Finite_Diff_v7

Finite_Diff_v7 - Determination of the Reaction Order:...

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Determination of the Reaction Order: Differential Method via Finite Differences Dr. Kalju Kahn UC Santa Barbara, 2008-2010 When a concentration-time profile for a reaction with unknown order is available, one of the two approaches integral methor or differential method—are commonly taken to elucidate the reaction order. In the integral method, an initial hypothesis is made concerning the reaction order, and the corresponding integrated rate equation is fitted to the data. The reaction order is determined by finding which integrated equation provides the best fit (both statistically and in terms of physically reasonable parameters) This approach works well if the researcher can a priori choose a small number of plausible kinetic mechanisms based on theoretical consid- eration about the reaction. In the differential method, reaction rate is calculated at several times through the course of the reaction, and the power coefficient is determined directly by fitting rate vs. concentration data. The advantage of the differential approach is that it is applicable to complex reactions with rate equations that are difficult to integrate. The differential method requires rate values, i.e. derivatives of concentration with respect to time. One can estimate slopes grphically as tangent lines of the progress curve but this method tends to have low accuracy. A numeric method that gives derivatives based on discrete data points is known as the finite difference method. The familiar rise-over-run way of calculating the slope represents the simplest finite difference method. For significantly curved progress curves a more accurate slope at point (x,y) can be obtained via the following central finite difference formula: dy dx = y - 2 h - 8 y - h + 8 y + h - y + 2 h 12 h where h is the spacing between x-values in the data. This central difference formula can be used if there are at least two points before and two points after the point for which the slope is being calculated. Note that this formula assumes that the data points are equally spaced; thius is usually true in kinetic studies. Finite differ- ence methods can be also used to numerically calculate second and higher order derivatives of functions. Repeated use of finite difference formula may seem tedious but computers are quite good at repetitive tedious
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This note was uploaded on 01/09/2011 for the course CHEM 111 taught by Professor Kahn during the Fall '08 term at UCSB.

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Finite_Diff_v7 - Determination of the Reaction Order:...

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