Study on the reaction rate and mechanism of synthesis of tert-butyl ethyl ether in liquid phase based on synergetics

Received on Sep. 20, 2003. Supported by the National Natural Science Foundation of China (No. 200176044)

1 INTRODUCTION
Generally, most chemical reaction processes are in the state of non-equilibrium ^[1]. Therefore, the dynamic processes of chemical reaction should belong to the category of non-equilibrium statistical mechanics. The latest theory for researching such a non-steady state is synergetics presented by H.Haken ^{[2, 3]}. In synergetics, Haken described the behavior of systems that was near the critical point and elaborated the conception of slaving principle and order parameter. Synergetics considered that the evolution of a system is controlled by the order parameter. The final organization and the degree of order are decided by the order parameter. The physical meaning of the order parameter is deferent in different systems. For example, in laser system, the intensity of light field is order parameter; in chemical reaction, the concentration or the number of particles is chosen as order parameter. In this theory, Haken induced the fundamental evolution rule of the non-equilibrium systems and the most important is that it makes people describe the disorder-order transition of different systems with the same mathematical model. But it is difficult to solve the equation set of reaction rate with Haken's methods directly. It is a reason that why the adiabatic approximate method have not been used in the field of kinetics of chemical reaction ^[4].
    In this paper, a specified reaction rate equation is converted to Haken's form by using a non-singular linear transformation about kinetics variable directly and then the difficulty that involved in the employment of synergetics can be avoided. Thus, it is unnecessary to apply steady-state approximation that is frequently used in the investigation of chemical kinetics. With this method, the mechanism of synthesis of tert-butyl ethyl ether in the liquid phase ^{[5, 6]} was analyzed; the rate equation set was solved and the analytic expressions of order parameters and other variables were obtained. The calculated reaction rate compared favorably with the experimental data.

3 ANALYSIS OF THE REACTION KINETICS
In order to elucidate the application routine of the method, the reaction kinetics of the synthesis of ethyl tert-butyl ether in liquid phase were analyzed. The mechanism is shown as follows:

    In this process, tert-butyl alcohol is catalyzed to be carbocation free radical. Then, the carbocation combine with ethanol to form ETBE; meanwhile carbocation is decomposed to isobutene. If C₁=C_BuOH, C₂=C_C4H9⁺, C₃=C_H2O, C₄=C_C4H8, C₅=C_EtOH, C₆=C_ETBE express the concentration of each component then, the rate equation set that related to the previous stated mechanism will be written as the following:
₁ = -k₁C₁ + h₁                                                                                     (3.1)
₂= k ₁C₁ –k ₃C₂ + k ₄C₄ + k ₆C₆ + h₂                                               (3.2)
₃ = k ₁C₁ + h₃                                                                                     (3.3)
₄ = k ₃C₂ –k ₄C₄ + h₄                                                                       (3.4)
₅ = k ₆C₆ + h₅                                                                                     (3.5)
₆ = - k ₆C₆ + h₆                                                                                   (3.6)
    Here, r_i represents reaction rate of each reaction, h_i represents the non-linear part of each reaction rate equation. As catalyst, the concentration of H⁺ is high enough, and it is considered to be constant at any stage of the reaction.
    According to the known reaction mechanism of the above process ^[6], the production of C₄H₉⁺ from C₄H₈ can be ignored, the k ₄ in the above equation sets then become zero and h₁= k ₂C₂C₃, h₂= - k ₂ C₂ C₃- k₅ C₂ C₅, h₃= - k₂ C₂ C₃, h₄=0, h₅= - k₅ C₂ C₅, h₆= k₅ C₂ C₅.
    The boundary condition is:
    C₁(0)=A, C₅(0)=B, C_i(0)=0   (i=2, 3, 4, 6)
Equation (3.4) is not independent, only equations (3.1) (3.2) (3.3) (3.5) (3.6) need to be solved.

4 PRINCIPLE OF THE NON-SINGULAR LINEAR TRANSFORMATION
In order to obtain the equations of order parameters by using adiabatic approximate method, the most important step is to find the order parameters. Haken suggested a method to find the order parameters by distinguishing the changing rate in time of state variables, which is a distinctive and effective method. With the method, many variables can be represented by only one or a few ones of them. Then, it becomes possible to research their effect on the evolution of system in mathematics.
    Because the rate equation set of chemical reaction generally contains linear and non-linear part, the linear transformation can be used to convert it into Haken's form instead of using the method of variation parameters used by Haken.
    Suppose that the rate equation set is in the form as follows (the concentrations are dimensionless in the following discussion):
=AC+h (C₁, C₂, ..., C_n)                                                               (4.1)
where, is the time derivative of the concentration, c_j (j=1, 2, ..., n) represents the kinetics variable(concentration of component), A represents a constant matrix of n×n, and a_ij(i, j=1,2,...,,n) is its element. C is a column with C_j as its component. h is a column with h_j as its component and h_j=h_j(C₁, C₂, ... , C_ｎ) expresses a non-linear function of c_ｊ (without constant and linear item).
Suppose           
C =VC' or C'=V^{- 1}C                                                                             (4.2)
whereV is a matrix of n×n with v_ij (i, j=1,2,...,,n) as its element (it has inverse matrix V^{- 1}) and C'represents a column with C_j' as its component. Substituting (4.2) into (4.1), the following relational expressions are gotten:
'=V^{- 1}ＡVC+V^{- 1}h (VC')=V^{- 1}AVC+h ' (C')                                    (4.3)
whereｈ'(C') represents the transformed h (C). And if
V^-1A V=〔l₁,l₂, ..., l_ｎ〕                                                             (4.4)
where〔l₁，l₂，…l_ｎ〕is diagonal matrix,l_j (j=1, 2, ..., n) is matrix element, equation(4.3) can be turned to the following form:
j' =l_jC_j +ｈ_j' (C₁', C₂', ..., C_n')，(j=1, 2, ... , n)                        (4.5)
    Thus equation (4.5) is obtained as the Haken's form of equation(4.1). Therefore the above method can be considered as the process of finding the matrix element (V_ij) of the non-singular matrix V and the value of l_j. Since the equation set (4.4) can be written as the following form:
(j, k =1 ,2, ... ,n)                                                     (4.6)
the method can be further considered as the process of finding the characteristic value, l_k of equation (4.6) .

A＝ C＝                                          (5.1)
The characteristic equation of matrix A is:
=0                                                         (5.2)
The solution of this equation is:
l₁＝-k₁, l₂= -k₃, l₃=l₅= 0, l₆= -k₆                                                                                   (5.3)
The equation set corresponding to (4.6) is:
( - k₁ -l_K)V₁Ｋ= 0
k₁V₁Ｋ+(-k ₃-l_Ｋ)V₂Ｋ+ k₆V₆Ｋ=0
k₁V₁Ｋ+(-l_Ｋ)V₃Ｋ=0                                                                                                                      (5.4)
(-l_Ｋ)V₅Ｋ+ k ₆V₆Ｋ=0
(-k₆-l_Ｋ) V₆Ｋ=0
The following solution can be obtained from (5.3) and (5.4) with the orthonormal relation condition:
V₁₁=; V₂₁=; V₃₁= -V₁₁; V₅₁=V₆₁= 0; V₂₂= 1; V₁₂=V₂₃=V₂₅=V₂₆= 0;
V₃₃=V₅₃=; V₃₅=V₅₅=; V₆₆=; V₅₆= -V₆₆; V₂₆=; V₁₆=V₃₆=0
Matrix V can be written as the following form:
V=                                                                               (5.5)
And then
V^-1=    (5.6)
Therefore, the relationship corresponding with (4.2) becomes:

                                                                           (5.7)
With the transformation of (5.7), (3.1)-(3.6) can be written as follows:
₁^' = -k₁C₁^' + h₁^'
₂^' = –k ₃C₂^'+ h₂^'
₃^' = h₃^'                                                                                                                                       (5.8)
₅^' = h₅^'
₆^' = - k ₆C₆^' + h₆^'
Where, h₁^' =h₁, h₂^' =h₁+h₂+h_6;
h₃^'=h₁+h₃+h₅+h₆;
h₅^'=h₁+h₃+h₅+h₆;
h₆^'=h₆.
    In the equation set (5.8), the variables ₃^' and ₅^' contain the non-linear item only. According to the Haken's viewpoints of synergetics, they should belong to the slow relaxation parameter. This equation set has the Haken's form, so the adiabatic approximate method can be used:; ; . Then ₁=0, ₂=0, ₆=0 can be obtained from the transformation of (5.7), And just at the beginning of the reaction, C₁＝A, C₂＝0, and C₆＝0, which can be taken as the first approximate result. Substitute them into (3.1)- (3.6), the following results can be obtained:
C₁＝A e^－k1^At                                                                                                                                      (5.9)
C₂＝k₁/ k₂(A－1)[1/2－1/(1+e ^k1^At)]                                                                                               (5.10)
C₃=A－Ae^{- k}1^At                                                                                                                                 (5.11)
C₄＝(A－1)k₃/ k₂[(-k ₁/2)t＋(1/A)ln(1＋e ^k1^At)]                                                            (5.12)
C₅=e^{(M/2- k}6^)t (1+e ^k1^At)^{M/ k}1^A－2 ^k5^{/ k}2^(1-1/A)                                                                              (5.13)
According to the material balance of the reaction:
C₆=B-C₅                                                                                                                                             (5.14)
Thus, as shown in (5.9) - (5.14), the expressions of the change of the concentration in time in the reaction process are obtained.

6 RESULTS AND DISCUSSION
When a chemical reaction reached equilibrium, the rate of creation of free radicals or chain carrier is equal to the rate of consuming, namely that the variation rate of concentration with time is zero. If this reaction was researched by using steady state approximation, it must be provided that . But this is improper when the reaction is in the beginning phase or far from equilibrium state. This hypothesis is needless when the advantages of the adiabatic approximate method in the principle of synergetics were taken.
    The reaction rate constants as the function of temperature were also determined to be the following:
[m⁶/(mol·s·mol-H^＋)]                                                                      (6.1)
[m⁶/(mol·s·mol-H^＋)]                                                                        (6.2)
[m³/(s·mol-H^＋)]                                                                                 (6.3)
[m⁶/(mol·s·mol-H^＋)]                                                                                                    (6.4)
[m⁶/(mol·s·mol-H^＋)]                                                                    (6.5)
[m³/(s·mol-H^＋)]                                                                                  (6.6)
    Fig.1 and Fig.2 are the relations of concentration to time. Reaction temperature is 328K. The lines in Fig.1 are the calculated results with the genetics algorithm (GA) by using the results of synergetics method, the lines in Fig.2 shows the relations of concentration to time calculated with GA by using the results of Runge-Kutta as in the earlier article ^[7]. It can be seen that the results of Runge-Kutta are only slightly different to the results of synergetics method: the results of Runge-Kutta are a little lower than experiment data while the results of synergetics are a little higher. Fig.3 and Fig.4 are the relations of concentration to time. Reaction temperature is 318K. The fitted results of synergetics are better than that of Runge-Kutta. The method of Runge-Kutta is a kind of relatively precise method and the synergetics is an approximate method, but the results are similar to each other. Thus, the validity of the method based on the synergetics is proved.
  

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