Proceedings
of the Tomsk Polytechnic Institute named after S. M. Kirov,
holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Volume 235　　　　　　　　　　　　　　　　　　　　　　　　　1973

On the Determination of the Rate Constant of a First-Order Reaction under Linear Temperature Programming

V. M. Mironov, O. P. Lobas

When a chemical reaction is carried out under stationary nonisothermal conditions, in the case of a continuous rise in temperature, the curves of the change in the concentration of the reactants with time exhibit an inflection point corresponding to the maximum reaction rate, calculated from the change in the concentration of the given reactant.

Fig. 1.

Using the concept of the surface of a chemical reaction, defined by the dependence of the reactant concentration or the reaction rate on temperature and time (a spatial coordinate system), one can determine the value of its constant from the time at which the maximum value of the reaction rate is reached. This is most simply done for a first-order reaction.

Let us represent the surface of a chemical reaction in the coordinates

\[ y = f(T; \tau), \]

where

\(y\) — degree of conversion;
\(T\) — temperature;
\(\tau\) — time,

in the form of contour lines on the \(TO\tau\) plane (Fig. 1). Let the curve \(y_5\) determine the degree of conversion of the reactant corresponding to the inflection point-

the inflection of the reaction-path line in any plane \(yOR\). Under linear temperature programming, the reaction path will correspond to the line of intersection of the chemical-reaction surface and the plane \(yOR\), with

\[ \tg \alpha = a, \]

where

\(\alpha\) is the angle between the planes \(yOR\) and \(yO\tau\);
\(a\) is the rate of linear temperature rise.

To find the coordinates \(T\) and \(\tau\) of the existence of the conditional extremum of the function

\[ r = y' = f'(T;\tau) \tag{1} \]

(\(r\) is the reaction rate), it is necessary to use the Lagrange method. In this case the principal function will have the form (1), while the auxiliary function will be characterized by the expression

\[ T = a\tau . \]

But application of the Lagrange method even in this simplest case (a first-order reaction and a linear change in temperature) leads to a system of equations, generally unsolvable, with exponential and logarithmic terms. Therefore, by this method it is very difficult to determine the relation between the temperature and the time at which the maximum reaction rate is attained (\(T_m\) and \(\tau_m\)).

However, under practical conditions the values of the quantities \(T_m\) and \(\tau_m\) are known to us; therefore the relation between them is determined more simply. As can be seen from Fig. 1, the inflection point of the reaction-path curve in the plane \(yOR\) will correspond to the inflection point of the curve of intersection of the reaction surface and the plane

\[ \tau = \tau_m = \mathrm{const}. \tag{2} \]

In this connection we may seek the maximum of function (1) under condition (2).

For a first-order reaction:

\[ r = \frac{dy}{d\tau} = \kappa(1-y), \tag{3} \]

whence

\[ 1-y = e^{-\kappa \tau}. \tag{4} \]

Substituting (4) into (3), we obtain

\[ r = \kappa \cdot e^{-\kappa \tau}. \tag{5} \]

The condition for an extremum of function (5) is:

\[ \left(\frac{dr}{d\kappa}\right)_{\tau=\tau_m}=0,\ \text{i.e.} \]

\[ e^{-\kappa \tau_m} - \kappa \tau_m e^{-\kappa \tau_m} = 0,\ \text{whence} \]

\[ \kappa \tau_m = 1, \]

or, finally,

\[ \kappa_m = \frac{1}{\tau_m}, \tag{6} \]

where \(\kappa_m\) is the value of the reaction-rate constant at the point of maximum.

If the dependence of the reaction-rate constant on temperature obeys the Arrhenius law, then, carrying out the reaction at two heating rates and determining two values \(k_m\), one can find the activation energy of the process and the frequency factor, i.e., determine the law of the temperature dependence of the reaction rate.

It is quite remarkable that the value of the heating rate does not enter into expression (6); therefore, using the proposed method for calculating the reaction-rate constant, one can dispense with linear temperature programming and increase it according to any law (lines 1 and 2 in Fig. 1).

The proposed method for determining the principal kinetic parameters of a first-order reaction will be valid only in the case where the maximum reaction rate, irrespective of the heating rate, is observed at one and the same degree of conversion.

Conclusions

On the basis of an analysis of the kinetic equation for a first-order chemical reaction, the possibility has been shown of determining the rate constant of this reaction from the value of the time required to attain the maximum rate of conversion of the reactants under conditions of a stationary and nonisothermal process.