Rate Parameter Distributions for Isobutane Dehydrogenation and Isobutene Dimerization and Desorption over HZSM-5

1. Introduction

2. Kinetics of Steady-State, Low-Pressure Catalysis

2.1. Mechanism

Isobutane (IA_(g)) flows into the Knudsen cell and collides with the external surface of the zeolite catalyst. Adsorption (IA-S) may occur at non-specific sites on the surface (S) with rate constant, k₁. The reverse step, desorption with rate constant, k₋₁ will also be prevalent.

(R1)

Isobutane diffusion to and adsorption at active sites or into the zeolite micropores (IA-S*) with rate constant, k₂ and reverse desorption and diffusion (k₋₂) will control the density of reactants bound to active sites.

(R2)

Dehydrogenation to adsorbed isobutene and hydrogen(IE-S*), with rate constant k₃ then proceeds. The reverse hydrogenation step may also occur in the confines of the zeolite channels.

(R3)

An alternative is dimerization of isobutene to adsorbed 2,4,4-trimethyl-2-pentene and 2,4,4-trimethyl-1-pentene, (IE₂-S*).

(R4)

Competition then is apparent between direct product desorption (P_(g)) and diffusion to non-specific surface sites (PS). Each of these steps is reversible:

(R5)

(R6)

Both reactant and product gas-phase molecules then exit from the Knudsen cell and into the mass spectrometer with escape rate constants, k_escIA and k_escIE, respectively.

(R7)

Escape rate constants are calculated from Equation (23), below [1,2,27].

2.2. Rate Laws

The rate of escape of isobutene from the Knudsen cell, under steady-state conditions is simplified to desorption from non-specific sites:

(1)

Steady-state isobutene concentrations on the surface of the catalyst at non-specific sites [IE-S] and specific active sites [IE-S*] are shown in Equations (2) and (3). Equation (2) has been reduced by assuming that the reverse step in the numerator does not make a significant contribution to the concentrations.

(2)

For product isobutene coverage at specific active sites, dimerization (R4) is chosen as the more prevalent reaction when compared with hydrogenation (reverse of R3):

(3)

Steady-state concentrations of adsorbed isobutane at specific active sites [IA-S*] are shown in Equation (4). Here dimerization (R4) is taken to be negligible.

(4)

Finally, steady-state isobutane concentration on non-specific catalyst sites, [IA-S] is approximated by:

(5)

The overall rate law for escape from the Knudsen cell can be approximated by Equation (6) and the apparent rate constant, k_app:

(6)

This can be further deconvoluted to three rate laws. The first is when neither dehydrogenation nor dimerization is rate-determining, while the rate is dependent on product desorption from non-specific sites. That is, k₄[IE-S*] ≪ k₅[S], k₋₂[S] ≪ k₃, and k₋₅[S*] ≫ k₆

(6a)

The second rate law is when dehydrogenation is dominant, and product desorption and dimerization are not rate determining: k₋₂[S] ≫ k₃, k₋₅[S*] ≪ k₆ and k₄[IE-S*] ≪ k₅[S].

(6b)

The third rate law is when dimerization (R4) is causing an apparent decrease in isobutene partial pressure, while dehydrogenation and product desorption do not affect the rate: k₋₂[S] ≪ k₃, k₋₅[S*] ≪ k₆ and k₄[IE-S*] ≫ k₅[S].

(6c)

Isobutene mass spectral abundances at m/e = 56, I₅₆ is proportional to the mass spectral sensitivity factor, a₅₆, escape rate constant from the Knudsen cell and steady-state concentration [IE_(g)] in the Knudsen cell.

(7)

The concentration of isobutane at sites on the surface or in the pores of the catalyst is unlikely to be directly proportional to the steady-state isobutane pressure [IA_(g)] in the Knudsen cell and hence the associated mass-spectral abundance, I₅₈. As a consequence, an adjustable correction exponent, n, is included in the optimization:

(8)

Including Equations (7) and (8) into Equation (6) gives:

(9)

Taking logarithms and rearranging

(10)

Writing the rate constant expression with apparent Arrhenius prefactors and activation energies yields:

(11)

The simulations undertaken in this paper of the temperature-dependent reactant and product mass-spectral abundances give three temperature-dependent parameters, A(T), n(T) and E_app(T). The exponent n(T) is a correction for differences in the steady-state pressure of reactant in the cell and concentrations at the external surface and within the zeolite pores.

(12)

By assuming intrinsic reaction, diffusion and desorption prefactors are equal and so cancel in the deconvoluted rate laws (6a, 6b and 6c), apparent prefactors can be associated with reactant collision frequencies on the surface of the zeolite. They can be calculated from:

(13)

3. Results and Discussion

Figure 1. Apparent activation energies (blue circles) and prefactors (red circles) for the first temperature-programmed run for isobutene evolution over HZSM-5.

Figure 2. Apparent activation energies (blue circles) and prefactors (red circles) for the second temperature-programmed run for isobutene evolution over HZSM-5.

Figure 3. Apparent activation energies (blue circles) and prefactors (red circles) plotted against temperature for the third temperature-programmed run for isobutene evolution over HZSM-5.

Table 1. Apparent activation energies and prefactors for each peak, plateau and shoulder plotted in Figure 1. Site-1are assigned to isobutene desorption. Site-2a, Site-2b and Site-2c are isobutane dehydrogenation and Site-3a and Site-3b are isobutene dimerization parameters.

Table 2. Apparent activation energies and prefactors for each peak, plateau and shoulder plotted in Figure 2. Site-1 are assigned to isobutene desorption. Site-2a, Site-2b and Site-2c are isobutane dehydrogenation and Site-3a to Site-3b are isobutene dimerization parameters.

Table 3. Average apparent activation energies and prefactors for each peak, plateau and shoulder in Figure 3. Site-1 are assigned to isobutene desorption. Site-2a and Site-2b are isobutane dehydrogenation and Site-3a, Site-3b and Site-3c are isobutene dimerization parameters.

3.1. Isobutene Desorption

Low- and high-temperature sites (Site-1) for isobutene desorption are identified in Figure 1, Figure 2 and Figure 3, by the low apparent activation energies, less than 50 kJ mol⁻¹. The dehydrogenation products at low-temperatures are likely to have formed during the initial extended exposure at 100 °C. As a consequence isobutane desorption and diffusion are not rate-limiting and k₋₁ ≪ k₂[S*]. Equation (6a) becomes:

(14)

That is, the rate constant for desorption, multiplied by the apparent isobutane collision frequency and divided by the rate constant for diffusion of adsorbed isobutene from the external surface to either an active site or into the micropores of the zeolite. This can be expanded as apparent prefactors

(15)

and activation energies

(16)

Apparent activation energies for the low-temperature sites over the three runs range from 7.9 ± 0.3 to 41.3 ± 0.8 kJ mol⁻¹, with mean 24 ± 14 kJ mol⁻¹. The high-temperature sites range from 8.5 ± 0.6 to 31.1 ± 1.6 kJ mol⁻¹. The mean is 17 ± 9 kJ mol⁻¹.

Adsorption and diffusion kinetic parameters for branched-chained alkenes in zeolites are not widely reported in the literature. Hufton et al. [28] report adsorption and diffusion energies for isobutane and isobutene in large silicalite crystals at low concentrations, measured by chromatographic techniques. Silicalite has the same structure as HZSM-5 and adsorption energies have been found to be independent of the Si/Al ratio [25]. For isobutene in silicalite the heat of adsorption and diffusion are 72.8 and 56.5 kJ mol⁻¹ respectively [28].

Table 4. Intrinsic isobutene desorption activation energies (E₆) and collision frequencies (log₁₀A₁) calculated from Equations (15) and (16) for the three runs. Isobutene diffusion activation energy is E₋₅ = 56.5 kJ mol⁻¹ [1] and E₁ = 0, and the ratio log₁₀A₆: A₋₅ ≈ 0.

Table 4. Intrinsic isobutene desorption activation energies (E₆) and collision frequencies (log₁₀A₁) calculated from Equations (15) and (16) for the three runs. Isobutene diffusion activation energy is E₋₅ = 56.5 kJ mol⁻¹ [1] and E₁ = 0, and the ratio log₁₀A₆: A₋₅ ≈ 0.

	Site-1 (Low T)	Site-1 (High T)
	E6 (kJ mol-1)	E6 (kJ mol-1)
Run 1	64.4 ± 0.3; 90.5 ± 12	87.6 ± 1.6; 70.2 ± 1.7
Run 2	67.2 ± 1.8; 97.8 ± 0.8	65.0 ± 0.6; 78.3 ± 1.9
Run 3	83 ± 6	66.8 ± 0.7
Average	77 ± 12
	log10A1(s-1)	log10A1(s-1)
Run 1	–2.49 ± 0.03; 0.6 ± 1.4	0.42 ± 0.10; –0.69 ± 0.15
Run 2	–1.9 ± 0.2; 1.72 ± 0.07	–1.11 ± 0.07; 0.04 ± 0.15
Run 3	0.1 ± 0.6	–0.82 ± 0.06
Average	–0.4 ± 1.2

Table 4 lists isobutene average desorption activation energies, E₆ calculated using Equation (16) and with E₋₅ = 56.5 kJ mol⁻¹ and E₁ = 0, i.e., non-activated isobutane adsorption. The overall average of all energies is 77 ± 12 kJ mol⁻¹. The mean value is within 4.2 kJ mol⁻¹ of the heat of adsorption of isobutene in silicalite (72.8 kJ mol⁻¹) [28]. Isobutene desorption from Sites-1 are assigned to desorption from non-specific sites within the zeolite micropores, because energies are close to the literature heat of adsorption [28], silicalite would not have strong acid sites.

A first approximation for the desorption and diffusion prefactors are log₁₀A₆ ≈ log₁₀A₋₅ ≈ 13 [29] and [S] ≈ [S*]. Hence the apparent prefactor is the isobutane collision frequency, A_app ≈ A₁, and are also listed in Table 4. The overall mean collision frequency is log₁₀A₁(s⁻¹) = –0.4 ± 1.2. Low and negative values demonstrate that isobutane collisions are not rate-determining.

3.2. Isobutane Dehydrogenation

Due to the large and positive apparent activation energies the rate parameters at Sites-2a, 2b and 2c are assigned to isobutane dehydrogenation. For runs 1 and 2 the average parameters at Site-2a and 2b are paired. That is, each pair is linked with increasing rate parameters. For run 3, Site-2a shows a linked pair at lower temperatures then for the first two runs, while Site-2b parameters are not linked. For dehydrogenation isobutane adsorption and diffusion is also expected to play a part. Both isobutane desorption and diffusion to active sites may affect the rate. The heat of adsorption of isobutane has been measured in silicalite in low concentrations to be 49.0 kJ mol⁻¹ [1]. Calculated adsorption energies for isobutane on HZSM-5 are 38 kJ mol⁻¹ for primary carbons and 48 kJ mol⁻¹ for tertiary carbons [25]. These magnitudes would be similar to the isobutane desorption energy, E₋₁, assuming adsorption is not activated (E₁ = 0). In comparison the heat of diffusion of isobutane in silicalite is less by only 16.8 kJ mol⁻¹, i.e., 32.2 kJ mol⁻¹ [28]. Equation (6b) can, however be simplified by setting k₋₁ ≪ k₂[S*] and the apparent rate constant, becomes independent of the site concentration:

(17)

This can be expanded as apparent Arrhenius prefactors

(18)

and activation energies

(19)

Dehydrogenation activation energies, E₃ are calculated from the E_app values in Table 2 and by setting E₋₂, isobutane diffusion activation energy from active sites to 32.2 kJ mol⁻¹. These dehydrogenation energies are listed in Table 5 for each of the three Sites and three temperature-programmed runs.

Site-2a intrinsic activation energies in Table 5 increase with increasing temperature and decrease with consecutive runs. Decreasing activity (i.e., increasing E₃) with increasing temperature demonstrates that the most active sites take priority in dehydrogenation catalysis. Increasing activity (i.e., decreasing E₃) maybe related to increasing dehydration due to repeated heating to 500 °C, and hence increasing exposure of Lewis acid sites. The overall average of the Site-2a dehydrogenation activation energies is 125 ± 30 kJ mol⁻¹ and energy barriers range from 86 ± 3 to 170.7 ± 1.2 kJ mol⁻¹.

The Site-2b intrinsic dehydrogenation activation energies also show decreasing activity with increasing temperature. However, in contrast to Site-2a, increasing E₃ is apparent for consecutive runs. This suggests that the active sites at Site-2b are Brønsted acid sites with the most active sites deactivating first, as the zeolite becomes dehydrated. Overall average activation energy for isobutane dehydrogenation at Site-2b is 152 ± 31 kJ mol⁻¹ and the energy barriers range from 131.6 ± 1.6 to 235.2 ± 1.0 kJ mol⁻¹.

For Site-2c, a sharp E₃ positive peak at around 380 °C is observed for both runs 1 and 2. The magnitude of the first run dehydrogenation energy of 122.2 ± 2.0 kJ mol⁻¹ is similar in magnitude to the possible Lewis acid sites of Site 1, while 172 ± 7 kJ mol⁻¹ for the second run is a closer match to the possible Brønsted acid site activation energy at Site 2. However, the average of E₃ for the two runs (147 ± 35 kJ mol⁻¹) is similar to the overall average Brønsted acid Site-2b energy (152 ± 31 kJ mol⁻¹). A sharp peak suggests hydrogenation is occurring in confined spaces within the zeolite micropore structure, followed by release to the gas phase. The average of all the proposed intrinsic dehydrogenation activation energies is 150 ± 42 kJ mol⁻¹.

Table 5. Intrinsic isobutane dehydrogenation activation energies (E₃) and collision frequencies (log₁₀A₁) calculated from Equations (15) and (16) for the three runs. Isobutane diffusion activation energy is E₋₂ = 32.2 kJ mol⁻¹ [1] and E₋₁ = 0, and the ratio log₁₀A₃: A₋₂ ≈ 0.

Table 5. Intrinsic isobutane dehydrogenation activation energies (E₃) and collision frequencies (log₁₀A₁) calculated from Equations (15) and (16) for the three runs. Isobutane diffusion activation energy is E₋₂ = 32.2 kJ mol⁻¹ [1] and E₋₁ = 0, and the ratio log₁₀A₃: A₋₂ ≈ 0.

	Site-2a	Site-2b	Site-2c
	E3 (kJ mol−1)	E3 (kJ mol−1)	E3 (kJ mol−1)
Run 1	125 ± 6 to 170.7 ± 1.2	131.6 ± 1.6 to 137.2 ± 1.0	122.2 ± 2.0
Run 2	106 ± 3 to 143 ± 2	148 ± 9 to 203 ± 6	172 ± 7
Run 3	86 ± 3 to 116.2 ± 1.4	202 ± 12 to 235.2 ± 1.0	–
Range	86 ± 3 to 170.7 ± 1.2	131.6 ± 1.6 to 235.2 ± 1.0	122.2 ± 2.0 to 172 ± 7
	log10A1(s−1)	log10A1(s−1)	log10A1(s−1)
Run 1	7.2 ± 0.6 to 11.80 ± 0.15	7.09 ± 0.12 to 7.44 ± 0.10	5.05 ± 0.18
Run 2	5.3 ± 0.3 to 9.0 ± 0.3	8.9 ± 0.9 to 13.8 ± 0.5	9.1 ± 0.5
Run 3	2.9 ± 0.3 to 5.91 ± 0.15	16.7 ± 1.4 to 20.03 ± 0.15	–
Range	2.9 ± 0.3 to 11.80 ± 0.15	7.09 ± 0.12 to 20.03 ± 0.15	5.05 ± 0.18 to 9.1 ± 0.5

The apparent prefactors are again equivalent to effective isobutane collision frequencies, when it is assumed that desorption and diffusion prefactors are log₁₀A₃ ≈ log₁₀A₋₂ ≈ 13 [29]. The paired and single average apparent collision frequencies from the three dehydrogenation sites are: log₁₀A₁(s⁻¹) =7.8 ± 2.6, 10.8 ± 4.3 and 7.1 ± 2.9. The large ‘errors’ reflect the large range of parameters, rather than experimental errors. Values significantly greater than zero demonstrate that isobutane collisions and coverage are affecting the dehydrogenation rate. Sites-2a and 2c show similar average collision frequencies, although the Site-2c log₁₀A₁ values increase between the first and second runs, equivalent to the Site-2b trend. Site-2c reactions need to be considered with the dimerization reactions at Sites-3a, 3b and 3c, below.

Zheng and Bowers [23] used ab initio and density functional theory methods to investigate the conversion reactions of isobutane over zeolites. For dehydrogenation, calculated activation energies varied from 220 to 263 kJ mol⁻¹, depending on zeolite proton acidity (high to low). The equilibrium value is 242 kJ mol⁻¹. These activation energies are similar in magnitude to the highest Site-2b values (Brønsted acid sites): 235.2 ± 1.0 kJ mol⁻¹. Measured activation energies for isobutane dehydrogenation over three acidic zeolites (one MWW and two MFI) range from 150 ± 9 to 178 ± 9 kJ mol⁻¹ [13]. These also fall most comfortably within the Site-2a, Brønsted acid range listed in Table 2. Other reported experimental dehydrogenation activation energies over HZSM-5 are 100 kJ mol⁻¹ [14] and 136 ± 6 kJ mol⁻¹ [1]. The latter was calculated using the same data sets and at the highest temperatures (>395 °C). These studies do not distinguish between acid sites; however, they are within the range of dehydrogenation activation energies associated with Site-2a (Lewis acid sites). Activation energies for the dehydrogenation of isobutane at Lewis acid sites (extra-framework aluminum species) have been calculated using density functional theory to be in the range 163 to 172 kJ mol⁻¹ [24]. Like the Brønsted site calculations [23], these magnitudes are similar to the highest Site-2b values.

An alternative mechanism for one or more of the low-activation energy sites is low-temperature activation of isobutane to isobutene and hydrogen, caused by weak hydrogen bonding between C–H and O on the zeolite [4,30]. This measured average and apparent activation energy was previously calculated to be 62.8 ± 1.0 kJ mol⁻¹ for this same isobutane/HZSM-5 system [4]. The corresponding intrinsic activation energy for dehydrogenation, from Equation (19) is 95.0 ± 1.0 kJ mol⁻¹, and is most closely aligned with the Site-2a values (86 ± 3 to 170.7 ± 1.2 kJ mol⁻¹).

3.3 Isobutene Dimerization (or Hydrogenation)

The Site-3a, 3b and 3c regions of the profiles in Figure 1, Figure 2 and Figure 3 are dominated by negative activation energies are associated with dimerization, oligomerization or hydrogenation of isobutene. This is because they represent a decrease in the pressure of isobutene within the Knudsen cell. Dimerization is chosen for this analysis, but hydrogenation could equally be causing the isobutene pressure drop. A simplification of rate constant, Equation (6c) is when isobutene diffusion from active sites is competing with dimerization, and again, k₋₁ ≪ k₂[S*], because isobutane diffusion and desorption are next expected to be rate limiting:

(20)

Expanded as apparent Arrhenius prefactors, which is dependent on availability of non-specific sites squared and the coverage of isobutene at active sites:

(21)

and activation energies:

(22)

Again using the isobutene diffusion activation energy of E₅ = 56.5 kJ mol⁻¹ [28] and E₁ = 0, energies for dimerization can be calculated and are listed in Table 6. Three sites are observed, the first, Site-3a has E₄ ranging from 48.3 to 267 kJ mol⁻¹, the second, Site-3b from 67.6 to 81.5 kJ mol⁻¹ and the third site, Site-3c has a single sharp peak in run 3, with an activation energy of 104 ± 15 kJ mol⁻¹. In contrast with the dehydrogenation sites, energies decrease with increasing temperature for the linked pairs. This trend of increasing activity within a run maybe a consequence of E₅, the isobutene diffusion energy, increasing with increasing temperature, rather than a significant change to E₄. That is, as the isomers oligomerize and aromatize there may be blockage and hence isobutene is trapped within the micropores of the zeolite leading to decreased apparent activation energies. This blockage is further demonstrated by the increasing activation energies between succeeding runs; isobutene has reduced access to the most active sites. Site-3b demonstrates a much narrower range of activation energies across the three runs. In this case the size of the cavities maybe larger and active sites, where dimerization is occurring shows higher density, than for Site-3a. The single sharp peak observed for the third run at Site 3c, occurs at the same temperatures as the positive dehydrogenation peaks in the first and second runs and immediately following the dimerization sites. This suggests that Sites-3b, 2c and 3c are at equivalent locations within the HZSM-5 micropores. At lower temperatures dimerization or hydrogenation of isobutene dominates, while at higher temperatures (Sites-2c) decomposition of the 2,4,4-trimethylpentene products or dehydrogenation takes over. For Site-3c, where there may be pore blockage, decomposition or dehydrogenation leads to further dimerization or hydrogenation. These are similar arguments to the transition state, reactant and product shape selectivities reviewed by Smit and Maesen [8].

Kinetic modeling of isobutene dimerization over a commercial ion-exchange resin catalyst in a continuous stirred tank rank at 60–120 °C gives apparent activation energies of 30 ± 3 kJ mol⁻¹ [16,17]. Other experimental and theoretical studies [18,19] reported higher activation energies using different models: 51.49, 52.59, 47.91 and 89.34 kJ mol⁻¹. Hauge et al. [20] also investigated isobutene dimerization in the liquid phase over a range of zeolites, including HZSM-5. Rapid deactivation was observed and an apparent activation energy of 48 kJ mol⁻¹ was reported. These apparent energies are likely to include isobutene adsorption terms at two active sites [16], i.e., one for each monomer, and so are in agreement with the very broad range of intrinsic activation energies listed in Table 6.

Table 6. Intrinsic isobutene dimerization activation energies (E₄) and collision frequencies (log₁₀A₁) calculated from Equations (21) and (22) for the three runs. Isobutene diffusion activation energy is E₅ = 56.5 kJ mol⁻¹ [1] and E₁ = 0, and the ratio log₁₀A₅: A₄ ≈ 0.

Table 6. Intrinsic isobutene dimerization activation energies (E₄) and collision frequencies (log₁₀A₁) calculated from Equations (21) and (22) for the three runs. Isobutene diffusion activation energy is E₅ = 56.5 kJ mol⁻¹ [1] and E₁ = 0, and the ratio log₁₀A₅: A₄ ≈ 0.

	Site-3a	Site-3b	Site-3c
	E4 (kJ mol−1)	E4 (kJ mol−1)	E4 (kJ mol−1)
Run 1	88 ± 2 to 60.0 ± 0.7	76.9 ± 0.7	–
Run 2	136 ± 10 to 48.3 ± 0.7	81.3 ± 1.9	–
Run 3	267 ± 5 to 223 ± 4	81.5 ± 1.6 to 67.6 ± 0.5	104 ± 15
Range	48.3 ± 0.7 to 267 ± 5	67.6 ± 0.5 to 81.5 ± 1.6	104 ± 15
	log10A1(s−1)	log10A1(s−1)	log10A1(s−1)
Run 1	5.7 ± 0.2 to 2.91 ± 0.06	4.30 ± 0.11	–
Run 2	10.0 ± 0.9 to 1.71 ± 0.06	4.50 ± 0.19	–
Run 3	22.2 ± 0.5 to 17.5 ± 0.5	4.2 ± 0.4 to 3.25 ± 0.04	5.6 ± 1.1
Range	1.71 ± 0.06 to 22.2 ± 0.5	3.25 ± 0.04 to 4.50 ± 0.19	5.6 ± 1.1

Approximate isobutene collision frequencies listed in Table 6 give an indication of the location of the sites. The broad range for site-3a, 1.71 ± 0.06 to 22.2 ± 0.5 s⁻¹ suggests a range where isobutene density is independent of collision frequencies, i.e., deep within the micropores of the zeolite to direct involvement of collisions at the external surface of the HZSM-5 (Run 3). Sites-3b and 3c also show low to medium values of log₁₀A₁ and may also indicate dimerization within the micropores. The dependence on [IE-S*] in Equation (21) and the assumption that log₁₀A₅ = log₁₀A₄ = 13 for biomolecular processes may further complicate the interpretation of these data.

4. Experimental and Data Analysis

4.1. Experimental Technique and Materials

The apparatus and technique of temperature-programmed, low-pressure, steady-state catalysis has been previously described [1,2] and the data analysed in this paper was taken from work reported by Le Minh and Brown [1]. A mixture of isobutane and inert argon flow over ca. 0.3 g of HZSM-5 located in a Knudsen cell reactor at 100 °C. After typically 2 hours when molecular flow is at steady-state a linear heating program (5 °C/min) is initiated to 500 °C. Collisions of isobutane with zeolite, leads to adsorption and catalysis for the conversion of isobutane. Elementary steps can be identified due to the short residence times within the reactor, thus minimizing the opportunity for multiple reaction steps. The rates of escape of both isobutane and isobutene from the Knudsen cell are monitored by quadrupole mass spectrometry. For isobutane, the m/e = 58 peak abundance (I₅₈) characterized the reactant and for isobutene, the m/e = 56 peak abundance (I₅₆) was recorded. A small correction was applied to the product abundances at (I₅₆), due to a contribution from isobutane (Table 7). Figure 4 is a plot of isobutene mass spectral abundance against temperature for the first exposure of isobutane to HZSM-5. Two subsequent and identical runs from 100 to 500 °C were carried out on this same zeolite sample on consecutive days.

Key details of the experimental technique are summarized in Table 7. The zeolite, NH₄HZSM-5 (CBV 3024G) was obtained from Zeolyst International, Kansas City, KS, USA). Unit cell formula is (NH₄)_5.48(Si_90.52Al_5.48)O_189.3, SiO₂/Al₂O₃ ratio is 33.0 ± 1.8 and surface area is 358 ± 10 m² g⁻¹. The zeolite was pre-heated in air at 500 °C to remove ammonia. Instrument grade isobutane, isobutene and argon were obtained from BOC gases and used without further purification.

The escape rate constants from the Knudsen cell are [1,2,27]

(23)

where A_h is the area of the escape aperture in cm², V the volume of the reactor in cm³, c the mean molecular speed of isobutane or isobutene and M is the molar mass. The constant A_esc is a combination of these constants and has units K^−1/2 s⁻¹.

Figure 4. Raw mass spectral abundances (black circles) and wavelet denoised abundances (dashed line) for isobutane and isobutene plotted against temperature for the first temperature-programmed run over HZSM-5.

Table 7. List of specifications for the Knudsen cell reactors and mass spectrometer calibration and correction factors. Maximum isobutene flow rate is estimated by equating the largest cell dimension (ca. 40 mm) with the isobutane mean free path [1].

Table 7. List of specifications for the Knudsen cell reactors and mass spectrometer calibration and correction factors. Maximum isobutene flow rate is estimated by equating the largest cell dimension (ca. 40 mm) with the isobutane mean free path [1].

Knudsen cell (5 reactors)
Volumes	8,000 to 10,000 mm3
Internal surface areas	2,000 to 3,000 mm2
Exit aperture areas	0.8 to 1.6 mm2
Maximum isobutane flow rate (500°C)	4.2 × 1015 molecules s−1
Pressure in cell	7 to 10 Pa
Collisions/molecule	2500 to 1900
Escape rate constant, kesc	0.12 to 0.18 × (T/M)1/2s−1
Mass spectrometer calibration
Isobutane mass spec. sensitivity factor, α58	2.3 ± 1.5 × 10−16 I58 s molecule−1
Isobutene mass spec. sensitivity factor, α56	3.5 ± 2.3 × 10−16 I56 s molecule−1
I56 correction for isobutane	0.1377 ± 0.0028 × I58

4.3. Time-Varying Flexible Least Squares

Time-varying flexible least squares [3] is a generalization of ordinary linear regression by including time-variant regression parameters. For a series of observations y₁, y₂, …, y_T obtained at successive times 1, 2, …, T, there are known predictor functions h₁, h₂, …, h_T that are dependent on N unknown parameters b_m,1, b_m,2, …, b_m,T which evolve slowly over time. Minimizing both the ordinary linear measurement specification:

(24)

and a parameter dynamic specification optimizes these parameters:

(25)

The N time-dependent parameters are ultimately calculated by minimizing the following cost function:

(26)

Here μ is a weighting that minimizes the difference between subsequent parameters (b_n,t+1 and b_n,t). As μ approaches infinity the parameter determination approach the zero dynamic cost (ordinary least squares) solution [3,33]. When μ is set close to zero, priority is given to the dynamic specification and to estimates of the time-varying parameters [34], and hence a closer fit to the data (μ = 0.001 was used).

For temperature-programmed rate data

(27)

and

(28)

with

(29)

These temperature-dependent parameters are calculated using code provided by Kalaba and Tesfatsion [3] and adapted for the statistical computing language R [31]. Flexible least-squares simulations were undertaken for three-parameter (log(A(T)), n(T) and E_app(T)) fits to the isobutene evolution during the three temperature-programmed runs. Windows of 20 data points (ca. 33 °C) were fitted then averaged and moved one data point across the full range of temperatures. Optimized temperature-dependent parameters log₁₀(A_app(T)) from Equation (13) and E_app(T are plotted against temperature in Figure 1, Figure 2 and Figure 3 and summarized in Table 1, Table 2 and Table 3. Flexible least-squares simulations are a near perfect fit to the wavelet denoised data.

5. Conclusions

Isobutene desorption is linked to non-specific sites within the zeolite micropores and have average intrinsic rate constants:

(30)

For isobutane dehydrogenation the first pathways is assigned to Lewis acid sites or is related to weak hydrogen bonding between C–H and O on the zeolite. The intrinsic rate parameters cover the range:

(31)

The second dehydrogenation pathway is linked to Brønsted acid sites, and the intrinsic rate parameters also cover a broad range:

(32)

The third sites are associated with isobutene evolution at Lewis or Brønsted acid sites within the zeolite micropores. The increase in isobutene pressure immediately follows a period of dimerization or hydrogenation of isobutene. The intrinsic rate constants are:

(33)

For isobutene dimerization (or hydrogenation) the rate constants for the first pathway cover a very wide range and are linked to active sites within the HZSM-5 micropores:

(34)

The second dimerization pathway is associated with different active sites again within the HZSM-5 micropores; however in this case the range of the intrinsic parameters is narrow:

(35)

The final dimerization pathway is a sharp fluctuation in apparent rate parameters, similar temperature range, but opposite in sign to the Site-2c peaks. This peak is associated with dimerization or hydrogenation within the zeolite mircopores:

(36)

Nomenclature:

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