Integral rate methods, fixed-time

Sensitivity The sensitivity for a one-point fixed-time integral method of analysis is improved by making measurements under conditions in which the concentration of the monitored species is larger rather than smaller. When the analyte s concentration, or the concentration of any other reactant, is monitored, measurements are best made early in the reaction before its concentration has substantially decreased. On the other hand, when a product is used to monitor the reaction, measurements are more appropriately made at longer times. For a two-point fixed-time integral method, sensitivity is improved by increasing the difference between times t and f2. As discussed earlier, the sensitivity of a rate method improves when using the initial rate. 

FIGURE 21-2 Fixed-time (teft) and variable-time (right) integral methods of measurement of reaction rates. 

Equation 18.12 is the basis for the derivative approach to rate-based analysis, which involves directly measuring the reaction rate at a specific time or times and relating this to [A]fl. Equation 18.11 is the basis for the two different integral approaches to kinetic analysis. In one case, the amount of A reacted during a fixed time is measured and is directly proportional to [A]o ( fixed-time method) in the other case, the time required for a fixed amount of A to react is measured and is also proportional to [A]o variable-time method). Details of these methods will be discussed in Section... 

Two types of techniques are employed for analyzing a single-component system. The most straightforward is the derivative or slope method in which one obtains the derivative of the electrical signal by electronically differentiating the signal from the transducer. The second approach uses the integral forms of the rate equations, and one of two possible types of measurement the fixed-time or constant-time method... 

The variable-time method, like the fixed-time method, is an integral method which, for short measurement times and small changes in concentration, also gives results approaching the instantaneous reaction-rate. 

One of the most common catalytic reactors is the fixed-bed type, in which the reaction mixture flows continuously through a tube filled with a stationary bed of catalyst pellets. Because of its importance, and because considerable information is available on its performance, most attention will be given to this reactor type. Fluidized-bed and slurry reactors are also considered later in the chapter. Some of the design methods given are applicable also to fluid-solid noncatalytic reactions. The global rate and integrated conversion-time relationships for noncatalytic gas-solid reactions will be considered in Chap. 14. 

The differential and the integral method are compared in Figure 4.11.4 for a fixed bed reactor where, usually, the modified residence time (ratio of catalyst mass to total feed rate) is used. 

Isothermal Method 2. This method is necessary to obtain cure data at low temperatures where the rate of heat evolution is too small for method 1 to be reliable. It is also recommended for nth-order reactions where the maximum rate of cure occurs at f = 0, and to obtain simultaneous Tg and conversion data to construct Jg-conversion plots. The conversion-time data can be fit to integrated forms of the rate equation, such as Eqs. (2.83)-(2.85). Several samples are cured isothermally, for example, in an oven, in the calorimeter or at ambient temperature, for various times until no additional curing can be detected. The samples are subsequently scanned in the DSC at a fixed heating rate, from which Tg and the residual heat of cure (A//res) the heat evolved during completion of the reaction, are measured, as illustrated in Fig. 2.69. 

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