J . Phys. Chem. 1987, 91, 4809-4813
4809
agreement, there are unresolved quantitative discrepancies. The results from these light scattering experiments have not been included here in obtaining the temperature dependence of the tensions. This method has been the method of choice in several studies of low interfacial tensions near critical points and is likely in the future to allow the more extensive measurements in the vicinity of the tricritical point that are needed for the detailed description of the asymptotic behavior of the interfacial tensions.
and density difference, respectively, at the a@critical end point. These quantities as functions of the temperature are shown as logarithmic plots in Figures 17 and 18. ApaB,ris found to be well described by an expression of the form Appolg,r = Apl(Tt - T)0.48M@', with Apl a constant. This is in good agreement with the (Tt T)'12 behavior predicted by t h e ~ r y .From ~ this and the earlier , the behavior of a$r can be inferred. analysis for u , ~ , ~clearly For the case of ua8 = uBr = bo, the behavior of the capillary constants and density differences are very different when the two interfaces are compared. For the CY@interface, the variation in a$o is very strong and well described by atg,o (Tt - T)'.9. AQ,O is seen to show a very small increase at first before changing to a small decrease with rise in temperature. In other words, at the a/3 interface, essentially the entire change in the interfacial tension a,@ = uo as a function of temperature comes from the capillary constant. Ap8r,ois seen to vary as ApBr,o (Tt and again, the behavior of a;,,? can be inferred. Here, in contrast to the CY@ interface, the variation in density difference is seen to contribute substantially to that in uBr = uo. These observations may have some bearing on the question of symmetry earlier discussed. The theory predicts that the isothermal variation of urraand up., is symmetric about uaB = uBr = uo, and such symmetry would be predicted to hold more generally for various other properties, in particular the density differences and hence also the capillary constants. But clearly, from the data, at each temperature, a$o > a;,,o and A P , ~ < , ~A P ~ ? , ~Then . from the discussion presented about the variation with temperature of each of these quantities and from the data in Table VI11 it is clear that this asymmetry decreases with increasing temperature; Le., with increasing temperature the ratio Ap8y,o/Apug,o,and hence a&o/a;,,o, decreases from about 6.9 at 21 OC to about 1.9 at 46 OC. In collaboration with Professor C. M. Knobler and Dr. M. C. Goh, we have also attempted to apply the technique of inelastic light scattering off thermally excited surface waves to determine the interfacial tensions, primarily with a view to approaching the tricritical point more closely. For samples of the same compositions at 46 and 47 OC, the two methods yield results that are generally in accord with each other. A few samples at 30 O C were also looked at by this method. Here the results from capillary-rise measurements are very reliable, and while there is again qualitative
-
Summary The interfacial tensions in the three-liquid-phase region of the system ammonium sulfate water ethanol benzene have been measured by capillary rise between 21 and 47 OC with the primary intention of determining the behavior of the tensions on approach to the tricritical temperature Tt of 49 OC. Two paths of approach have been considered, one where at each temperature the tensions a,@ and ubyare equal, and the second the locus of a/3 critical end ~ determined. points, along which the noncritical tension u , ~ ,was Along both these paths, the relevant interfacial tensions are found to vary as (Tt - T)fi with I.( = 1.9 f 0.15. The density differences and hence the capillary constants are found to behave very differently depending on the path. Along the line of a@critical end points, the limiting density difference is found to vary as ( Tt T)048M.04, which is in accord with the behavior (Tt - T)'12expected from t h e ~ r y .On ~ the other hand, along the path uaB= ubr, the behavior is very different for the two interfaces. At the a@interface, shows very little variation over this temperature This range, while at the 07 interface, AP&,~varies as (Tt observation may be related to the fact that simple theory predicts symmetric behavior for various properties in the three-phase region, which is not found to be the case in the experiments, but which may hold true very close to the tricritical point.
+
-
+
+
Acknowledgment. This work was supported by the National Science Foundation and the Cornel1 University Materials Science Center. We are grateful to Professor C. M. Knobler and Dr. M. C. Goh for their collaboration in the light-scattering measurements that were mentioned in the text. Registry No. Ethanol, 64-17-5; benzene, 71-43-2; ammonium sulfate, 7783-20-2.
A Theoretical Study of the Reaction between NH, and HOP Claude Pouchan,*+Brenda Lam, and David M. Bishop* Department of Chemistry, University of Ottawa, Ottawa, Canada KIN 9B4 (Received: June 16, 1986; In Final Form: February 6, 1987)
-
-
-
It has been proposed that the reaction NH2 + H0, N H 2 0 0 H * H2N-OzHl HNO + HzO might play an important role in the photooxidation of ammonia. To see if this is feasible, we have carried out ab initio calculations of the energies of the species involved. Our mast accurate results show that the energy difference between the cis-NH,OOH addition complex and the transition-state complex is very large (more than 60 kcal-mol-') and that the energy difference between the reactants and the transition state is approximately 16 kcal-mol-'. It would therefore appear that, as a channel in the photooxidation of ammonia, this reaction will not be significant.
I. Introduction Ammonia is an important constituent of the atmosphere and the way in which it is photooxidized is thus equally important. Kinetic studies suggest'.2 that in the flash photolysis of the NH3-N2-02 system, the following reactions occur
+ + +
+
+
4
-
+
-
4
'Permanent address: Laboratoire de Chimie Structurale, UA 474, Universitd de Pau, 64000 Pau, France. *Member of the Ottawa-Carleton Chemistry Institute.
+ +
NH3 h~ NH2 H H 0 2 M -.+ HO2 M NH2 NH2 M NzH4 M N H 2 HOz .+ products HO2 + HO2 HzO2 + 0 2 H + H 0 2 products and the combination of the amidogen radical (NH,) with the
0022-3654/87/2091-4809$01.50/00 1987 American Chemical Society
+
4810 The Journal of Physical Chemistry, Vol. 91, No. 18. 1987
Pouchan et al. H4
TABLE I: SCF Geometries, Frequencies, and Ewrgies for the Reactants and Products geometric harmonic parameters vibratal freq, cm-I energy (Eh) NH2 rjqH = 1.0117 A 3315, 3224, 1547 -55.499246 = 104.2O roo = 1.3129 A rOH = 0.9536 A BHm = 105.4' rOH = 0.9480 OHOH = 105.3' rNO = 1.1735 A rNH = 1.0322 A OHNO = 108.8'
HE;.:.
I
BHNH
H02 H2O HNO
a
3578, 1459, 1020
-150.023878
3726, 3620, 1646
-75.939003
2967, 1774, 1565
-129.662261
+
-
HO~
-
NH,OOH*
/
0 2
-
H--N\-o '\
0,-
IT
peroxy radical (HO,) is thought to be the principal reaction. The mechanism for this reaction, using NH2 + NO as an a n a l ~ g u e , ~ has been postulated by Lesclaux4 as NH,
,
V
-__. .."N
\
H---0-H HNO
+
H20 (1)
m
The reason for suggesting this reaction is that the formation of H N O would then allow for rapid N 2 0 production via 2HN0 N20 H20
-
+
and account for its observed presence5 which would not be explained by the perhaps more likely reaction NH2 HO2 NH, + 0 2
+
-+
However, since the intermediate N H 2 0 0 H * and the transition-state complex H-N-0 '\,
\
H---0-H
have never been investigated theoretically, it has not been possible to assess the feasibility of this proposed mechanism. In the work reported here, we have calculated the energies of all the species involved in (1) and are able to show that this mechanism is extremely unlikely. Our studies have been based on the self-consistent field (SCF), multiconfigurational SCF (MCSCF), and configuration interaction (CI) methods. At the SCF level, as well as finding the energies, we have also determined the optimal geometries and harmonic vibrational frequencies.
II. Method and Results
+ HO2
+
NH200H
+
[NHZOOH]'
-
HNO
+ HzO
(2) were optimized by using analytic gradient procedures.6 Split valence polarization 4-31G* basis sets [with d(N) = 0.80 and d ( 0 ) = 0.851 were used in these optimizations within a restricted Hartree-Fock (RHF) procedure. The equilibrium configurations of the ground states of the reactants (NH,, H 0 2 ) and products (HNO, H 2 0 ) which we have found are shown in Table I; they are in good agreement with previous experimental and theoretical results.' For the addition compound NH,OOH there are no experimental data and our calculations were made without imposing ~~~
TABLE E SCF Geometries, Frequencies, and Energies for the Addition Complexes and Transition State in the NHz + HOz Reaction' structure 111. structure I, structure 11, cis-NH202H rruns-NH20zH transition state r I.7
1.401
1.382
0.95 1 1.369 1.004 1.004 108.79 107.13
0.952 1.388 1.003 1.002 107.53 103.66
106.72 102.07 117.62
105.89 103.06 71.21
57.86 57.12 3677, 3402, 3312 1679, 1391, 1348 1228, 1004,940 509, 364, 123 -205.548413
100.1 1 146.26 3657, 3407, 3311 1652, 1432, 1340 1195, 1009,969 542,295, 220 -205.545873
r15
r16 r23 r34 r35 8123
8351 BS12
e615
Stationary points on the SCF potential energy surface for
NH2
H6
Figure 1. Structure of cis-NH200H (I), truns-NH,OOH (11) and transtion-state complex (111).
~~~~
(1) Cheskis, S . G.; Sarkisov, 0. M.Chem. Phys. Lett. 1979, 62, 72. (2) Lesclaux, R. Rev. Chem. Intermed. 1984, 5, 347. (3) Abou-Rachid, H.; Pouchan, C.; Chaillet, M.Chem. Phys. 1984, 90, 243. (4) Kurasawa, H.; Lesclaux, R. Paper presented at the XIVth Informal Conference on Photochemistry, Newport Beach, CA, 1980. (5) Jayanty, R. K. M.; Simonaitis, R.; Heicklen, J. J . Phys. Chem. 1976, 80, 433. (6) Schlegel, H. B. J. Comput. Chem. 1982, 3, 214. (7) (a) Sutton, L. E. Chem. Soc., Spec. Publ. 1965, 18. (b) Whiteside, R. A.; Frisch, M.J.; Binkley, J. S.; Defrees, D. J.; Schlegel, H. B.; Raghavachari, K.; Pople, J. A. Carnegie-MellonQuantum Chemistry Archive, 2nd ed.; July 1981.
e612 861-23 851-23 e43-21 853-21
3
'r
1.583 1.152 0.958 1.357 1.012 1.422 94.42 75.88 113.93 75.54 109.82 101.68 102.44 3.71 3570, 3210, 2079 1901, 1516, 1465 981, 903, 835 772, 479, 28 1 -205.423381
in angstroms, B in degrees, frequences in cm-I, and energies in
hartree. any symmetry constraint. We found that there are two local minima on the potential energy surface, corresponding to the cis and trans isomers; the geometric parameters for these two structures (see Figure 1) are given in Table 11. The transition state [NHzOOH]' was detected with the methodology of Schlege16and the unconstrained C,symmetry was used. Once the transition state had been found (the geometric parameters are given in Table 11; see also Figure l), the reaction could be traced forward along the intrinsic reaction coordinate (IRC)* to the products or backward to the reactants. This pro(8) (a) Fukui, K. J . Phys. Chem. 1970,74, 4161. (b) Ishida, K.; Morokuma, K.; Komornicki, A. J . Chem. Phys. 1977, 66, 2153.
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987 4811
Reaction between N H 2 and HO2 TABLE III: MCSCF and CI (See Text) Energies (in atomic units) for Species Involved in the Reaction between NH2 and H02 species
NH2 H02 H20 HNO c ~ s - N H Z O ~ (I) H
truns-NH202H(11) transition state (111)
MCSCF energy
CI energy
-55.548 200 -150.1 11 930 -75.991 330 -129.794360 -205.718 352 -205.7 14 13 1 -205.626210
-55.653 117 -150.348 710 -76.129 387 -129.995 800 -206.086519 -206.080 405 -205.977 152
TABLE IV: Values (in kcalmol-') of Certain Thermodynamic Functions for the Reaction between NHz and HOz
111. Discussion
LiE
-
process
+ HO, + NHI + H02 NH,
AG0298
298
MCSCF
CI
MCSCF
-78.8
-77.4
-17.6
-36.5
-53.1
-33.9
-49.3
21.3
15.5
CI
MCSCF
CI
-76.2
-77.4
-76.0
-31.8
-48.4
-21.7
-38.3
-29.1
-44.5
-19.0
-34.4
H20 HNO
cis-NH202H NH2 + H02 --c trans-NH202H NH2 HO2 transition state
+
+
'Activation energies ( E , =
u0298*
have been evaluated using the S C F harmonic frequencies and structural parameters. These have been used to obtain the AH'298 and results given in Table IV. The internal energy change for reactants to transition state is 63.1, 21.9, and 16.1 kcabmol-I for the SCF, MCSCF, and C I levels of approximation, respectively. With respect to the process occurring via a four-center transition state, as can be expected, correlation effects lower drastically the activation energy, greatly overestimated at the SCF level. The three program packages used in this work GAUSSIAN 82,15 GAMESS,]' and C I P S I ' ~were run on the University of Ottawa's AMDAHL 470 or the CCVR's CRAY 1 computer.
21.3'
15.5Q
+ nRT).
cedure was carried out using an improved automated process for determining the I R C 9 The stationary points were characterized (minimum, saddle point etc.) by calculating at each one the second derivatives of the energy with respect to nuclear displacement. These derivatives were also used to find the harmonic frequencies. Since it is well established'O that S C F vibrational frequencies are approximately 10% too large, the values given in Tables I and I1 have been reduced by 10% so as to give a more realistic estimate of the actual values. In order to make more precise energy predictions, single point calculations were made for the reactants, products, cis and trans addition compounds, and transition-state complex with MCSCF and C I wave functions and the 4-31G* basis set. The MCSCF calculations were performed with the GAMESS computational package" using as an initial guess the orbitals of the R H F calculations in a 4-31G* basis. Configurations were built from all single and double excitations in the full orbital valence space (this leads to 1326 spin-adapted-antisymmetrized products for the transition and addition complexes); energies were corrected by Davidson's methodlZand are given in Table 111. The configuration interaction (CI) calculations were performed by taking for the configurations all single and double excitations from the appropriate S C F reference state. The CIPSI algorithmI3 was used for successive diagonalization of the more and more important parts of the selected subspace (up to 5000 determinants). The final extrapolated energies14 were again corrected by the Davidson method', for quadruple excitations and are shown in Table 111. The principal limitations of the present study at the higher levels of theory are the use of geometries calculated at a level which does not include electron correlation. To allow for direct comparison with the experimental thermodynamic properties, zero point energies and partition functions (9) Dupuis, M.; Schmidt, M. W.; Gordon, M. S. J . Am. Chem. SOC.1985, 107, 2585. (10) (a) Yamaguchi, Y.; Schaeffer 111, H. F. J . Chem. Phys. 1980, 73, 2310. (b) Pouchan, C.; Dargelos, A.; Chaillet, M. J . Chim. Phys. 1978, 75, 595. (1 1) Dupuis, M.; Spangler, D.; Wendoloski, J. J. NRCC Cut. Prog. QGOl 1980. (12) Langhoff, S. R.; Davidson, E. R. Int. J . Quantum Chem. 1974,8, 61. (13) Huron, B.; Malrieu, J. P.; Rancurel, P. J . Chem. Phys. 1973, 58, 5745. (14) Buenker, R. J.; Peyerimhoff, S. D.; Butscher, W. Mol. Phys. 1978, 35, 771.
The changes in geometry which ensue in the course of the reaction NH2
+ H02
-
N H 2 0 0 H (I, 11)
-
-
[ N H 2 0 0 H ] * (111) HNO + H20 (2)
(see Tables I and I1 and Figure 1) are relatively straightforward. Two bonds remain unbroken (0,-H6 and N3-H4), the rI6length remaining practically constant and the r34 length decreasing slightly in the addition complex but then increasing to its larger value in HNO. Two bonds are broken (01-O2and N3-H,); the r12length increases during each step of the reaction but the r35 length decreases slightly in the addition complex before increasing. Two new bonds are formed (Ol-H5 and N3-02), and the r I 5 distance decreases as does the r3, distance in a logical fashion. The H6-01-H5 angle is '5 larger in the transition-state complex than in HzO; the H4+3-02 angle changes irregularly from 107' in the addition complex to 102' in the transition-state complex to 109' in HNO. In the cis and trans addition complexes, the bond lengths and angles, except for the dihedral ones, are all essentially the same. In the transition-state complex, the ring 01-02-N3-H5 is almost planar since the dihedral angle 851-23 is only 3.7'; the 861-23 dihedral angle lies between the corresponding angle in the cis and trans compounds. The vibrational frequencies for the cis and trans forms of N H 2 0 0 H are, in general, very similar. For both forms the highest frequency can be attributed to an OH stretch with uOH = 3677 and 3657 cm-' for the cis and trans isomers, respectively. These values are 80 to 100 cm-' smaller than the corresponding value for HO,, reflecting a shortening of the OH bond in the addition complexes. For the modes associated with the NH, group, the ~is higher ~ than ~ the , symmetric ~ ~ uNH2,+; this pheasymmetric u nomenon is also true for NH2O2,I6NH2N0,3*17 and NH2N0218 and is explained by the fact that the interaction force constant FNH~,+,NH as is negative. The frequencies lying between 1200 and 1680 cm-t can be associated with N H deformations ~ (aNHl,+,rNH2, uNH2) and the hydroxyl deformation aOH, this latter is found at 1391 and 1432 cm-' for the cis and trans complexes, respectively, and is not much different from the value we have calculated for HOz (1459 cm-I) or that given by JacoxI9 for HO, (1 397 cm-I). The two modes in the 1000-cm-' region are essentially 0-0 and N-0 bond stretches which are coupled differently for the two isomers; it might be noted that we find uG0 for H 0 2 to be in this same region (Le., at 1020 cm-I). Lower still, there is a mode between 510 and 540 cm-' which corresponds to a deformation of the N O 0 skeleton and is compatible with a similarly observed mode in N H 2 0 2at -560 cm-'.I6 Finally, the last two frequencies can be attributed to Y~~ and skeleton torsion and are in line with the frequencies calculated for H02N0220and NH20,.I6 (15) Binkley, J. S.; Frisch, M. J.; Defrees, D. J.; Krishnan, R.; Whiteside, R. A.; Schlegel, H. B.; Fluder, E. M.; Pople, J. A. GAUSSIAN 82; CarnegieMellon University: Pittsburg, PA, 1983. (16) Pouchan, C.; Chaillet, M. Chem. Phys. Left. 1982, 90, 310. (17) Abou-Rachid, H.; Pouchan, C. THEOCHEM, 1985, 22, 288. (18) Nonella, M.; Miiller, R. P.; Huber, J. R. J . Mol. Spectrosc. 1985, 112, 142.
(19) Jacox, M. to be submitted for publication. (20) Saxon, R. P.; Liu, B. J . Phys. Chem. 1985, 89, 1227.
4812
The Journal of Physical Chemistry, Vol. 91, No. 18, 1987
In going from the addition complex through the transition state to H 2 0 and HNO, there is the simultaneous formation of an 0-H bond and the breaking of an 0-0bond. As this happens the other bond lengths and angles adjust such as to lead to the stable geometric structures of the products. The reaction coordinate (v* = 2079i cm-I) gives a transition vector X,,,
= 0.66r12- 0.57r15- 0.29r23+ 0.3203,,
-
+ HO2
+
HNO
+ H2O
+ O2H
78.4
, 57.8,
68.6
(3)
if we consider that the most accurate experimental values of the heat of formation are respectively 45.3 or 45.9 kcabmol-' for NH2,21v22 3.0 f 0.4 kcal-mol-' for H02,23-57.8 kcal-mol-i for H20,24and 24.7 f 1 kcabmol-I for H N 0 2 5 thus the heat of reaction 3, AH3'298K, is within the range -79.4-83.4 kcal-mol-I. This result is consistent with the values (-80 and -8 1 kcal-mol-'), respectively, reported by Lesclaux, and calculated by Melius et from bond additivity corrections applied to fourth-order Mdler-Plesset perturbation theory (BACMP4). For the addition process NH2
*
+
The reliability of the present energy calculations is perhaps best assessed through a comparison with the and theoretical26enthalpies of formation. For the reaction NH2
Pouchan et al.
--T I
-
-- - - - -
15.9,36.5, 53.1 -+
NHzOOH (I, 11)
(4)
no experimental or theoretical determination of heat of reaction is actually available and, in order to obtain this enthalpy, it is necessary to estimate a heat of formation for the N H 2 0 2 Hspecies. According to Lesclaux2 and Shum et al.,27we can assume that the bond dissociation energy (&H2OrH) in N H 2 0 2 His comparable to the values adopted for the isoelectronic molecule CH,COOH (DCH,O2-H 90 kcal-mol-I) and R02-H species (DHO2+= 89.6 kcal-mol-'). Moreover, the value of 41.2 kcalqmol-I determined by Melius et a1.26from the BAC-MP4 method appears to be the most reasonable estimation of the heat of formation for N H 2 0 2 . Thus, taking Ahf0298K(H) N 52 kcal-mol-', these estimations yield a value of 3.2 kcal-mol-' for Ahfozss (NH202H). Consequently, the heat of reaction in the addition process 4 appears to be about -46 kcal-mol-I. From Figure 2 and Table IV, we see that the calculated values of AHozssfor process 3 are quite good at the MCSCF and C I levels (-77.6 and -76.2 kcal-mol-], respectively), though extremely poor at the S C F level (-47.8 kcalvmol-I); this is to be expected when applying the restricted Hartree-Fock method to open-shell species. Moreover, our correlated values appear in good agreement with those of Melius et a1.26 (-81 kcal-mol-') calculated at the MP4 level with bond additivity corrections applied to fit experimental data. For process 3 it should be emphasized that our correlated values are in error by about 4 to 5 kcal-mol-' with respect to the average experimental heat of reaction. This accuracy can be expected for the others reactions studied. For the second process (4), the MCSCF value is approximately -30 kcal-mol-I (-3 1.8 for cis and -29.1 for trans) and the CI value is approximately -46 kcal-mol-' (-48.4 for cis and -44.5 for trans). This speaks eloquently in favor of the latter calculations and implies that with the MCSCF method more than just single and (21) Sutherland, J. W.; Michael, J. V.; Paper presented at the 9th International Symposium on Gas Kinetics, Bordeaux, July 1986). (22) Hack, W.; Rouveirolles, P.; Wagner, H . Gg. J. Phys. Chem. 1986, 90, 2505. (23) Hillis, A. J.; Howard, C. J. J . Phys. Chem. 1984, 81, 4458. (24) Benson, S. W. Thermochemical Kinetics, 2nd ed.;Wiley: New York, 1976. (25) (a) Clement, M . J. Y.; Ramsay, D. A. Can. J . Phys. 1961, 39, 205. (b) "JANAF Thermochemical Tables"; Natl. Srand. Re$ Data Ser. ( U S . ) Natl. Bur. Stand 37; U S . Government Printing Office: Washington, DC, 1971. (c) Bruna, P. J. Chem. Phys. 1980, 49, 39. (26) (a) Melius, C. F.; Binkley, J. S. ACS Symp. Ser. 1984, 249, 103. (b) Melius, C. F.; Binkley, J. S. Chem. Phys. Proc. Combust. 1983, 39. (c) Melius, C. F.; Binkley, J. S., private communication. (27) Shum, L. G. S.; Benson, S. W. J . Phys. Chem. 1983, 87, 3479.
NH202H
---A!--H2O + HNO Figure 2. Relative energies (AE) of species involved in the reaction of NH2 and H 0 2 in k ~ a l . m o l - ~The . first, second, and third numbers refer to SCF, MCSCF, and CI calculations, respectively.
double excited configurations in the valence space are needed for the supersystem, though this suffices for the reactants and the products. Hence, considering our C I calculations as the most accurate, we predict that the energy difference bewtween cisN H 2 0 0 H and [ N H 2 0 0 H ] *to be 64.5 kcal-mol-]. There is a relatively good correspondence between this high predicted barrier and the activation energies obtained via a four-membered transition state as in the mechanism of formic acid decarboxylation28 ( E , = 77.6 kcal-mol-') and in the dehydration process of acetic ( E , N 67.5 kcal-mol-]). This value is much too large acid29*30 therefore for reaction 2 to be a viable channel in the photooxidation of ammonia since it leads to a transition-state energy 16 kcal.mol-' higher than the energy of the reactants. This does not, however, exclude the possibility of breaking N H 2 0 0 H * into two radicals ( N H 2 0 and O H ) with further reactions being possible. In all three methods (SCF, MCSCF, and CI) we find the cis addition compound to be of lower energy than the trans, the difference being 1.6, 2.6, and 3.8 kcal-mol-,, respectively. Since the energy differences between the two isomers are little affected by electron correlation, we have calculated the inversion and rotation barriers of N H 2 0 0 H only at the S C F level. We find that internal rotation requires less energy (7.8 kcal-mol-I) than that for inversion (1 1.4 kcal.mol-'). These results are similar to those recently obtained by Gordon et aL3I for the species X H 2 0 H ( X = N, P). (28) Ruelle, P.; Kesselring, U. W.; H6 Nam tran. J. Am. Chem. SOC.1986, 108, 371.
(29) Bamford, C. H.; Dewar, M. J. S. J . Chem. SOC.1949, 2877. (30) Ruelle, P. Chem. Phys., to be submitted for publication. (31) (a) Schmidt, M. W.; Yabushita, S.;Gordon, M. S. J . Phys. Chem. 1984,88, 382. (b) Yabushita, S.;Gordon, M. S . Chem. Phys. Lett. 1985, 117, 321.
4813
J . Phys. Chem. 1987, 91, 4813-4820 Acknowledgment. C.P. acknowledges the award of a Canada-France Scientific Exchange grant. D.M.B. thanks the Natural Sciences and Engineering Research Council of Canada for financial support. We are indebted to Martin Laplante for computational advice and help. We thank the computing center at
the University of Ottawa and the Conseil Scientifique du CCVR. Finally, we are grateful to Professor R. Lesclaux for several helpful suggestions. Registry No. NH,, 13770-40-6; HOz, 3 170-83-0.
Bimodality in the Cooperative Binding of Ligands to Molecules with Multiple Binding Sites Johannes Reiter and Irving R. Epstein* Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254 (Received: October 23, 1986; In Final Form: April 2, 1987)
In an isolated system the equilibrium distribution of the number of ligands bound to a molecule with multiple binding sites can have two maxima if the binding is sufficiently cooperative. If the minimum between the two maxima is deep enough, the observed behavior resembles bistability and the kinetic behavior differs significantly from that expected in systems with a unimodal distribution of bound ligands.
Introduction For many years a sizable majority of chemists believed that chemical reaction systems always evolve to a unique stationary state. Although oscillating chemical reactions were observed in the first half of this century,' it is only in the past 20 years that the existence of nonlinear chemical phenomena* like bistability, chemical oscillations, and chaos has been widely accepted by the scientific community. From the work of Onsager it follows that for an ideal reversible chemical system with detailed balance there exists a unique equilibrium point with all its eigenvalues negative and rea1.3,4 Thus, in such a system, no unstable steady state is possible. This observation, which holds so long as the system follows ideal mass action kinetics, Le., the rate constants are indeed constant, rules out bistability in an isolated system. If the activity coefficients vary strongly with the concentration of the species, bistable behavior is in principle possible in an isolated ~ y s t e m . ~We , ~ consider in this paper only ideal systems. An isolated ideal chemical system that is detailed balanced in each step must evolve as stated to a unique equilibrium point. If one violates the condition of detailed balance, however, steady states can be unstable, making bistability and other exotic behavior possible. This is most easily achieved by an influx of matter. Energy fluxes have also been employed.' Nonsteady and multistable (ideal) chemical behavior is synonymous in the chemical literature with open systems. In this paper we show that the kinetics of even an ideal isolated chemical system which is detailed balanced in every reaction step may resemble bistability (at low population numbers). The cooperative binding of a ligand to a molecule (lattice) with multiple binding sites can show a bimodal density distribution in the extent of binding at equilibrium.8 First we examine the conditions for the distribution to be bimodal. We then investigate the binding kinetics and show (1) Bray, W. C. J . Am. Chem. SOC.1921, 43, 1262. (2) Vidal, C.; Pacault, A,, Eds. Nonlinear Phenomena in Chemical Dynamics; Springer-Verlag: Berlin, 1981. ( 3 ) Onsager, L. Phys. Reu. 1931, 37, 405. (4) Bauer, H. J. In Physical Acoustics; Mason, W. P., Ed.; Academic: New York, 1965; Vol. IIA, p 47. (5) Ebeling, W.; Sandig, R.Ann. Phys. (Leipzig) 1973, 28, 289. (6) Gitterman, M.; Steinberg, V. Chem. Phys. Lett. 1978, 57, 455. (7) Nitzan, A.; Ross, J. J . Chem. Phys. 1973, 59, 241. Zimmerman, E. C.; Ross, J. J. Chem. Phys. 1984, 80, 720. (8) Bimodality was first demonstrated for the helix-coil equilibrium in DNA. Applequist, J.; Damle, V. J . Am. Chem. SOC.1965, 87, 1450.
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that in the limit of very strong cooperativity the bimodal distribution becomes bistable under an approximation which replaces the distribution by its maxima. The model we introduce initially is somewhat artificial but serves to show the general principles. We then discuss more realistic binding equilibria. Long-Range Cooperativity A Simple Model. We consider a large molecule with m identical binding sites and a small ligand that binds to and covers a single site. We refer to the large molecule as the lattice. We have the following reaction scheme
G,
+L
-
G,+l
q = 0, ..., m
(1)
where G, denotes a lattice with q ligands bound and L stands for the free ligand. We use the same symbols in italics to represent species concentrations. The binding constant of the ligand to an individual binding site depends on the number of ligands already bound. We assume that the (q + 1)th ligand binds more strongly to an individual binding site than the qth ligand by a cooperativity factor fl, which is independent of the site to which the (q 1)th ligand binds. There are thus ( m - q ) identical reactions for the binding of the (q 1)th ligand. Physically, this model would correspond to a long-range interaction between the ligands, independent of distance, or to a gradual conformation change of the large molecule as q varies. In order to keep the model somewhat realistic, we should restrict ourselves to no more than about 20 binding sites. Also, the large molecule should preferably have spherical symmetry. In experimental systems the kinetics of dissociation is often anticooperative, which makes the binding cooperative. None of our conclusions are affected by whether the cooperativity appears in the association step, in the dissociation step, or in both. We arbitrarily take the cooperativity to occur only in the association step and obtain the following rate equations for the binding reactions:
+
+
dG,/dt = -kfWL(m - q)G, - ( k f / K ) q G , ~ W - +I ( k f / K ) ( q + l)Gq+l
+ k f W I L ( m- q + = 0, ..., m ( 2 )
The fundamental association rate constant for binding to an individual site of the bare lattice Go is kf. The association rate constant for a ligand to a single site of the large molecule with q ligands already bound is k f W . The dissociation rate constant for any bound ligand is k f / K ,independent of the number of ligands 0 1987 American Chemical Society
