To obtain different experimental data, the reaction must be carried out with different initial concentrations of the reactant. These methods are described in more detail in this chapter and the resulting final equations to be used for the mathematical treatment of the kinetic data are highlighted. It is recommended for relatively simple kinetic expressions corresponding to elementary reactions or when the experimental data are so scattered that reaction rates cannot be calculated with sufficient accuracy to allow the use of the differential method.
Plot the experimental data and if they fit a straight line, the assumed reaction order is correct, and the reaction rate coefficient is determined from the slope of the straight line. The value ofk can be determined directly from the slope of the straight line. k can also be obtained from Eq. Figure 2.2 Graphical integral method for irreversible reactions of one component of the first order. By combining these last two equations, the following expression of the straight line is obtained:
Furthermore, the value of CA is not necessary if ki is calculated using either of the two equations, which only occurs for the first order of the reaction. In both cases, the slope of the straight line (kCAo) gives the value ofk, which can also be calculated with:. To use this equation, the value of the initial concentration of CA is needed, which is calculated using Eq.
Starting with the order of elementary reaction (n = 1), Eq. 2.18) as a function of xA is used to calc.
Differential Method
At constant density, k can be determined from the intersection of the straight line: ln k for the equation as a function of concentration (equation 2.31), and ln (kCAon−1) for the equation as a function of conversion (equation 2.32). With variable density, k is also calculated from the intersection of the straight line in the case of the equation as a function of the concentration (Eq. 2.33). For the equation as a function of conversion (equation 2.34), however, it is not possible to calculate from the intersection of the straight line.
The application of these equations requires knowledge of the data from ln(−dCA/dt)orln(dxA/dt)andlnCAorlnxA. The evaluation of the derivatives can be done by differentiating the data of concentration CA or conversionxA and time. 1 +εAxA n−1 +nln 1−xAp 2 38 whereCApandxA represent the average values of the two values ofCA. andxAwhich were used to calculateΔCAandΔxA. Table 2.4 illustrates the procedure using data from CA to determine the parameters needed in Eqs.
The method of finite differences can use the following formulas of three-point differentiation to evaluate the derivative (−dCA/dt). 2Δt CAn−2−4CAn−1+ 3CAn 2 41 Table 2.5 presents an example of the formulas used to apply the method of finite differences for the case of five experimental points, including the initial date at zero time. If the experimental data are not measured at regular time intervals, it is recommended before applying the method of finite differences to plot the data often against CA, and from this graph to generate a different set of concentration data at each time taking into account regular intervals (ie with the same Δt).
The order of the polynomial is determined based on the correlation coefficient rm between the experimental data (tvs.CAorxA) and that calculated by the polynomial. A polynomial order showing a higher value of r would be more suitable for use in Eqs. We must remember that the order of the polynomial cannot exceed the number of data.
The graphical differentiation method with surface compensation is an extension of the numerical differentiation method by approximating the derivatives (−dCA/dt) to (ΔCA/Δt) or (dxA/dt) to (ΔxA/Δt), because the values of (ΔCA/Δt) or (ΔxA/ Δt), obtained numerically (Table 2.4), should be drawn accordingly in the form of rectangles. Then the area of the formed rectangle must have as its basis the difference between t1 and t2 when plotted against (ΔCA/Δt). The next step is to draw an average curve over the area of the resulting rectangles, trying to ensure as much as possible that the area that is omitted is similar to the area that does not belong to the rectangle, but that the area under the curve is actually considered inside (ie the areas are compensated).
Method of Total Pressure
In this method, the data of time and total pressure are used to determine the kinetic model. To have variable pressure data, the system must operate at constant density and temperature, with a gas phase reaction in which there is a change in the number of moles (Δn 0). The procedure is similar to that of the integral method, but it varies the integrated equations.
2 49 the general reaction rate equation as a function of total pressure is:. which∗is not the reaction rate coefficient, but just a constant by which the value ork with Eq. Finally, determine either ∗from the slope of the straight line obtained by plotting t against the left side of Eq. 2.58) and see if they are similar without trend (if they are, calculate the average value, which will be the final value of k∗). Example 2.4 The dimerization of trifluorochloride-ethylene in gas phase (2A R) was studied by changes in the total pressure.
The best fit is achieved forn= 2, since forn= 1 there is a decreasing trend of the values, whereas forn= 2 there is an increasing trend, so that the average value of k∗ is sec−1KPa−1. The slope of the straight line directly gives the value of and the intersection the value of k∗, which is further used with Eq. For a reaction where the total pressure decreases as the reaction progresses (Po > P), the expression [Po(1+εA)−P] is negative in some cases, depending on stoichiometry and feed composition (i.e. on the value of εA). 2.59) cannot be used because it would involve the logarithm of a negative number.
To evaluate the derivative (dP/dt), the method of finite differences will be used, since Δ in this case is the same for all the data, Δt= 100 sec. The actual reaction order is approximated to an integer value, which is n= 2, and with this the value of the reaction rate coefficient is calculated as done in Example 2.4.
Method of the Half-Life Time
Example 2.6 The thermal decomposition of nitrogen oxide (N2O) was studied at 1030 K, and the data reported in Table 2.12 were often1/2 obtained at different initial partial pressures of N2O. It is observed that for the first order of reaction, the value of kha has an increasing trend, while the second order of reaction accurately matches the experimental data, with an average value ofk= 0.8954 lt/gmol sec. It is important to mention that for reactions that follow the first order, the half-life is the same to calculate; in other words, the initial concentration has no effect at n= 1.
With the slope (1−n) the reaction order is calculated, and with the intercept the value ork, as illustrated in Figure 2.20. The whole value of the reaction order is 2, and the reaction rate coefficient is evaluated with the integrated equations used in Example 2.6. The half-life method can be extended to any fractional lifetime of the limiting reactant.
For example, T1/3 is the time required to reduce the initial concentration to one third of its value. In general, t1/m is the time required for the initial concentration to reduce to 1/m of its value. The intercept is useless because it is a function of the reaction order and frequency factor A, as illustrated in Figure 2.22.
The value of Akan can only be calculated if the reaction order has been determined by another method. Example 2.8 The decomposition of N2O5 was carried out at different temperatures in an isothermal reactor at constant density.
