Order, determination Differential methods

Differential temperature method. A differential method has been applied to a study of the iodination of acetone, a pseudo-zeroth-order reaction when [(CHj)2CO] [I2].26 It allows the determination of AW to much higher accuracy than otherwise. The reaction rate is expressed mathematically as... 

Use (a) the differential method and (b) the integral method to determine the reaction order, and the value of the rate constant. Comment on the results obtained by the two methods. 

The mathematical relationships using the differential method to determine a, P and k are linear, and any commercial spreadsheet or scientific software package will have a suitable fitting program. It is desirable that the fitting software gives the standard deviations of the fitted parameters, so that the statistical errors of reaction orders and rate coefficients are known. 

The methods used to determine the order of the reaction and its rate constant may be divided into two groups (1) differentiation method and (2) integration method. 

Another technique of the differentiation method is the initial rate measurement. A series of experiments are carried out for different initial concentrations over a short time period (5 to 10% or less conversion). This approach is different from the experimental run discussed in Figure 5.7. Each rate measurement requires a new experiment with a different initial concentration. The initial rate of the reaction is determined from the curve of the concentration vs. time, as shown Figure 5.9a. The log of the initial rate is then plotted against the log of the initial concentration (Figure 5.9b). If the order of the reaction calculated from the concentration-time curve is different from the one determined by initial rate experiments, interference by the reaction products is expected, leading to complex reaction kinetics. 

In the analysis of H-NMR spectra of compounds elaborated from aliphatic fatty acids and polyamines it is rather difficult to determine the linear or branched nature of the fatty acid. In order to differentiate, well-known oxidative degradation reactions (e.g., HN03) are recommended in addition to spectroscopic methods. 

Figure 5-1 Differential method to determine reaction order.

We found the plot of lnC2F,( /(3F,o - F,)] versus f was linear, indicating that the reaction is first order (i.e., ot = 1). If we try zero, first, or second order as shown cm the CD-ROM, and they do not seem to describe the reaction ritte equation, it is usually best to try some other method of determining the reaction order, such as the differential method. 

The use of the differential method of data analysis to determine reaction orders and specific reaction rates is clearly one of the easiest, since it requires only one experiment. However, other effects, such as the presence of a significant reverse reaction, could render the differential method ineffective. In these cases, the method of initial rates could be used to determine the reaction order and the specific rate constant. Here, a series of experiments is carried out at different initial concentrations, C q, and the initial rate of reaction, is determined for each run. The initial rate, can be found by differentiating the data and extrapolating to zero time. For example, in the tfi-tert-butyl peroxide decomposition shown in Example 5-1, the initial rate was found to be... 

Table IX presents the results for stoichiometric mixtures (Po = 3 torr) when the temperature of the cold trap is higher than —195° and therefore for increased partial pressures of carbon dioxide. Apparent orders in Table IX were determined by the differential method (plot of log dPjdt as function of log P). They show that as the partial pressure of carbon dioxide increases, the autoinhibition increases. The rate of the reaction is also greatly decreased and the order increased if a constant activity catalyst has adsorbed carbon dioxide previous to the...

Pigs. 18 and 21). As in the case of Ni0(200°), the initial total order is close to zero when NiO(250°) is used as a catalyst and the reaction rate on the fresh sample decreases with time according to the kinetics of order one (74). Kinetics of order one are not followed, however, on regenerated catalysts. Reaction orders were determined in this case by the differential method and were found to vary from 1 (fresh catalyst) to 0.77 (constant activity). Since the initial total order is, in all cases, zero, it was concluded that, as in the case of the same reaction on NiO(200°), the reaction order with respect to time is apparent and results from the inhibition of the catalyst by carbon dioxide, the reaction product. Modification of the apparent order with the runs indicates that regenerated samples of Ni0(250°) are less inhibited than the fresh catalyst. 

The reaction order with respect to time was determined by the differential method. A fractional order (1.3) is obtained for the catalytic reaction on both doped samples. However, as in the case of the same reaction on pure oxides, the initial reaction rate does not depend upon the pressure of either reagent (order zero). Since these results are similar to those obtained on pure samples, NiO(200°) and NiO(250°), we believe that the order with respect to time is, as in the former case, apparent and that it results from the inhibition of surface sites by carbon dioxide, the reaction product. The slowest step of the reaction mechanism on doped oxides should occur, therefore, between adsorbed species. 

From this information determine the first-order and second-order specific rates, and kj, assuming that the reaction is irreversible over the conversion range covered by the data. Use both the integration and the differential method, and compare the results. Which rate equation best fits the experimental data ... 

Initial Rate Method For reversible reactions, we use a modified differential method—the initial rate method. In this case, a series of experiments are conducted at selected initial reactant compositions, and each run is terminated at low conversion. From the collected data, we calculate (by numerical differentiation) the reaction rate at the initial conditions. Since the reaction extent is low, the reverse reaction is negligible, and we can readily determine the orders of the forward reaction from the known initial compositions. The rate of the reversible reaction is determined by conducting a series of experiments when the reactor is charged with selected initial product compositions. The initial rate method is also used to determine the rates for complex reactions since it enables us to isolate the effect of different reactants. 

The structure and interrelationship of the batch conservation equations (population, mass, and energy balances) and the nucleation and growth kinetic equations are illustrated in an information flow diagram shown in Figure 10.8. To determine the CSD in a batch crystallizer, all of the above equations must be solved simultaneously. The batch conservation equations are difficult to solve even numerically. The population balance, Eq. (10.3), is a nonlinear first-order partial differential equation, and the nucleation and growth kinetic expressions are included in Eq. (10.3) as well as in the boundary conditions. One solution method involves the introduction of moments of the CSD as defined by... 

Propose a plausible rate equation and then, possibly in conjunction with an isolation method, use the differential method to determine whether the proposal is correct and, if so, establish the partial order with respect to each individual reactant. (Question 5.1 and Exercise 5.1)... 

Table Q.2 Data for using the differential method to determine the partial order of reaction with respect to NO2 for the gas-phase thermal decomposition of this compound at300°C...

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