J. Phys. Chem. 1994,98, 136-140
136
Unimolecular Dissociation of Very Large Polyatomic Molecules V. Bernshtein and I. Oref Department of Chemistry, Technion- Israel Institute of Technology, Haifa 32000, Israel Received: May 5, 1993; In Final Form: October 12, 1993"
The unimolecular decomposition of large polyatomic hydrocarbon (720 and 1000 modes) and protein (540 modes) molecules and molecular ions is investigated by applying RRKM, effective R R K (ERRK), and Arrhenius models. It is found that the calculated values of the R R K M unimolecular rate coefficients, k(E), agree well with fragmentation lifetimes of molecular ions in a mass spectrometer. Calculations of the effective number of modes a t a given temperature and excitation energy together with E R R K expression for k ( E ) yield values which are similar to those calculated by R R K M . With the assumption that a 1000-mode supermolecule is its own heat bath a t a vibrational temperature commensurate with its total internal energy, an Arrhenius expression for the rate coefficient is obtained which yield values for the rate coefficient which are in a rough agreement with values calculated by the two methods described above.
Introduction The unimolecular decomposition of large polyatomic molecules (up to 1000 modes) is of interest in biochemistry and mass spectroscopy. The question being addressed is, do such molecules behave ergodically and, if so, can rate coefficient be calculated by the statistical methods? (An interesting paper' analyzes the unimolecular dissociation of large polyatomic ions in mass spectrometry.2) In other words, is a situation possible where a molecule is large enough such that its intramolecular vibrational energy redistribution (IVR) is slow compared with the rate of ion dissociation? Since very large ions are found to dissociate with rate coefficients larger than the ones calculated from classical statistical RRK theory,'g2 the question arises whether there is a threshold size above which statistical theories fail and the energy is localized in a moiety a la Slater. The classical RRK treatment, now available in any basic physical chemistry book, gives the following expression for the number of ways of obtaining m quanta in a single mode when there are j quanta distributed in the rest of the s - 1 modes.
+
+
w ' = j ! 0'- m s - l)!/(j- m)! 0' s - l)! (1) The well-known compact RRK expression is obtained by dividing w' from eq 1 by the number of ways, w, of dividing j quanta among s modes (letting m = 0 in eq l), by using Stirling's approximation, and by making the judicious assumption that j -m>>s-1
the threshold energy for dissociation of the molecular ion Eo was taken as 46 kcal/mol (2 eV), and the excitation energy varied from 1 to 50 photons of 4.7 eV each. It was found that neither of the cases gave reasonable results. The values of k ( E ) found for s = 1000 at the experimental values of the number of exciting photons were much lower then expected for fragmentation in a mass spectrometer. In the calculation, it was assumed that the thermal energy even at 300 K is negligible and does not contribute significantly to the available excitation energy which leads to decomposition. The conclusions drawn from these dramatic results are that in a large molecule there is an energy bottleneck, IVR is not complete, and therefore statistical RRK theory fails. The solution recommended is some form of Slater's theory with restricted IVR. In an additional study,4 the same authors discuss specifically the ionization of large molecules or clusters in which an electron is removed in a charge separation process. They also apply statistical theories to the ionization process by convoluting the electronic density of state in the Rydberg states of the electron with the core densities of vibrational states. They compare their results with (another) statistical model of bulk thermionic emissions and suggest that ionization does not occur from a fully energetically-equilibrated molecule. We next analyze the above results and conclusions and provide an alternative explanation to the "puzzle" why very large ions are observed to decompose in a mass spectrometer.
Vibrational Temperature where E = jhu and EO = mhv. A simple-minded approach is to use the classical expression = keT/h. We will return to this point later. For a large polyatomic, j - m < s - 1, eq 2 fails and the full expressions for w and w' (eq 1) are the ones to use.' A modified RRK theory is explored in ref 1 where only quantum states in which no individual oscillator is in its ground state are considered. Again, the nonconventional case is considered where j - m < s - 1. As mentioned before, this is obviously true for very large molecules with hundredsof degrees of freedom. Finally, another modification of RRK theory is considered in ref 1 in which the semiclassical approximation of Marcus and Rice3 is used in the expression for the energy-dependent rate coefficient, k(E). In a comparative study, the values of k(E) obtained from the three versions of RRK theory were calculated in ref 1 for very large systems with s = 300 and s = 1000. To emulate a real case, Abstract published in Advance ACS Abstructs, December 1, 1993.
0022-3654/94/2098-0136%04.50/0
A large polyatomic molecule with internal energy E (a microcanonical ensemble) can be looked upon as a canonical ensemble of oscillators at temperature, TV,6s7which is calculated by iterating the equation (3) where Q is the vibrational-rotational partition function. This definition of T,leads naturally to a new definition of an effective number of modes, serf7
= EIRT" (4) In order to check the unimolecular rate coefficient of the fragmentation of a large (s 720, 1008) polyatomic molecule, we have taken a model polyatom made of units of octane less two hydrogens H-(CsH,&H. The octane normal modes were grouped into 15 groups around the average of each group (Table 1). For example, 90, 110, and 130 cm-' were grouped into the Serf
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0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 1, 1994 137
Unimolecular Dissociation of Polyatomic Molecules
TABLE 1: Frequencies (in cm-l) Used in RRKM Calculations of Octane and Polyoctane Fission (Degeneracies in Parentheses) Moleculeb (m = 1," s = 72) 101 (4) 201 (4) 1101 (2) 1201 (3) ComplexC(m = 1,* s = 72) 51 (1) 1301 (8) 3001 (18) 101 (2) 201 (3) 1001 ( 5 ) 801 (1) 1501 (5) 1101 (2) 1201 (3) 1071 (1) 63 (1) 651 (1) 722 (1) 151 (1) Moleculef (m = 10, s = 720) 51 (21) 1301 (71) 3001 (162) 101 (41) 201 (32) 1001 (61) 801 (11) 1501 (72) 1101 (21) 1201 (31) ComplexC(m = 10, s = 720) 51 (20) 1301 (70) 3001 (162) 101 (39) 201 (32) 1001 (60) 801 (11) 1501 (72) 1101 (20) 1201 (31) 1071 (1) 63 (1) 651 (1) 722 (1) 151 (1) Moleculec (m = 14, s = 1008) 51 (30) 1301 (100) 3001 (226) 101 (58) 201 (44) 1001 (86) 801 (16) 1501 (100) 1101 (30) 1201 (43) ComplexC(m = 14, s = 1008) 51 (29) 1301 (99) 3001 (226) 101 (56) 201 (44) 1001 (85) 801 (16) 1501 (100) 1101 (29) 1201 (43) 1071 (1) 63 (1) 651 (1) 722 (1) 151 (1) a m indicates the number of Octane units. From ref 15. See text for choice of frequencies. 1301 (8) 801 (2)
51 (2) 1001 (6)
TABLE 2
3001 (18) 1501 (6)
301 (1) 1401 (8)
901 (4) 501 (2)
701 (2)
301 (1) 1401 (8) 61 (1)
901 (4) 501 (1)
701 (2) 71 (1)
301 (21) 1401 (81)
901 (42) 501 (22)
701 (31)
300 (20) 1401 (80) 61 (1)
901 (42) 501 (22)
701 (31) 71 (1)
301 (30) 1401 (113)
901 (58) 501 (30)
701 (44)
301 (29) 1401 (112) 61 (1)
901 (58) 501 (30)
701 (44) 71 (1)
Frequencies (in cm-l) Used in RRKM Calculations of Andotensine Fission (Degeneracies in Parentheses)
51 (40) 2681 (1) 1351 (17) 951 (11) 551 (11) 151 (12)
3671 (2) 1701 (2) 1301 (17) 901 (13) 501 (11) 101 (13)
3501 (10) 1651 (10) 1251 (9) 851 (9) 451 (18) 3051 (13)
51 (39) 2681 (1) 1351 (17) 951 (11) 551 (11) 151 (12) 651 (1)
3671 (2) 1701 (2) 1301 (16) 901 (13) 501 (11) 101 (11) 722 (1)
3501 (10) 1651 (10) 1251 (9) 851 (9) 451 (18) 3051 (13)
Molecule 3451 (4) 3351 (3) 1601 ( 5 ) 1551 (2) 1201 (10) 1151 (15) 801 (12) 751 (16) 401 (8) 351 (15)
3001 (7) 1501 (21) 1101 (13) 701 (10) 301 (14)
3951 (19) 1451 (30) 1051 (15) 651 (9) 251 (9)
2901 (30) 1401 (21) 1001 (15) 601 (9) 201 (9)
Complexb 3451 (4) 3351 (3) 1551 (2) 1601 ( 5 ) 1201 (10) 1151 (15) 801 (12) 751 (16) 351 (14) 401 (8) 151 (1) 61 (1)
3001 (7) 1501 (21) 1101 (12) 701 (10) 301 (14) 71 (1)
2951 (19) 1451 (30) 1051 (15) 651 (9) 251 (9) 1071 (1)
2901 (30) 1401 (20) 1001 (15) 601 (9) 201 (9) 63 (1)
Reference 16. See text for choice of frequencies.
100-cm-1 group with a degeneracy of 3 and so on. Using eq 3, we first calculated the internal thermal energy at 300 K which was found to be -60 and 80 kcal/mol for s = 720 and 1008, respectively. Adding this thermal energy to the photon energy of five 4.7-eV photons (=540 kcal/mol) suggested in ref 1 and iterating eq 3, we find Tv = 1026 and 867 K for s = 720 and 1008, respectively. This gives s,fffromeq 4 equal to 294 and 362 modes, respectively. These values are about a third of the number of normal modes of the molecules. The case of s = 720 was chosen to resemble the melittin molecule mentioned in ref 1. We have also used eqs 3 and 4 to calculate T, and serfat various numbers of absorbed photons for the protein angiotensine* using the vibrational frequencies in Table 2.
Models for k(E) RRKM. If one is to use statistical theories to calculate the microcanonical &(E) rate coefficient, it is better to use RRKM theory with direct counting of states. The energy-dependent RRKM microcanonical rate coefficient k(E) is given by
W E )= WE+)/hP(E)
(5)
A modified version of eq 5 which gives (after solving the appropriate master equation) the transition-state expression for the rate coefficient at infinite pressure is8*
where L is the statistical factor, the number of ways a molecule can break, Qr is the rotational partition function, W(E) is the
number and p(E) is the density of vibrational-rotational states, indicates the transition state, and h is Planck's constant. In a large molecule, the transition state affects only a small part of the molecule; the rest are unaffected by the process which is taking place. Therefore, the ratio Qr+/Qr is constant and identical for 500-, 700-, and 1000-modemolecules which basically means that the A factor for such a series depends mostly on the value of L. Effective RRK (ERRK). It is well-known that classical RRK theory in its original form of eq 2 never worked, does not work, and will never work for normal molecules.sb It does not work because the original expression uses unrealistic values for s and i. In any hydrocarbon, CnHb+2,there are 2n + 2 C-H stretches that have, at normal temperatures or energy content, heat capacity C,, which is negligible. Other high-frequency vibrations contribute also only partially to the value of C,. Thus, it is, and was for quite sometimes, the custom to define an effective number of modes, serf. (In a simple-minded approach, senis taken as half the number of modes in the molecule, e.g., sen= s/2.9 Another modification of the classical expression, eq 2, is replacing = kBT/h by = A where A is the frequency factor in the Arrhenius expression.9 The value of k(E) improves substantially when the above two modifications are used. We therefore rewrite eq 2
+
(7)
where serfis calculated by eq 4. The value of A used in eq 7 will be discussed later on. Arrhenius. In the model developed below (called the Arrhenius
Bernshtein and Oref
138 The Journal of Physical Chemistry, Vol. 98, No. 1, 1994
model) we assume that a large supermoleculecan serve as its own heat bath. Intermolecular, as well as intramolecular, energy transfer takes place. This dense internal heat bath is akin to the high-pressure limit in unimolecular decomposition, and the usual definition applies
TABLE 3: Vibntiolrpl Tem ntures, IUfective Number of Modes,RRKM, Effective RRE and Arrheniw Ex mion Rate Coefficients for the Decomposi on of a 72rMode Long-chain Hydrocarbon (See Text for Definitions of Psrameters)
l;lpe
wheref(E) is the Boltzmann distribution. Using RRKM theory and f ( E ) , an expression for the thermal rate coefficient at temperature Tv is obtained
8 10 12
921 1134 1353
Molecule (EO= 69.8 kcal/mol) 1326 16.Ic 350 5.14X 10' 1.37 X 10, 3.94X 10' 1512 16.2 377 1.92X 10" 6.12X 1 0 1.29X 106 1692 16.2 402 2.87X 106 8.77X 106 1.53X lo7
(9) Using a RRKM theory expression from eq 6 instead of eq 5 for &(E),the transition-state theory (TST) expression for the highpressure canonical rate coefficient is obtained
3 4 5
129 237 345
Molecular Ion (EO=I 21.9 kcal/molq 456 15.7c 142 3.43 X 10' 1.57 X 10, 1.60X 1 0 621 15.9 192 9.76 X 106 6.50X IO' 1.56X 1 0 757 16.0 229 4.42X 1 0 2.94X 109 4.76X 109
3 4 5
129 237 345
5 8 10 12
345 669 885 1101
Molecular Ion (Eo = 46.0 kcal/moV) 757 16.0s 229 1.84 X 101 5.86 X 1095 16.1 307 9.86X 1 0 3.87 X 1294 16.1 344 3.68 X 10' 1.32 X 1482 16.2 374 4.90 X 1 0 1.86 X
5 8 10 12
345 669 885 1101
757 1095 1294 1482
Equation 10 provides the means for calculatingthe TST frequency factor A which will be used later in various calculations of the rate coefficient. It is interesting to note that expanding eq 7 in series and remembering that scff>> m (remember m = Eo/keT in eq l), we obtain A simple Arrhenius expression as befitting such a canonical ensemble. The expressions given in eqs 9-1 1 have a pleasant interpretation of a molecule which is also its own heat bath, and intersegment collisions as well as IVR effect the system equilibrium.
Results and Discussions Two groups of calculations were performed, one on hydrocarbons whose frequencies are given in Table 1 and the other on the protein angiotensine (Table 2) where we havecalculated k ( E ) for a number of absorbed photons. In each case full harmonic and anharmonic RRKM calculations were performed. Values of k(E)calculated by other models given above are reported and compared with RRKM theory values. We report the results of calculations for each molecule separately. HydrocarbonDissociation. We have chosen or calculated the parameters used in the various models for k(E) in the following way. For the RRKM calculations we have used the frequencies given in Table 1, which were obtained as described in the Vibrational Temperature section. In the case of a supermolecule there is no major difference between the frequenciesof the normal modes of the molecule and the molecular ion because the ionization site affects only a small number of modes. This is known for other systems as well. For example, the value of RRKM theory k(E) is practically identical for the azulene molecule and azulene molecular ion dissociations.I0 The reason is that any changes in the frequencies between molecule and ion will appear in the excited ion and in the transition state simultaneously, and the ratio of number to density of states which appears in the expression for k(E), eq 5 , will have a cancellation effect." Two sets of frequencies were used for the transition state. In one the recipe given in ref 12 for the decomposition of three hexyl. radicals was used. A 1000-cm-l C-C stretching vibration is the reaction coordinate. One skeletal bend ( 140 70 cm-I), one torsion ( 120 60 cm-I), and three bends (1 300 700,1400 700,300 150 cm-1) are all associated with the breaking bond. No bond contraction due to a double-bond formation occurs here (unlike the hexyl radical case) so the two frequencieswhich are associated with bond contraction given in ref 12 were not used here. The A factor thus obtained is the TST value of A = S-I. In a second set, the frequencies of the transition state were adjusted
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-. -
- -
456 14.5' 621 14.7 757 14.8
14.W 14.9 14.9 14.9
142 6.39 X 102 9.93 X 101 1.01 X 10, 192 1.34X 106 4.10X 106 9.83 X 106 229 5.23X l o 7 1.85X 10 3 . W X 10'
229 307 344 374
102 106 1(r 109 2.24 X 100 3.52 X 100 3.31 X 10' 9.51 X 10, 2.44 X lo5 5.23 X 105 3.45X 106 8.35X 106 1.35X lo7 4.14X lo7 9.34X IO7 1.31 X l(r 10' 106 l(r 109
5.24X 8.29X 2.14X 2.61 X
Including ionization potential in the case of molecular ions. One photon = 4.7 eV. b Em = excitation energy + thermal energy. e From ref 13. Fromref 14. Afrom transition-statetheory,fr#luenciesfromTable 1.fFrom ref 1. to give the experimental value for the dissociation of n-hexane to give two CJH, radicals, log A = -16.3 at shock tube temperatures of 1500 K.I3 These we felt are the best values that can be used for the large hydrocarbon which is our model molecule. The ratio of the moments of inertia is I+/I = 1.2. Since the molecule can decompose from either end, the paths degeneracy was taken as two. In one set of calculations, the activation energy for the molecular ion dissociation is taken as EO = 0.95 eV (21.9 kcal/mol) (calculated by subtracting the ionization potential 10.95 eV of propane from the appearance potential of CzHs+ fragment, 11.90 eV14). This is the value for the dissociation of the propane molecular ion to CzHs+ and CH3.14 Similar, or lower, values for EOare reported for the dissociations of n-butane molecular ion.I4 Another value for the activation energy of a supermolecular ion is 2 eV (46 kcal/mol) for the supermolecule melittin with -700 normal modes (taken from ref 1). The results of our calculations are given in Tables 3 and 4, together with values reported in ref 1. All our calculations were done at total energy, E,, which includes internal energy which is a function of the excess photons plus 80 or 60 kcal/mol thermal energy at 300 K for s = 1008 and s = 720, respectively. The vibrational temperature is calculated from eq 3, and the effective number of modes is calculated from eq 4. As can be seen, the values obtained by RRKM theory (eq 6), by the effective RRK theory (eq 7), and by the Arrhenius model (eq 9) with EOvalue s-I as in ref 1 but with a more realistic A factor ( A m ) , (exact temperature-dependent values are in Tables 3 and 4), give values which are 24&1600 larger than any of the modified RRK theory in ref 1. Using experimental A values13in the calculation yields values of k(E) which are lOs-l@ larger than using any of the modified RRK theory in ref 1. Using the experimentally accepted value of Eo = 21.9 kcal/mol, dissociation is affected by as little as two and three excess photons for a 720- and 1008mode ions, respectively. Our calculations show that, with very reasonable parameters, large molecular ions can be observed to decompose in a mass spectrometer and will do so in some cases with very little excess photons. Contrary to the facile decomposition of molecular ions in a mass spectrometer, molecular fission
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The Journal of Physical Chemistry, Vol. 98, No. 1, 1994 139
Unimolecular Dissociation of Polyatomic Molecules
TABLE 4 Vibrational Temperatures, Effective Number of Modes,RRKM, Effective RRK, and Arrhenius Type Expression Rate Coefficients for the Decomposition of a 1008-Mode Long-chain Hydrocarbon (See Text for Definitiolrs of Parameters)
Moltcule (EO= 69.8 kcal/molc) 9
10 12 15
1053 1161 1377 1701
1172 1243 1379 1576
16.2C 16.2 16.2 16.3
452 470 503 543
1.54X 1.08 X 2.32 X 7.07 X
102 103 104 10
5.40X 3.49 X 7.04X 2.61 X
102 1.53 X 102 103 8.46 x 103 104 1.37 x 105 106 4.17 X 106
molecular ion (EO= 21.9 kcal/mol’) 152 260 368
417 15.7C 183 3.47 X 102 2.02X lo3 1.64 X 104 548 15.9 239 7.32 X 1 0 5.97 X 106 1.47 x 107 655 16.0 283 3.69 X lo7 2.93 X 1 V 4.93 x 108
152 260 368
417 1 4 3 183 6.36 X 10’ 1.28 X 102 1.05 x 103 548 14.6 239 9.90 X 104 2.99 X lo5 7.34 x 10’ 655 14.7 283 4.25 X 106 1.47 X lo7 2.47 x 107
Molecular Ion (EO= 46.0 kcal/moo 5 8 10 12
368 655 16.W 692 921 16.1 908 1075 16.2 1124 1219 16.2
283 378 425 464
1.20 X 10-1 4.01 X 10-I 4.48 1.45 X 104 6.37 X 104 1.53 X 10’ 8.24 X 10’ 4.01 X 106 7.05 X 106 1.43 X lo7 6.01 X lo7 8.97 x 107
5 8 10 12
368 655 14.7’ 692 921 14.8 908 1075 14.9 1124 1219 14.9
283 378 425 464
1.42 X 10-2 2.01 X le22.25 X 1.32 X lo3 3.19 X 103 7.66 X 6.88 X 104 2.01 X 1 0 3.53 x 1.12X 106 3.01 X 106 4.49 x
10-1 103 105 106
a Including ionization potential in the case of molecular ions. One photon = 4.7 cV. Et& = excitationenergy + thermal energy. From ref 13. Fromref 14. e A from transition-statetheory,frequenciesfromTable 1.fFrom ref 1.
NUMBER OF PHOTONS
3
5
7
9
III
’
: : I0 4
10
2 0 3 0 4 0 5 0 6 0 ENERGY (eV 1
Figure 1. Unimolecular rate coefficient vs energy for molecular ion dissociation with 720 and 1008 modes, EO = 21.9 kcal/mol, and for molecular fissions of hydrocarbon molecules with the same number of s-l (see modes and EO = 69.5 kcal/mol both with A factor of
Tables 3 and 4).
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has an Arrhenius activation energy of 72.1 kcal/mol (from which the threshold energy for dissociation Eo is calculated to be 69.5 kcal/mol) and therefore requires higher energies for dissociation in the mass spectrometer “time window”. Twelve and 15 photons are needed for 720- and 1008-mode molecules, respectively. The situation is depicted in Figure 1 where the unimolecular rate coefficients are given as a function of internal energy for molecular and molecular ion fission. It is clear that ionization followed by dissociation is the energetically preferred route and that it is done in a time window which is perfectly suited for observation in a mass spectrometer. The idea of a molecule being its own heat bath such that k,, Le. Arrhenius equation, applies is very appealing and easy to use provided Tv is known. The latter is easy to calculate via eq 3. To check whether the idea of inter-intramolecular vibrational relaxation (IIVR) is reasonable, we have made preliminary calculations of possible configurations of a CIIHz1molecule by
molecular mechanics. We have sorted out those configurations in which segments are van der Waals distances apart. Our preliminary results indicate that indeed it is possible to obtain IIVR in long-chain molecules. There is no qualitative difference in the numerical results between the effective RRK expression and Arrhenius equation, and either one can be used when exact RRKM calculations are not possible. Thevalueof k(E)calculated by the approximate methods is, on the average, larger by only a factor of 2-6 than thevalue of I E ~ K M . This is better than expected considering the different expressions for k ( E ) which are used. A factor which can easily increase the value of ~ ( E ) R R Ior ( MkmT by more than an order of magnitude is the reaction path degeneracy, L. We have taken a very conservative value of L = 2, but a 1000-mode molecule can break in various places, making L much larger and thus increasing the value of k ( E ) . Since the molecule is so large, it does not make a difference if fission occurs at carbon 50, 48, or 72. Eo and A are identical since the C-C bond energy is the same and the transition state involves only the modes around the breaking bond. Carbon 97 is not affected when the bond between its distant relatives carbons 31 and 32 is breaking. Reaction path degeneracies are important for the hydrocarbon molecules discussed here. In cases where there is a limited number of weak bonds, L will be the number of those bonds. For example, in cyclopropane isomerization L = 3 since all three bonds are identical. In cyclopropene L = 2 because only two bonds are identical and the weakest. We have also performed RRKM calculations with the anharmonic state counting with anharmonicity factor of 0.03. The values of k ( E ) obtained varied only by a little from the harmonic values of k ( E ) (see next section). The small difference between thevalues of harmonic and anharmonic k(E) comes about because the value of anharmonic number of states increases but so does the density, and since k(E) includes the ratio between the two, there is a cancellation effect and the difference between the two is minimal.I1 From the results for protein dissociation discussed later and presented in Table 5, it can be seen that the RRKM theory value for k(E) with five photons absorbed and Eo = 46 kcal/mol is 7 X 104 s-1 for harmonic counting of states and 4.5 X lo4 s-I for anharmonic counting. Protein Dissociation. We have applied the methods and procedures reported above to evaluate k ( E ) for the protein angiotensine with 540 modes15 which are listed in Table 2. The thermal energy content of angiotensine at 300 K is 59 kcal/mol. The reaction coordinate is a C-C bond rupture, and the choices of the transition-state frequencies are similar to those discussed in the previous section. Calculations with harmonic and anharmonic frequencies were done and are reported in Table 5 for two values of Eo as described previously for hydrocarbon dissociation. RRKM calculations indicate that absorption of 4-5 photons by the molecule provides enough energy for dissociation in the “time window” of a mass spectrometer. Here again we find that the number of effective modes is less than 50% of the actual number of modes in the molecule and that the value of the TST frequency factor is -2.5 X 1014s-1. The value of k ( E ) calculated by the effective RRK expression varies by a factor 1-3, depending on the energy, from the value of k ( E ) obtained by exact RRKM calculation. An interesting quasi-equilibrium theory of the rate of ionization was developed in ref 4. The energy was assumed to be statistically distributed among the normal modes of the molecule, the “core”, and an electron in Rydberg states. Marcus’ expression (eq 5) was used, and the densities and number of states of the molecule and transition state were used by convoluting the core classical densities with the density of the electronic states obtained from Bohr’s expression for an electron energy in a Rydberg system. With the reasonable assumption that any loss of the core angular momentum is gained by the electron (the sum of the total angular momenta of the two is null), an expression for the rate coefficient
140 The Journal of Physical Chemistry, Vol. 98, No. 1, 1994
Bernshtein and Oref
TABLE 5: Vibrational Temperatures, Effective Number of Modes, RRKM, Effective RRK, and Arrheniw Type Expression Rate Coefficients for the Decomposition of Andotensine Eo = 21.9b(kcal/mol) EO= 46.W (kcai/mol) N E~ot photons' (kcal/mol) T,(K) log A self ~(E)RRKM 6-l) ~(E)ERRK (PI) k ( E ) h (s-') ~(E)RRKM (6') k(E)@wtK(bl) &(E)- (s-') 200 7.17 X lo6 697 14.36 3.1 1 X lo7 1.61 X lo7 5.12X 1W2 277 8.63 X 10-I 3.86 X 10-2 2 3 4 5
385 494 602
846 984 1 1 15
14.41 14.44 14.46
229 252 272
1.67 X 1.27 X 5.47 X
lo8 lo9 lo9
5.68 X lo8 3.77X lo9 1.47X 1Olo
3.85 X lo8 2.92 X lo9 1.22X 1Olo
4.53 X 10' 3.29 X lo3 7.04X 10'
3.36 X 102 1.67X 10, 2.78 X 105
5.68 X 10' 5.20X 10) 1.20X 105
Anharmonic Model Calculations
N Etot photons (kcal/mol) 2 277 3 386 4 494 5 602 a
T,(K) log A 688 839 978 1110
14.35 14.41 14.44 14.46
sslt
203 231 254 273
EO= 21.9 (kcal/mol) ~(E)RRKM (8-I) ~(E)ERRK (s-I) 5.65 X lo6 2.47 X lo7 1.24X lo8 5.05 X lo8 8.90 X lo8 3.52X lo9 3.64X lo9 1.41 X 1Olo
EO
~(E)RRKM (8l) lo7 3.08 X 1W2 lo8 2.97X 10' lo9 2.15 X lo3 1Olo 4.47 X l@
&(#)A,, (s-l)
1.27X 3.44X 2.72 X 1.17X
= 46.0(kcal/mol jk(E)emx (s-l) &(E)- (8-I) 5.46 X 10-1 2.35 X 10-2 2.68 X lo2 1.45X 10, 2.53 X lo5
4.48 X 101 4.47X 103 1.09 X lo5
Photons above ionization potential. One photon = 4.7 eV. From ref 14. From ref 1.
for ionization, k@), is obtained which is identical to the RRK expression for dissociation of a molecule with A given by (2R/ h)(IP/R)3/2,4 where R is Rydberg's constant and IP is the ionization potential. To check whether this model applies, we have used an expression for k1(E) which is a modification of the one given in ref 4, i.e., eq 7 with IP replacing Eo. If internal rotations are present in the core, they should be included in eq 7 as well. The value of A is -3 X 1015 s-1, larger than the value of A used in ref 4 or used in eq 7 for molecular ion dissociation and given in Table 5. However, the ionization potential is >>EO and therefore k ( E ) > k@); molecular dissociation will be much faster than ionization even when five photons are absorbed by the molecule, contrary to experimental findings. In conclusion, using conventional RRKM calculations or treating a very large molecular ion as a canonical ensemble of oscillators with temperature T, and using various statistical models show that it is possible to observe unimolecular fission of such a molecular ion in a mass spectrometer.
Acknowledgment. 1.0. thanks Ms. Hagit Denekemp for bringing the problem to his attention and Professors R. D. Levine, E. W. Schlag, and Chava Lifshitz for very helpful and interesting discussions. This work was supported by the P. and E. Nathan Research Fund and by the Technion Vice President for Research
Fund (to LO,), by thecenter for Absorption in Science, Ministry of Immigrant Absorption, and by the Wolfson Family Charitable trust (to V.B.).
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