how to find rate constant

Reaction Rate Constant

Reaction rate constants are predicted based on transition state theory (Eyring, 1935) from the activation barrier ΔG‡ between the reactants and the transition state.

From: Computer Aided Chemical Engineering , 2019

29th European Symposium on Computer Aided Process Engineering

Christoph Gertig , ... André Bardow , in Computer Aided Chemical Engineering, 2019

2.1 Prediction of Reaction Kinetics and Thermodynamic Properties

Reaction rate constants are predicted based on transition state theory ( Eyring, 1935) from the activation barrier ΔG ^‡ between the reactants and the transition state. The activation barrier ΔG ^‡ is computed in two steps for a reaction in an inert solvent (Kröger et al., 2017):

1.

Geometries of reactants and transition states are optimized and vibrational frequencies are determined with b3lyp and TZVP basis set using the software Gaussian 09 (Frisch et al., 2009). Electronic energies are calculated with DLPNO-CCSD(T) and aug-cc-pVTZ basis set using the software ORCA (Neese, 2018). Next, the activation barrier ΔG ^‡ in an ideal gas state is computed with the statistical thermodynamics package TAMkin (Ghysels et al., 2010).

2.

Solvation effects are accounted for based on Gibbs free energies of solvation G ^solv of reactants and transition states. We predict G ^solv using the COSMO-RS method (Eckert and Klamt, 2002) implemented in the COSMOtherm software (Eckert and Klamt, 2017).

Step 1 is the computationally most demanding step, but is independent of the solvent and thus needed only once per reaction. COSMO-RS is also used for predicting normal boiling points and vapor pressures as well as activity coefficients used to fit parameters for the NRTL model (Renon and Prausnitz, 1968). NRTL is used to efficiently compute activity coefficients in process optimization.

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Polycondensation

R. Rulkens , C. Koning , in Polymer Science: A Comprehensive Reference, 2012

5.18.1.4.3 Kinetics

The reaction rate constant k for the amidation reaction is generally split into a noncatalytic part (k′) and a catalytic part (k″), where k  = k′   + k″. The general order of reactivity for the formation of an amide decreases as indicated in Figure 7 . The formation of aliphatic PAs easily takes place without a catalyst. Steppan et al. ¹⁹ have published an excellent polycondensation model applied to PA 66.

Figure 7. Reactivity of formation of fully aliphatic, semiaromatic, and fully aromatic amide groups.

The amidation kinetics for forming an amide link between an aromatic carboxylic acid and an aliphatic amide was studied in detail by EMS-Chemie. ^20,21 By performing experiments on PA 6T/6I copolyamide in comparison with PA 66, it was found that aromatic dicarboxylic acids have an amidation reaction rate constant k ₁ that is about an order of magnitude lower in the reaction with aliphatic diamines compared to adipic acid. For that reason, PAs based on aromatic dicarboxylic acids generally require a catalyst for their formation. Due to the slower reaction kinetics and high melting points, side reactions are generally more pronounced in semiaromatic PAs. Also the amidation equilibrium constant K is about a magnitude lower for aromatic carboxylic acids. Because K  = k ₁/k ₋₁, the hydrolysis rate constants k ₋₁ of aliphatic and semiaromatic amide links are about the same. Fully aromatic PAs are generally produced from the reactive acid chlorides.

In the production process of PA 6, amine and carboxylic end groups are generated by high-pressure hydrolysis of the CL and/or backmixing of PA oligomers. For high-molecular-mass PA 6 applications such as in film and in industrial yarns, a solid-state postcondensation (SSPC) is carried out, typically during 5–20   h at 160–180   °C, till a relative viscosity η _rel of typically 2.3–3.6, as measured in a 1   wt.% aqueous formic acid solvent (90   wt.%) at 25   °C, is reached.

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Modeling of ozonation

Tatyana I. Poznyak , ... Alexander S. Poznyak , in Ozonation and Biodegradation in Environmental Engineering, 2019

2.7.2 Estimation of the reaction rate constants considering the pH effect

The reaction rate constants estimation technique, presented in (2.24), is also valid, when the pH effect is considered. Indeed, it is sufficient to extend the vector of parameters and the regressor structure as follows:

(2.26) $\begin{array}{c}\hfill \left[\begin{array}{c}\hfill k_i^{⁎}\left(t\right)\hfill \\ \hfill k_{OH,d}^{⁎}\left(t\right)\hfill \end{array}\right]={\mathrm{\Gamma }}_1(t)\cdot G_1(t),\hfill \\ \hfill {\mathrm{\Gamma }}_1^{-1}(t):=\underset{\tau =0}{\overset{t}{\int }}\left[\begin{array}{c}\hfill -\frac{Q\left(\tau \right)}{V_{liq}}{\hat{c}}_i\left(\tau \right)\hfill \\ \hfill -c_i\left(\tau \right)c_{O{H}^{\cdot }}\left(\tau \right)\hfill \end{array}\right]{\left[\begin{array}{c}\hfill -\frac{Q\left(\tau \right)}{V_{liq}}{\hat{c}}_i\left(\tau \right)\hfill \\ \hfill -c_i\left(\tau \right)c_{O{H}^{\cdot }}\left(\tau \right)\hfill \end{array}\right]}^{\top }d\tau ,\hfill \\ \hfill G_1(t)=-\underset{\tau =0}{\overset{t}{\int }}(\frac{d}{d\tau }{c}_i\left(\tau \right)\cdot \left[\begin{array}{c}\hfill \frac{Q\left(\tau \right)}{V_{liq}}{c}_i\left(\tau \right)\hfill \\ \hfill c_i\left(\tau \right)c_{O{H}^{\cdot }}\left(\tau \right)\hfill \end{array}\right]d\tau ),\hfill \end{array}\}$

(2.27) $\begin{array}{c}\hfill \left[\begin{array}{c}\hfill k_{OH,f}^{⁎}\left(t\right)\hfill \\ \hfill k_{OH,d}^{⁎}\left(t\right)\hfill \end{array}\right]={\mathrm{\Gamma }}_t\cdot G_t,\hfill \\ \hfill {\mathrm{\Gamma }}_t:={\left\{\underset{\tau =0}{\overset{t}{\int }}\left[\begin{array}{c}\hfill c_{O{H}^{-}}(\tau )Q(\tau )\hfill \\ \hfill -c_i\left(\tau \right)c_{O{H}^{\cdot }}\left(\tau \right)\hfill \end{array}\right]{\left[\begin{array}{c}\hfill c_{O{H}^{-}}(\tau )Q(\tau )\hfill \\ \hfill -c_i\left(\tau \right)c_{O{H}^{\cdot }}\left(\tau \right)\hfill \end{array}\right]}^{\top }d\tau \right\}}^{-1},\hfill \\ \hfill G_t=\underset{\tau =0}{\overset{t}{\int }}(\frac{d}{d\tau }{c}_{O{H}^{\cdot }}\left(\tau \right)\cdot \left[\begin{array}{c}\hfill c_{O{H}^{-}}(\tau )Q(\tau )\hfill \\ \hfill -c_i\left(\tau \right)c_{O{H}^{\cdot }}\left(\tau \right)\hfill \end{array}\right]d\tau ),\hfill \end{array}\}$

(2.28) $\begin{array}{c}\hfill {\tilde{k}}_{OH,f}^{⁎}(t)=\tilde{\mathrm{\Gamma }}(t)\cdot \tilde{G}(t),\hfill \\ \hfill {\tilde{\mathrm{\Gamma }}}^{-1}(t):=\underset{\tau =0}{\overset{t}{\int }}{(c_{O{H}^{-}}(\tau )Q(\tau ))}^{2}d\tau ,\hfill \\ \hfill \tilde{G}(t)=\underset{\tau =0}{\overset{t}{\int }}\frac{d}{d\tau }{c}_{O{H}^{-}}\left(\tau \right)c_{O{H}^{-}}(\tau )Q(\tau )d\tau .\hfill \end{array}\}$

Eq. (2.26) is the generalization of the previous integral scalar expression to the matrix case. This is difficult to realize in practice and its equivalent realizable form is required. The next subsection gives such a representation in differential form.

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CO2 capture by Novel Amine Blends

Prakash D. Vaidya , Eugeny Y. Kenig , in Proceedings of the 1st Annual Gas Processing Symposium, 2009

4.1 CO₂ - DEEA reaction kinetics

To determine the reaction rate constant, it is essential that the system belongs to the fast reaction regime, without depletion of the amine at the gas-liquid interface ( Danckwerts, 1970). In the fast pseudo-first-order reaction regime, the rate of absorption is independent of the liquid-side mass transfer coefficient and hence it should not depend on the agitation speed. We studied this effect experimentally and found practically no change in the absorption rate, while varying the stirring speed in the range 50-100   rpm at 303   K. All further experiments were conducted at 70   rpm.

When CO₂ concentration in the bulk liquid is negligible, it can be shown, based on the two-film theory of mass transfer, that the following relation holds (Danckwerts, 1970):

(18) ${\mathrm{R}}_{{\mathrm{CO}}_2}={\mathrm{P}}_{{\mathrm{CO}}_2}{\mathrm{H}}_{{\mathrm{CO}}_2}\sqrt{{\mathrm{D}}_{{\mathrm{CO}}_2}{\mathrm{k}}_2{\left(\mathrm{DEEA}\right)}_{\mathrm{o}}}$

where k₂ is the second order reaction rate constant (m³/(kmol s)) and ${\mathrm{H}}_{{\mathrm{CO}}_2}$ is the Henry's constant (kmol/(m³ kPa)). The CO₂ absorption rates into aqueous DEEA solutions at various temperatures and values of the parameter ${\mathrm{H}}_{{\mathrm{CO}}_2}\sqrt{{\mathrm{D}}_{{\mathrm{CO}}_2}{\mathrm{k}}_2}$ are represented in Table 1. Littel et al.(1990) reported correlations for the physical properties of aqueous solutions containing DEEA (<   2.4 kmol/m³). In our work, the viscosity and N₂O solubility of an aqueous 2 kmol/m³ DEEA solution were estimated by using these correlations. With the aid of the N₂O analogy and a modified Stokes-Einstein relation (Versteeg and van Swaaij, 1988), the values of DCO₂ and HCO₂ at 303   K were found to be 9.67 x 10^{−   10}  m²/s and 2.72 x 10^{−   4} kmol/(m³ kPa), respectively. Using these values, an average value of k₂ at 303   K was found to be 173   m³/(kmol s). Littel et al. (1990) reported a considerably lower value of 61   m³/(kmol s) at 303   K. However, the k₂ value of Littel et al. (1990) at 323   K is also much lower than that presented by Kim and Savage (1987). Once the value of k₂ was obtained in our work, the condition of fast reaction regime (Danckwerts, 1970) was checked and found to be satisfied.

Table 1. CO₂ absorption rates $\left({\mathrm{R}}_{{\mathrm{CO}}_2}\right)$ into aqueous DEEA solutions

Temperature K	(DEEA)o kmol/m3	${\mathrm{P}}_{{\mathrm{CO}}_2}\mathrm{kPa}$	${\mathrm{R}}_{{\mathrm{CO}}_2}\times {10}^6\mathrm{kmol}/\left({\mathrm{m}}^2\mathrm{s}\right)$	${\mathrm{H}}_{{\mathrm{CO}}_2}\sqrt{{\mathrm{D}}_{{\mathrm{CO}}_2}{\mathrm{k}}_2}\times {10}^7{\mathrm{kmol}}^{1/2}/\left({\mathrm{m}}^{1/2}\mathrm{skPa}\right)$
298	2.25	15.62	1.86	0.79
	2.50	14.10	2.49	1.12
	3.0	14.26	2.79	1.13
303	2.0	4.54	0.77	1.20
	2.0	6.36	0.90	1.0
	2.0	7.58	1.30	1.21
	2.0	12.74	1.84	1.02
	2.25	5.30	0.98	1.23
	2.25	7.73	1.47	1.27
	2.25	10.46	1.80	1.15
	2.25	15.62	2.58	1.10
	2.50	2.27	0.48	1.34
	2.50	4.99	1.22	1.55
	2.50	8.94	2.19	1.55
	2.50	13.19	2.98	1.43
	3.0	3.94	0.98	1.44
	3.0	6.82	1.91	1.62
	3.0	10.16	2.66	1.51
	3.0	14.41	3.52	1.41
308	2.0	16.38	3.19	1.38
	2.25	15.62	3.47	1.48
	2.50	14.86	4.17	1.77

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New Developments and Application in Chemical Reaction Engineering

Yasuhiko Takuma , ... Toshinori Kojima , in Studies in Surface Science and Catalysis, 2006

3.2 Relations between decomposition rate and temperature

Second-order reaction rate constants for the three compounds at 20, 35 and 50°C were evaluated as in the methodology section of 2.2. Also, theoretical frequency factors are evaluated by Eq.(1). To calculate the frequency factors, we used the value shown in Table 1.

Table 1. Molecular radius and molecular mass

	Radius [10-8m]	Mass [10-25 kg]
Parathion	5.372	4.836
Fenitrothion	5.255	4.604
Diazinon	5.709	5.054
CH3CH2O-	2.557	0.7482

Second-order decomposition rate constants experimentally determined and frequency factors theoretically predicted are plotted on Fig. 2. Good linear expressions were demonstrated, which suggest the reasonability of assumption of limiting elementary steps. From this figure, activation energies for three compounds, parathion, fenitrothion and diazinon are calculated. The obtained activation energies are shown in Table 2 together with theoretical frequency factors.

Fig. 2. Arrhenius plot

Table 2. Frequency factors and activation energies

	Parathion	Fenitrothion	Diazinon
Second-order reaction rate constant at 20 °C	[1·(mol ·s)-1] 8.28 x 10-5	2.26 x 10-4	1.52 x 10-4
at 35 °C	[1·(mol ·s)-1] 4.34 x 10-4	1.03 x 10-4	4.17 x 10-4
at 50 ° C	[1·(mol ·s)-1] 1.43 x 10-4	5.20 x 10-3	1.67 x 10-3
Frequency factor	[1 · mol·s)-1] 4.74 x 109	4.60 x 109	5.14 x 109
Activation energy	[kJ· mol-1] 77.2	74.5	76.8

From the difference among the reaction rates, organophosphorus insecticides used in this work were divided into two groups. One group consists of the compounds with high decomposition rates such as phenthoate and malathion, the other with low rates such as parathion, fenitrothion and diazinon. The difference of reaction rate comes from the difference of these structures. The former compounds with high rates have a structure with central sulfur atom between the phosphoric acid functional group and the rest part and the latter, central oxygen. The mean bond enthalpy between carbon and sulfur is 259kJ/mol, and between carbon and oxygen is 360kJ/mol [5]. The former indicate the weaker bond than the latter, which explains the easier decomposition of malathion and phenthoate.

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Theory of Steady, One-Dimensional, Laminar Flame Propagation

F.A. WILLIAMS , in Modern Developments in Energy, Combustion and Spectroscopy, 1993

Symbols

a	temperature exponent in specific reaction-rate constant, Eq. (7)
B	constant prefactor in specific reaction-rate constant, Eq. (7)
b	a constant related to stoichiometry in Eq. (52)
C i	molar concentration of species i
c p	average specific heat at constant pressure
D i	the i′th Damköhler number
D ij	binary diffusion coefficient for species pair i and j
E	activation energy
ho i	standard specific enthalpy of formation of species i
K r	equilibrium constant for reaction r
k r	specific reaction-rate constant for reaction r
m	mass burning rate
p	Pressure
R°	universal gas constant
T	Temperature
v	Velocity
Wi	molecular weight of species i
wi	mass rate of production of species i
Xi	mole fraction of species i
x	space coordinate
Yi	mass fraction of species i
y	nondimensional fuel concentration variable defined above Eq. (51)
z	nondimensional CO concentration variable defined above Eq. (52)
a	water–gas constant defined in Eq. (36)
δ	small parameter measuring the size of the fuel-consumption zone
ɛ	small parameter measuring the size of the zone of H2 and CO oxidation, Eq. (33)
ɛi	mass flux fraction of species i
ζ	stretched variable defined above Eq. (51)
η	stretched variable defined above Eq. (52)
λ	thermal conductivity of the mixture
ξ	nondimensional space coordinate, Eq. (32)
ρ	Density
τ	nondimensional temperature
ω	molar reaction rate

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21st European Symposium on Computer Aided Process Engineering

Eirini Siougkrou , ... Claire S. Adjiman , in Computer Aided Chemical Engineering, 2011

4 Results

In figure 2 , the reaction rate constant is given as a function of the mole fraction of CO ₂ in the mixed solvent. An increase of the amount of CO₂ causes an increase in the reaction rate constant, reaching a maximum for a CO₂ mole fraction close to 0.95, except for the case of methanol, where, although the rate is increasing, the reaction is so slow that the maximum rate constant is at pure supercritical CO₂. In figure 3, vapour-liquid equilibrium (VLE) data and predictions using GC-VTPR for the binary solvent systems are shown. It is evident that the higher the CO₂ content, the higher the pressure of the system. The predictions of GC-VTPR EoS are satisfactorily accurate. Another issue that needs to be taken under consideration is the solubility of anthracene in the mixed solvent and how this is influenced with the change in pressure. Our phase equili-brium calculations, shown in figure 4, indicate that the solubility decreases when the mole fraction of CO₂ (or equivalently, the pressure) increases. GC-VTPR successfully captures the solubility of anthracene in the pure organic solvents. In order to design a continuous stirred-tank reactor, for the Diels-Alder reaction, one must find a trade-off between increasing the amount of CO₂, which increases the reaction rate constant but decreases the solubility of anthracene and results in higher pressures, which increases the cost considerably. For a fixed production rate (42kg/min), the dependence of the reactor volume on the mole fraction of CO₂ is shown in figure 5. The volume of the reactor when methanolis the co-solvent is not given, as it turns out to be very large (around 40m³), since the reaction in methanol + CO₂ is very slow compared to the other solvents (figure 2). The total cost of the reactor [13], including the cost of the organic solvents, follows the same trend as the volume of the reactor and is given in figure 6. When the co-solvent is acetonitrile a minimum cost occurs for a mole fraction of CO₂ (x_CO2) equal to 0.47, while, when the co-solvent is acetone the cost always increases with increasing x_CO2.

Figure 2. Calculated reaction rate constants at T =40 °C. Continuous line:acetonitrile, dashed line:acetone, dash-dot line: methanol. Symbols: experimental data for acetonitrile + CO2 [6] .

Figure 3. VLE at T = 40 °C with GC-VTPR EoS. Symbols are experimental data, diamonds: acetonitrile [14], circles: acetone[15], triangles:methanol [16]. Lines as figure 2.

Figure 4. Solubility of anthracene at T = 40 °C as a function of pressure in the mixed solvents, with GC-VTPR. Symbols indicate experimental data[17–18] and lines as perfigure 2.

Figure 5. The volume of the reactor as a function of the mole fraction of CO2. Continuous curve is acetonitrile, dashed curve is acetone.

Figure 6. The cost of the reactor as a function of the mole fraction of CO2. Continuous curve is acetonitrile, dashed curve is acetone.

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Ceramic and Carbon Matrix Composites

F. Langlais , G.L. Vignoles , in Comprehensive Composite Materials II, 2018

5.4.3.3 Chemical Reactions

Knowledge of the complex sequence of the gas-phase and heterogeneous reactions involved in CVD/CVI is central to the control of the overall process. Experimental data are mostly obtained in specific lab-scale CVD/CVI hot-wall reactors with diagnostics like mass uptake measurements, using a microbalance, ex-situ thickness measurements on samples, either flat or porous, and gas-phase analysis, either in-situ or at the reactor outlet, by Gas-Phase Chromatography (GPC), possibly coupled to Mass Spectrometry (GC/MS) or Fourier-Transform Infra-Red spectroscopy (FTIR). Results concern the main detectable chemical species; most of the radicals and less stable species remain concealed to direct observation.

It is therefore highly desirable to couple the experimental determinations to models of the gas-phase and heterogeneous chemistry, featuring thermodynamic and kinetic aspects. Indeed, thermodynamics are valuable in predicting the equilibrium composition of a given gas phase in the reactor, but it often happens that the residence and contact times are too short to allow this equilibrium to fully occur; so, kinetic data has also a large interest. The global flowchart of computational approaches of CVD/CVI reactor chemistry is depicted in Fig. 21. The final goal of the approach is the determination of the critical gaseous species responsible for the deposit quality and deposition rate. To achieve this, thermochemical and reaction kinetics computations are carried out, possibly inserted in a CFD reactor model. They are based on the existence of a sound, consistent database describing the properties of all reactants and the rate laws of all reactions. This in turn relies on the production of thermochemistry and reaction rate data. If experimental or literature values are unavailable, theoretical chemistry computations are necessary. First, from quantum chemical computations, equilibrium geometries, harmonic vibration frequencies and potential energy surfaces of the chosen set of chemical species are computed. Popular quantum chemistry software are often DFT-based, working with rather high levels of theory. Then, partition functions, entropies and heat capacities are obtained from standard statistical equations. ²⁰²

Fig. 21. Complete strategy for chemical modeling of CVI, starting from quantum chemical computations and going to reactor-scale global results.

Reproduced from Reinisch, G., Leyssale, J.-M., Patel, S., et al., 2013. Studies of gas-phase reactivity during chemical vapor deposition of boron carbide. In: Kriven, W.M., Gyekenyesi, A.L., Westin, G., Wang, J. (Eds.), Developments in Strategic Materials and Computational Design III. In: Halbig, M., Mathur, S. (Vol. Eds.), Ceramics Engineering and Science Proceedings, Vol. 33, Issue 10. New York: John Wiley &amp; Sons, pp. 105–116.

Central to the prediction of reaction rate constants is the calculation of the activation energies. Indeed, a variation of a few kcal/mol in the potential energy barrier between reactants and transition state can change a reaction rate by an order of magnitude. Calculations of energies thus require numerical methods of high accuracy. The Transition State Theory (TST) may be applied as a post-processing after energy and vibration data have been obtained along a reaction path from reactants to products. However, this classical theory is not enough to account correctly for some specific types of reaction (see Fig. 22):

Fig. 22. Example of some difficulties arising in the study of the B/H/3Cl system.

•

Unimolecular decomposition reactions have a dependency to total pressure, since they only can occur through collisions of the molecule with other ones. The Rice–Ramsperger–Kassel–Markus (RRKM) theory has to be employed, ²⁰³ or at least light versions of it like the Quantum Rice–Ramsperger–Kassel (QRRK). ^204,205 This is particularly relevant for systems using MTS as a precursor, since MTS decomposition is the initial step of the whole CVD chemistry in that case.

•

Reactions without an energy barrier should be treated by the Variational Transition State Theory, ²⁰⁶ in order to locate the Gibbs energy barrier that exists instead. This is a more involved technique since the reaction path itself has to be optimized with respect to the Gibbs energy maximum, using a Variable Reaction Coordinate (VRC) framework. ^207,208

•

The rate of reactions involving a single hydrogen may be affected by the tunnel effect, and therefore undergo a reassessment from the estimation of the tunnel effect frequency. For reaction paths leading to multiple valleys of products from a single transition state, the respective amounts of the products from both valleys must be determined by other methods. One of them is the Transition Path Sampling (TPS) method, which consists in scanning the trajectory space by generating new trajectories from known ones, including systematically a passage by the potential energy barrier. ²⁰⁹

The contribution of some loose vibration modes (rocking and wagging) to the partition functions of several molecules and transition states may be critical. If a harmonic approximation and a free-rotor approximation suffice at respectively low and high temperatures, the values in the intermediate domain may be tricky to estimate, requiring again specific methods, ²¹⁰ among which the recently developed 1DQ method. ²¹¹

5.4.3.3.1 Thermodynamics

The first theoretical approach of value in obtaining a preliminary knowledge of a CVD/CVI chemical system is thermodynamics which permits derivation of data on the heterogeneous gas-solid equilibrium. For any experimental condition (i.e., temperature, pressure, and initial gas phase composition), it is possible to calculate, under the equilibrium hypothesis, (1) the theoretical composition of the deposit (which can be a single phase or made of several phases) and the gaseous phase and (2) the thermodynamic yields. The most thermodynamically probable reactions can be deduced, giving information about the deposit formation, the presence of gaseous by-products, and the possible etching of the substrate. Such a thermodynamic approach is usually carried out according to the method of minimizing the overall free enthalpy of the system. ²¹³ As an example, CVD of zirconia (which was infiltrated as a matrix of CMC ⁷⁹ from the ZrCl ₄–CO ₂–H ₂–Ar system), was investigated from a thermodynamic point of view. ²¹⁴ On an inert substrate, pure ZrO ₂ can be deposited with a 100% yield when water vapor is in excess at equilibrium while carbon-zirconia co-deposition can be obtained for H ₂-rich initial compositions. Under conditions for ZrO ₂ yields close to 100%, a carbon substrate is found to be oxidized while a mullite one is thermodynamically stable. Such predictions, which were confirmed experimentally, are very important because they permit, for instance, an appropriate choice of fibrous preform for preparing new CMC by CVI. Another example of a thermodynamic study included the calculation of both the heterogeneous and homogeneous equilibria in the CH ₃ SiCl ₃–H ₂ system for the CVD of silicon carbide. ²¹⁵ First, it was shown that in a hot-wall reactor, the precursor molecule CH ₃ SiCl ₃ is not the actual source species, but is decomposed into intermediate species SiCl ₂ and CH ₄, possible source species for Si and C, respectively, in the most general case (Fig. 23), according to the following reaction sequence:

Fig. 23. Calculated partial pressures of the gaseous species vs. initial composition ratio α=n°(H ₂)/n°(CH ₃ SiCl ₃) at 1200 K and 10   kPa for the CH ₃ SiCl ₃–H ₂ system in the hypothesis of (a) homogeneous equilibrium and (b) heterogeneous equilibrium. ²¹⁵

[1] $\begin{array}{l}C{H}_{3}SiC{l}_3+H_2\to C{H}_4+SiC{l}_2+HCl\mathrm{homogenous}\mathrm{reaction}\hfill \\ C{H}_4+SiC{l}_2\to SiC+2HCl+H_2\mathrm{heterogenous}\mathrm{reaction}\hfill \end{array}$

The thermodynamic approach also permits evaluation of the supersaturation of the gas phase, which represents the difference between the actual state and the equilibrium and whose high values can be correlated to a nucleation regime. ²¹⁶ Finally, on the basis of thermodynamic data, an estimation of the concentrations of various chemical species chemisorbed on SiO ₂ or β–SiC surfaces was proposed; the importance of the chemisorption of SiCl ₃ and H radicals on C atoms and of CH ₃ and Cl radicals on Si atoms was pointed out and different mechanisms for the formation of SiC-based ceramics were derived. ²¹⁷

The necessary thermochemical data are usually extracted from the available tables. ^218–220 However, sometimes the values may be doubtful; accordingly there is a growing interest in evaluating these constants on the basis of quantum chemical computations. Numerous studies have been carried out, as summarized in Table 2; the obtained data is generally in good agreement with available experiments.

Table 2. Some thermochemical databases relevant to CVI studied by quantum chemistry

Chemical system	Method	References	Molecules
General study	G2, G3	221	148
	BAC-G3B3	222	155
B–Cl–H	BAC-MP4	223	BH i Cl j (i+j ≤ 3)
			+N-containing molecules
	G2	224	BH i Cl j (i+j ≤ 3)
	G3B3	225
Si–C–Cl–H	CCSDT	226	50 light species
	G3B3	227	163 light species
B–C–Cl–H	G3B3/G3MP2	228	154 light species
Si–B–Cl–H	BAC-MP4	229	Si i B j HCl (i ≤ 2, j ≤ 2)

5.4.3.3.2 Kinetics

In a CVD/CVI reactor, owing to the continuous flow of precursor species, heterogeneous equilibrium is never reached except under very specific conditions. Kinetic factors must be taken into account. An experimental study of the variations of the deposition rate as a function of the various experimental parameters usually is carried out in order to acquire a more exhaustive knowledge of the chemical system (Table 3).

Table 3. Some kinetic gas-phase reaction mechanism databases

Chemical system	Type of work	References	Context
C–H	Detailed compilation	230–232	C 3 H 8 pyrolysis and CVD
	Reduced model	233
	Reduced model	44	CH 4 pyrolysis and CVD
	Detailed compilation	234	CH 4 pyrolysis and CVD
C–H–Cl	Compilation	235	C 6 H 5 Cl pyrolysis
	Compilation	236	CH 3 Cl–O 2 pyrolysis
	Compilation	237	CH 4–Cl 2 pyrolysis
B–H–Cl	Theory – G2 level	238	BCl 3–H 2 CVD
	31 reactions
	Theory – G3B3 method	212,225	BCl 3–H 2 pyrolysis and CVD
	N reactions
	Exptl. measurements	239	BCl 3–H–O–OH
B–C–Cl–H	Exptl.	73,240
	Theory	225
Si–C–Cl–H	Theory – CCSDT	241	CH 3 SiCl 3–H 2 CVD
	Theory – BAC-MP4//RRKM	242	CH 3 SiCl 3 decomposition
	Experimental	243,244
Si–B–Cl–H	Theory – DFT	245	CVD of B-doped Si from
			SiCl 2 H 2–B 2 H 6–H 2
Si–B–C	Exptl.	76–78
–Cl–H

The first step of this investigation is to define the conditions of a kinetic process controlled by the chemical reactions. The influence of deposition temperature reported as an Arrhenius plot is frequently used to determine chemical and mass transfer regimes. When the activation energy is very low, i.e., lower than 20   kJ   mol⁻¹, the kinetics is limited by mass transfer while for higher activation energy, usually observed in a lower temperature range, the kinetic process is governed by the chemical reactions, as shown in Fig. 24 for the deposition of SiC from CH ₃ SiCl ₃–H ₂ precursor. ²⁴⁶

Fig. 24. SiC-based ceramic deposition rate as a function of reciprocal temperature for a composition ratio α=3, a total flow rate Q=200 sccm, and three total pressures in the CH ₃ SiCl ₃–H ₂ system. ²⁴⁶

The transition between these two kinetic regimes can also be revealed by varying the total flow rate (Fig. 25). If the deposition rate increases with the total flow rate, which is usually the case for the lowest values of this parameter, mass transfer is the controlling step of the process. If at higher total flow rate, the growth rate becomes constant or decreases with increasing total flow rate, the kinetic process is considered to be controlled by the chemical reactions. The chemical process giving rise to solid deposit can include homogeneous reactions. A possible consequence of homogeneous reactions is a decrease of deposition rate with increasing total flow rate or decreasing residence time of the gaseous precursor in the hot reaction area, as evidenced in the case of pyrocarbon deposition from various hydrocarbons (Fig. 26). ^{18,30–33,42} In this system, an increase of the residence time to a greater or lesser extent favors the transformation of the precursor by gas phase reactions, called 'maturation,' giving rise to more or less efficient intermediate species for the formation of various types of pyrocarbons. ^{18,19,28,44,45}

Fig. 25. SiC-based ceramic deposition rate as a function of total flow rate for a composition ratio α=3 in the CH ₃ SiCl ₃–H ₂ system. ²⁴⁶

Fig. 26. Pyrocarbon deposition rate as a function of residence time for a propane total pressure P=2   kPa and different temperatures: (a) T=905   °C, (b) T=950   °C, (c) T=992   °C, and (d) T=1035   °C. ¹⁹

The values of activation energies result from the overall chemical pathway and the more sequential steps this pathway includes, the higher is the apparent activation energy. In specific cases, when the deposition rate does not depend on the total flow rate or residence time, the kinetic process is only controlled by heterogeneous reactions. Under such conditions, the experimentally determined activation energy can be associated directly with the surface chemical process and kinetic laws can be derived by varying the partial pressure of the various precursor species and determining the reaction orders with respect to these species. As an example, such kinetic data were acquired for the deposition process of SiC from CH ₃ SiCl ₃–H ₂ precursor. ²⁴⁷ Moreover, by adding to the precursor increasing concentrations of species produced by the deposition reactions, for example, HCl for SiC growth or H ₂ for pyrocarbon growth, it is possible to estimate inhibition effects, which can play an important role in the CVI process because such inhibitor species can greatly limit the deposition rate within the preform and result in marked infiltration gradients. ^{30–32,39,247–250} In atmospheric pressure CVD from CH ₃ SiCl ₃–H ₂, adding HCl may give rise to a sudden transition to pure pyrocarbon growth; decreasing the HCl amount allows going back to SiC growth, with a clear hysteresis. ²⁵¹ The effect of hysteretical chemistry on multilayer deposition has also been discussed theoretically in the case of pyrocarbon deposition. ²⁵²

To have a better insight in the pre-deposition homogeneous reaction steps, the gas phase surrounding the substrate can be analyzed. Truly in-situ analyzes at high temperatures are very difficult to perform. Few experimental investigations by FTIR spectroscopy can be mentioned in the cases of SiC and PyC deposition. ^18,253

More popular analyzes are made at the hot zone outlet: in the case of the hydrocarbons/H ₂ systems, gas phase chromatography has often been used; ^{30–33,42,230–232,254} MALDI-ToF characterizations of the heavy molecules ⁴⁵ gave interesting complementary information about the textural transition as a function of experimental parameters. ²⁸ FTIR has also been used, first through the whole reactor, averaging over the upstream zone, the hot zone, and the downstream zone, ¹⁸ then only at the outlet in the case of the B–C–Cl–H system. ²⁴⁰

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Enzyme-enhanced CO2 absorption

N.J.M.C. Penders-van Elk , G.F. Versteeg , in Absorption-Based Post-combustion Capture of Carbon Dioxide, 2016

10.2.3.2 N,N-Dimethylethanolamine (DMMEA)

In Fig. 10.13 the forward-reaction rate constants of the enzyme-catalyzed CO ₂ hydration at the temperatures 278, 283, 298, 308, and 313   K are presented. It is shown that $k_{{\mathrm{H}}_2\mathrm{O}}^{\ast }$ increases with increasing enzyme concentration and with increasing temperature. Again, the increase in $k_{{\mathrm{H}}_2\mathrm{O}}^{\ast }$ decreases with increasing temperature.

Figure 10.13. Forward-reaction rate of the enzyme-catalyzed CO₂ hydration reaction $k_{{\mathrm{H}}_2\mathrm{O}}^{\ast }$ as function of the enzyme concentration for aqueous N,N-dimethylethanolamine (DMMEA) solutions at five different temperatures.

Using nonlinear regression, the kinetic constants $k_3^{\ast }$ and $k_4^{\ast }$ have been determined for each temperature (see Table 10.2). Next to the determined values for the kinetic constants, the pKa values calculated from data published by Hamborg and Versteeg (2009) are presented in Table 10.2.

Table 10.2. pKa values for N,N-dimethylethanolamine (DMMEA) and kinetic constants for DMMEA solution with enzyme at the temperatures evaluated

T [K]	pKa [–]	$k_3^{\ast }$ m6/g/mol/s	$k_4^{\ast }$ m3/g
278	9.65	0.033	0.20
283	9.54	0.040	0.21
298	9.22	0.054	0.21
308	9.02	0.062	0.30
313	8.92	0.062	0.28

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SAFER SOLVENTS AND PROCESSES

AYDIN K. SUNOL , ... Guillermo Reglero , in Handbook of Solvents (Second Edition), Volume 2, 2014

Enhanced reaction rate

The effect of pressure on the reaction rate and equilibrium constant at high pressures is described in Section 20.1.2.4. As can be perceived from this section, supercritical fluids that exhibit very high negative activation volumes for certain reactions will improve the rate and equilibrium conversion of the reaction.

Homogenization

Reactions that otherwise would be carried out in more than one phase (heterogeneous reactions) can be transformed to homogeneous ones with the aid of supercritical fluids, so that inter-phase transport limitations are eliminated. This is realized due to enhanced solubilities of the reaction components in the supercritical fluids. Typical examples are reactions in water (supercritical water can solubilize organic compounds), homogeneous catalytic reactions, and reactions of organometallic compounds. Homogenizing one compound more than the others in a system may also affect relative rates in complex reactions and enhance the selectivity.

Enhanced mass transfer

In many instances, reaction rates are limited by diffusion in the liquid phase. The rate of these reactions can be increased if the reaction is carried out in the supercritical phase. Typical examples are enzyme catalyzed reactions as well as some very fast reactions such as certain free radical reactions. Selectivity considerations usually dominate in complex reactions. If some steps of the complex reaction are controlled by diffusion, changing the diffusivity changes the relative rates of the reaction steps and affects the selectivity.

Ease of down-stream separation

Another reason for using supercritical fluids as the reaction medium is to fractionate products, to purify the products or to remove unreacted reactants from the product stream. Supercritical fluids can be used as either a solvent or anti-solvent in these instances.

Increased catalyst activity

Some heterogeneous catalytic reactions are carried out in the supercritical phase, in order to increase catalyst activity and life through in-situ regeneration of surfaces with tuning of operation conditions. For example, supercritical fluids are capable of dissolving carbon that may otherwise be deposited on the catalyst in the absence of the supercritical solvent.

Tunable reaction rates through dielectric constant

Some properties of supercritical fluids can be monitored (manipulated) continuously by adjusting the density of the fluid. Dielectric constant is such a property and the solvent's dielectric constant can influence the rate of the reaction.

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how to find rate constant

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