Theoretical study of the pressure dependent rate constants of the thermal decomposition of β-propiolactone

1 Introduction

The thermal decarboxylation of β-lactones (2-oxetanones) is a well-established methodology for the stereo-specific synthesis of substituted olefins. The reaction is also interesting from the mechanistic point of view, since it is one of the very few examples of a [2+2]-cyclo-reversion process taking place with retention of configuration (Fig. 1) (Moyano et al., 1989).

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Figure 1 Thermal decarboxylation of β-lactones (2-oxetanones).

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β-Propiolactone was a small molecule and offered the opportunity for the following fall-off in unimolecular rate constant with respect to pressure. Moreover, it can be decomposed by three different ring cleavages (Fig. 2) (James and Wellington, 1969). In pyrolysis studies on β-propiolactone, the activation energy for reaction 1 has been found to be 191.7 or 180 kJ mol⁻¹ (Frey and Pidgeon, 1985; James and Wellington, 1969). Considering enthalpy changes, reactions 2 and 3 are presumed to have much higher activation energies than reaction 1.

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Figure 2 Three different ring cleavages of β-propiolactone.

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The kinetics of the gas-phase thermal decomposition of β-propiolactone has been experimentally studied by James and Wellington (1969) and Frey and coworkers (Sugita et al., 2000; Frey and Watts, 1983). The presented results demonstrate that the reaction is homogeneous, unimolecular and obeys a first-order rate law. In particular, the rate coefficient for the gas-phase decomposition at low pressure to form carbon dioxide and ethylene has been determined (Sugita et al., 2000; Frey and Watts, 1983) and expressed as a function of temperature by the following Arrhenius-type equation: $$\log k_{\text{obs}}({\text{s}}^{-1})=\frac{(14.86\pm 0.3)-(180.46\pm 3.2)\text{kJ}{\text{mol}}^{-1}}{2.303\mathit{RT}}$$ The kinetic of thermal decomposition of β-propiolactone has been studied previously but in this work, the rate constants and fall-off pressures were computed for the thermal decomposition reactions of β-propiolactone by using high-level ab initio methods and RRKM theory. Based on the obtained results, thermal decomposition pathways were compared according to kinetic and energetic points of view and available experimental data. Finally, using NBO analysis, reasonable insights about the reaction pathways mechanism were given.

2 Computational details

The ab initio molecular orbital calculations have been performed with the gaussian 98 program (Frisch et al., 1998), implemented on a Pentium-PC computer with 2.4 GHz processor. Equilibrium geometries and harmonic vibrational frequencies of β-propiolactone molecule and the transition states in the considered reactions were optimized at the MP2/6-311+G^∗∗ level of theory. The thermodynamic functions have been calculated at the same level. The MP2/6-311+G^∗∗ level of theory has been used for characterization of stationary points, zero-point vibrational energies (ZPVE) and RRKM computations. Initial estimates of the geometry of β-propiolactone and related products in each reactions were obtained by using MMX method in a molecular-mechanics program pc-model (88.0) (Serena Software, 1991) followed by full minimization using semi-empirical calculations which were carried out using the PM3 method with the MOPAC 6.0 computer program (Stewart, 1990a, 1990b). The gaussian 98 was finally utilized to perform ab initio calculation at the MP2/6-311+G^∗∗ level of theory in order to obtain the energy minimum and energy maximum structures and B3LYP/6-311+G^∗∗//MP2/6-311+G^∗∗, and MP4/6-311+G^∗∗//MP2/6-311+G^∗∗ levels of theory for single point calculations.

Energy minimum molecular geometries were located by minimizing the energy, with respect to all geometrical coordinates. The structures of the transition state geometries were obtained using the optimized geometries of the equilibrium structures and the procedure of Dewar et al. (keyword SADDLE) (Dewar et al., 1984). For minimum state structures, only real frequency values and for the transition states, only a single imaginary frequency value is accepted. All energy-minima and energy-maxima geometries obtained in the present work are calculated to have 3N-6 and 3N-7 real vibrational frequencies (Ermer, 1975; Mclever, 1974).

In order to calculate the unimolecular rate constant on the basis of the RRKM theory, the critical energy was computed using ground state and transition state vibrational frequencies, temperature of the thermal decomposition and E_o. The selected pressure ranges for the decomposition of β-propiolactone are 10⁻¹²–10³ mmHg. The RRKM algorithm and the Q-Basic package were used to carry out the unimolecular rate constants and fall off pressure calculations for reactions 1–3 in the various pressures.

3 Results and discussion

3.1

3.1 Energetic parameters

The ZPVE, total electronic (E_el) energies and total internal energies at 0 K (E_o = E_el + ZPVE) for reactant and transition states of reactions 1–3 are summarized in Table 1. MP2/6-311+G^∗∗//MP2/6-311+G^∗∗, B3LYP/6-311+G^∗∗//MP2/6-311+G^∗∗ and MP4/6-311+G^∗∗//MP2/6-311+G^∗∗ results show that the energy barrier of thermal decomposition in reaction 1 is 49.95, 35.83 and 44.36 kcal mol⁻¹, respectively. In reaction 2, the energy barrier of thermal decomposition is 66.52, 50.89 and 58.48 kcal mol⁻¹, respectively. While, in reaction 3, it is 93.44, 77.25 and 81.76 kcal mol⁻¹, respectively at the same calculations. These results reveal that reaction 1 has the lowest energy barrier among the considered reactions. Clearly reaction 1 has the lowest energy barrier and the result of this reaction is in good agreement with the computed activation energy and that obtained by Frey and Pidgeon (43.11 kcal mol⁻¹) (Frey and Pidgeon, 1985). Indeed, this agreement is achieved at the MP4/6-311+G^∗∗//MP2/6-311+G^∗∗ levels of theory. Further improvement in the quality of the basis set or in the computational level does not result in a better agreement. Energy profile for thermal decomposition reactions 1–3 is depicted in Fig. 3.

Table 1 Calculated energies (in hartree) for ground state and transition state structures for thermal decomposition reactions of four-membered rings: β-propiolactone [oxetane-2-one].

Method	Geometry
	(GS)	(TS1)	(TS2)	(TS3)
MP2/6-311+G∗∗//MP2/6-311+G∗∗
ZPVE	0.0694	0.0651	0.0638	0.0608
Eel	−265.7104	−265.6265	−265.5988	−265.5529
Eo	−265.6410	−265.5614	−265.5350	−265.4921
ΔEoa	0.0000 (0.0000)b	0.0796 (49.9497)b	0.1060 (66.5161)b	0.1489 (93.4362)b
B3LYP/6-311+G∗∗//MP2/6-311+G∗∗
ZPVE	0.0694	0.0651	0.0638	0.0608
Eel	−267.2356	−267.1742	−267.1489	−267.1039
Eo	−267.1662	−267.1091	−267.0851	−267.0431
ΔEoa	0.0000 (0.0000)b	0.0571 (35.8308)b	0.0811 (50.8911)b	0.1231 (77.2465)b
MP4/6-311+G∗∗//MP2/6-311+G∗∗
ZPVE	0.0694	0.0651	0.0638	0.0608
Eel	−266.5541	−266.4791	−266.4553	−266.4152
Eo	−266.4847	−266.414	−266.3915	−266.3544
ΔEoa	0.0000 (0.0000)b	0.0707 (44.3649)b	0.0932 (58.4839)b	0.1303 (81.7645)b
Ea (kcal/mol)c		43.11	–	–

a Relative to the best configuration.

b Numbers in parenthesis are the corresponding ΔE values in kcal mol⁻¹.

c Experimental values are given in parenthesis (James and Wellington, 1969).

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Figure 3 Energy profile for β-propiolactone unimolecular thermal decomposition reactions 1–3.

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The changes of thermodynamic functions i.e. Gibbs free energies (ΔG_r), enthalpies (ΔH_r), and entropies (ΔS_r), enthalpies of activation (ΔH^≠), Gibbs activation free energies (ΔG^≠) and activation entropies (ΔS^≠) of reactions are given in Table 2. The thermal decomposition of compounds 1–3 is an endothermic process; ΔH^≠ and ΔS^≠ values are positive. As can be seen, ΔS^≠ values are relatively small, so that the calculated ΔH^≠ and ΔG^≠ parameters are close to the ΔE_o values.

Table 2 Theoretical activation parameters and change of thermodynamic functions for the three reactions of the thermal decomposition of β-propiolactone at the MP2/6-311+G^∗∗ level of theory.

Reaction	ΔHr (kcal mol−1)	ΔSr (kcal mol−1 K−1)	ΔGr (kcal mol−1)	ΔH≠ (kcal mol−1)	ΔS≠ (kcal mol−1K−1)	ΔG≠ (kcal mol−1)
R1	−17.761670	0.036649	−28.677874	41.553084	0.002611	40.774972
R2	27.696408	0.044721	14.363076	54.765307	0.003024	53.864203
R3	−2.692017	0.041791	−15.151856	82.927956	0.005843	81.186616

3.2

3.2 RRKM study and thermal rate constants

RRKM calculations have been performed to determine the rate constants of the unimolecular decomposition reactions 1–3 by using the ab initio calculated harmonic frequencies and the energy barrier heights. The reaction 1 involves the lowest activation barrier; therefore, it has the fastest rate over the whole energy ranges considered. In order to calculate the unimolecular rate constants on the basis of modern RRKM theory, several parameters such as temperature, collision diameter, reduce mass and vibrational frequencies of ground state and transition state structures should be used. The harmonic vibrational frequencies of β-propiolactone and transition states at the MP2/6-311+G^∗∗ level of theory are presented in Table 3.

Table 3 The calculated harmonic vibrational frequencies (in cm⁻¹) at the MP2/6-311+G^∗∗ level of theory for the thermal decomposition reactions of β-propiolactone.

States	Geometry
	(GS)	(TS1)	(TS2)	(TS3)
1	163.8436	−1039.3274	−655.5682	−992.1818
2	499.6887	105.0030	173.1409	102.4405
3	518.5293	323.3872	227.7582	132.3231
4	753.3876	459.3902	357.0684	384.0039
5	809.4455	565.3825	470.6599	438.7145
6	910.7025	659.4909	627.1011	531.7378
7	968.2425	779.9780	722.1626	572.6190
8	1039.5582	839.5812	869.4198	768.8294
9	1064.5471	931.4578	890.6937	980.5825
10	1114.2314	1000.7532	1013.7005	1017.9513
11	1180.4820	1066.0180	1086.3011	1031.1339
12	1213.7873	1200.9404	1186.0083	1230.0656
13	1230.8474	1253.4061	1239.7614	1289.4975
14	1367.6988	1325.3584	1419.2081	1349.0152
15	1476.6096	1485.7128	1472.6945	1456.7215
16	1546.0812	1581.7495	1605.2539	1526.1187
17	1901.0079	1994.2972	2126.5492	2058.4005
18	3134.9825	3190.8998	3012.2454	2380.7898
19	3140.2356	3198.9206	3081.1217	3050.3459
20	3203.5321	3287.4598	3126.9494	3135.7020
21	3217.9141	3310.1744	3295.4099	3258.0571

We have also presented the fall-off data for the unimolecular thermal decomposition of β-propiolactone at 523.15 K and the pressure range, 10⁻¹²–10³ mmHg, respectively, in Fig. 4. The rate constant of the fastest reaction, reaction 1, approaches at atmospheric pressure limit about 10^0.38 mmHg (P_1/2 = 2.415 mmHg). For reaction 1, the rate constant, k₁, can be represented as k₁= 1.1 × 10¹⁴ exp(−82.82/T) s⁻¹, while k₂ for reaction 2 is 1.35 × 10¹⁴ exp(−111.65/T) s⁻¹ and k₃ for reaction 3 is 5.61 × 10¹⁴ exp(−156.16/T) s⁻¹. The Arrhenius factors, A, in k₁, k₂ and k₃ are almost of the same order. The exponential factors of k₂ and k₃ are much lower than that of k₁, which means the activation energies in reactions 2 and 3 are much higher than that of reaction 1. This indicates again that k₁ is much higher than k₂ and k₃ and that the decomposition of the β-propiolactone molecule is dominated by reaction 1.

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Figure 4 Fall-off pressure curve of the rate constants for reactions 1–3 at the 523.15 K.

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In the neighborhood of atmospheric pressures within the lower stratospheric region it is readily seen that the rates of all three reactions are well outside the fall-off region. The pressure in which k_∞ is reduced to k_∞/2 is determined using the obtained fall-off pressure curve. The fall-off pressures for the decomposition of β-propiolactone are found to be 2.415, 9.423 × 10⁻² and 3.676 × 10⁻³ mmHg, respectively. The fall-off curves of the decomposition of β-propiolactone show that the decomposition of reaction 1 is faster than reactions 2 and 3 and is in agreement with activation energies. A small fall-off pressure value indicates that collision between molecules and their activated forms occur at low pressures; consequently the reaction would be slow.

4 Conclusion

Theoretical calculations provide a picture from structural and energetic points of view for three reactions of the thermal decomposition of β-propiolactone. The results calculated by MP2/6-311+G^∗∗//MP2/6-311+G^∗∗, B3LYP/6-311+G^∗∗//MP2/6-311+G^∗∗ and MP4/611+G^∗∗//MP2/6-311+G^∗∗ levels showed that the barrier height of reaction 1 is the lowest of the three reactions. The fall-off curves of the above mentioned reactions 1–3 have been drawn by calculating the unimolecular rate constants at the 523.15 K and the pressure range of 10⁻¹²–10³ mmHg and the fall-off pressures are found to be 2.415, 9.423 × 10⁻² and 3.676 × 10⁻³ mmHg, for reactions 1–3, respectively. The obtained small fall-off pressure values indicate that collision between molecules and their activated forms occur at low pressure; consequently the reaction would be slow.