vibrational-rotational) states of the activated state with energy less than or equal to E+;E+equals E* - Eowhere Eo is the critical energy; h is Planck’s constant; the reaction path degeneracy, L*,was set equal to n. Sums and densities of states were calculated according to eq 7-10 in ref 27. The vibrational states were counted with the Beyer-Swinehart algorithm.28 The effect of overall rotations were not considered in the present calculation, and these degrees of freedom were not included in the density and sum of states in eq 41. The frequencies given in Table IV were used. The water ligands were treated as rotating freely around the ionoxygen bond. The number of internal rotors were thus 0, 1, and 3-6 for n = 1-6, respectively. As is common for ion-molecule dissociations, the activated state (H20),-1K+...0H2was treated as loose. Both fragmentswere assumed to tumble freely, and four of their rotations were included in the sum of states calculation. The vibrational frequencies, for the activated state, were those of a free water ligand plus those given in Table IV, for the K+(H20)n-1complex. ~

~~~

(26) P.J. Robinson and K. A. Holbrook, “UnimolecularReactions”, Wiley-Interscience,London, 1972. (27) K.M. Maloney and B. S. Rabinovitch,J. Phys. Chem., 73,1652 (1969). (28)S.E.Stein and B. S. Rabinovitch,J.Chem. Phys., 68,2438(1973).