CORNELIUS E. KLOTS
1526
factorily for all the observed effects.”31 However, this study has shown that the interaction between histone f-1 and DNA is extraordinarily sensitive to salt and solvent effect’s. This interaction results in changes in the CD spectrum of the DNA, implying a conformational change of state. Thus, small alterations in sol-
vent medium may be a selective means of affecting the state of aggregation of t,hese complexes. Both t,he conformational control and aggregative phenomena have relevance to the purported role that histones may play in association with DNA in the chromatin of eukaryotes.
Reformulation of the Quasiequilibrium Theory of Ionic Fragmentation1 by Cornelius E. Klots Health Phgsics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee $7890 (Received December 2, 1970) Publication costs assisted by Oak Ridge National Laboratory
An alternative to the transition-state formalism of unimolecular decomposition theory is further developed. When the conservation of angular momentum is heeded, the content of this formulation is seen to be that of phase-space theory. Application to the fragmentation processes of methane ions proves generally successful, although some difficulties are delineated. Tunneling is implicated as a source of the metastable decompositions occurring in methane.
Introduction The work of Butler and Kistialtowsky2 has strongly suggested that the kinetics of unimolecular fragmentation are determined by the energy content of an activated species but not otherwise by its mode of preparation. Since then chemical activation techniques have constituted an important branch of chemical kinetics, and have been instrumental in the ascension of those theories which explicitly predict such behavior and likewise the demise of a t least one theory which explicitly does not. Thus, the quasiequilibrium theory of unimolecular decompositions specifically describes an activated species in terms of microcanonical ensemble theory; hence its average lifetime can be a function of only a small number of “constants of motion” and not a t all of its past history-precisely as the Butler and Kistiakowsky paper implied. The essential content of quasiequilibrium theory has found its most common expression within the formalism of the transition The first-order rate constant for fragmentation of a species with total energy ( E Eo*) is given in this formalism by
+
x=E
k,
=
x=O
+
Q ~ * ( z , ~ J / P (Eo*,aOh E
(1)
where g,*(x) is the internal Etatistical degeneracy factor for the transition state of energy (x);the summation extends by energy conservation over the range 0 5 x I E. Eo* is the activation energy. These degenerThe Journal of Phgsical Chemistry, Vol. 76, N o . IO, 1971
acy factors must further be chosen to be compatible with any additional constraints ai. Similarly p(E Eo*,ai) is the density of states of the activated species, again evaluated with the indicated constraints. This expression has found wide use in calculating the rates of unimolecular reaction in the absence of collisions. When coupled with the strong collision hypothesis, the Rice-Ramsperger-Kassel-Marcus description, applicable a t all pressures, is obtained.6 There is a second formalism available for calculating a fragmentation rate constant
+
where gf(x,af)is the degeneracy factor for the separated fragments, u(f,i) is the cross section for their association to form the pertinent activated species, and X is the deBroglie wavelength associated with their relative kinetic energy (E - x). Equation 2 follows from the quasiequilibrium hypothesis and microscopic re(1) Research sponsored by the U. s. Atomic Energy Commission under contract with Union Carbide Corporation. (2) J. N. Butler and G. B. Kistiakowsky, J . Amer. Chem. SOC., 82, 759 (1960). (3) H.M.Rosenstock, Advan. Mass Spectrom., 4, 523 (1968). (4) B. S. Rabinovitch and D. W. Setser, Advan. Photochem., 3 , 1 (1964). (5.) M.L. Vestal in “Fundamental Processes in Radiation Chemistry,” P. Ausloos, Ed., Wiley, New York, N. Y., 1968, pp 59-118. (6) E . V. Waage and B. S. Rabinovitch, Chem. Rev., 70,377 (1970).
REFORMULATION OF THE QUASIEQUILIBRIUM THEORY OF IONIC FRAGMENTATION
1527
versibility. It has been used in the contexts of both unimolecular fragmentations7and autoionization.8r9 We shall make extensive use of this latter formalism in the present paper. At this point however we should clarify the relation between eq 1 and 2 and enumerate whatever advantages accrue to the latter. This is most simply done by noting that Marcuslo and, more recently, Karplus and coworkers’l have shown that the relationship
how a collision couple will fragment. MiesZ0has observed that phase-space theory is equivalent to transition state theory for “loose transition states.” This also follows from the above remarks. Actually a stronger statement is possible. Phase-space theory is equivalent to quasiequilibrium theory; they are each equivalent to transition state theory whenever the latter is correct.
(3)
We shall illustrate the use of eq 2 in the context of the fragmentation processes of the CH4+ ion. The simplicity of this system lends itself to a scrutiny of the quasiequilibrium hypothesis relatively unobscured by ancillary assumptions. Thus only three reactions need to be considered. These are, together with their activation energiesz1&
defines the thermodynamic properties of the transition state if the latter is to provide numerically consistent answers. Whether or not these properties will correspond to those calculated from the potential energy surface remains an open question. This hinges upon such ancillary matters as whether the concept of a separable reaction coordinate is meaningful and whether transmission coefficients are unity. Recent work has tended to cast some doubt on each of these.l1,l2 Hence we see that eq 2 rests upon the quasiequilibrium hypothesis; eq 1 presupposes both this and the usual further assumptions associated with the theory of the t,ransition state. Any work which purports to test the quasiequilibrium hypothesis itself is evidently more safely rooted in eq 2 . Further advantages of eq 2 may also be noted. Its evaluation requires the thermodynamic properties of the separated fragments. These are amenable to experimental observation, while properties of the transition state are not. The latter can be constructed only after the potential energy surface in the neighborhood of the saddle point has been mapped. Despite impressive recent advances in this direction,13 some impatience on the part of the experimentalist is underst andable. These advantages are not obtained, however, without some cost. One does need good cross-section data for the evaluation of eq 2. But again these are amenable to experimental observation or, again for the impatient, estimation from the interaction potential. This latter procedure is exemplified by the Langevin treatment whereby collision cross sections are obtained from the long-range forces. l4 We shall use this method below. Before doing so, however, some further general remarks are appropriate. It is well knownl6Pl6that bimolecular association rate constants obtained with Langevin cross sections are equal to those obtained from the “loose transition state” models of Gorin17 and Eyring, Hirschfelder, and Taylor. l8 This is merely a reflection of the fact that eq 3 is fulfilled for such “loose transition states.” We conclude then that the GorinE H T method for constructing transition states is the correct one for reactions governed by Langevin cross sections. Now, scrutiny of eq 2 shows that it is equivalent to the phase-space methodology19 for estimating
Illustrative Calculations
+ H, Eo CH4++CH2+ + Hz, Eo CHI+ +CH2+ + H, Eo CH4+-+ CH3+
=
1.71 eV
(I)
=
2.65 eV
(11)
=
5.42 eV
(111)
The following observations must be accounted for by any unimolecular decomposition scheme. (1) The appearance potentials for these processes correspond well to the above-indicated activation energies, as determined by independent means.21a Thus there is little, if any, kinetic shift22for these reactions. (2) Metastable ions have been observed to undergo reaction I for all isotopic ~ a r i a n t s and , ~ ~with ~ ~ comparable ~ in(7) C. E. Klots, J . Chem. Phys., 41, 117 (1964). (8) C. E. Klots, ibid., 46, 1197 (1967). (9) Provided one assigns the photon a polarization degeneraoy factor of 2, eq 2 also gives the usual Einstein relation between emission lifetimes and photoabsorption cross sections. This is of more than
purely formal interest. The energy randomization frequently thought of as underlying unimolecular decompositions is also one of the factors which complicates the use of the Einstein relation; see, for example, A. E. Douglas, ibid., 45, 1007 (1966). (10) R. A . Marcus, ibid., 45, 2138, 2630 (1966). (11) K . Morokuma, B. C. Eu, and M. Karplus, ibid., 51, 5193 (1969). (12) R. A. Marcus, ibid., 41, 2614 (1964). (13) I. Shavitt, R. M. Stevens, F. L. Minn, and M. Karplus, ibid., 48, 2700 (1968). (14) P. Langevin, Ann. Chim. Phys., 5, 245 (1905). (15) B . Mahan, J . Chem. Phys., 32, 362 (1960). (16) K. Yang and T. Ree, ibid., 35, 588 (1961). (17) E. Gorin, Acta Physicochim. URSS,9, 691 (1938). (18) H. Eyring, J. 0. Hirschfelder, and H. S. Taylor, J . Chem. Phys., 4, 479 (1936). (19) P. Pechukas and J. C. Light, ibid., 42,3281 (1965) ; J. C. Light, Discuss. Faraday Soc., 44, 14 (1967). (20) F. H. Mies, J . Chem. Phys., 51, 798 (1969). (21) (a) “Ionization Potentials, Appearance Potentials, and Heats of Formation of Gaseous Positive Ions,” NSRDS Publication, National Bureau of Standards Report NBS-26, 1969; (b) the structure of CH4+ has been estimated by F. A. Grimm and J. Godoy, Chem. Phys. Lett., 6 , 336 (1970). (22) W. A. Chupka, J. Chem. Phys., 30, 191 (1959). (23) Ch. Ottinger, 2. Naturforsch. A , 20, 1232 (1965). (24) J. H. Futrell, private communication.
The Journal. of Physical Chemistry, Vol. 76,No. 10, 1971
CORNELIUS E. KLOTS
1528
tensity. This is not easily reconciled with the predictions of the usual transition state theory.2S (3) Rotational energy can assist in overcoming the activation energy of reaction I. (4) The breakdown curves, a function of internal energy of the parent ion, are known from the work of vonKoch.26 We shall see that the present formalism is able to account for each of these observations. The thermodynamic properties of HZare well known. Those of the hydrocarbon ions are not21b and so were estimated in the following way. The structures of CH4+, CHs+, and CH2+ were taken as tetrahedral, planar symmetric, and linear symmetric, respectively. For computations of moments of inertia, all their bond lengths were arbitrarily assigned the value 1 A. The vibrational frequencies of CH4+were taken as those of CH4,27from which were derived in the simplest fashion possible those of the fragment ions. These are summarized in Table I in units hvt = et(eV), together with their degeneracy factors. Also included in the table are the rotational constants which define rotational l), where J is the rotaenergy via E,(eV) = B J ( J tional quantum number. For CH4+, CHz+, and Hz these follow from the assumed structures. Since CHa+ will be treated below as a spherical top, its rotational constant was taken as the geometrical mean of those obtained from the three principal moments of inertia.
+
~
~~
~
Table I : Vibrational Energies (with Degeneracy Factors) and Rotational Parameters in Units of eV CH4
+
0.361 (1) O.lSQ(2) 0.374 (3) 0.162(3) 0.771" a
B
x
CHa+
CHa +
0.361 (1) 0.189(1) 0.374(2) 0.162 (2)
0.361 (1)
1-09"
Ha
0.374 (1) 0.162(2) 1.03"
J
JO
Figure 1. Allowed L, J, JO combinations and limitations imposed by Langevin cross sections.
initial angular momentum, indexed as Jo,must appear as orbital angular momentum L or as product angular momentum J . The cross section ( u / d z ) for an ( L J -+ Jo) association is (2L 1) times the fraction of such collisions which result in JO, specifically (2J0 1 ) c (2L l)(2J 1). At this point it is useful to consider Figure 1; the L,J combinations compatible with a given Jo are contained within the rectangular area defining L = J o J . , . I J o - J 1 . Energy conservation, and the use of the Langevin model for the total cross section, poses the further restriction
+
+
+
+
+
+
+
(Lax
=
?[(E-
X)
- B J ( J + 1)1'/% (4)
where (E - x) is the energy not appearing in the vibrations of the products, B is the rotational constant of the product sphere, given by
B = fi2/21 0.545 (2) 3.750
108.
where I is the moment of inertia. The parameter y is given by the Langevin model for ion-molecule collisions y = (2p/m)( [ Y / R U ~ ~ ) ' / *
One further preliminary datum, the statistical reaction-path factor for each decomposition, must be discussed. For reactions I, 11, and I11 there are, respectively, four, six, and three equivalent reaction paths. But for the reverse reaction, the total cross section appearing in eq 2 is apportioned among two, two, and one routes, respectively. The net statistical reactionpath factors for these reactions, then, are two, three, and three, respectively. Note that, by a theorem due t o Laidler,28these are also the ratios of the respective symmetry factors for the several reactants. The manner by which one includes conservation of angular momentum in fragmentation kinetics has been amply illustrated by Nikitinz0and Light,lg and so need not be belabored here. Reaction I will be treated as a sphere dissociating to a sphere plus an atom. The The Journal of Phyeical Chemistry, Vol. 76,No. 10, 1971
where p is the reduced mass of the separating pair, m that of an electron, [Y the polarizability of the neutral fragment, a0 the Bohr radius, and R the Rydberg energy. With reference again to Figure 1, curve (a) is typical of the restriction obtained with B = 0. This is exactly the approximation used in some earlier work of the present author.' But, as NikitinZQhas pointed out, it is extremely limited in its applicability. I n fact, a (25) M. L. Vestal, J . Chem. Phys., 41, 3997 (1964). (26) H. vonKoch, Ark. Fys., 28, 529 (1964). (27) G. Herrberg, "Infrared and Raman Spectra," Van Nostrand, New York, N. Y., 1946. (28) D. M. Bishop and K. J. Laidler, J . Chem. Phys., 42, 1688 (1965). (29) E. E. Nikitin, Theor. Exp. Chem., 1, 144 (1966).
REFORMULATION OF THE QUASIEQUILIBRIUM THEORY OF IONIC FRAGMENTATION quite different limiting approximation, illustrated by curves (b) and (e), is more appropriate. On expansion of eq 4, with L J as will always be the case for not too large Jo, and assuming
-
(E
- X ) -l are too high. suggest that rate constants This difficulty might be ameliorated in the following fashion. As Rice30 has discussed, a fast association reaction implies an abnormally high frequency factor (33) W.A. Chupka, J . Chem. Phys., 48, 2337 (1968). (34) R.L. LeRoy, ibid., 53, 846 (1970); A. Henglein, ibid., 53, 458 (1970). (35) D.L. Bunker, ibid., 40, 1946 (1964). (36) 0. K.Rice, J . Phys. Chem., 65, 1588 (1961).
REFORMULATION OF THE QUASIEQUILIBRIUM THEORY OF IONIC FRAGMENTATION
1531
hance rotational temperatures significantly. Hence, for the reverse dissociation reaction, and this in it is not immediately evident that a tunneling mechaturn implies drastic “anharmonicity” in the activated nism can account for the observed metastable intensispecies. The density-of-states for this species, appearties. This can be established, however, by observing ing in the denominator of eq 1 and 2, must then be acthat for an ion to have rotational energy E, and a total cordingly increased. This will have the desired effect energy between Eo and Eo e 1 ( ~ , / y 2 B Othe 2 ) , width of of reducing the implied rate constants, perhaps to more ~). possible vibrational energies is also er( ~ ? / y ~ B o When plausible magnitudes. At the same time this need not averaged over a rotational equilibrium distribution, the affect the apparent achievements obtained above. average such width is seen to be Thus, the results of Figure 3 hinge only upon relative rate constants and so will remain unaltered. Several authors have introduced anharmonicities into density-of-states expressions, usually as minor perturb a t i o n ~ . ~ ’ -These ~~ treatments seem unsatisfactory. Now only parent ions with vibrational energy less than The role of the long-range potential, as implied by the 1.71 eV are eligible for the tunneling mechanism. If use of the Langevin cross sections, would appear to rethese are roughly equally distributed over this range, quire a more drastic model. It is not presently clear then the probability that such an ion will fall into the how this is best done. Note that Ottinger40 has average tunneling width is -Ax/1.71; with kT = achieved a similar effect by allowing the flow of energy 0.03 eV and y2B02= 2.5 eV, this gives 6 X loF4 as a between vibrations and rotations. The treatment, due predicted metastable to parent ion intensity ratio. originally to Whitten and Rabinovitch141does not conThe agreement with Ottinger is quite satisfactory. serve angular momentum and hence is not satisfactory. Two other predictions follow from the above model. Figure 2 contains an intimation of still a second difI n the sequence CH4+, CH3D+, CHzD2+,CHD3+, the ficulty. A role for metastable decompositions in methaverage moment of inertia increases, as then does ane is indeed predicted by the monotonic behavior of of eq 11. The metastable intensity should increase rate constants near threshold. We would emphasize accordingly, as indeed Ottinger observes.41 Note too here that this monotonic behavior is not an artifact that such intensities should go as the square of temintroduced by the use of continuous approximations to perature. This could be easily checked. densities-of-states. It results rather from the monoThe above discussion makes no reference to the rate tonic behavior of a/nX2 in the Langevin model. Nevof tunneling and indeed it is crucial that this rate fall ertheless, it is also clear that the energy width leading within the metastable range. We shall not attempt to rate constants characteristic of metastable behavior, any elaboration here other than to observe that tunneland hence -lo6 (sec)-l, is vanishingly small. The ing through such centrifugal barriers is of some current predictea abundance of such metastables is then much interest in bimolecular ~ c a t t e r i n g . ~Its ~ probable role too small to account for Ottinger’s o b ~ e r v a t i o n s . ~ ~in the present context would seem to merit further inIndeed, as a referee has noted, there is a fundamental vestigation. Thus, the small but finite intensity of inconsistency between the absence of significant, kinetic CD4+ metastables interestingly reflects its relative shifts and the presence of metastables in detectable tunneling width and with no indication of the usual amounts. mass dependence of tunneling.43 Again, a possible explanation can be proffered. The Having noted earlier a consistency between the lower limit for the integrations in eq 7, for example, is quasiequilibrium calculations and the breakdown curves ~ B ~ ~E, )is the strictly not zero but rather E , ( E , / ~ where of vonKoch, we should also point out that Brehm44 rotational energy of the activated complex and Bo its rotational constant. This can be understood by refer(37) M.L. Vestal and H. M. Rosenstock, J . Chem. Phys., 3 5 , 2008 (1961). ence to region (d) of Figure 1. Conservation of angular (38) K.A. Wilde, ibid., 41, 448 (1964). momentum and the use of the Langevin model suffice (39) E.W. Schlag, R. A. Sandsmark, and W. G. Valence, ibid., 40, to prohibit decomposition unless Lmax22 Jo2. Except 1461 (1964); W. Forst and Z. Prhlil, ibid., 53, 3065 (1970). in the neighborhood of threshold, this is of little conse(40) Ch. Ottinger, 2. Nuturforsch. A , 22, 20 (1967). quence. But when the excess energy E lies in the (41) G.2. Whitten and B. S. Rabinovitch, J. Chem. Phys., 41, 1883 (1964). range 0 i E 5 E , ( E , / ~ ~the B ~reaction ~), cannot pro(42) R. E. Roberts, R. B. Bernstein, and C. F. Curtiss, ibid., 50, Now Rosenstock3 has suggested ceed classically. 5163 (1969). tunneling as a source of Ottinger’s metastables. In(43) L. P.Hills, M. L. Vestal, and J. H. Futrell (to be submitted deed, when E lies in this range, it is the only available for publication) have confirmed the observations of Ottinger and have interpreted the metastables in terms of finite reflection coefmechanism. The great preference for H us. D fragmenficients above the potential barrier. This seems unnecessary, but tation and the vanishingly small kinetic energies assoalso unlikely for ion-molecule systems. See E. Vogt and G. H. Wannier, Phys. Rev., 9 5 , 1190 (1954). On the other hand the inciated with such metastables40 each suggest such an sensitivity of tunneling through centrifugal barriers to mass is now interpretation. understood: C. E. Klots, Chem. Phys. Lett., in press. Ionization with light particles is not expected to en(44) B. Brehm and E. van Puttkamer, Z . Nuturjorsch. A , 22,8 (1967).
+
ax
The Journal of Physical Chemistry, Vol. 76, No. 10, 1971
DENISJ. BOGANAND CLIFFORD W. HAND
1532 has obtained breakdown data which do not agree with those of vonKoch. Brehm’s data can only be interpreted as implicating CH4+ ions with energy greatly in excess of that required for reaction I and yet not fragmenting. Such a result would require “isolated st,ates” and hence is incompatible with the fundamental premise of the quasiequilibrium model. Again this would warrant the most careful scrutiny. Experiments such as those of Brehm and of Ottinger are taking much of the guess work out of unimolecular fragment,at,ion st,udies. The formulation of quasiequilibrium theory that has been presented here is hoped to have a similar result by obviating any need
to estimate transition-state parameters. Our illustration of its use has, by contrast, involved only the most transparent of approximations. And, of course, there is the not inconsiderable advantage that this formalism satisfies microscopic reversibility. Accordingly, while discrepancies between theory and experiment persist, one can be reasonably certain that these are not artifacts of the theory but rather worthy of a more measured study.
Acknowledgment. Discussions with Professors Ch. Ottinger and J. H. Futrell and with Dr. H. C. Schweinler are gratefully acknowledged.
Mass Spectrum of Isocyanic Acid1
by Denis J. Bogan Carnegie-itfellon University and, the University of Alabama, University, Alabama
36486
and Clifford W . Hand* Department of Chemistry, University of Alabama, University, Alabama
36486
(Received January 4, 1971)
Publication costs assisted by The University of Alabama
Energies of several possible structures for the HNCO+ ion have been calculated by INDO methods. Compared with the linear HNCO+, a cyclic structure was found to be more stable by 80 kcal. A fragmentation pattern and appearance potentials for the major fragment ions are reported, and a mechanism involving the cyclic structure is presented. The mechanism explains the observed features, including a metastable transition, of the mass spectrum of HNCO. The successful use of INDO calculations in this problem suggests that semiempirical methods may be generally useful in the interpretation of mass spectra.
Introduction The major product of the thermal decomposition of cyanuric acid has the empirical formula CHON, corresponding to the two possible structures HOCK’ (cyanic acid) and HNCO (isocyanic acid). Compelling evidence of the true structure comes from an infrared study2in which all the spectral features were accounted for by the structure HNCO, with a linear KCO group and an HNC angle of ca. 130”. The structural constants were later very accurately determined by millimeter wave t e c h n i c ~ e s . ~Thus, although some chemiCa1 evidence4 suggests the existence, in solution, of a tautomeric with 1301, H°CN present’ and photolytically generated HOCN has been observed in low temperature matriceSJ6the conclusion is inescapable that gaseous samples at room temperature consist almost exclusively of HNCO. The mass spectrum Of HNC06’7shows a large peak at m/e 29, corresponding to an ion of formula HCO+;
-
The Journal of Physical Chemistry, Vol. 76, No. 10, 1&71
this species is the second most abundant fragment ion in the spectrum. This is a curious feature inasmuch as the HCO+ fragment cannot be formed without extensive rearrangement of the parent structure. Mass peaks corresponding to CH+, OH+, and NO+ also appear, and on this basis Smith and Jonassen’ suggested that HNCO+ might have a “triangular structure.” (1) Presented in part at the Southeast-Southwest Regional Meeting of the American Chemical Society, New Orleans, La., Dec 2, 1970. (2) G.Heriberg and C. Reid, Discuss. Faraday Soc., 9, 92 (1950); C.Reid, J . Chem. Phys., 18, 1544 (1950). (3) R. Kewley, K. L. V. N. Bastry, and M. Winnewisser, J. Mol. Spectrosc., io, 418 (1963). 78, 6234 (1956); (b) N. (4) (a) A. Amell,, J . Amer. Chem. SOC., Groving and A. Holm, Acta Chem. Scand., 19, 1768 (1965). (5) M. E.Jacox ahd D. E. Milligan, J . Chem. Phys., 40,2467 (1964). (6) J. M. Ruth and R . J. Philippe, Anal. Chem., 38, 720 (1966). (7) 8. R. Smith and H.B. Jonassen, J. Inorg. Nucl. Chem., 29, 860 (1967).
