Variable-time methods

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. 

Noncnzymc-Catalyzcd Reactions The variable-time method has also been used to determine the concentration of nonenzymatic catalysts. Because a trace amount of catalyst can substantially enhance a reaction s rate, a kinetic determination of a catalyst s concentration is capable of providing an excellent detection limit. One of the most commonly used reactions is the reduction of H2O2 by reducing agents, such as thiosulfate, iodide, and hydroquinone. These reactions are catalyzed by trace levels of selected metal ions. Eor example the reduction of H2O2 by U... 

The kinetics of the addition of aniline (PI1NH2) to ethyl propiolate (HC CCChEt) in DMSO as solvent has been studied by spectrophotometry at 399 nm using the variable time method. The initial rate method was employed to determine the order of the reaction with respect to the reactants, and a pseudo-first-order method was used to calculate the rate constant. The Arrhenius equation log k = 6.07 - (12.96/2.303RT) was obtained the activation parameters, Ea, AH, AG, and Aat 300 K were found to be 12.96, 13.55, 23.31 kcalmol-1 and -32.76 cal mol-1 K-1, respectively. The results revealed a first-order reaction with respect to both aniline and ethyl propiolate. In addition, combination of the experimental results and calculations using density functional theory (DFT) at the B3LYP/6-31G level, a mechanism for this reaction was proposed.181... 

Kinetics of the addition of PI13P to p-naphthoquinone in 1,2-dichloromethane, using the initial rate method, revealed the order of reaction with respect to the reactants the rate constant was obtained from pseudo-first-order kinetic studies. A variable time method using UV-visible spectrophotometry (at 400 nm) was employed to monitor this addition, for which the following Arrhenius equation was obtained log k = 9.14- (13.63/2.303RT). The resulting activation parameters a, AH, AG, and Aat 300 K were 13.63, 14.42 and 18.75 kcalmol-1 and —14.54 calmol 1K 1,... 

In the variable time method the concentration is measured as a function of / = o9-t and all other variables are kept constant. 

Fig. 8.2 Theoretical diagram for homogeneous, incremental centrifuge technique (variable time method).

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... 

With variable-time methods, standards should be chosen that have concentrations near those of the samples. Variable-time techniques are most suited to measurements where the signal is nonlinear with concentration and to the measurement of concentrations of catalysts such as enzymes, where A[I] represents a constant fraction of a reaction (the catalyst being regenerated). 

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 variable-time method, as employed for first- or pseudo-first-order reactions, also uses the integral form of the first-order rate equation (Eqn. 18.15). Solving for [A]o yields... 

In the variable-time method, A[P] is held constant and At is the measured parameter that is related to [A]o. If measurements are carried out during the first 1-2% of the overall reaction (initial-rate conditions), the exponential term, c a i, is approximately equal to unity and if the measurements are begun very near zero reaction-time. Equation 18.35 becomes... 

Figure 18.4. Variable-time method for uncatalyzed reactions, (a) Variation of as a function of time for various different initial concentrations of A. (b) Variation of If At required to reach a fixed value of AS/ as a function o/[/l]o.

Since fS]2, [S]i, and (thus) A[S], are held constant in the variable-time method, a linear relation between [E]q and 11 At always exists because the bracketed term is a constant. Again, if [S] Km. and A[S]/[S]i is kept small (initial-rate conditions). Equation 18.38 can be simplified to... 

The variable-time method, however, is not as well suited for determining substrate concentrations as the fixed-time method there will be a nonlinear relationship between [S]o and 1/Ar unless pseudo-zero-order conditions are used during the interval At and the measurements are begun very close to r = 0. 

For other types of catalyzed reactions, the variable-time method is well suited for determining catalyst concentrations. Initial-rate procedures and linearity of instrumental response are not necessary. 

Determination of a singie species Pseudo-first-order reactions Initial rate method Fixed-time method Variable-time method Second-order reactions Identical reactant concentrations Unequal reactant concentrations Multipoint methods Curve-fitting methods Predictive methods Error-compensated methods... 

The mathematical treatments used by the fixedtime and variable-time methods rely on the exponential version of eqn [1], namely... 

Figure 1 Kinetic profiles (left) and determination of a single species based on pseudo-first-order reactions. (A) Initial rate, (B) fixedtime, and (C) variable-time methods. Curves l-V in the kinetic profiles correspond to 1-5 x 10 mol I , respectively. Simulated data...

Vicinal glycols 104 1.4x10 -7.0x 10 Malaprade reaction. Use of variable-time method... 

Phenols Bromination reaction 10 moll Variable-time method... 

Differential variable-time method This method, also known as the fixed- or constant-concentration method , entails measuring the time required for a preset change in the reaction medium to take place. Solving eqn [4] for 1/At yields... 

Figure 3 Implementation of the variable-time method (Reproduced with permission from Perez-Bendito D and Silva M (1988) Kinetic Methods in Analytical Chemistry. Chichester Ellis Norwood.)...

The time step chosen is variable and changes throughout the simulation. It can be infinitely small or infinitely large and depends both on the random probabihty and the overall calculated rate. The variable-time method is mathematically an exact approach and there are no concerns about accuracy due to time step size. Systems which contain fast events have very small time steps and are thus dominated by the time scale scales of the fastest processes. Faster rates of reaction lead to higher probabilities that these steps are chosen. This leads to problems for systems with disparate rates since the simulation will spend nearly all of its time simulating the faster rates without ever simulating the slower processes. This is esp>ecially a problem for systems where diffusion is fast and reaction is slow and systems which contain fast processes which are nearly equihbrated together with slow processes. 

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