Differential rate methods

Direct-Computation Rate Methods Rate methods for analyzing kinetic data are based on the differential form of the rate law. The rate of a reaction at time f, (rate)f, is determined from the slope of a curve showing the change in concentration for a reactant or product as a function of time (Figure 13.5). For a reaction that is first-order, or pseudo-first-order in analyte, the rate at time f is given as... 

Mottola, H. A. Catalytic and Differential Reaction-Rate Methods of Chemical Analysis, Crit Rev. Anal. Chem. 1974, 4, 229-280. Mottola, H. A. Kinetic Aspects of Analytical Chemistry. Wiley New York, 1988. 

In 1950 French " and Wideqvist independently described a data treatment that makes use of the area under the concentration-time curve, and later authors have discussed the method.We introduce the technique by considering the second-order reaction of A and B, for which the differential rate equation is... 

Frost and Pearson treated Scheme XV by the eigenvalue method, and we have solved it by the method of Laplace transforms in the preceding subsection. The differential rate equations are... 

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... 

Since the measurements of conductance change are not directly related to the composition of the solution, as an alternative method numerical integration of the differential rate equations implied by the proposed mechanism was employed. The second order rate coefficients obtained by this method are... 

In the DSC method, the sample and reference are maintained at the same temperature and the heat flow required to keep the equality in temperature is measured. Hence DSC plots are obtained as the differential rate of heating (in units of watts/second, calories/second, or Joules/second) against temperature. The area under a DSC peak is directly proportional to the heat absorbed or evolved by the thermal event, and integration of these peak areas yields the heat of reaction (in units of calories/second gram or Joules/second gram). 

Because of the complexity of biological systems, Eq. (1) as the differential form of Michaelis-Menten kinetics is often analyzed using the initial rate method. Due to the restriction of the initial range of conversion, unwanted influences such as reversible product formation, effects due to enzyme inhibition, or side reactions are reduced to a minimum. The major disadvantage of this procedure is that a relatively large number of experiments must be conducted in order to determine the desired rate constants. 

However, the average rates calculated by concentration versus time plots are not accurate. Even the values obtained as instantaneous rates by drawing tangents are subject to much error. Therefore, this method is not suitable for the determination of order of a reaction as well as the value of the rate constant. It is best to find a method where concentration and time can be substituted directly to determine the reaction orders. This could be achieved by integrating the differential rate equation. 

ISO 15105-1 2002 Plastics - Film and sheeting - Determination of gas-transmission rate -Part 1 Differential-pressure method... 

The differential method of analysis deals directly with the differential rate equation to be tested, evaluating all terms in the equation including the derivative dCJdt, and testing the goodness of fit of the equation with experiment. 

Another method to obtain kinetic data is to use a differential reactor in which the concentration does not change much from the initial concentration Cao- this case the differential rate expression... 

We are interested in solving equation (7.19) to obtain an expression which describes how the concentration of A varies with time, subject to the boundary condition that the concentration of the reactant at time t = 0 is [A]0 (note that the differential rate law above tells us only how the rate depends on [A]). Thus, using the separation of variables method, equation (7.19) is first rearranged to ... 

The acid-base properties, and hence ionic character, of peptides and proteins also can be used to achieve separations. Ion-exchange chromatography, similar to that described for amino acids (Section 25-4C), is an important separation method. Another method based on acid-base character and molecular size depends on differential rates of migration of the ionized forms of a protein in an electric field (electrophoresis). Proteins, like amino acids, have isoelectric points, which are the pH values at which the molecules have no net charge. At all other pH values there will be some degree of net ionic charge. Because different proteins have different ionic properties, they frequently can be separated by electrophoresis in buffered solutions. Another method, which is used for the separation and purification of enzymes, is affinity chromatography, which was described briefly in Section 9-2B. 

Rate equations like 2.27 and 2.28, obtained from a proposed set of elementary reaction steps, are differential equations. Although for our purposes in this book we shall require only differential rate equations, it is usually more convenient in interpreting raw experimental data to have the equations in integrated form. Methods of integration of rate equations can be found in the literature.34... 

Although not very commonly used (with the exception of the initial rate procedure for slow reactions), the differential method has the advantage that it makes no assumption about what the reaction order might be (note the contrast with the method of integration, Section 3.3.2), and it allows a clear distinction between the order with respect to concentration and order with respect to time. However, the rate constant is obtained from an intercept by this method and will, therefore, have a relatively high associated error. The initial rates method also has the drawback that it may miss the effect of products on the global kinetics of the process. 

Neither (a) the differential method, based on rate/concentration data, nor (b) the integrated rate method, based on concentration/time data, is easily applicable. 

The Use of Differential Rate Expressions According to this method, which was devised by van t Hoff, the rate of an nth-order reaction is given by... 

For the first three types it is possible to write down differential rate equations for reactants, products, and intermediates as follows. (For more complicated cases it becomes increasingly difficult— often even impossible—to obtain exact integrated solutions, and approximate methods have been developed. These methods fall outside of the scope of this book). 

The most common method for directly determining the form of the differential rate law for a reaction is the method of initial rates. The initial rate of a reaction is the instantaneous rate determined just after the reaction begins (just after t = 0). The idea is to determine the instantaneous rate before the initial concentrations of reactants have changed significantly. Several experiments are carried out using different initial concentrations, and the initial rate is determined for each run. The results are then compared to see how the initial rate depends on the initial concentrations. This procedure allows the form of the rate law to be determined. We will illustrate the method of initial rates by using the following reaction ... 

The most common method for experimentally determining the differential rate law is the method of initial rates. In this method several experiments are run at different initial concentrations, and the instantaneous rates are determined for each at the same value of t as close to t = 0 as possible. The point is to evaluate the rate before the concentrations change significantly from the initial values. From a comparison of the initial rates and the initial concentrations, the dependence of the rate on the concentrations of various reactants can be obtained—that is, the order in each reactant can be determined. 

Rate methods are much less suited for evaluation of results from batch, tubular, and differential recycle reactors because for these the rate must be obtained by a finite-difference approximation (see eqns 3.1, 3.5, and 3.8). In particular the method based on eqn 3.11 should not be used for such a purpose because of its high sensitivity to even minor experimental errors. 

---