Methods of integration

The second method, the differential method, employs the rate equation in its differential, unintegrated, form. Values of dcjdt are obtained from a plot of c against by taking slopes, and these are directly compared with the rate equation. The main difficulty with this method is that slopes cannot always be obtained very accurately, but in spite of this drawback the method is on the whole the more reliable one and unlike the integration method it does not lead to any particular difficulties when there are complexities in the kinetic behavior. 

These two methods will now be considered in further detail. 

Suppose that at the beginning of the reaction (f 0) the concentration of A is that of X is zero. If after time t the concentration of X is x, that of A is — x. The rate of formation of X is dx/dt, so that for a first-order reaction 

One procedure for testing the applicability of equations (9.17) to (9.19) to a chemical reaction would be to make determinations of x at various times during the course of reaction. At each time a value of k could then be calculated using (9.17), and if k were truly constant during the course of the reaction the conclusion would be that the reaction was of the first order. If the values of k were found to drift, the reaction is not of the first order, and other equations have to be tried. 

Alternatively, the first-order equations can be tested and the constant evaluated using a graphical procedure. It follows from (9.17) that a plot of 

Reaction in A Differential rate equation Integrated rate equation 

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The UK government enquiry into the Piper Alpha disaster in the North Sea in 1988 has had a significant impact on working practices and equipment and has helped to improve offshore safety around the world. One result has been the development of a Safety Management System (SMS) which is a method of integrating work practices, and is a form of quality management system. Major oil companies have each developed their own specific SMS, to suit local environments and modes of operation, but the SMS typically addresses the following areas (recommended by the Cullen Enquiry into the Piper Alpha disaster) ... 

Methods of Integration All the methods of integration available for the indefinite integral can be used for definite integrals. In addition, several others are available for the latter integrals and are indicated below. 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

Thus the Laplace transformation constitutes a method of integration, and a table of Laplace transforms plays a role in this process that is analogous to a table of... 

Some methods of integration of definite integrals are covered in Numerical Methods. ... 

An alternative method of integrating the Gibbs-Duhem equation was developed by Darken and Gurry [10]. In order to calculate the integral more accurately, a new function, a, defined as... 

Fluorescence quantum yields are usually determined by integration of the fluorescence spectrum (and subsequent normalization using a standard of known fluorescence quantum yield in order to get rid of the instrumental factor k appearing in Eqs 3.17 or 3.18 see Chapter 6). In practice, attention should be paid to the method of integration. 

Here ti/2 is the half-life and C-ao the initial concentration of reactant. Now, half-life data (versus CAo) may be analyzed by the same techniques used in the method of integration. 

The method of integration recommended depends on the situation. In this problem probably the numerical method is the quickest and simplest and gives a good enough answer for most purposes. 

These factors have led researchers to examine novel methods for tire degradation of these contaminants, including liquid-phase photocatalytic oxidation. The treatment of wastewaters contaminated with agents like chlorinated aromatics has also prompted research into the optimum methods of integrating chemical oxidation techniques with existing water treatment methods, particularly biological treatment and ozonation techniques [37],... 

Numerical integration of the radial Schrodinger equation is generally necessary if realistic interaction potentials are selected. For this purpose, efficient computer codes exist based, for example, on Cooley s method of integration. Other widely available codes written to compute scattering... 

The method of integration of this kind of expression is given in Jeans, Dynamical Theory of Gases, Appendix A. The result is... 

The method of integration will be found in any text-book of the Integral Calculus which deals with Gamma Functions. The value of the expression under the integral sign... 

This is a first-order linear differential equation and can be solved using the method of integrating factors that we show below. Multiplying both sides by e1 2, we have... 

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... 

This no-chaos conclusion can be validated further by trying different methods of integration with different complexities in MATLAB. Each integrator ode.. of MATLAB shows that a different tolerance bound is needed in order to obtain the correct unchaotic results without numerical instability. Moreover, when each method is executed with the appropriately tight tolerance, then all methods give the same correct and unchaotic result. 

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. 

Jaenchen, R., Malsy,). Analysis of water Penetration Through Hydrophobic Filters as a Method of Integrity Testing. Pall Corporation, Dreieich, Nov. 1992... 

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