Concentration Versus Time The Integrated Rate Equation

16-4 Concentration Versus Time The Integrated Rate Equation 

Often we want to know the concentration of a reactant that would remain after some specified time, or how long it would take for some amount of the reactants to be used up. 

The equation that relates concentration and time is the integrated rate equation. We can also use it to calculate the half-life, 1/2, of a reactant—the time it takes for half of that reactant to be converted into product. The integrated rate equation and the half-life are different for reactions of different order. 

We will look at relationships for some simple cases. If you know calculus, you may be interested in the derivation of the integrated rate equations. This development is presented in the Enrichments at the end of this section. 

This relates the half-life of a reactant in 2i first-order reaction and its rate constant, k. In such reactions, the half-life does not depend on the initial concentration of A. This is not true for reactions having overall orders other than first order. 

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CONCENTRATION VERSUS TIME THE INTEGRATED RATE EQUATION... 

The integrated rate equation can help us to analyze concentration-versus-time data to determine reaction order. A graphical approach is often used. We can rearrange the integrated first-order rate equation... 

The values of the rate constants are estimated by fitting equations 1.4a and 1.4b to the concentration versus time data. It should be noted that there are kinetic models that are more complex and integration of the rate equations can only be done numerically. We shall see such models in Chapter 6. An example is given next. Consider the gas phase reaction of NO with 02 (Bellman et al. 1967) ... 

Equation (8) is the differential rate expression for a first-order reaction. The value of the rate constant, k, could be calculated by determining the slope of the concentration versus time curve at any point and dividing by the concentration at that point. However, the slope of a curved line is difficult to measure accurately, and k can be determined much more easily using integrated rate expressions. 

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. 

Again plotting concentration versus time using these integrated second-order rate laws gives linear plots only if the reaction is a second-order process. The rate constants can be determined from the slopes. If the concentration-time plots are not linear, then the second-order rate equations do not correctly describe the kinetic behavior. There are integrated rate laws for many different reaction orders. 

It has become customary to classify evaluation methods as "differential" or "integral." These terms stem from a time when practically all experiments were conducted in batch reactors, so that rates had to be found by differentiation of concentration-versus-time data, and the calculation of concentrations from postulated rate equations required integration. The terms do not fit the work-up of data from gradientless reactors such as CSTRs, in which rates and concentrations are related to one another by algebraic equations requiring no calculus, and are therefore avoided here. 

Simultaneous integration of these two equations gives the explicit solution as a multiexponential equation, the exponents being expressed as a function of the distribution (a) and elimination rate constants (jS), and factoring in the volumes of the compartment (Vc). The following equation (eqn(3)) represents the concentration versus time for a drug which follows a two-compartment model ... 

Kinetic methods can be classified according to how the measurement is made. Differential methods compute the rate of reaction and relate it to the analyte concentration. Rates are determined from the slope of the absorbance versus time curve. Integral methods use an integrated form of the rate equation and determine the concentration of analyte from the absorbance changes that occur over various time intervals. Curve-fitting methods fit a mathematical model to the absorbance versus time curve and compute the parameters of the model, including the analyte concentration. The most sophisticated of these methods use the parameters of the model to estimate the value of the equilibrium or steady-slate response. These methods can provide error compensation because the equilibrium position is... 

A special application of the first-order integrated rate equation is in the determination of decimal reduction times, or D values, the time required for a one-logio reduction in the concentration of reacting species (i.e., a 90% reduction in the concentration of reactant). Decimal reduction times are determined from the slope of logio([A(]/[Ao]) versus time plots (Fig. 1.4). The modified integrated first-order integrated rate equation can be expressed as... 

There are two procedures for analyzing kinetic data, the integral and the differential methods. In the integral method of analysis we guess a particular form of rate equation and, after appropriate integration and mathematical manipulation, predict that the plot of a certain concentration function versus time... 

The dependent variable y is most frequently the reaction rate independent variables are the concentration or pressure of reaction components, temperature and time. If in some cases the so-called integral data (reactant concentrations or conversion versus time variable) arc to be treated, a differential kinetic equation obtained by the combination of a rate equation with the mass balance equation 1 or 3 for the given type of reactor is used. The differential equation is integrated numerically, and the values obtained arc compared with experimental data. 

The first term on the right-hand side of Eq. (7-50) is the molar feed rate of the components, which can be different for each component, hence the subscript i, and can vary with time. A typical concentration profile versus time for a reactant whose concentration is kept constant initially by controlling the feed rate is shown in Fig. 7 21). Knowledge of the reaction kinetics allows these ordinary differential equations to be integrated to obtain the reactor composition versus time. 

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