Variable-time integral method measurement

An alternative to a fixed-time method is a variable-time method, in which we measure the time required for a reaction to proceed by a fixed amount. In this case the analyte s initial concentration is determined by the elapsed time, Af, with a higher concentration of analyte producing a smaller Af. For this reason variabletime integral methods are appropriate when the relationship between the detector s response and the concentration of analyte is not linear or is unknown. In the one-point variable-time integral method, the time needed to cause a desired change in concentration is measured from the start of the reaction. With the two-point variable-time integral method, the time required to effect a change in concentration is measured. 

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

Integral methods Variable time In the variable-time method of measurement of the initial slope, the concentration of the indicator substance I is measured twice, and the time interval At required to bring about a preselected change in concentration A[I] is the important quantity (Figure 21-2, right). Since the change in concentration is a fixed preselected value, it can be incorporated with the constant in Equation (21-5) to give... 

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... 

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. 

The quasiclassical trajectory method was used to study this system, and the variable step size modified Bulirsch-Stoer algorithm was specially developed for recombination problems such as this one. Comparisons were made with the fourth order Adams-Bashforth-Moulton predictor-corrector algorithm, and the modified Bulirsch-Stoer method was always more efficient, with the relative efficiency of the Bulirsch-Stoer method increasing as the desired accuracy increased. We measure the accuracy by computing the rms relative difference between the initial coordinates and momenta and their back-integrated values. For example, for a rms relative difference of 0.01, the ratio of the CPU times for the two methods was 1.6, for a rms relative difference of 0.001 it was 2.0, and for a rms relative difference of 10 it was 3.3. Another advantage of the variable step size method is that the errors in individual trajectories are more similar, e.g. a test run of ten trajectories yielded rms errors which differed by a factor of 53 when using the modified Bulirsch-Stoer... 

The Ziegler and Nichols closed-loop method requires forcing the loop to cycle uniformly under proportional control. The natural period of the cycle—the proportional controller contributes no phase shift to alter it—is used to set the optimum integral and derivative time constants. The optimum proportional band is set relative to the undamped proportional band P , which produced the uniform oscillation. Table 8-4 lists the tuning rules for a lag-dominant process. A uniform cycle can also be forced using on/off control to cycle the manipulated variable between two limits. The period of the cycle will be close to if the cycle is symmetrical the peak-to-peak amphtude of the controlled variable divided by the difference between the output limits A, is a measure of process gain at that period and is therefore related to for the proportional cycle ... 

The integrated DLS device provides an example of a measurement tool tailored to nano-scale structure determination in fluids, e.g., polymers induced to form specific assemblies in selective solvents. There is, however, a critical need to understand the behavior of polymers and other interfacial modifiers at the interface of immiscible fluids, such as surfactants in oil-water mixtures. Typical measurement methods used to determine the interfacial tension in such mixtures tend to be time-consuming and had been described as a major barrier to systematic surveys of variable space in libraries of interfacial modifiers. Critical information relating to the behavior of such mixtures, for example, in the effective removal of soil from clothing, would be available simply by measuring interfacial tension (ILT ) for immiscible solutions with different droplet sizes, a variable not accessible by drop-volume or pendant drop techniques [107]. 

Variables Affecting Measurement Flow measurement methods may sense local fluid velocity, volumetric flow rate, total or cumulative volumetric flow (the integral of volumetric flow rate with respect to elapsed time), mass flow rate, and total mass flow. 

Batch reactors are used primarily to determine rate law parameters for homo, geneous reactions. This determination Ls usually achieved by measuring coa centration as a function of time and then using either the differential, integral, or least squares method of data analysis to determine the reaction order, a, and specific reaction rate, k. If some reaction parameter other than concentration i s monitored, such as pressure, the mole bMance must be rewritten in terms of the measured variable (e.g., pressure). 

The ICP-OES-FIA technique allows a rapid and routine method of analysis for both major and trace levels of metals in aqueous and non-aqueous solutions in most samples provided that the sample is in solution form. The flow injection method can be used to correct for baseline drift that may originate from uncontrollable thermal and electronic noise during analysis. However, these errors can be corrected if the peak obtained is measured over at least three points, i.e. immediately before the peak, at the peak and immediately after the peak and the height or area is integrated over these points. The elaborate time consuming correction procedures required for batch operations are not required for FIA methods and the baseline is defined by the emission obtained from the carrier liquid and is reproduced between each sample injection. A typical FIA analysis of signals for standards and samples is shown in Figure 7.11 for triplicate injections of variable concentrations of boron. 

Rubinow defined a normalized cell maturation variable, p. such that a cell will divide when / = 1. Now /x is related to the biochemical events occurring during the cell cycle in some undetermined fashion. It is, therefore, a semiempirical variable and is operationally defined in terms of measured cell cycle times. This implies that all cells divide after the cell cycle time of t hours. It is observed, however, that cell division times are scattered about a mean value. This randomness must be accounted for suitably. Most models account for this with an explicit operation in the mathematical solution by averaging cell division times over the entire population. This scheme leads to the solution of a rather difficult integral equation (see, e.g., Trucco (4)). Recently Subramarian et al. (8) have considered weighted-residual methods for more easily solving these problems. 

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