Computational studies on thermo-kinetics aspects of pyrolysis of isopropyl acetate and its methyl, bromide and hydroxyl derivatives

Abstract

The gas-phase decomposition kinetics of isopropyl acetate (IPA) and its methyl, bromide and hydroxyl derivatives into the corresponding acid and propene were investigated using density functional theory (DFT) with the ωB97XD and M06–2x functionals, as well as the benchmark CBS-QB3 composite method. Transition state theory (TST) and RRKM theory calculations of rate constants under atmospheric pressure and in the fall-off regime were used to supplement the measured energy profiles. The results show that the formation of propene and bromoacetic acid is the most dominant pathway at the CBS-QB3 composite method, both kinetically and thermodynamically. There was a good agreement with experimental results. Pressures greater than 0.01 bar, corresponding to larger barrier heights are insufficient to ensure saturation of the measured rate coefficient when compared to the RRKM kinetic rates.

Natural bond orbitals (NBO) charges, bond orders, bond indices, and synchronicity parameters all point to the considered pathways taking place via a homogenous, first-order concerted, as well as an asynchronous mechanism involving a non-planar cyclic six-membered transition state. The calculated data exhibit that the elongation of the C_α−O bond length and subsequent polarization of the C_α^+δ…O^−δ bond is the rate-determining step of the considered reactions in the cyclic transition state, which appears to be involved in this type of reaction.

Isopropyl acetates; Pyrolysis; Rate coefficients; DFT; NBO; Reaction mechanisms.

1. Introduction

Ester pyrolysis has received a lot of attention [1, 2, 3, 4]. The corresponding acid and alkene are the products of esters containing a β-hydrogen in the alkyl group. Hurd and Blunk [3] proposed that alkyl esters pyrolysis by simultaneously breaking the O−R bond and forming an O–H bond between the carbonyl oxygen and the hydrogen of the ester's alkyl portion (see Scheme 1). They proposed a mechanism involving a "cyclic hydrogen bridge" to explain the pyrolysis of esters into acid and alkene. Kinetic studies display that this is a first-order homogeneous reaction. All of these point to a mechanism involving a quasi-cyclic intermediate as.

Scheme 1.

Pyrolysis mechanism of esters through six-membered cyclic intermediate.

It is reasonable to believe that the ortho-substituents involved may delay the formation of an O–H bond, thereby stabilizing these esters by preventing the formation of the postulated six-membered cyclic intermediate [5].

Many biodiesel models [6, 7, 8, 9] received a lot of attention both theoretically and experimentally. A few experimental and computational studies [10, 11, 12, 13, 14] have been conducted on isopropyl acetate (IPA) and its derivatives. IPA as the simplest branched biodiesel ester molecule is commercially produced by esterifying acetic acid and isopropanol and is widely used in the production of fats, oil, plastics, printing inks, perfumes, varnishes, adhesive agents, and synthetic resins [15, 16, 17]. The decomposition of three secondary acetates (isopropyl propionate, isopropyl glycolat, and isopropyl bromoacetate) were measured in a static system at temperatures ranging from 563 to 623 K and pressures ranging from 54 to 250 Torr, as well as isopropyl acetate, has been experimentally studied [18, 19] at pressures ranging from 1.03 to 2.28 mmHg and temperatures ranging from 714.7 to 801 K, indicating positive activation energy. The rate constants of these secondary acetates were discovered to be independent of their initial pressure.

The kinetics of the gas-phase unimolecular pyrolysis of IPA and its methyl, bromide and hydroxyl derivatives [X–CH₂CO₂CHMe₂ with X = Me, OH, and Br] into CH₃CH = CH₂ and X–CH₂COOH were experimentally investigated in a static system at pressures ranging from 54 to 250 Torr and temperatures ranging from 563 to 623 K. Chemical reactions in seasoned vessels are homogeneous, unimolecular, and follow a first-order law [18].

Emovon [20] showed an important relationship between the pyrolysis rate constant of several tert-butyl-substituted acetates and the acid's pK_a values, viz. CC1₂ > C1> H > Me. Using electronic structure theory, the mechanism of this reaction was interpreted [21] as a six-membered ring transition state in which C–O and C–H bonds are simultaneously broken, followed by olefin formation [22].

Figure 1 depicts an Arrhenius plot of the calculated rate coefficients against the considered temperature ranges. The rate constants of pyrolysis of secondary acetates such as isopropyl acetate (1, X = H), isopropyl propionate (2, X = Me), isopropyl glycolate (3, X = OH), and isopropyl bromoacetate (4, X = Br) exhibit positive temperature dependences with activation energies of (−47.5 ± 1.9) (−45.4 ± 0.2) (−43.0 ± 0.7), and (−43.3 ± 1.4) kcal mol⁻¹, respectively [18, 19]. Chuchani et al. [18], and Blades [19], demonstrated for the considered reactions, all obtained temperature-dependent rate constants and fitted modified Arrhenius expressions obtained approximately by the least squares procedure were as follows:

$$\begin{array}{l}\log k_1\left({\sec }^{-1}\right)=\left(13.85\pm 0.02\right)-\left[\frac{\left(47.5\pm 1.9\right)\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{m}\mathrm{o}\mathrm{l}}{2.303RT}\right];\left(T=440-630\mathrm{K}\right)\\ \log k_2\left({\sec }^{-1}\right)=\left(13.06\pm 0.09\right)-\left[\frac{\left(45.4\pm 0.2\right)\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{m}\mathrm{o}\mathrm{l}}{2.303RT}\right];\left(T=583.2-624.1\mathrm{K}\right)\\ \log k_3\left({\sec }^{-1}\right)=\left(12.56\pm 0.28\right)-\left[\frac{\left(43.0\pm 0.7\right)\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{m}\mathrm{o}\mathrm{l}}{2.303RT}\right];\left(T=573.2-623.1\mathrm{K}\right)\\ \log k_4\left({\sec }^{-1}\right)=\left(12.84\pm 0.02\right)-\left[\frac{\left(43.3\pm 1.4\right)\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{m}\mathrm{o}\mathrm{l}}{2.303RT}\right];\left(T=563.5-623.1\mathrm{K}\right)\end{array}$$

Figure 1.

Arrhenius plot of the experimental rate constant for the pyrolysis of IPA and its derivatives [18].

In the considered temperature ranges, the Arrhenius rate constants increase with increasing temperature, indicating positive activation energy. As a result, propene and related carboxylic acids are formed via a reaction path involving a six-membered ring. Experiments on real biofuels are difficult, both experimentally and computationally, due to the large molecules involved. As a result, model system studies are more efficient and effective in simulating the thermochemical and kinetic parameters of the considered compounds. The current study is the first to consider the unimolecular decomposition of IPA and its derivatives, which have the potential to be used as a biodiesel additive and model biodiesel [22]. The purpose of this research is to look into the activation energies, rate constants, and molecular mechanisms of the decomposition processes depicted in Scheme 2.

Scheme 2.

Unimolecular decomposition processes of isopropyl acetate and its derivatives through the six-center reaction of the type (atom numbering of the studied compounds).

For this purpose, transition state theory (TST) [23, 24, 25, 26, 27, 28, 29, 30] will be used, along with the ωB97XD [31] and M06–2x functionals [32], and 6–311++G(d,p) diffuse function basis set [33, 34]. A comparison with more accurate energies from the CBS-QB3 method will be made for clarity of these reaction mechanisms and to determine which functional provides the most accurate reaction energies and energy barriers [35, 36]. Furthermore, to unravel the detailed available experimental data over the temperature range 563–623 K, rate constants are estimated using transition state theory (TST) at the high-pressure (HP) limit [23, 24, 25, 26, 27, 28], and their fall-off behavior is studied using RRKM theory at lower-pressures (LP) [37, 38, 39]. The M06–2x functional is frequently regarded as the best exchange-correlation functional for both thermodynamics and kinetics [32]. Finally, we will work to provide additional qualitative chemical insights into the involved reaction mechanisms by analyzing natural bond orbital (NBO) results [40, 41].

2. Theory and computational details

All molecular structures involved in pyrolysis processes were performed using DFT [42, 43] in combination with the ωB97XD and M06–2x functionals, with the split valence 6–311++G(d,p) basis set [44], due to its high accuracy in achieving geometries, zero-point vibrational energy (ZPVE) [45], and frequencies [46], combined with computational efficiency [47, 48]. The thermodynamic properties were determined using the vibrational analysis obtained previously for each stationary point. The presence of only one imaginary frequency was demonstrated in the case of the transition state structures. All molecules and radicals carried out in this study are considered neutral with charge zero while the multiplicities of the saturated molecules are 1 and 2 for radicals.

Because the studied DFT energies are insufficiently accurate, the electronic energies were further optimized using the CBS-QB3 composite method [49]. All quantum calculations were done with the Gaussian 09 package of programs [50], and all molecular structures were visualized with ChemCraft [51]. The CBS-QB3 method [52, 53, 54], is a benchmark approach for calibrating the accuracy of DFT methods. Harmonic vibrational frequencies were calculated to determine whether the identified stationary points were local minima or saddle points. Furthermore, the intrinsic reaction coordinate (IRC) calculation [55] was carried out with 61 points along the reaction path in both directions (forward and backward) to determine whether the discovered TS structure is connected to the reactant and corresponding products [56].

The temperatures used in the experiment [18] were used to calculate the unimolecular rate constants for the investigated reaction pathways. ZPVE contributions were considered when calculating activation energies. The rate constants for unimolecular reactions in the HP limit are defined by transition state theory as given in Eq. (1) [57, 58, 59].

$$k_{\mathrm{u}\mathrm{n}\mathrm{i}}=\kappa \left(T\right)\frac{\sigma k_{\mathrm{B}}T}{h}\frac{Q_{\mathrm{T}\mathrm{S}}^{†}}{Q_{\mathrm{A}}}\exp \left(-E_a/RT\right)$$ (1)

where k_B and h denote Boltzmann’s and Planck’s constants, correspondingly. Here, Q^†_TS and Q_A represent the total partition function of the TS and the reactant, respectively, and σ denotes the reaction-path degeneracy. By integrating the probability of transmission, p(E), which is identical to the over Boltzmann distribution of energies, the Eckart tunneling factor, κ(T), is used to correct the calculated TST rate constants (Eq. (2)) [60]:

$${\kappa }_{\mathrm{E}\mathrm{c}\mathrm{k}\mathrm{a}\mathrm{r}\mathrm{t}}\left(T\right)=\frac{\exp \left(\Delta H_f^{\ne ,0K}/k_{\mathrm{B}}T\right)}{k_{\mathrm{B}}T}{\int }_{\mathrm{o}}^{\infty }p\left(E\right)\exp \left(-E/k_{\mathrm{B}}T\right)dE$$ (2)

where ΔH_f^≠,0K denotes the zero-point corrected energy barriers in the forward direction computed by the KiSThelP package [61], to calculate rate coefficients at various temperatures [62, 63, 64, 65, 66, 67, 68, 69, 70]. In all calculations, a scaling factor of 0.99 was applied to the calculated frequencies using the CBS-QB3 method [71].

All rate constant values provided are the results of calculations performed using the TST and RRKM theory implementations. The energy-dependent microcanonical rate constants k(E) are given by the standard RRKM expression (Eq. (3))

$$k\left(E\right)=\frac{\sigma N^{†}\left(E\right)}{h\rho \left(E\right)}$$ (3)

where σ is the reaction pathway degeneracy, N^†(E) represents the total number of energy states of the TS, and ρ(E) denotes the density of reactant states. Canonical rate constants k(T) (Eq. (4)) are determined thereafter by state integration and Boltzmann averaging [72]:

$$k\left(T\right)=\frac{1}{Q\left(T\right)}{\int }_{E_{\mathrm{o}}}^{\infty }k\left(E\right)\rho \left(E\right)\exp \left(-\beta E\right)dE$$ (4)

where β = 1/k_BT and Q(T) is the partition function of the reactants as obtained by Eq. (5) [73].

$$Q\left(T\right)={\int }_{\mathrm{o}}^{\infty }\rho \left(E\right)\exp \left(-\beta E\right)dE$$ (5)

The quantum mechanical tunneling contribution was modeled using an Eckart potential barrier height [74, 75]. With ω = β_c.Z_LJ [M] denoting the effective collision frequency, Z_LJ the Lennard-Jones collision frequency [M] the total gas concentration, and β_c = 0.2 the collisional efficiency, the strong collision approximation is used. Table 1 shows the collision frequencies (Z_LJ) calculated using the LJ parameters for argon (as a diluent gas) [76] and reactants [77].

Table 1.

Lennard-Jones potential parameters.

Species	LJ potential parameters
	σ (Å)	ε/kB (K)
isopropyl acetate (1)	5.6	436.8
isopropyl propionate (2)	5.9	454.4
isopropyl glycolate (3)	5.7	505.4
isopropyl bromoacetate (4)	5.9	504.5
Argon	3.465	113.5

3. Results and discussion

3.1. Energetic and thermodynamic parameters

Tables 3 and 4 summarize the activation and reaction parameters for the thermal decomposition of the studied compounds. According to the available experimental activation energies [18], decomposition of compounds 1–4 exhibits endothermic processes (ΔH > 0) [with ΔH≈ 11.39–12.30, 11.80–11.83, 11.36–11.74, and 11.53–11.79 kcal mol⁻¹ at the studied DFT methods, while ΔH is nearly equal to 11.73, 11.57, 11.54, and 10.70 kcal mol⁻¹ when X = H, Me, OH, and Br, respectively at the CBS-QB3 method). Gibbs free reaction energies, ΔGs are positive (endoergic) for the decomposition of Isopropyl acetate (1) at all studied methods while ΔGs are negative for other pathways [2–4], hence these reactions are exergonic processes (ΔG < 0) at a pressure of 1 bar and T = 298 K. According to the energy profiles in Table 2, the formation of CH₃CH = CH₂+CH₂(Br)CO₂H species (P4) via pathway 4 appears to be more favorable in terms of kinetics and thermodynamic control than the other reactions at the CBS-QB3 method.

Table 3.

Activation parameters (energy, enthalpy, and Gibbs free energy) (in kcal mol^-1), and activation entropy (in cal mol^-1K^-1) of compounds 1–4 at the studied DFT levels and the CBS-QB3 method (P = 1 bar).

Method Reaction	ωB97XD/6–311++G(d,p)				M06–2x/6–311++G(d,p)				CBS-QB3				Experiment (ΔE†)
	ΔE0K†	ΔH298K†	ΔG298K†	ΔS298K†	ΔE0K†	ΔH298K†	ΔG298K†	ΔS298K†	ΔE0K†	ΔH298K†	ΔG298K†	ΔS298K†
H–CH2CO2CHMe2 → TS1	37.139	37.218	36.570	2.174	39.324	38.862	39.591	−2.446	40.268	40.291	39.703	1.971	(47.5 ± 1.9) b
Imaginary frequency TS1 (cm−1)	1317.86i				1426.77i				1261.81i
CH3–CH2CO2CHMe2 → TS2	36.251	36.371	35.635	2.469	38.316	38.342	37.647	2.332	39.384	39.370	38.882	1.636	(45.4 ± 0.2)a
Imaginary frequency TS2 (cm−1)	1315.36i				1415.89i				1245.16i
OH–CH2CO2CHMe2 → TS3	35.203	35.254	34.675	1.942	37.578	37.577	37.129	1.506	38.579	38.564	38.074	1.648	(43.0 ± 0.7)a
Imaginary frequency TS3 (cm−1)	1256.18i				1382.79i				1230.76i
Br–CH2CO2CHMe2 → TS4	34.179	34.133	33.768	1.225	36.947	36.822	36.817	0.019	37.530	37.513	36.875	2.141	(43.3 ± 1.4)a
Imaginary frequency TS4 (cm−1)	1155.09i				1323.89i				1098.31i

^a Experimental values: Refs.: a [18], and b [19].

Table 4.

Bond order analysis using NBO for reactions 1–4.

Reaction	Bond	C2–O3	C2–O7	O3–C4	C4–C5	C5–H6	O7–H6	δBav	Sy
Isopropyl acetate (1)	Bi(R)	1.0096	1.7789	0.8461	1.0192	0.9396	0.0000	0.491	0.90
	Bi(TS)	1.4208	1.3762	0.3013	1.3716	0.4822	0.2855
	Bi(P)	1.7767	1.0274	0.0000	1.9884	0.0000	0.7518
	%EV	53.60	53.59	64.39	36.36	48.68	37.98
Isopropyl propionate (2)	Bi(R)	1.0107	1.7761	0.8458	1.0182	0.9386	0.0000	0.486	0.90
	Bi(TS)	1.4140	1.3771	0.3013	1.3680	0.4875	0.2812
	Bi(P)	1.7761	1.0246	0.0000	1.9884	0.0000	0.7509
	%EV	52.69	53.09	64.38	36.05	48.06	37.45
Isopropyl glycolate (3)	Bi(R)	1.0035	1.8040	0.8453	1.0190	0.9411	0.0000	0.512	0.88
	Bi(TS)	1.4522	1.3620	0.2894	1.3722	0.4888	0.2784
	Bi(P)	1.7535	1.0642	0.0000	1.9884	0.0000	0.7505
	%EV	59.83	59.75	65.76	36.43	48.06	37.10
Isopropyl bromoacetate (4)	Bi(R)	1.0136	1.7996	0.8373	1.0201	0.9385	0.0000	0.505	0.86
	Bi(TS)	1.4659	1.3677	0.2611	1.3724	0.5072	0.2593
	Bi(P)	1.7724	1.0464	0.0000	1.9884	0.0000	0.7471
	%EV	59.61	57.34	68.82	36.38	45.96	34.71

Table 2.

Reaction parameters (energy, enthalpy, and Gibbs free energy) (in kcal mol⁻¹) of compounds 1–4 at the studied DFT levels and the CBS-QB3 method (P = 1 bar).

Parameter Reaction	ωB97XD/6–311++G(d,p)			M06–2x/6–311++G(d,p)			CBS-QB3
	ΔE0K	ΔH°298K	ΔG°298K	ΔE0K	ΔH°298K	ΔG°298K	ΔE0K	ΔH°298K	ΔG°298K
Isopropyl acetate → P1 [propene + Acetic acid]	11.488	11.385	0.282	11.926	12.299	0.162	11.440	11.734	0.277
Isopropyl propionate → P2 [propene + Propionic acid]	11.475	11.829	−0.584	11.392	11.800	−1.572	11.366	11.571	−0.469
Isopropyl glycolate → P3 [propene + Glycolic acid]	11.052	11.361	−0.919	11.374	11.744	−1.148	11.294	11.541	−0.841
Isopropyl bromoacetate → P4 [propene + Bromoacetic acid]	11.370	11.529	−0.464	11.628	11.791	−0.257	10.596	10.698	−1.120

Activation parameters namely Gibbs free energy (ΔG^†) and enthalpy (ΔH^†) for reactions 1–4 are positive, indicating that activation processes require energy and are not spontaneous (see Table 3). It is expected that replacing the H atom with Me, OH, or Br will lower the energy barrier while increasing the associated rate constants. The barrier energy (ΔE_0K^†) for the decomposition of compound 4 is lower by nearly 2.74, 1.85, and 1.05 kcal mol⁻¹ at the CBS-QB3 method, respectively, than the energy barrier for the other species [1–3] which is consistent with the replacement of the H, Me, and OH groups by Br atom. Similar observations can be made when Gibbs free activation energies are considered: despite slightly lower entropies, the Gibbs free energy for the pyrolysis of isopropyl bromoacetate (nearly 33.76, 36.82, and 36.88 kcal mol⁻¹ at the ωB97XD, M06–2x, and CBS-QB3, respectively) is lower than for isopropyl acetate (approximately 36.57, 39.59, and 39.70 kcal mol⁻¹, respectively at the same methods).

The kinetics of the decomposition is determined by the activation energy barrier and the activation entropy (ΔS^†) required to reach the TS structure [78, 79]. The activation entropies for the studied compounds [X–CH₂COOCHMe₂ with X = H (1), Me (2), OH (3), and Br (4)] are equal to 1.97, 1.64, 1.65, and 2.14 Cal mol⁻¹K⁻¹, respectively, at the CBS-QB3 method (see Table 3). Compound 4 has a higher activation entropy than compounds 1–3, which could be a result of higher charge separation and thus a less restricted TS, or to larger restriction on rotations in this compound's initial state.

Compounds 1–4 have unimolecular decomposition processes that are concerted, six-center, and homolytic. The substitution of methyl, hydroxyl, and bromine groups for a hydrogen atom attached to a –CH₂ group is expected to lower the activation energy and thus increase the kinetic rate constant of the reaction. The influence of an ortho-bromine could be due to the halogen depolarizing the carbonyl group, which reduces the attraction of a hydrogen atom in β position to the carbonyl oxygen.

The OH and Br substituents tend to enhance the pyrolysis of the investigated esters in comparison to the parent isopropyl acetate, while the α-methyl group appears to decrease the kinetic rate constant. At room temperature, the sequence of kinetic rate constants as Br > OH > H > CH₃ has a direct relationship with acid strength [80]. Because electron withdrawal by the bromine substituent tends to enhance the ester's decomposition with respect to the parent compound isopropyl acetate, the kinetic rate constant for the decomposition of isopropyl bromoacetate (compound 4) to the formation of propene and bromoacetic acid at T = 623.25 K is most favorable from a kinetic viewpoint, and the RRKM calculations show that low pressures around 0.01 bar are required to restore the validity of the TST approximation for this reaction channel.

Given that these secondary acetates' polar substituents appear to influence reaction rate constants, it is reasonable to believe that the polarization of the alkyl side's C–O bond length in the sense –O=C–O^−δ….C^+δ is the determining factor in the TS of the considered pathways [18].

3.2. Transition state and mechanism

The geometrical properties of the reactants (R1−R4), transition states (TS1−TS4), propene, and related acids (P) of the considered pathways are optimized using the CBS-QB3 method (see Scheme 2). The studied reactions lead to the cleavage of the O₃–C₄ and C₅–H₆ bonds through TS structures to produce propene and the related acid with reaction energy up to ∼11.44 kcal mol⁻¹ at the CBS-QB3 composite method. The transition state structure forms a six-membered ring. The O₃–C₄ bond length increases correspondingly up to ∼0.71 Å, showing that this bond is broken in the TS structures. Similarly, the C₅–H₆ bond length increases by only ∼0.2 Å. The O₃–C₄ and C₅–H₆ bond lengths are elongated by 2.157 and 1.293 Å in the transition states [TS1−TS4], respectively, while simultaneously shrinking the O₇–H₆ distance to form a new O₇–H₆ bond by 0.965 Å. Also, the bond lengths in the TS structures decrease from ∼1.52 to ∼1.41 Å that indicating a shift in the C₄–C₅ bond lengths from single to double bond characters. The bond distances of C₂–O₇ show that a shift from double bonds (1.271–1.273 Å) in the TS structures to single bonds (1.333–1.349 Å) in the related acid. The C₅–H₆ bond length is elongated by ∼18% in such compounds to the calculated equilibrium structures for R1−R4 molecules (1.093 Å). On the contrary, the O₇–H₆ bond forms quite naturally, and at a greater distance than in the reactants (see Figure 2).

Figure 2.

Stationary points that are involved in reactions 1–4 using the CBS-QB3 method [Bond lengths are given in angstroms (Å) and bond angles in degrees (^o)].

Dihedral angles [φ (O₃–C₂–O₇–H₆)] for the chemical pathways 1–4 in TSs are 125.22° (112.77 to –12.45°), 129.67° (113.41 to −16.26°), 142.91° (126.65 to –16.26°), and 117.85° (105.38 to –12.47°), respectively at the CBS-QB3 method (see Table 3 and Figure 2 for atom numbering). These findings indicate that the TS structures for the decomposition reaction of compounds 1–4 are non-planar. The imaginary frequencies for the transition states of chemical channels 1–4 were also found to be 1262i, 1245i, 1231i, and 1098i cm⁻¹, respectively.

The n_T parameter (Eq. (6)) defines Hammond’s postulate behavior along the reaction coordinate [81, 82].

$$n_{\mathrm{T}}=\frac{1}{2-\left(\Delta G_{\mathrm{r}}/\Delta G^{†}\right)}$$ (6)

Eq. (6) states that the position of the TS structure along the reaction coordinate is solely determined by the ΔG_r and ΔG^† as thermodynamic and kinetic quantities, respectively. When n_T < 0.5 (early TS), the TS structure is similar to reactant, while when n_T > 0.5, the TS structure is similar to product (late TS) [82]. In the thermal decomposition of compounds 1–4 [R_i → propene + P_i (i = 1–4)], the TS structures are identical to the studied reactants and products, and the n_T values are around 0.5 using the CBS-QB3 method.

3.3. Bond order analysis

Bond order (B), a concept used to study the molecular mechanism of many reactions, provides a more balanced measure of the extent of bond formation/breaking along the reaction [83]. Wiberg bond indices [84] were calculated using an NBO analysis [85]. Several bond-forming/breaking processes occur during fragmentation, and the overall nature of the pyrolysis reaction can be monitored using the synchronicity (S_y) index as given in Eq. (7) [86].

$$S_y=1-\frac{\left[\sum _{i=1}^n\frac{\left|\delta B_i-\delta B_{av}\right|}{\delta B_{av}}\right]}{2n-2}$$ (7)

where n is the number of direct bonds in the chemical process, δB_i denotes the bond index's relative variation at the TS for a bond i, and δB_av represents the average change in bond orders. It is noted that the synchronicity parameter, S_y ranges from 0 (fully asynchronous process) to 1 (fully concerted synchronic process) [87].

Bond indices were calculated for the bonds involved in the studied reactions, i.e. C₂–O₃, C₂–O₇, O₃–C₄, C₄–C₅, C₅–H₆, and O₇–H₆ (see Scheme 2). All other bonds are essentially unchanged. The calculated Wiberg bond indices, B_i allow us to examine the progress of the reaction and assess the position of the TSs between reactants and products (Table 4). The pyrolysis of the IPA and its derivatives [X–CH₂CO₂CHMe₂ with X = Me, OH, and Br] implies the cleavage of the O₃–C₄ and C₅–H₆ bonds yielding products located at 11.44, 11.37, 11.29, and 10.60 kcal mol⁻¹ above the reactants, respectively, at the CBS-QB3 method. The formation of the C=C double bond results in the elongation of the bond breaking of C₅–H₆ and O₃–C₄, as well as the simultaneous shortening of the bond lengths of C₄–C₅ (Table 4).

The most advanced reaction coordinate for pyrolysis of compounds 1–4, according to Wiberg bond indices, O₃–C₄ bond breaking (%EV = 64.38–68.82). The progress in C₅–H₆ bond breaking is also significant (%EV = 45.96–48.68). For the investigated reactions, there is less progress in the formation of C₄–C₅ double bonds (%EV = 36.05–36.38). The synchronicity indices for reactions 1–4 are 0.90, 0.90, 0.88, and 0.86, indicating that the studied reactions are concerted and slightly asynchronous, respectively.

The distribution of electrons during the decomposition reactions of IPA and its derivatives can also be studied using charges. In this regard, NBO charges have proven useful [78]. In all reactants, the O₃–C₄ bond is polarized (O₃ charge is −0.5783 to −0.5702 and C₄ is 0.1468–0.1485, charge separation Δq = 0.7187–0.7251) in the sense O₃^δ−–C₄^δ+. The following partial charges occur as the considered reaction progresses from reactants to transition states: an increase in negative charge δ⁻ in oxygen O₃ [(−0.5783 to −0.5702) in reactants compared to (–0.6573 to –0.6305) in TSs] as O₃–C₄ bond breaks, an increase in positive charges δ⁺ in carbon C₄ [(0.1468–0.1485) in the reactants associated with (0.1793–0.1983) in TS] are summarized in Table 5.

Table 5.

NBO charges of reactants (R) and transition states (TS) for the pyrolysis of reactions 1–4 at the CBS-QB3 method.

atom	1 (X = H)		2 (X = Me)		3 (X = OH)		4 (X = Br)
	R1	TS1	R2	TS2	R3	TS3	R4	TS4
C1	0.8294	−0.6711	−0.4899	−0.4740	−0.1331	−0.1112	−0.5415	−0.4994
C2	0.8294	0.8446	0.8324	0.8492	0.8077	0.8214	0.8092	0.8174
O3	−0.5759	−0.6521	−0.5758	−0.6573	−0.5783	−0.6305	−0.5702	−0.6401
C4	0.1485	0.1793	0.1482	0.1828	0.1468	0.1850	0.1474	0.1983
C5	−0.5824	−0.7264	−0.5813	−0.7260	−0.5820	−0.7239	−0.5824	−0.7180
H6	0.2033	0.3964	0.2056	0.3951	0.2043	0.3962	0.2072	0.3930
O7	−0.5682	−0.6894	−0.5742	−0.6911	−0.5402	−0.6990	−0.5434	−0.6869

The C_α−O bond elongation [C₄–O₃] and successive polarization in the C_α^+δ…O^−δ denotes the rate-determining step in the studied reactions (Scheme 2). Polarization of the C_α^+δ…O^−δ bond can result from electron transmission toward the more electronegative group, and to a lesser extent, larger π-electron delocalization of the C=O group. As a result, the partially negatively charged oxygen atom of acetate (O₃) would face greater difficulty in stabilizing its p electrons toward the C=O group's adjacent more π-bonded carbon [5]. Thus, the more electronegative the substituent, the less stable the oxygen p electrons of the C_α−O bond toward the carbonyl carbon and the slower the rate constant [88].

3.4. Bonds dissociation energies (BDEs)

Estimating BDEs can provide early insights into bond strength and fuel behavior at high ignition temperatures [89, 90, 91, 92, 93]. The CBS-QB3 energies were used to calculate the BDEs of the considered compounds at T = 298 K as depicted in Figure 3.

Figure 3.

BDEs (in kcal mol⁻¹) of the studied compounds at the CBS-QB3 method.

The C₄−X bond is the weakest bond compounds 2, 3, and 4, with 78.3, 78.9, and 60.4 kcal mol⁻¹, correspondingly, followed by comparable C₃–C₄ bond breaking with 81.8 and 82.6 kcal mol⁻¹ for compounds 3 and 4, individually, while in compound 1, C₁–C₂ bond is the weakest bond with 90.7 kcal mol⁻¹, followed by C₂–O₁ bond energy 92.2 kcal mol⁻¹. For all investigated molecules, the internal C₄–H bond is the most accessible for H-atom removal with different oxidizing agents, while the terminal C₁–H bond is the most difficult.

3.5. Kinetic parameters

The CBS-QB3 method was used to calculate the TST and RRKM rate coefficients for the pyrolysis reactions of compounds 1–4 at P = 1 bar and the available experimental temperatures [18] (see Table 6 and Tables S1a−d of the Supplementary material contains additional RRKM data for the same temperatures computed at lower and higher pressures). The rate coefficient for the production of bromoacetic acid and propene species [via R4] is larger than that obtained for the acetic acid, propionic acid, and glycolic acid species [via R1−R3, respectively], which corresponds to a decrease in energy barrier of 2.74, 1.85, and 1.05 kcal mol⁻¹ on the related reactions. Indeed, the TST results reveal that reaction 4 has higher rate constants than the other pathways. These rate constants gradually increase with increasing temperature due to the positive energy barriers involved. Furthermore, the entropy loss of the studied cyclic TSs at the CBS-QB3 method is low, as expected.

Table 6.

Unimolecular rate constants (in s⁻¹) for the studied reactions using TST and RRKM (P = 1 bar).

Path T (oC)	TST				RRKM
	R1 → P1	R2 → P2	R3 → P3	R4 → P4	R1 → P1	R2 → P2	R3 → P3	R4 → P4
290 (563 K)	5.39 × 10−1	1.17×10°	1.91×10°	1.24×101	3.66 × 10−1	7.95 × 10−1	1.32×10°	4.57×10°
295 (568 K)	7.16 × 10−1	1.54×10°	2.51×10°	1.61×101	4.90 × 10−1	1.06×10°	1.74×10°	5.99×10°
300 (573 K)	9.48 × 10−1	2.02×10°	3.28×10°	2.09×101	6.52 × 10−1	1.39×10°	2.29×10°	7.81×10°
305 (578 K)	1.25×10°	2.64×10°	4.27×10°	2.70×101	8.64 × 10−1	1.83×10°	3.00×10°	1.01×101
310 (583 K)	1.64×10°	3.43×10°	5.53×10°	3.47×101	1.14×10°	2.40×10°	3.91×10°	1.31×101
315 (588 K)	2.13×10°	4.44×10°	7.13×10°	4.45×101	1.50×10°	3.13×10°	5.07×10°	1.68×101
320 (593 K)	2.77×10°	5.73×10°	9.16×10°	5.67×101	1.95×10°	4.05×10°	6.55×10°	2.16×101
325 (598 K)	3.58×10°	7.35×10°	1.17×101	7.20×101	2.54×10°	5.24×10°	8.42×10°	2.75×101
330 (603 K)	4.62×10°	9.41×10°	1.49×101	9.12×101	3.29×10°	6.74×10°	1.08×101	3.50×101
335 (608 K)	5.92×10°	1.20×101	1.89×101	1.15×102	4.25×10°	8.63×10°	1.38×101	4.43×101
340 (613 K)	7.57×10°	1.52×101	2.39×101	1.44×102	5.46×10°	1.10×101	1.75×101	5.59×101
345 (618 K)	9.64×10°	1.92×101	3.01×101	1.81×102	6.98×10°	1.40×101	2.21×101	7.02×101
350 (623 K)	1.22×101	2.42×101	3.78×101	2.25×102	8.90×10°	1.77×101	2.79×101	8.79×101

According to Table 6, TST overestimates the RRKM rate constants by 2.78 orders of magnitude, indicating that RRKM theory should be used to investigate the fall-off behavior of these rate constants towards the low-pressure limit. Based on the computed CBS-QB3 energy profiles, all calculated temperature-dependent TST rate constants for the considered reactions, confirm that the production of propene and bromoacetic acid species (via R4) is the fastest under atmospheric pressure at all studied temperatures (see Figure 4). The same holds for pressures ranging from 10⁻¹⁰–10² bars (see Tables S1a−d in the Supplementary material). Because the energy barriers involved are high, the rate constants for the formation of the “acetic acid”, “propionic acid” and “glycolic acid” species are lower at the studied temperatures than the rates for the formation of the “bromoacetic acid” species: the rates describing the pyrolysis of isopropyl bromoacetate into propene + bromoacetic acid species are up to two orders of magnitude lower (reaction 4). As shown in Figure 4 and Table 6, the temperature dependence of the RRKM rate constants for reactions 1–4 over temperature ranges 563–623 K was investigated with an interval of 5 K, revealing that rate constants increase with increasing temperature.

Figure 4.

Arrhenius plot of the measured TST rate constants [for R_i (i = 1–4) →propene + related acid] (P = 1 bar).

The RRKM equation modeling revealed that at atmospheric and higher/lower troposphere pressures, the rate constants of reaction are in the fall-off regime. Figure 5 shows additional fall-off curve plots for the RRKM rate constants (k_uni). Pressures greater than 0.01 bar appear to be required for ensuring saturation to the high-pressure limit of the RRKM rate constants, by rather larger energy barriers ranging from 37.53 to 40.27 kcal mol⁻¹. However, when compared to RRKM data, the TST approximation breaks down at pressures less than 0.01 bar for the kinetic rate coefficient of reaction 4.

Figure 5.

Fall-off plots for the rate constants as a function of pressure for the thermal decomposition processes of compounds 1–4 using RRKM theory calculated at the CBS-QB3 method.

Table 6 shows that kinetic rates calculated at a pressure of 1 bar using the TST and RRKM approaches with the same energy profiles do not vary significantly. It is thus better to use the RRKM approach to evaluate rate constants because the strong pressure dependence of rate constants can be accounted for at pressures less than 0.01 bar. The effect of Eckart tunneling appears to be (almost) negligible and has only a minor impact on the computed rate constants (see Table S2 of the Supplementary material). Based on the computed CBS-QB3 energy profiles and B3LYP/6–311++G(d,p) vibrational frequencies, TST calculations of the rates that characterize the studied reaction pathways yielded κ(T) values ranging from 1.3 to 1.5. As the temperature increases from 563 to 623 K, the ratios of TST and RRKM estimates of rate constants for reaction 4 decrease from 2.71 to 2.56. These variations are caused by tunneling corrections to the TST rate constants.

4. Conclusions

The thermal decomposition of isopropyl acetate and its derivatives [X: Me, OH, and Br] in the gas phase was investigated at the ωB97XD/6–311++G^∗∗ and M06–2x/6–311++G^∗∗ levels of theory as well as CBS-QB3 composite method. At the high-pressure limit, unimolecular rate coefficients were calculated using transition state theory (TST). RRKM theory was used to investigate their pressure dependence in the pressure range 10⁻¹⁰−10² bar and at available experimental temperatures. According to the data provided, the reaction [isopropyl bromoacetate → propene + bromoacetic acid] is more favorable than the other reactions in terms of both thermodynamics and kinetics. The calculated energy barriers and unimolecular rates agree reasonably well with the experiment. These reactions' transition states correspond to non-planar six-membered cyclic structures. The use of synchronicity indices to analyze the computed pathways reveals concerted and slightly asynchronous processes. TST and RRKM theories were used to calculate rate constants under atmospheric pressure and in the fall-off regime, respectively, to supplement the calculated energy profiles. According to the RRKM calculations, pressures greater than 0.01 bar are sufficient to ensure the validity of the TST approximation for all studied reactions.

The most significant difference is seen in the O₃–C₄ bond length, which is elongated in the TSs (at 2.039–2.993 Å) when compared with the reactant distances (1.437–1.442 Å). When compared to other reaction changes, the TS structures show more progress in breaking the O₃–C₄ bond. According to the NBO charges analysis, the polarization of this bond in the sense O₃^δ−–C₄^δ+ is the determining factor in the pyrolysis process. A six-membered cyclic transition state is used to determine the rate of these reactions. The analysis of NBO charges shows that as the O₃–C₄ bond breaks, carbon C₄ becomes more positively charged, and oxygen atom O₃ becomes more negatively charged in TSs.

Declarations

Author contribution statement

S.H. Reza Shojaei, Abolfazl Shiroudi, Mohamed A. Abdel-Rahman: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability statement

Data included in article/supplementary material/referenced in article.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.

Appendix A. Supplementary data

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