Integrated form of rate law

The rate laws with simple reaction order, such as first-order Equation 7.13 and second-order Equation 7.14, can be easily integrated to get the respective integrated forms. Equation 7.15 and Equation 7.16, of rate laws. 

In the special case in which the concentrations of A and B at t = 0 are identical (i.e., [Ag] = [Bg] = [Rg]), the rate law is given by Equation 7.17, in which [R] is the remaining concentration of A or B at any time t. The integrated rate law is given by Equation 7.18, which is also valid for second-order reaction involving a single reactant such as A and, in this case, the elementary reaction is represented as 2A P with second-order rate constant k. 

The integrated form of rate law shows the relationship between the concentration of either reactant or product as a function of rate constant and reaction time (t). It is generally easier and convenient to verify experimentally the integrated form of rate law (i.e.. Equation 7.15), than the rate law (i.e.. 

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In contrast to the differential method, integral methods take advantage of integrated forms of rate laws, such as those shown by Equations 29-6, 29-7, and 29-9. 

SECTION 14.4 Rate laws can be used to determine the concentrations of reactants or products at any time during a reaction. In a first-order reaction the rate is proportional to the concentration of a single reactant raised to the first power Rate = fc[A]. In such cases the integrated form ofthe rate law is In [A], = —kt + ln[A]o,where [A],isthe concentration of reactant A at time t, k is the rate constant, and [A] is the initial concentration of A. Thus, for a first-order reaction, a graph of In [A] versus time yields a straight line of slope —k. 

The integrated form of the rate law for equation 13.4, however, is still too complicated to be analytically useful. We can simplify the kinetics, however, by carefully adjusting the reaction conditions. For example, pseudo-first-order kinetics can be achieved by using a large excess of R (i.e. [R]o >> [A]o), such that its concentration remains essentially constant. Under these conditions... 

Several additional approaches for analyzing mixtures have been developed that do not require such a large difference in rate constants.Because both A and B react at the same time, the integrated form of the first-order rate law becomes... 

In this section we review the application of kinetics to several simple chemical reactions, focusing on how the integrated form of the rate law can be used to determine reaction orders. In addition, we consider how rate laws for more complex systems can be determined. 

Proceeding in the same manner as for a first-order reaction, the integrated form of the rate law is derived as follows... 

Demonstrating that a reaction obeys the rate law in equation A5.11 is complicated by the lack of a simple integrated form of the rate law. The kinetics can be simplified, however, by carrying out the analysis under conditions in which the concentrations of all species but one are so large that their concentrations are effectively constant during the reaction. For example, if the concentration of B is selected such that [B] [A], then equation A5.11 simplifies to... 

Equation (5.11) is the differential form of the rate law which describes the rate at which A groups are used up. To test a proposed rate law and to evaluate the rate constant it is preferable to work with the integrated form of the rate law. The integration of Eq. (5.11) yields different results, depending on whether the concentrations of A and B are the same or different ... 

Integr ation may lead to a relation for rate constant with temperature dependency in the form of Arrhenius law ... 

Before discussing such theories, it is appropriate to refer to features of the reaction rate coefficient, k. As pointed out in Sect. 3, this may be a compound term containing contributions from both nucleation and growth processes. Furthermore, alternative definitions may be possible, illustrated, for example, by reference to the power law a1/n = kt or a = k tn so that k = A exp(-E/RT) or k = n nAn exp(—nE/RT). Measured magnitudes of A and E will depend, therefore, on the form of rate expression used to find k. However, provided k values are expressed in the same units, the magnitude of the measured value of E is relatively insensitive to the particular rate expression used to determine those rate coefficients. In the integral forms of equations listed in Table 5, units are all (time) 1. Alternative definitions of the type... 

Rather than the use of instantaneous or initial rates, the more usual procedure in chemical kinetics is to measure one or more concentrations over the timed course of the reaction. It is the analysis of the concentrations themselves, and not the rates, that provides the customary treatments. The concentration-time data are fitted to an integrated form of the rate law. These methods are the subjects of Chapters 2, 3, and 4. 

After 575 min. ( 4 half lives, hence, we expect 6.25% remains), use integrated form of the rate law ... 

Figure 3.4 Linear integrated form of nth-order rate law ( rA) = k c for constant-volume BR (n 1)...

Figure 3.5 Linear integrated form of first-order rate law for constant-volume BR...

Equations 3.4-9, -10 or -11, and -13 are only three examples of integrated forms of the rate law for a constant-volume BR. These and other forms are used numerically in... 

The first part of equation 9.2-19 corresponds to the integrated form of Fick s law in equation 9.2-5, and the second part incorporates equation 9.2-7 equation 9.2-20 applies in similar fashion to species B. The rates NA and NB are related through stoichiometry ... 

Graphical Assessment Using Integrated Equations Directly. Another way to ascertain mechanistic rate laws is to use an integrated form of Eq. (2.7). One way to solve Eq. (2.7) is to conduct a laboratory study and assume that one species is in excess (i.e., B) and therefore, constant. Mass balance relations are also useful. For example [A] -I- [Y] = A0+ Y0 where Y() is the initial concentration of product. One must also specify an initial... 

A second kind of rate law, the integrated rate law, will also be important in our study of kinetics. The integrated rate law expresses how the concentrations depend on time. As we will see, a given differential rate law is always related to a certain type of integrated rate law, and vice versa. That is, if we determine the differential rate law for a given reaction, we automatically know the form of the integrated rate law for the reaction. This means that once we determine either type of rate law for a reaction, we also know the other one. 

Which rate law we choose to determine by experiment often depends on what types of data are easiest to collect. If we can conveniently measure how the rate changes as the concentrations are changed, we can readily determine the differential (rate/concentration) rate law. On the other hand, if it is more convenient to measure the concentration as a function of time, we can determine the form of the integrated (concentration/time) rate law. We will discuss how rate laws are actually determined in the next several sections. 

For 1-butene pyrolysis, the calculated first-order rate constants decreased significantly with increasing conversion at each temperature. Reduction of the data by using the integrated form of the second-order rate law provided specific rate constants that were satisfactorily independent of conversion. 

The integrated form of this second-order rate law is... 

In principle, equation (3.1) may be integrated for any combination of rate laws for nucleation and for growth to give a rate expression of the form g(a) = kt. In practice, this cannot be performed generally because there is no functional relationship between the nucleation and the growth terms. 

This integrated form of the rate law gives the concentration of A as a function of the initial concentration [A]q, the rate constant k, and the time t. A plot of this relationship is depicted in Figure 29-1. Example 29-1 illustrates the use of this equation in finding a reactant concentration at a particular time. 

Integral methods Kinetic methods based on an integrated form of the rate law. 

Table 2.2 is a summary of the differential and integrated forms of some simple rate laws. Also listed are expressions for the reaction half-times corresponding to several of the integrated rate expressions. These are the times at which the concentration of a reactant is half its initial value. 

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