5291
J. Phys. Chem. 1995,99, 5291-5298
Reaction Kinetic Modeling from PM3 Transition State Calculations Nelson Malwitz Sealed Air Corporation, 10 Old Sherman Tumpike, Danbury, Connecticut 06810 Received: April 22, 1994; In Final Form: September 13, 1994@
Semiempirical PM3 calculations systematically overstate the free energy of the second-order-reaction transition state of aromatic isocyanate reactions with monofunctional active hydrogen compounds. The corresponding reaction rate predicted by transition state theory is far lower than observed. It is found that PM3 results accurately predict quantitatively isocyanate reaction rates when calibrated calculated Gibbs free energies are applied to activated complex theory. Modeling is accomplished by fitting experimental data to the Arrhenius reaction rate constant expression obtained in PM3 calculations. Calibrating semiempirical transition state results with experimental data of phenyl isocyanate and carboxylic acids with varied ligands, it was possible to distinguish likely atomistic mechanisms from among several transition states that were fully characterized.
Introduction Semiempirical quantum-mechanical methods are available in computational packages in the public domain and several commercial packages. From the chemist’s or engineer’s perspective, quantum methods are important because of the potential to determine reaction rates. This capability is particularly useful in cases where competing or parallel reaction mechanisms determine reaction speed, yield, or for copolymerization, structure. Such is the case for the commercial reactions of isocyanates producing polyurethane elastomers, films or foams. A formulation may, for example, contain any combination of active hydrogen compounds such as primary or secondary hydroxyls, amines, carboxylic acids, and water. All compete for the attention of the isocyanate. Further, the water reaction involves sequential and parallel reaction paths. To more fully understand this complex reaction system, experimental data for the reaction of monofunctional alcohols and carboxylic acid with phenyl isocyanate are compared to computationalresults. By evaluation of data for various carboxylic acid ligands, it is possible to suggest a likely atomistic transition state mechanism from among several that can be successfully characterized by semiempirical methods. PM3 results, calibrated by a proposed scheme to known kinetic data, allow quantitative extrapolation of individual forward and reverse reaction rates in complex reaction mixtures that are impractical to determine experimentally.
gas constant, 0.001987 kcal/mol K;T = temperature, K; AG* = free energy to activated complex from starting state, kcal/ mol; where
*
The superscript designates TS; subscripts R and P designate reactants and products, respectively. HR, Hp, @ = AHHfe (fl - H@) = standard enthalpy of formation at T. AHfe is the standard heat of formation at 298.15 K. fl and H@ represent enthalpy contributions for the increase of temperature from 0 K to T and 298.15 K, respectively. Similarly SR,Sp, 9represent the (absolute) entropy at temperature, T. AHR* and A&* are standard enthalpy and entropy changes, respectively, for formation of the activated complex from the reactants at temperature, T. For a liquid or solid phase reaction the temperature dependence on collision frequency is weak. The reaction rate equation can be approximated by fixing temperature in the nonexponential portion of eq 1 at the reference temperature, p,298 K. Then ( k p l h ) has a constant value of 6.235 x lo’*s-l. For modeling purposes, it is convenient to set K and u as constants equal to unity, effectively combining these coefficients into the fitting constants of a model. The form of the kinetic expression will then yield the familiar straight line Arrhenius plot.
+
Theory According to activated complex theory, altemately referred to as collision or transition state theory, it is the change in free energy from a ground state to the transition state (TS) that determines the rate of a reaction and its equilibrium temperature. For gas phase reactions, the one molar reaction rate constant is derived from collision theory and may be calculated using the Eyring equation:’-3
k = Ko(kT/h)e‘-AG*’Rn where K = collision energy transmission coefficient (often = 1); o = relative stericlcollision diameter factor, dimensionless; k = Boltzmann constant, J K ; h = Planck constant, J s; R = @
Abstract published in Advance ACS Abstracts, March 15, 1995.
0022-365419512099-529 1 $09.0010
Calibration factors for the expanded form of AG* to fit experimental results are constants conveniently added to the exponential term because they are large, as discussed below.
Modeling The semiempirical PM3 parameter set used for this study is reported to have an average accuracy of &7 kcallmol for heats of f ~ r m a t i o n . ~Heat of reaction is predicted well by the difference
For the second-order reactions of isocyanates the enthalpy and entropy of the TS are found to be systematically large. Subsequently, the & and A 9 predictions are overstated, 0 1995 American Chemical Society
5292 J. Phys. Chem., Vol. 99, No. 15, 1995
Malwitz
resulting in the understatement of predicted reaction rates. However, the results are consistent within a family of reactions allowing calibration and comparison. The computational reactions are simulated gas phase models. It is useful, but not necessary, that the absolute reaction rates be determined experimentally in a low-dielectric solvent for the calibration. It is proposed that a useful general model equation for predicting the free energy of the activated complex using semiempirical calculations take the form
AG* = a
+ DAH,'
- T(Y
+ CAS,')
(5)
Combining eq 5 with eq 3, an expression to model the reaction rate from semiempirical calculations can be written as
where the pre-exponential constant is
TABLE 1: Experimental Reaction Rate Data for Phenyl Isocyanate Reactions with Lower Alcohols in Toluene (Uncatalyzed) reactant k4 x IO4 forward5 reverse' alcohol temp "C l/(mol s) E, kcal/mol E, kcaYmol T,, "C methanol 20 1.2 IO ethanol 30 0.6 9.5, 11 2-propanol 20 1.4 10 I-butanol 20 1.4 I-butanol 25 1.8 9.6 13.9 300 TABLE 2: Experimental Reaction Rates: Carboxylic Acid Reaction Kinetics with Phenyl Isocyanate (Solvent, Toluene; Catalyst, None; kl x lo4, V(mo1 s) temperature, "C carboxylic acid 60 70 80 E, kcal/mol acetic 2.15 3.84 5.55 11.1 n-butyric (pri) 1.66 3.00 4.43 11.5 i-butyric (sec) 1.16 2.18 3.16 11.6 DMBA" (tert) 1.03 2.00 2.88 11.8 benzoic (aromatic) 1.36 2.20 3.02 9.3 2.2-Dimethylbutyric acid.
A = (kp/h)e'Y'R'
(7)
in toluene.
Experimental-Computational Chemistry Version 6.2 of MOPAC was used for this study as adapted by CAChe Scientific, Inc., for an accelerated Macintosh computer. For the examination of isocyanate reactions with hydroxyl and amine compounds the PM3 parameter set was selected since it produced expected results. For the isocyanate reaction AM1 failed to predict the observed order of reactivity of amines and primary and secondary hydroxyl compounds. All studies were done at the single-determinant restricted HartreeFock level. The program was used to obtain heat of formation, optimum and transition state geometries, bond orders, atom partial charges, vibrational spectra, and enthalpy and entropy of molecules including TSs. Semiempirical methods are parametrized at 298 K. Calculations of enthalpy and entropy at temperatures other than 25 "C were adjusted by FORCE calculations. Transition states were located using the SADDLE method in MOPAC and subsequently refined minimizing the gradient using the nonlinear least squares and Eigenvector Following methods, respectively. The free energies of the TS and each optimized reactant and product molecule were determined by a FORCE calculation. The temperature range 0-300 "C was specified using the THERM0 keyword to calculate enthalpy and entropy at 25 "C intervals needed to determine the free energy of each reaction participant, including the TS. The number of free rotations and, in the case of the TS, the negative vibration were specified using the "TRANS =" keyword. The molecule symmetry was also specified. TSs were fully characterized. Each had a large single negative vibration. Intrinsic reaction coordinate calculations performed with a positive, then negative, direction along the negative energy vibration mode led to the expected reactants and products. Experimental- Wet Chemistry Models using monofunctional reactants are the most appropriate to consider because they are the only ones for which reliable experimental data can readily be obtained. In the present study the second order reactions of phenyl isocyanate with carboxylic acids and with lower alcohols were explored
H
0 amide
W'
N=C=O
+ R-OH alcohol
kq_ k5
H R./N,C/O,
II
R
(9)
0 carbamate
To fit the calibration constants of eq 5 for the forward and reverse reaction at least three, preferably four, elements of experimental data are required: (1) the reaction rate of either the forward or reverse reaction at a known temperature in the solvent of interest, if any; the activation energy (Ea)for the ( 2 ) forward and (3) reverse reactions, and (4) T q , the point at which the forward and reverse reaction rates are equal. At Tes,AG* and the rates for the forward and reverse reactions are equal. It is also useful to know the relative reaction rates of reactants in an analogous series to allow a generalized model expression applicable to the reaction type under scrutiny. Quantitative studies on isocyanate reactions in toluene were used to calibrate the model reaction of lower alcohols with phenyl isocyanate.5 Kinetic data of phenyl isocyanate with lower alcohols at equal molar ratios and in 1 M or less reactant concentrations without catalyst in toluene are summarized in Table 1. Data also exists in the literature using dioxane and alkyl ethers as solvent5 The reactions in toluene cannot be compared to studies in ether solvents, which reduce the partial positive charge on the isocyanate carbon, thereby altering the reaction rate. Unfortunately, all early data found in the literature for the carboxylic acid reactions were obtained by following the liberation of C02. This is not useful to determine ki since the intermediate mixed anhydride can readily reverse to starting materials or decarboxylate to an amide. The mixed carbamic/ carboxylic anhydride can also disproportionate to an anhydride,
PM3 Transition State Calculations
J. Phys. Chem., Vol. 99, No. 15, 1995 5293
TABLE 3: Mechanistic Study Using Phenyl Isocyanate with nlutvric Acid (Reference, Figure 1) heat of formation, AH, kcal/mol PM3 kcal/mol ref state, 298 K calca A P reaction HR HI HP scheme reactants transition state product (forward) AH” -77.7 -110.8 A -100.0 22.3 -10.8 -55.0 -110.8 B -100.0 45.0 -10.8 c -100.0 -50.7 -101.9 49.3 -1.9 -176.0 -222.6 D -211.7 35.7 -10.8 This is the uncorrected calculated @, kcal/mol. A P from experimental data = 11.5 kcal/moL8 a urea, and C02. However, these individual reaction rates are difficult to determine experimentally. Recently, kinetic data were reported using gas chromatography as an analytical tool to monitor progress of the reaction.6 This method is suspect,
however, as the reaction continuing in the high-temperature column is not accounted for. To find kl, then, it was necessary to monitor the reaction by following the disappearance of isocyanate in conventional initial reaction rate studies. All chemicals used in this study were reagent grade. Phenyl isocyanate and benzoic acid were used as received. Acetic acid, butyric acid, isobutyric acid, and 2,2-dimethylbutyric acid (DMBA) were dried by molecular sieve and purified by distillation. The reactions were performed in toluene to minimize solvent interaction effects. Alcohols, acids, and isocyanates associate with ethers, for example, inhibiting the reaction rate. Toluene was dried by CaH2 and purified by distillation. The carboxylic acidisocyanate reactions were studied uncatalyzed. The kinetics of the reactions between phenyl isocyanate and the acids were investigated by observing the disappearance of
TABLE 4: Transition State Geometries in XYZ Coordinates H C
C C 0 0 C C C H C H C C H H H H H C
N 0
c H H H H H
H C C C
0 0 N C H C 0 H H C
Mechanism A 3.0210 -3.5867 1.9690 -3.7456 1.2205 -2.4362 -2.5662 3.3335 1.9485 -1.3301 -0.0408 -2.3759 -1.1932 3.5486 -0.3077 2.4773 1.9246 -4.3900 1.5181 -4.4197 -0.8078 1.1698 -0.4578 -1.2816 -2.1954 0.9507 -3.0627 2.0334 -3.2553 4.1832 -0.801 1 4.5710 0.771 1 2.6743 -2.6014 -0.0679 -4.1440 1.8626 1.3776 0.0000 0.0000 0.0000 2.3162 0.7591 2.6624 -5.7097 2.2188 -6.4222 0.8735 -4.5432 2.3707 -3.7084 3.7178 -5.5854 2.6370 -6.1757
-0.3247 -0.0088 -0.0065 -0.0867 0.0056 -0.0194 -0.0714 -0.0430 1,3714 -0.7653 -0.0302 -0.0139 -0.0457 -0.0736 -0.1092 -0.0816 -0.031 4 -0.0355 -0.0856 0.0000 0.0000 0.0000 1.3684 0.6602 1.6869 2.1224 1.091 1 2.3618
Mechanism D -2.7370 3.0941 -2.4628 2.0386 -1.3461 1.6691 -4.7921 1.5365 -0.2024 1.3368 -1.5520 1.7555 0.0000 0.0000 1.3565 0.0000 -2.1024 2.0104 -3.6650 1.1204 2.3112 -0.7485 -0.9243 1.2243 -3.3711 0.0728 3.9762 5.5241
-3.2647 -3.4690 -2.5245 -4.2143 -2.9536 -1.2396 0.0000 0.0000 -4.5177 -3.2966 0.0000 -0.6596 -3.5064 -1.1632
Mechanism B -2.1405 -1.2674 -2.6108 -0.3865 -1.9093 -0.8150 -0.7280 -2.1 154 -0.2490 -2.9792 -0.9620 -3.2470 -2.6981 -0.9390 -3.5387 0.6299 -2.2894 1.3903 1.4725 -2.4557 0.6752 -3.9970 -0.5942 1.3684 0.0000 0.0000 0.0000 2.3832 -0.6458 1.6658 3.2822 2.2592 2.2779 3.6763 1.8467 4.4827 2.8951 5.9008 2.4204 0.1775 1.4640 4.1506 1.6187 3.6710 0.8879 3.9994 3.1224 4.4845 3.8497 6.4180 2.2221 5.9314 1.4940 6.4867 3.1737
C -2.5593 C C C C C H H H
0 H H C
N 0 0 C C C C H H H H H H H H
2.5521 2.3434 1.5285 0.91 09 1.1226 1.9431 3.1957 2.8231 1.3704 -0.1851 0.641 3 2.1099 0.0000 0.0000 0.0000 -1.2274 -0.9180 -1.1575 -1.9089 -2.1329 -0.1568 -0,1800 -1.7181 -2.8798 -1.3464 -1.1845 -2.7221 -2.6752
Mechanism D Continued 1.1309 -2.2421 H -4.0073 H 4.2287 2.7802 -1.0655 C 3.2518 3.1776 -0.7178 2.0980 -0.9496 C 2.2210 1.3599 0.0796 0 1.9115 1.9193 -2.0810 0 1.6851 C 2.9190 4.4773 -1.4347 H 3.3743 3.3609 0.3708 H 0.7385 1.3588 -2.3302 C -2.9173 -2.1425 0.7175 C -2.1278 -1.0548 0.3720 C -0.7265 -1.1740 0.3602 0.7124 C -0.1459 -2.4015 C -0.9512 -3.4792 1.0587
,
MechanisnI C 4.5238 C 0.2398 3.0323 C 0.3372 2.5559 C 1.7577 H 0.7363 -2.6135 3.1351 0 2.8081 1.2209 0 2.0661 N 0.0000 0.0000 C 1.2959 0.0000 H -0.1476 2.7876 2.4529 H -0.2203 0 2.3548 -0.7424 H 3.0729 0.4872 H -0.8244 -4.5272 C -0.8027 -1.1 81 1 H -3.2294 -4.2075 C -0.3223 -2.4531 C -1.1994 -3.531 5 C -2.5451 -3.3534 C -3.0215 -2.0878 C -2.1604 -0.9986 H -4.0817 -1.9492 H -2.5441 -0.0048 4.9640 C -1.2059 4.7634 H 0.7446 5.0964 H 0.7848 H -1.7244 4.7688 H -1.7625 4.4421 H -1.2826 6.0402
C H H H H H H H H H H H H H
0.5770 0.2923 0.2551 -0.6001 0.3875 0.0083 0.0000 0.0000 -0.6762 1.0589 0.0000 0.0421 -0.6471 -0.0292 -0.1094 -0.3617 -0.3879 -0.0874 0.2415 0.2719 0.4771 0.5295 0.6277 1.5337 -0.1993 -0.3206 1.4176 0.8283
Mechanism D Continued -2.3355 -3.3585 1.0598 -4.0070 -2.0399 0.7231 -2.6191 -0.1065 0.1248 -4.4911 1.5034 -5.2700 0.9425 -2.5344 0.7272 -0.4864 -4.4315 1.3342 -2.9628 -4.2124 1.3314 -5.1325 2.5589 -4.0023 -5.6579 0.8719 -4.0993 2.8294 4.3027 -2.5254 1.9278 4.8469 -1.1066 4.0591 5.7512 -0.0919 4.9676 5.1990 -1.5062 3.7417 6.4632 -1.6803
5294 J. Phys. Chem., Vol. 99, No. 15, 1995 Mechanism
Malwitz of 0.25 N dibutylamine (DBA)/toluene solution and then back titrated with 0.1 N HC1 solution to determine NCO content. At recorded time intervals, 4 mL of solution was taken from the reaction mixture to a 40 mL DBAkoluene solution in a 250 mL flask for the determination of the NCO concentration.8 Each reaction indicated was performed in triplicate to 10% completion, and the results were averaged. Using classical calculation methods for second-order reactions, experimental initial reaction rates were obtained, as listed in Table 2. The experimental reaction rate for the tertiary DMBA bears further investigation, as it is unexpectedly high in view of the report that tertiary carboxylic acid 2,2-dihydroxymethylpropionic acid is reported to be very slow to react with isocyanate~.~
Schematic
R\'
N=C=O
H.O-$+O H.O-$+O R
A
Six membered rmg attack
pi
N=C=O
H-?
C-R
d'
Discussion The computer simulation is done by postulating a TS geometry. In many cases the reaction mechanisms historically proposed are subject to challenge by such a computer study. For the reaction of carboxylic acid with isocyanates identification of the atomistic mechanisms has not been achieved. Recently, Stamer hypothesized the acyl oxygen attacks the isocyanate carbon but offers no discussion or proof.* The computational strategy must be limited to consider one reaction coordinate at a time. The carboxylic/isocyanate interaction is a multistep reaction involving the initial attack and subsequent decarboxylation or disproportionation. For this study only the first reaction step is considered. To establish the pathway of the initial carboxylic acid reaction with isocyanates, four mechanisms were postulated and explored. Interestingly all four TSs were successfully achieved and fully characterized. These are summarized in Figure 1. However, semiempirical PM3 thermodynamic results using these proposed mechanisms differ as shown in Table 3. Notice that the enthalpy changes from the reactants to TS, A@, are 2-4 times larger than the 11.5 kcal/mol expected. The uncatalyzed polyurethane reaction (eq 9) has been shown to have a TS geometry as postulated in Figure 2.6,'0 All alcohol reaction studies considered this mechanism exclusively.
Attack as a hydroxyl group
R;
N=C=O
0-H
R+\;
0
Alternate hydroxyl attack to unstable intermediate
Attack as conventional dimer acid
Figure 1. Postulated carboxylic acidisocyanate reaction mechanisms.
the isocyanate groups (by a titration method). The kinetic studies were carried out at temperatures in the range 50-90 "C. In each reaction, 0.023 mol of phenyl isocyanate in 50 mL of toluene was charged into a 250 mL three-neck round bottom flask, equipped with a nitrogen inlet, magnetic stirrer, condenser, and thermometer. The solution was kept under a dry nitrogen atmosphere, at the desired temperature in an oil bath. An equal molar quantity of acid was added rapidly into the flask. Immediately, an initial 4 mL sample was withdrawn from the reaction mixture to a 250 mL flask containing 40 mL
Applied Computer Results Ideally, once calibration factors are applied to calculated AG* values, PM3 TS calculations can be used to distinguish competing or parallel reaction rates. However, for each of the four proposed carboxylic acid mechanisms a transition state can be successfully achieved and characterized. Atom lists for the
AG+ = pmf - ETAS+
AG* = pmr - ETAST
n
1217 = 9 9 6 + 2 2 1 , Kcal/mol
o
14 27
/
Reactants to TS
f
,/ nBuOH
KcaVmol
S e ,cal/mol/"K
14 55 - 28
Product to TS
\
/
AI+*,
=
'BU
-69.45 68.98
+
~ N C O
11.83
81.46
-+T S -+
-17.46 99.27
'I
H t$
--N-c
-0Bu
-76.13 92.84
Estimated Teq calibrated = PAWeAS = 272'K Figure 2. Isocyanatehydroxyl transition state at standard temperature, 25 "C.
(AH,,,= -18.21) (ASRm,= -57.60)
J. Phys. Chem., Vol. 99, No. 15, 1995 5295
PM3 Transition State Calculations
lOOO/T, OK 2.10
2.30
2.50
2.70
Methyl Experimental -Methyl Forward x i Propyl Experimental i Propyl Forward m n Butyl Forward Experimental n Butyl Forward 0 n Butyl Reverse Experimental n Butyl Reverse
-10
--
-12
I
3.30
3.10
2.90
4
\
.. \
*\
.
\
\
*
\
\
.
4
\
I
-.I
-14
3.50
0
'
.\
.
\
\
,
Data for forward reaction: Ref 6 Data for reverse reaction: Ref 8 Figure 3. Phenyl isocyanate/alcohol reactions, MOPAC PM3.
four transition state geometries of the phenyl isocyanate reaction with n-butyric acid are given in Table 4. Calibration factors for each mechanism, then, need to be generated separately and contrasted with experimental data. Significantly, it is found that experimental data must be obtained for multiple carboxylic acid ligands in order to determine which of the successfully characterized transition states is likely to be correct. Isocyanate/Hydroxyl Reaction. To calibrate the n-butanol/ phenyl isocyanate reaction the necessary data to fit the proposed calibration equation is available (Table 1). Fitting eq 5 with PM3 results obtained using the mechanism of Figure 2, a = 0, second-order experimental data fits Ea adjusting p at a = 0; p = 0.25, to fit experimental Ea of the forward reaction at a = 0; y = -34.75 (caV(mo1 K)),to fit to the absolute reaction rate in toluene solvent; E = 0.145, to fit Teqat 285 "C for n-butanol/ phenyl isocyanate. Comparison of calibrated and experimental results is shown graphically in Figure 3 for lower alcohols with phenyl isocyanate. The constants were determined by a fit of the data for n-butanollphenyl isocyanate forward and reverse reactions. The minimum of the square root of the sum of the squares of the deviation between the predicted value and data points was used to establish all calibrating constants. Use of the entropy term cannot be neglected in cases when elevated temperatures are of interest or in a condensation
reaction, for example, where the molecule count is different for products and reactants. The Teqcan therefore be estimated from fitted PM3 results at 25 "C to be 272 "C. The log of the functions used to calculate H and S at elevated temperatures has some nonlinearity in semiempirical methods. For the hydroxyllisocyanate reaction in toluene, AG* values for the forward and reverse reactions are calibrated to Tes = 285 "C by interpolation between 275 and 300 "C. The semiempirical methods with PM3 pxameters, then, will predict the forward or reverse alcohol/isocyanate reaction rate by the following expression:
k = (1.587
105)~(-[(.25)AHi-(.145)rA~*l/Rn
(10)
A quantitative prediction of relative reaction rates of isocyanate with any active hydrogen compound used in commercial polyurethane systems may now be possible by applying these calibration factors to calculated A@ using semiempirical PM3 parameters. To verify this postulate, an alternate active hydrogen compound reaction with phenyl isocyanate was examined in a similar manner. IsocyanatdCarboxylic Acid Reaction. For the reaction of carboxylic acids with an isocyanate the two intuitively anticipated transition states, atomistic mechanisms A and B of Figure 1, might be expected, where the labile hydrogen is captured by
5296 J. Phys. Chem., Vol. 99, No. 15, 1995
Malwitz
lOOO/T, OK 2.70 -7
2.80
2.90
3.00
3.10
3.20
-8
8
? 0)
I
0
E
-9
3
J 4
-Acetic o
Acid
Acetic Experimental
- - - n Butyl -10
n Butyl Experimental i Butyl A i Butyl Experimental t Dimethylbutyl x t Dimethylbutyl Experimental - - . - Benzyl o Benzyl Experimental
I
I
-1 1 Figure 4. Phenyl isocyanatekarboxylic acid kinetics, semiempirical PM3 as calibrated.
the isocyanate nitrogen. The oxygen attachment to the isocyanate carbon could either come from the carbonyl oxygen or the hydroxyl oxygen of the acid. Mechanisms A and B are impossible to judge experimentally using tagged oxygen in the carboxylic acid, as the carboxylhydroxyl oxygens readily exchange. Inspection of the raw thermodynamic result, as shown in Table 3, would indicate that the six-membered-ring mechanism is highly favored. The immediate reaction product, an aromatic carbamate-aliphatic carboxylic acid anhydride, is impossible to isolate, as it is subject to rapid irreversible decarboxylation. Consequently, it is not possible to find Teq experimentally. Taking a = 0 and E = 0.145 as in the alcohol reaction, and fitting experimental results for n-butyric acid, the calibration factors for the six-membered-ring mechanism are found to be
p = 0.515 y = -34.75 The fit of data can be inspected in Figure 4. With the entropy calibration factor, E , matched to the alcohol reaction, the preexponential factor, y , used for the alcohol reaction fits. It is shown, therefore, that calibrated PM3 results for the six-
membered-ring mechanism can be calibrated to represent experimental data for the n-butyric acid reaction using mechanism A. However, the prediction of relative reaction rate among ligands is not as comfortable. The rate for benzoic acid is incorrectly predicted to be faster than acetic acid, for example. Similarly for the diacid attack represented by mechanism D, with a = 0 and E = 0.145, enthalpy factor ,b = 0.31 and y = -28.85. Again the rate predicted for benzoic acid is incorrect this time, at 2 orders of magnitude below the experimental rate. For carboxylic acidisocyanate reaction mechanism B, PM3 results can also be fitted to the data for n-butyric acid. In this case the hydroxyl group sits on the active N=C double bond in a four-membered-ring arrangement identical to the mechanism for the alcoholhsocyanate reaction. Again, taking entropy factor E = 0.145, and a = 0, as for the alcohol reaction, the experimental data for the n-butyric acid forward reaction gives the best fit with y = -34.75 and ,b = 0.2502 when the data is fit according to the minimum of the square of the deviation method. Figure 5 shows the results of the calibrated PM3 calculations with the experimental data when the calibration factors are set identical to those for the alcohollphenyl isocyanate reaction. For this mechanism the PM3 prediction fits the data
J. Phys. Chem., Vol. 99, No. 15, 1995 5297
PM3 Transition State Calculations 1OOO/T OK 2.70 -7
2.80
2.90
3.00
I
I
3.10
3.20
,
i
i
\\
0
-8
u
f
I
0
5
-9
&€ E
.d
-10
-
n Butyl n Butyl Experimental i Butyl A i Butyl Experimental t Dimethylbutyl -..- Benzyl o Benzyl Experimental o
I
I
--11
Figure 5. Phenyl isocyanate/carboxylic acid kinetics, semiempirical PM3 as calibrated.
for acetic acid, isobutyric acid, and benzoic acid as well. This finding suggests mechanism B is a likely electronic route for the reaction. Similarly for mechanism C the data fits PM3 results with /3 = 0.23 and the remaining calibrating factors at values determined for the alcohollphenyl isocyanate reaction. This enthalpy factor is within 10% of the factor for mechanism B and the alcohol reaction. Also the prediction of the aliphatic and aromatic acids with phenyl isocyanate fits the experimental data. It is not possible to distinguish between mechanisms B and C using this technique. This result may indicate that both fourmembered-ring mechanisms apply.
Summary Calibrating semiempirical transition state results requires exploration of all electronically reasonable mechanisms. Using experimental data of phenyl isocyanate and carboxylic acids with varied ligands, it vjas possible to distinguish likely atomistic mechanisms from amring several transition states that can be successfully characterized. For the case of isocyanate/carboxylic reactions, the lower TS energy path predicted by semiempirical methods did not
correspond to experimental data when comparison is made using various aliphatic and aromatic carboxylic acid ligands. It may be possible to extend this technique to resolve ambiguous atomic mechanisms for other reactions. A model for the semiempirical PM3 parameter set has been applied to experimental second order reaction rate data with effectively only three fitting constants to obtain Ahrrenius reaction rate expressions. Using experimental data to calibrate the computational results for and Asf to obtain AG* and the pre-exponential Arrhenius frequency factor, a quantitative expression is developed to obtain Arrhenius dependent reaction rate constants of phenyl isocyanate with hydroxy and carboxylic acid compounds directly from semiempirical calculations using PM3 parameters. The modeling expression (eq 10) fits second-order experimental data of two phenyl isocyanate reactions and may extend to the reaction of other aryl isocyanates with other active hydrogen compounds, such as water and amines. If so, a scheme to predict polymer structure and reaction heat history during polyurethane formation is possible. The relative reaction rates still need to be empirically adjusted for catalyst selectively. However, kinetics determined by semiempirical methods serve
Malwitz
5298 J. Phys. Chem., Vol. 99, No. 15, 1995
as a useful starting point. Early efforts utilizing this scheme indicate that reaction modeling appears to predict well the relative rate of reaction of polyurethane formulation ingredients as the reaction proceeds adiabatically. It is reasonable to postulate that this calibration scheme is applicable to other second-order organic reactions. However, it is prudent to follow the calibration procedure as indicated above for each reaction type in the solvent of choice, if used. Since Arrhenius rate expressions can be determined by semiempirical methods, quantum mechanical chemistry is shown to be capable of offering insight useful toward minimizing unwanted side reactions, optimizing yields, suggesting reaction conditions, and determining polymer composition under adiabatic conditions or reaction circumstances with programmed temperatures. References and Notes (1) Atkins, P. W., Physical Chemist?, 4th ed.; W. H. Freeman and Co.: New York, 1990.
(2) Morrison, R. T.; Boyd, R. N. Organic Chemist?, 6th ed.; Prentice Hall: New Jersey, 1992. (3) Scudder, P. H. Electron Flow in Organic Chemistry; John Wiley and Sons, Inc.: New York, 1992. (4) Stewart, J. J. P. Optimization of Parameters for Semiempirical Methods 11. Applications. J. Comput. Chem. 1989, 10 ( 209). ( 5 ) Saunders, J. H.; Frisch, K. C. Polyurethanes, Chemistp and Technology, John Wiley and Sons: New York, 1962, and references therein. (6) Starner, W. E. The Synthesis of N-Aryl Amides and Amines from Aromatic Isocyanates. Ph.D. Thesis, Drexel University, 1988. (7) Lateef, A. B.; Reeder, J. A,; Rand, L. The Thermal Dissociation of Aryl Carbanilates in Glyme. J. Org. Chem. 1971, 36 (16). (8) Xiao, H.; Xiao, H. X.; Frisch, K. C.; Malwitz, N. Kinetic Studies of the Reactions Between Isocyanates and Carboxylic Acids. High Perform. Polym. 1994, 6. (9) Milligan. C. L.; Hoy, K. L. Water-dilutable Polyurethanes. U. S. Patent 3,412,054, 1968. (IO) Borkent, G. Kinetics and Mechanism of Urethane and Urea Formations. In Advances in Urethane Science and Technologies; Frisch, K. C., Reegen, S . L., Eds.; Technomic Publishing: CT, 1974; Vol. 3. JP941015K
