On a Method for Determining the Kinetic Parameters of Monomolecular Reactions under Nonisothermal Conditions
Abstract Generated abstract
This paper examines determination of kinetic parameters for first-order unimolecular reactions under nonisothermal conditions, with particular attention to linear heating. The authors integrate the rate equation with an Arrhenius rate constant, present exact and approximate expressions involving the exponential integral, and note that numerical integration becomes straightforward once the activation energy and pre-exponential factor are known. They propose using the shift of the maximum in the differential reaction curve at different heating rates to derive the activation energy and then calculate the pre-exponential factor, requiring in principle experiments at only two heating rates.
Full Text
On the Methodology for Determining the Kinetic Parameters of Unimolecular Reactions under Nonisothermal Conditions
V. G. Kriter, L. T. Butaenko, L. B. Serzhiv
(Presented by a scientific seminar of the Department of Radiation Chemistry)
The paper discusses the solution of the equation of a unimolecular reaction under nonisothermal conditions and the possibility of determining all unknown parameters from the displacement of the maximum of the differential curve.
The equation for a first-order reaction for the consumption of some product, with the rate constant in Arrhenius form, has the form:
\[ \frac{dC}{dt}=-k_0\exp\left(-\frac{E}{RT}\right)\cdot C, \tag{I} \]
where \(C\) is the concentration of the consumed product at the time \(t\).
\(E\) is the activation energy of the given reaction,
\(k_0\) is the pre-exponential factor,
\(T\) is the absolute temperature.
The aim of the solution is to construct the dependences \(C(t)\), \(\frac{dC}{dt}\), and to determine the unknown parameters \(E\) and \(k_0\).
In the case when the temperature in (I) does not depend on time, this presents no difficulty. But if the temperature is an explicit function of time, the solution of (I), as a rule, is not expressible through elementary functions, and the determination of \(E\) and \(k_0\) presents definite difficulties. Quite a number of works have been devoted to this question (see, for example,
[1–3], [5]). The greatest number of developed methods is devoted to determining the activation energy. It is most often determined by constructing the dependence \(E = E(T)\) in the corresponding rectifying coordinates. The drawback of these methods is their low accuracy, since, in determining \(E\), one has to use either methods of graphical integration or differentiation, or else make additional assumptions about the form of the dependence \(E(T)\) in order to be able to straighten the curve. A method for determining \(E\) for a monomolecular reaction without introducing additional assumptions is given in [2].
To determine \(k_0\), as a rule, still more various assumptions have to be made, and these are by no means always valid.
In the present work, for integrating (1) and determining \(E\) and \(k_0\), no additional assumptions are made except for their constancy for the given reaction in a definite temperature interval.
Let us consider the solution of equation (1) in the case when the temperature increases with time. The simplest dependence of this kind is linear heating:
\[ T = \beta t . \tag{2} \]
where \(\beta\) is the heating rate.
This case is often used experimentally. In addition, integration of (1) in the case of a more complicated dependence \(T(t)\) (for example, of polynomial type) ultimately reduces to the same case. In the case of linear heating, a necessary condition for the fulfillment of (1) is a small deviation of the sample temperature from the prescribed law during the reaction.
Let the temperature in (1) vary according to the law. Then (1) can be written in the form:
\[ \frac{dC}{dT} = -\frac{k_0}{\beta}\exp\left(-\frac{E}{RT}\right). \tag{I'} \]
Integrating (1), and considering \(E\) and \(k_0\) constant for the given reaction, we obtain:
\[ C = C_0 \exp\left(-\frac{k_0}{\beta}\int_0^T \exp\left(-\frac{E}{RT}\right)\,dT\right) \tag{3} \]
where \(C_0\) is the initial concentration.
Consider the integral \(A = \int_0^T \exp\left(-\frac{E}{RT}\right)\,dT\). By substituting \(x=-\frac{E}{RT}\), \(-\infty < x < 0\), it reduces to the expression:
\[ A=\frac{E}{R}\left(-\frac{\exp(x)}{x}+Ei(x)\right), \tag{4} \]
where
\[ Ei(x)=\int_{-\infty}^{x}\frac{\exp(x)}{x}\,dx, \]
then solution (3) takes the form:
\[ C=C_0 \exp\left[\frac{k_0}{\beta}\frac{E}{R}\left(\frac{\exp(x)}{x}-Ei(x)\right)\right] \tag{5} \]
For large \(x\), using the expression for \(Ei(x)\) in the form of the series [6]
\[ Ei(-|x|)=\exp(-|x|)\sum_{k=1}^{n}(-1)^k\frac{(k-1)!}{x^k}+R_n \]
one can obtain the approximate expression for \(A\):
\[ A=\frac{E}{R}\frac{\exp(x)}{x^2}, \tag{6} \]
where, as before, \(x=-\frac{E}{RT}\), and the relative error will not exceed:
\[ \frac{2}{|x|}. \]
Then the approximate solution (3) is written as
\[ C=C_0 \exp\left(-\frac{k_0}{\beta}\frac{E}{R}\frac{\exp x}{x^2}\right) \tag{7} \]
In view of the fact that solutions (5) and (7) are very cumbersome, in order to construct the dependences \(C(T)\) and \(\dfrac{dC}{dT}\) one often more simply solves the initial equation (1) numerically on a computer; but for this it is necessary to know the parameters \(E\) and \(k_0\).
To find these parameters, let us use the fact that the curve \(\dfrac{dC}{dT}\) has an extremum at a certain temperature \(T_m\). Indeed, differentiating (1) with respect to \(T\) and setting \(\dfrac{d^2 C}{dT^2}=0\), we obtain:
\[ \frac{\beta}{T_m^2}\exp\left(\frac{E}{RT_m}\right)-\frac{k_0R}{E}=0 \tag{8} \]
From (8) it is seen that for \(T<T_m\), \(\dfrac{d^2 C}{dT^2}>0\), while for \(T>T_m\), \(\dfrac{d^2 C}{dT^2}<0\), i.e., at this point the curve \(\dfrac{dC}{dT}\) reaches a maximum.
Let us note that the second term in (8) does not depend on the heating rate, and for the given reaction one may write:
\[ \frac{\beta}{T_m^2}\exp\left(\frac{E}{RT}\right)=\frac{k_0R}{E}=\text{const}. \tag{9} \]
From (9) it is evident that, when the heating rate is changed, the position of the maximum will change; namely, with an increase in the heating rate the maximum of \(\dfrac{dC}{dT}\) will shift toward higher temperatures. Therefore, having obtained the experimental curve \(\dfrac{dC}{dT}\) at different heating rates, we may write:
\[ \frac{\beta_1}{T_1^2}\exp\left(\frac{E}{RT_1}\right) = \frac{\beta_2}{T_2^2}\exp\left(\frac{E}{RT_2}\right) \tag{10} \]
whence it is easy to obtain an expression for the activation energy:
\[ E=\frac{RT_1T_2}{T_2-T_1} \left(\ln\frac{\beta_2}{\beta_1}-2\ln\frac{T_2}{T_1}\right) \tag{11} \]
Moreover, the accuracy of determining \(E\) depends primarily on the accuracy of determining \(\Delta T=T_2-T_1\). Having found the values of \(E\) from (11) and substituting them into (9), one can calculate \(k_0\). Then
all the parameters in \((I')\) become known and, using the solutions (5), (7), or by numerically integrating \((I')\), one can find \(C(I)\) and \(\dfrac{dC}{dT}\). Thus, to find \(C(T)\) and \(\dfrac{dC}{dT}\) and to determine \(E\) and \(K_0\), it is sufficient, generally speaking, to carry out the experiment at two different heating rates.
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