Integrated Kinetic and Thermodynamic Model Describing the

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Ind. Eng. Chem. Res. 2004, 43, 2296-2301

Integrated Kinetic and Thermodynamic Model Describing the Nitrosation of Aniline and Its Derivatives under Reaction- and Encounter-Controlled Conditions Gabriel da Silva, Eric M. Kennedy,* and Bogdan Z. Dlugogorski Process Safety and Environment Protection Research Group, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia

A kinetic model has been developed that describes the rate of reaction (nitrosation) of aniline and its derivatives with a range of nitrosating agents. This model was derived by first developing a thermodynamic description of the system based on the basicity of the substrate and the nucleophilicity of the nitrosating agent. A kinetic model was obtained by combining the thermodynamic description of the system with experimentally observed kinetics via the transition state theory. The final model exhibited a good correlation with experimental results across a wide range of reactivities. The encounter-controlled regime was also modeled from mass-transfer theory, which was integrated with the kinetic model to describe the rate of nitrosation during the transition regime. Through application of the kinetic model, the intrinsic barrier to reaction was estimated as 15 kJ mol-1. Introduction Nitrosation reactions and their kinetics are of considerable industrial significance. For example, these reactions are involved in the production of azo dyes,1 hydroxylamine via the Raschig process,2,3 and -caprolactam (a nylon precursor),4 among other important chemicals. As such, the ability to predict the rate of nitrosation is of great interest and is explored here. Under conditions commonly encountered in many industrial processes, nitrosation proceeds according to eq 1, where ONX is a nitrosating agent (X- being a nucleophile), and S is a substrate. Nitrosating agents are formed from the equilibrium reaction of nitrous acid with certain nucleophilic species, including chloride,5 bromide,6 thiocyanate,7 iodide,8 thiourea,9 thiosulfate,10 and nitrite11 (to form N2O3).12 Nitrosating agent formation proceeds according to eq 2, with equilibrium constant KONX given by eq 3. This reaction mechanism is known to apply under mildly acidic conditions in the presence of high concentrations of X-, where the acidcatalyzed mechanism of nitrosation with the nitrosating agent H2NO2+ is negligible. kN

ONX + S 98 ONS+ + XKONX

HNO2 + X- + H+ y\z ONX + H2O KONX )

[ONX]aH2O [HNO2][X-][H+]

(1) (2) (3)

The rate expression for nitrosation under the above conditions is given by eq 4, which is consistent with the currently accepted mechanism of nitrosation.13 Here, [HNO2] represents the actual nitrous acid concentration. * To whom correspondence should be addressed. Tel.: (+61 2) 4921 6177. Fax: (+61 2) 4921 6920. E-mail: eric.kennedy@ newcastle.edu.au.

Under typical nitrosating conditions, the total nitrite added as sodium nitrite would be divided between its descendant species nitrite, nitrous acid, and dinitrogen trioxide. Similarly, the total substrate concentration would be divided between the protonated and unprotonated forms, although only the unprotonated form reacts in the current mechanism. Although the equilibrium (KONX) and reactivity (kN) terms are of equal importance in determining the overall reaction rate, KONX exhibits the greatest variance between catalysts and is therefore the key factor in determining the overall catalytic efficiency. Recently, a model was developed to predict KONX,12 which indicated that the equilibrium constant increased with increasing strength of the nucleophilic species as a linear function of Edwards’14 En parameter. However, no similar correlation exists for predicting kN values.

r ) kNKONX[S][HNO2][X-][H+]

(4)

Two different effects have been found to influence the rate of nitrosation under the above conditions. These are the basicity of the substrate (pKa)15-17 and the magnitude of KONX (i.e., the nucleophilicity of the species X-).18 It is also known that kN values can be affected by molecular diffusion, where they are constrained by the encounter-controlled limit. Because of their relatively high nucleophilicity, aniline-type compounds (and other amines) are often known to react at or near the encounter-controlled limit during nitrosation, as well as in other certain organic mechanisms such as nitration, halogenation, and carbocation reactions.19,20 Encountercontrolled limits are well-known and arise as a result of the need, in solution, for reactants to diffuse to each other from within their respective solvent cages. This has been an active area of research,19,21 and encountercontrolled rate constants can be predicted by masstransfer theory. In recent years, much of the research concerning nitrosation reactions has been directed at examining

10.1021/ie0304560 CCC: $27.50 © 2004 American Chemical Society Published on Web 04/07/2004

Ind. Eng. Chem. Res., Vol. 43, No. 10, 2004 2297

nitroso group reactivity, through the development of linear free energy relationships (LFERs)22-26 or through calculations of bond dissociation energies.27-30 The present manuscript utilizes LFER in deriving the first quantitative model capable of predicting kN values from both of the identified contributing factors (pKa and KONX). This model is formulated through a description of the thermodynamics of the system that is linked to the reaction kinetics via the transition state theory. The encounter-controlled regime is also modeled, using mass-transfer theory, and integrated with the reactioncontrolled model so as to predict kN values across both regimes.

comparison to protonation (eq 9, with the free energy change of the reaction given by eq 10).

Theoretical Section

Equation 11 can subsequently be substituted back into eq 7, yielding

Thermodynamic Modeling. Consider the nitrosation reaction presented in eq 1 and its corresponding free energy change (eq 5).

∆G°N ) ∆G°f,ONS+ + ∆G°f,X- - ∆G°f,ONX - ∆G°f,S (5) First, the effect of nucleophilic strength on the thermodynamics of nitrosation can be modeled employing a previously derived12 LFER between the free energies of a nucleophile and its corresponding nitrosating agent and Edwards’ nucleophilic constant, En.14 Upon rearrangement of this relationship, we obtain the relation

∆G°f,ONX - ∆G°f,X- ) -F(1 - Rn)En + 254 000 (6) where F is the Faraday constant (96 485 C mol-1) and Rn is a parameter that lies between 1 and 0. It has been postulated that Rn is a measure of the ON-X bond order relative to the bond found in the X-X species.12 The parameter Rn has been previously determined to equal 0.54 for nitroso transfer.12 Equation 6 can now be substituted back into eq 5 to yield

∆G°N ) ∆G°f,ONS+ - ∆G°f,S + F(1 - Rn)En - 254 000 (7) An equation is now needed to describe how the free energies of the two species ONS+ and S vary with changing substrates. Again, we can apply eq 6, where X- is now replaced by the amine nucleophile, S. The respective Edwards’ parameters of the nucleophile and the amine are now distinguished as En(X-) and En(S).

∆G°N ) F(1 - Rn)(En(X-) - En(S))

(8)

The difficulty with applying eq 8 is that En is not directly measurable for aniline-type compounds. Instead, approximate values are obtained by comparing the rate constants for reactions of the amines with known electrophiles. Edwards reported an En value of 1.78 for aniline,14 calculated from the reaction rate with methyl bromide; Radhakrishnamurti and Panigrahi calculated an En value of 1.32.31 However, En values have been calculated for few other aniline derivatives, thus making application of eq 8 to a wide range of data difficult. In addition, the En values that are known are not of sufficient accuracy to permit a detailed thermodynamic analysis. Another method of predicting the free energies of formation of S and ONS+ is provided by a

K-a

S + H+ y\z SH+

(9)

∆G°-a ) ∆G°f,SH+ - ∆G°f,S - ∆G°f,H+

(10)

We propose that an LFER exists between the free energies of the species ONS+ and S and the free energy change of protonation, taking the form of eq 11 (where Ca is some constant).

∆G°f,ONS+ - ∆G°f,S ) β(∆G°-a) + Ca

(11)

∆G°N ) β(∆G°-a) + F(1 - Rn)En + Ca - 254 000 (12) Whereas the above equation describes the free energy change of nitrosation as a function of the free energy change of protonation, it is more convenient to express this free energy change as a function of pKa. This is readily achieved through application of eq 13, which, upon substitution, produces eq 14.

pKa )

-∆G°-a 2.30RT

(13)

∆G°N ) -2.30βRT(pKa) + F(1 - Rn)En + Ca - 254 000 (14) Equation 14 provides a description of the free energy change of nitrosation as a function of the nucleophilic (electrophilic) strength of the nitrosating agent and the basicity of the substratesthe two effects that were identified earlier as being responsible for the rate of aniline nitrosation. Kinetic Modeling: Reaction Control. In attempting to model the kinetics of nitrosation, we first concern ourselves with intrinsic rate constants, solely within the reaction-controlled regime (designated k°N). Leffler32 showed that intrinsic rate constants and equilibrium constants for a reaction can often be related according to the expression

∆Gq ) R∆G° + ∆Gqo

(15)

where ∆Gq is the free energy of activation, ∆Gqo is the intrinsic barrier to reaction,33,34 and R is a parameter that lies between 0 and 1 and is a measure of the resemblance of the transition state to the reaction products. Application of eq 15 assumes that reaction follows a synchronous path between reactants and products.35 For the nitrosation reaction being studied, this would mean that cleavage of the ON-X bond would occur at the same rate as formation of the ON-S bond. Although this is thought to be the case,24 a more rigorous application of Leffler’s theory involves the use of two separate transition state parameters, RONX and RONS. Applying this more rigorous theory to eq 14, we obtain

∆GqN ) RONX[F(1 - Rn)En - 254 000] RONS[2.30βRT(pKa) + Ca] + ∆Gqo (16)

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Expressing eq 16 in terms of kN using eq 17 and grouping together all of the constant terms into one overall constant (C), we arrive at eq 18. The constant kT/h in eq 17 is equal to 6.213 × 1012 s-1 at 25 °C.

k)

(

)

kT -∆Gq exp h RT

(17)

ln k°N ) RONS[2.30βRT(pKa)] - RONX[F(1 - Rn)En] + C (18) RT Thus, a model capable of predicting intrinsic nitrosation rate constants has been derived. However, it is now desired that this model be expanded to account for both reaction-controlled (k°) and encounter-controlled (kD) rate constants. Kinetic Modeling: Combined Reaction/Encounter Control. The transition from reaction- to encountercontrol can be predicted by two different approaches. The first uses the Smoluchowski theory,36 which gives the transient solution

k(t) )

[

kDk° R k° 1+ kD + k° kD + k° (πDt)1/2

]

(19)

where kD is the diffusion-limited rate constant (M-1 s-1), D is an overall diffusion coefficient and R is the encounter radius for the reaction. At steady state, this solution simplifies to

k(∞)-1 ) k°-1 + kD-1

(20)

The second approach to integrating reaction- and encounter-controlled rate constants considers the bimolecular reaction of A and B, where in order to react, A and B must first approach from a large distance to form an encounter pair (AB) that subsequently reacts. This is represented by eq 21. From this mechanism, we obtain eq 22 describing the overall rate coefficient. kD

z (AB) f products A + B y\ k

(21)

-D

k)

kDk°

(22)

(k-D + k°)

As would be expected, eq 22 simplifies to the form of eq 20 upon assumption that kD ) k-D. Equation 20 is then substituted into eq 18, so as to enable prediction of reaction rates during the transition from reaction to diffusion control. kN )

{[ ( exp

)]

RONS[2.30βRT(pKa)] - RONX[F(1 - Rn)En] + C RT + kD-1

}

-1

Figure 1. Plot of En vs ln kN within the reaction-controlled regime for the nitrosations of (O) aniline, (b) p-chloroaniline, and (0) p-carboxyaniline with (1) ONBr, (2) N2O3, (3) ONSCN, (4) ONI, and (5) (NH2)2CSNO+.

refs 12, 15, 16, 19, and 37-40 and are for the nitrosation reactions of a wide range of aniline derivatives. Kinetic data for nitrosation with nitrosyl thiosulfate were found to deviate from the model in all cases and were subsequently excluded. Also, only aniline derivatives containing substituents that act through polar interactions with the aniline ring have been included, given that π-electron-withdrawing substituents such as nitro and cyano groups are well-known to be capable of inducing mechanistic change. Reaction Control. The parameters of eq 18 can be fitted by a comparison to experimental data. First, RONX is obtained from the slope of a plot of En vs ln kN. This is shown in Figure 1 for the substrates aniline, pchloroaniline, and p-carboxyaniline. We see that good linear relationships are observed, with an average slope of -14.9. Comparing this slope with eq 18 reveals that we are observing a relationship with an RONX value of 0.832. The product of RONSβ can next be obtained, in a fashion similar to R, by plotting ln kN against pKa for a number of different nitrosating agents. This is done for the nitrosating agents ONSCN and (NH2)2CSNO+ in Figure 2, giving a slope of 1.89. When this slope is compared to eq 18, we find that RONSβ ) 0.822. If we assume an RONS value of 0.832, then β can be estimated as ca. 1.0. The value of C can now be determined from Figure 1 by noting that the intercept of each line must equal C/RT + 1.89pKa. Accordingly, C was evaluated as 93.0 kJ mol-1. The above-determined parameters were then substituted back into the model for k°N, resulting in eq 24. At 25 °C, this simplifies to eq 25.

-1

(23)

Model Application It is now desired to apply the theoretically developed model to experimentally determined rate constants. This is performed first for experimental results obtained solely within the reaction-controlled regime and then expanded to include encounter-controlled data. The experimental measurements used were obtained from

ln k°N )

1.89RT(pKa) - 36 900En + 93 000 (24) RT

k°N ) exp(1.89pKa - 14.9En + 37.5)

(25)

Using the determined kinetic parameters, an estimate of the intrinsic barrier to reaction, an important fundamental parameter, can be made. Although calculations of intrinsic barriers based on intercept values can be questioned for not accounting for the curvature described by the Marcus equation,41 they do provide a useful estimate. First, we make an approximation of

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Figure 2. Plot of pKa vs ln kN within the reaction-controlled regime for the nitrosations of (1) p-carboxyaniline, (2) 4-bromo-1naphthylamine, (3) 4-chloro-1-naphthylamine, (4) p-chloroaniline, (5) 1-naphthylamine, (6) m-methoxyaniline, (7) aniline, (8) Nmethylaniline, (9) p-methylaniline and (10) p-methoxyaniline with (O) ONSCN and (b)(NH2)2CSNO+.

∆G°N for the nitrosation of aniline by the nitrosating agent N2O3 according to eq 8. Here, the En value of aniline was taken as the average of the two reported values, i.e., 1.55. Applying eq 8, the free energy change for the nitrosation reaction is estimated as 8 kJ mol-1 (the free energy of activation has been measured as 22 kJ mol-1). Assuming that RONS ) RONX ) 0.832, the intrinsic barrier to reaction is estimated as 15 kJ mol-1. Finally, the constant Ca can be determined, through a comparison of eq 24 (once ln kN has been converted to ∆GqN) with eq 16. This yields an approximate value of Ca ) 210 kJ mol-1. Encounter Control. The final parameter required for the model is the encounter-controlled rate constant, kD. The encounter-controlled rate constant for the bimolecular reaction of A and B in solution was derived by Smoluchowski36 by assuming that A and B must first diffuse to within a certain distance of each other for reaction to take place. Assuming molecules A and B to be rigid spheres of radii rA and rB, respectively, the steady-state rate coefficient under such conditions can be written as

4πNA(DA + DB)(rA + rB) kD ) 1000

reaction to proceed, the substrate and the nitrosating agent must collide in the correct orientation, i.e., at their respective nitrogen atoms. This introduces a steric effect into the calculation of kD, which can significantly reduce its value.45 Another important effect is brought about by the respective localized charges on the reacting molecules,46 where attractive forces work to increase the encounter-controlled rate constant. As such, the estimated value of 9.3 × 109 M-1 s-1 actually represents an upper limit to the encounter-controlled rate constant (k°D), where the reacting species are of significant opposite charge. For the lower limit, we must account for the fraction of collisions that are of the correct orientation. One method of predicting this quantity is via eq 27,47,48 where θo is the constraint angle of the reacting molecules’ so-called reaction cone (in radians). According to eq 27, with a typical θo value of 40°,44,47 we obtain a kD value of 1.5 × 109 M-1 s-1. Here, the probability that the reacting molecules collide with the correct orientation is about 16%. However, this value will only be valid if charge effects are negligible. In our case, the neutral nitrosating agents will typically possess a small positive charge on their nitroso nitrogens, whereas the substrates will possess a negative charge on their amine nitrogens. For the less basic amines, this charge will be small, and the resultant effect on kD will be minimal. However, as amine basicity increases, so does the magnitude of the negative charge, and one would expect the encounter-controlled rate constant to subsequently increase accordingly, approaching the estimated upper limit of ca. 9.3 × 109 M-1 s-1. As a first approximation, we have neglected charge effects and have thus used the kD value of 1.5 × 109 M-1 s-1 in our model. The choice of this parameter will, however, be reevaluated later.

kD ) (1 - cosθo)θok°D

(27)

Combined Encounter/Reaction Control. The estimated kD value is now substituted into the model for nitrosation (eq 24), so as to enable prediction of rate constants during the transition from reaction to diffusion control.

kN ) {[exp(1.89pKa - 14.9En + 37.5)]-1 + 6.67 × 10-10}-1 (28)

(26)

where NA is Avogadro’s number and DA and DB are the diffusion coefficients of A and B, respectively. An approximation of kD can be obtained by using either known or estimated diffusivities and collision diameters in eq 26. The diffusivity of aniline in water at 25 °C is known to be 0.92 × 10-9 cm2 s-1.42 For the nitrosating agent, we can estimate the diffusivity using the Wilke-Chang method. Taking nitrosyl chloride as a base case and using the additive volumes of Le Bas to estimate the molar volume as 48.5 cm3 mol-1, we arrive at an estimated diffusivity of 1.7 × 10-9 cm2 s-1. Nitrosyl chloride43 has a collision diameter of approximately 0.45 nm, whereas aniline44 has a collision diameter of 0.56 nm. Using these values, we estimate an encounter-controlled rate constant of 9.3 × 109 M-1 s-1 for the nitrosation of aniline with nitrosyl chloride. Two important effects have been ignored in deriving the above encounter-controlled rate constant. First, for

The above model is plotted in Figure 3 for the substrates aniline and p-carboxyaniline. Also included are the limits for the reaction- and diffusion-controlled regimes, as indicated by dashed lines. We see that the experimental data are well described by the model across both the reaction-controlled and encountercontrolled regimes. However, the transition from the kinetic- to the encounter-controlled regime appears to be more rapid than predicted by the model. It also appears that the encounter-controlled limit is lower for p-carboxyaniline than it is for aniline, possibly because of the reduced steric probability and decreased negative dipole on the amine nitrogen resulting from the carboxy substituent. The experimental result for the nitrosation of aniline by nitrosyl thiosulfate is also included in Figure 3,49 illustrating the significant deviation from the model that occurs at such low reactivities. Figure 4 shows a plot of ln kN as a function of pKa for a range of aniline derivatives and the nitrosating agents ONBr, N2O3, and ONI, as modeled by eq 28. The dashed

2300 Ind. Eng. Chem. Res., Vol. 43, No. 10, 2004

parameters were used to describe the rate of nitrosation under both reaction and encounter control. The encounter-controlled model was derived according to masstransfer fundamentals. For the reaction-controlled regime, we first developed a thermodynamic description of the system, which we linked to observed kinetic results using the transition state theory. The developed kinetic model describes the rate of nitrosation as a function of thermodynamic parameters related to substrate basicity and nitrosating agent nucleophilicity. In solving the model, the intrinsic barrier to reaction was evaluated as 15 kJ mol-1. Acknowledgment

Figure 3. Comparison of model with experimental data for the nitrosations of (O) aniline and (b) p-carboxyaniline with (1) ONCl, (2) ONBr, (3) N2O3, (4) ONSCN, (5) ONI, and (6) (NH2)2CSNO+. Also included is the deviant result for aniline nitrosation with S2O3NO- (7).49 Dashed lines represent the limits to the encounterand reaction-controlled regimes.

Figure 4. Comparison of model with experimental data for the nitrosations of (1) p-carboxyaniline, (2) o-chloroaniline, (3) 4-bromo1-naphthylamine, (4) 4-chloro-1-naphthylamine, (5) m-chloroaniline, (6) p-chloroaniline, (7) 1-naphthylamine, (8) m-methoxyaniline, (9) 7-hydroxy-1-naphthylamine, (10) aniline, (11) mmethylaniline, (12) N-methylaniline, (13) p-methylaniline, and (14) p-methoxyaniline with (O) ONBr, (b) N2O3, and (0) ONI. The dashed line represents the predicted encounter-controlled rate constant. The dashed-dotted line represents the predicted upper limit to the encounter-controlled rate constant.

line represents the sterically hindered encountercontrolled rate constant, which was used in the model, whereas the dashed-dotted line represents the predicted upper limit to kD. Reaction-controlled data are again well predicted by the model across a range of pKa values. The encounter-controlled data are generally observed to lie within the predicted range of kD values, increasing somewhat with increasing amine basicity (the same is also observed for nitrosation with ONCl). This result follows our earlier prediction that kD values would increase toward an upper limit of ca. 9.3 × 109 M-1 s-1 with increasing amine basicity. Conclusion A kinetic model has been successfully developed to describe the nitrosation of aniline and its derivatives by a range of nitrosating agents. Reaction and diffusion

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Received for review May 29, 2003 Revised manuscript received February 25, 2004 Accepted February 28, 2004 IE0304560