Step 1 Determine the Order of the Reaction

Transcription

1 Step 1 Determine the Order of the Reaction In this step, you fit the collected data to the various integrated rate expressions for zezro-, half-, first-, and second-order kinetics. (Since the reaction was performed using a large excess of water and HCl, only the rate dependence upon the benzenediazonium ion can be determined. Ideally, the reaction will have been monitored for five or more half-lives. In this experiment, significantly shorter reaction periods were monitored. Thus, it is crucial to examine the reaction profile for the dataset which contains the largest percent reaction, most likely the 40 C run, to determine the order. Once the order has been determined, the other datasets are analyzed simply to determine the observed rate constant. Below we try to fit some kinetics data, collected at 30 C, to both second and first order functions. We perform the fit first by specifying some poor seed values, to illustrate the importance of using good seed values. Sometimes, poor seed values will return an error during the fit analysis (such as, singular matrix. The best fit returned by the nonlinear least squares program is not necessarily a good fit - it simply means that the change in residual errors from two successive estimates for the coefficients of the tested function was less than some user-specified value. You need to examine the returned values to decide if the best fit is indeed a good fit. It is also important to realize that a poor fit does not always mean that the tested function does not hold for the data set. In many cases, it simply means that the seed values given for the analysis were poor, and the fitting routine proceeded down the wrong path toward finding good coefficients. It is necessary to give reasonably good seed values, for the fitting program to find the best values. These examples should help you appreciate that (1) the fitting routines require good seed values to give meaningful results; (2) a poor fit does not necessarily mean that the fitting function is incorrect; (3) some coefficients should be held constant (not allowed to be fitted); and (4) the returned values must be compared with any known physical measurements to confirm their validity. Chem Integrated Lab Kinetics help - 1 R. Trautman

2 In the fit shown to the right, we ve specified very poor seed values for the coefficients. In this particular case, the routine returned a best fit, but casual examination reveals that the coefficients do not fit the data at all. The various statistics values confirms that this is a horrible fit. However, this very bad fit does not necessarily mean that the tested function does not describe the data dependency, as the next fit will show. It is very important that you start the fitting routine on a good foot, by specifying appropriate seed values. NOTE: If you have difficulty reading the numbers within the Excel notebook figures, increase the page magnification within Acrobat Reader to 150%, 200%, etc., until the numbers are clear. Chem Integrated Lab Kinetics help - 2 R. Trautman

3 The fit results shown to the right are the result of fitting the same data to the same function as above, but this time we used decent seed values, obtained as described in the lab manual (but please read the important discussion about seed values and fitting vs holding a value below). This results in a fairly good fit. Note that the variance is 1.3 X 10-4, and that r-squared is Chem Integrated Lab Kinetics help - 3 R. Trautman

4 The fit shown to the right is the result of repeating the fitting routine several times, using the estimated coefficients from one fitting as the seed values for the subsequent fit. Note that the fit statistics reveal a better fitting than that shown above: a lower residuals erro and a better r-squared. Also compare the coefficient estimates for these two fits. The values for c1 and c2 are very different; but the values for c3 from the two fits are almost within the reported standard deviations: ± and ± The value for c2 obtained in this fit, 0.56, is unexpected, since we expect this to be equal to 1/A i. We should perform this fit again, and hold this coefficient during the fitting routine. Note also that the residuals are not randomly distributed, but rather show some systematic trend. This is often an indication that the fitting function is incorrect, even though the statistical results might indicate a good fit. Chem Integrated Lab Kinetics help - 4 R. Trautman

5 Now we analyze the same data, this time testing for first order kinetics. We use decent seed values, estimated as described in the lab manual, and have the program fit all coefficients. Note that c1 is estimated to be ± This coefficient represents the absorbance at infinite time, which should not be negative. (In some cases, estimated values for c1 will deviate even more from physically meaningful values.) Since we know what the final absorbance is (because we measured it!), this value should not be fitted, but should be held constant. That is, the absorbance at time infinity should be measured, and entered as the value for c1, and this value held constant. Chem Integrated Lab Kinetics help - 5 R. Trautman

6 Here we tell the fitting routine that it should hold c1 constant, and use the values retruned in the last fit as our seed values for c2 and c3. The output indicates that c1 was held constant because the standard deviation value is zero. These examples should help you appreciate that (1) the fitting routines require good seed values to give meaningful results; (2) a poor fit does not necessarily mean that the fitting function is incorrect; (3) some coefficients should be held constant (not allowed to be fitted); and (4) the returned values must be compared with any known physical measurements to confirm their validity. Chem Integrated Lab Kinetics help - 6 R. Trautman

7 Our final note on this example of analyzing kinetics data concerns the identification of the reaction order. Both of the functions fitted in this example give reasonably good fits, once proper seed values are supplied. So, how does one distinguish between the functions, and thus identify the order of the reaction? The answer is, go back into the lab and collect data such that a valid identification can be performed! Kinetics data should ideally be collected for at least four halflives; in the present example, less than one half-life of the reaction was monitored. With such poor experimental design, the order of the reaction cannot be unambiguously identified. (This is related to the observation, a circle is composed of many small straight lines. If reaction kinetics are collected for only a short period of the reaction, it is difficult to impossible to determine the reaction order.) Your experiment was not really designed so poorly: the kinetics data collected at ca. 40 C monitors significantly more than one half-life, from which a clear identification of order should be possible. Once the order is determined at that temperature, data collected at other temperatures can simply be processed using the identified order to determine the rate constants. The second part of your analysis, construction of an Arrhenius plot and determination of the activation energy, also serves as a check on the reaction order determination. If the plot of ln(k) versus 1/T is not linear, it is most likely that the rate constants were determined incorrectly (as they would be, if the wrong order was used for the analysis). It is also possible that the mechanism of the reaction changes over the temperature range investigated, but this is a much less common occurance. Chem Integrated Lab Kinetics help - 7 R. Trautman