Library of Chemical Kinetic Models for Scientist®
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1. T A PI P2 P3 0 0 3 1 1 4 0 3 2 0 2 46 1 16 1 51 0 57 4 0 2 01 1 3 1 6 0 79 6 0 1 65 1 41 1 67 0 98 8 0 1 35 1 5 1 73 1 13 10 0 1 1 1 57 1 78 1 25 12 0 0 9 1 63 1 82 1 35 14 0 0 74 1 68 1 85 1 43 16 0 0 61 1 72 1 88 1 5 18 0 0 5 1 75 1 9 1 55 20 0 0 41 1 78 1 92 1 6 We will begin our curve fitting from the parameter values that were used to construct the data set We omit the use of the simplex search because we only wish to demonstrate the method by which results may be obtained rather than trying to confirm Model 14 Parallel First Order Irreversible Reactions Page 71 of 75 these results The initial parameter values that we will use are Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization P10 1 0 INF Y N P20 1 4 0 INF Y N P30 0 3 0 INF Y N AO 3 0 INF Y N KI 0 03 0 INF N N K2 0 02 0 INF N N K3 0 05 0 INF N N Given these values we fix A0 P10 P20 and P30 since these values should remain constant and perform a least squares fit for K1 K2 and K3 The result of this fit are as follows K1 0 030026 K2 0 019959 K3 0 050007 We also find that the current sum of squared deviation for this fit is 0 00026357 which is not too bad considering the size of the errors in the data set We now check the statistical output of Scientist to determine just how well the simulated curve fits the data set The statistics are shown below Data Set Name M2. T 1 3 0 2 0 99 0 51 0 8 0 7 12 0 67 0 83 16 0 58 0 92 20 0 51 0 99 24 0 45 1 05 28 0 41 1 09 32 0 37 1 13 36 0 34 1 16 40 0 32 1 18 The initial parameter values used to generate the data set are also the values that will be used to begin the least squares curve fitting We do this only for demonstration A simplex search is recommended for other applications of this model The initial parameter values are as follows Model 4 Second Order Irreversible Reaction Page 23 of 75 Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization AO 1 3 0 INF Y N PO 0 2 0 INF Y N K2 0 03 0 INF N N The least squares curve fitting is performed by selecting only K2 for fitting and then starting the calculation The value that Scientist finds as the best fit solution is K2 0 029988 The current sum of squared deviations for this fit is 9 1983E 5 which indicates that the simulated points match the data points very well To see just how well they match we need to look at the summary of statistics which is shown below Data Set Name Model 4 Weighted Unweighted Sum of squared observations 14 692 14 692 Sum of squared deviations 9 1983E 005 9 1983E 005 Standard deviation of data 0 0020929 0 0020929 R squared 0 99999 0 99999 Coefficient of determination 0 99996 0 99996 Correlation 0 99998 0 99998 Model Selection Criterion 10 043 10 043 Confidence Intervals Parameter Name K2 Estim
3. Figure 1 1 Model 1 Zero Order Irreversible Reaction We conclude from the above calculations that we have found a good value for the reaction rate with confidence limits that are quite close to it We also see that the calculated curve fits the data set quite well Given the simplicity of the model and simulated accuracy of the data this result is about what we would expect Model 1 Zero Order Irreversible Reaction Page 13 of 75 Model 2 First Order Irreversible Reaction There are several possible uses for this model First and most importantly it can be used to find the reaction rate K1 given the initial concentration of A AO the initial concentration of P PO and a number of measurements of the concentration of the reagent A and the product P over some time interval Second it can be employed to simulate the concentration of P given the initial concentration of P PO the initial concentration of A A0 and a number of measurements of A over a period of time Third it can be used to simulate the concentration of A given the initial concentration of P PO the initial concentration of A A0 and a number of measurements of P over a period of time For Model 1 we produced output similar to the first case so for this model we will simulate the concentration of the product P The form of the model used to do this iS Page 14 of 75 Model 2 First Order Irreversible Reaction Model 2 First Order Irreversible Re
4. k Observed Figure 9 1 Model 9 pH Rate Profile Diprotic Acid Model 9 pH Rate Profile Diprotic Acid Page 51 of 75 Model 10 Arrhenius Equation Linearized Form The Arrhenius Equation as shown below allows the activation energy to be found from the temperature dependence of the reaction rate It is possible with the Scientist model constructed from this equation to find the parameters A and Ea which determine the reaction rate Ea is given in units of calories mole Ea k Axe RD With this model the best fit values of the parameters A and EA can be found given a number of measurements of the reaction rate and the inverse of the temperature measured in degrees Kelvin The last condition is necessary to obtain linear graphics To obtain nonlinear graphics use Model 11 This model could also be used to simulate the reaction rate given values of the parameters A and EA Since the determination of A and EA will be the most common use for this model this example will deal with the method used to obtain values for these parameters The model form of this equation is shown below Model 10 Arrehnius Equation Linearized Form IndVars TINV DepVars K Params A EA K A EXP EA TINV 1 987 As with any least squares fitting this example requires a set of data points The set used here was obtained by performing a simulation with some initial parameter values and the rounding the resulting data to p
5. AVA y SCIENTIFIC SOFTWARE Library of Chemical Kinetic Models for Scientist Scientist Chemical Kinetic Library rev A14E Copyright 1989 1990 1994 2007 Micromath Research All rights reserved Other brand and product names are trademarks or registered trademarks of their respective holders No part of this Handbook may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic or mechanical including photocopying recording or otherwise without the prior written permission of the publisher Page 2 of 75 Micromath Research 1710 S Brentwood Blvd Saint Louis Missouri 63144 Phone Fax 1 800 942 6284 www micromath com Page 3 of 75 MICROMATH SOFTWARE LICENSE AGREEMENT Micromath Research hereby grants the purchaser a nonexclusive license for use of the Scientist Chemical Kinetic Library This license agreement allows the purchaser to make copies of the data disk for archival purposes but the data may not be used concurrently on more than one computer Site licenses are available for concurrent use Customer support is only available to the original purchaser The Scientist program and Scientist Chemical Kinetic Library are protected by U S Copyright Law and International Treaty provisions By using the enclosed diskette the purchaser agrees to abide by the terms of this license agreement and acknowledges that the Micromath logo Scientist product name and informati
6. 0 9021 is probably not significant Skewness 4 5241E 013 is probably not significant Kurtosis 0 72217 is probably not significant Weighting Factor 0 Heteroscedacticity 4 7889E 015 Optimal Weighting Factor 4 885E 015 The above output suggests that we did not obtain as good a fit as we would like The Model Selection Criterion is less than nine which is good but not overly so We also see that the confidence limits for the parameters vary by around 1 which is about what Page 34 of 75 Model 6 First Order Reversible Reaction must be expected given that the errors in the data set can be as much as 0 5 and we are trying to fit two parameters to this slightly inaccurate data We therefore conclude that this model produces quite reasonable output and that the numbers that we obtained for the forward and reverse reaction rates are fairly well determined A plot of the calculated curve and the data set are shown in Figure 6 1 below Model 6 Chart m AvsT m PvsT A Calc vs T P Calc vs T 0 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Time Figure 6 1 Model 6 First Order Reversible Reaction Model 6 First Order Reversible Reaction Page 35 of 75 Model 7 pH Rate Profile Nonelectrolyte The equation that describes the pH rate profile for a nonelectrolyte is as follows Kobs Ki H k k OH where OH K H Ky is the ion product for water 1 0E 14 at 25 degrees Centi
7. 12 Eyring Equation Linearized Form IndVars TINV DepVars KDIVT Params S H KDIVT 1 3805E 16 EXP S 1 987 EXP H TINV 1 987 6 6255E 27 The data set to be used for this demonstration was generated by performing a simulation with set values of the parameters and rounding the resulting figures to three decimal places This produces small errors in each data point which approximate experimental measurements This data set is Page 60 of 75 Model 12 Eyring Equation Linearized Form TINV KDIVT 0 0027 43300 0 0 0028 26200 0 0 0029 15800 0 0 0030 9560 0 0 0031 5780 0 0 0032 3490 0 0 0033 2110 0 0 0034 1280 0 0 0035 772 0 0 0036 467 0 The initial parameter values to be used for curve fitting will be the values used to generate the data set These values are as follows Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization S 1 0 0 INF N N H 10000 0 INF N N The least squares fitting will be performed directly without being preceded by a simplex search since the data was generated from the initial parameter values For this fitting we will use a weighting factor of 2 0 since we have rounded numbers which vary over a large range to three significant digits The effect of this rounding is to produce errors which are roughly proportional to the inverse of the square of the magnitude of the value and thus the weighting factor of 2 0 We perform the least squ
8. 16001 Standard Deviation 3 6611 95 Range Univariate 15992 95 Range Support Plane 15990 Variance Covariance Matrix 0 00012821 0 041293 13 404 Correlation Matrix 1 0 9961 1 Model 13 Eyring Equation Nonlinear Form Weighted 10 2 0239E 005 0 0015906 1 1 1 13 985 1 2269 1 2346 16009 16011 Unweighted 2 6698E007 76 858 3 0996 1 1 1 11 999 Page 67 of 75 Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 0 67763 is probably not significant Skewness 2 851 indicates the likelihood of a few large positive residuals having an unduly large effect on the fit Kurtosis 2 0105 is probably not significant Weighting Factor 2 Heteroscedacticity 5 1927 Optimal Weighting Factor 7 1927 It is noteworthy that the results for the weighted case are much better than those for the unweighted case and that they are more meaningful in that all but the last few points of the data set are essentially ignored for the unweighted case since the errors were assumed to be equal This assumption is not true and therefore the weighting factor of 2 0 produces more significant results The fit for this case is very good The Model Selection Criterion is almost fourteen which is excellent and the confidence limits are good We find t
9. 98 32 Variance Covariance 0 0024214 0 00014747 9 1839E 005 4 1058 4 1129 0 1463 0 17601 98 834 98 98 8 3624E 006 5 2079E 006 1 7604E 005 2 2182E 005 1 3814E 005 5 1792E 005 0 00030489 5 7916E 005 3 6068E 005 0 00014011 0 001037 0 0073628 Correlation Matrix 1 0 31272 1 0 040504 0 12952 1 0 025817 0 082556 0 70695 1 0 013717 0 043863 0 38918 0 69215 1 Page 50 of 75 Model 9 pH Rate Profile Diprotic Acid Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 1 1861 is probably not significant Skewness 1 037 indicates the likelihood of a few large positive residuals having an unduly large effect on the fit Kurtosis 0 24815 is probably not significant Weighting Factor 2 Heteroscedacticity 0 72776 Optimal Weighting Factor 2 7278 The Model Selection Criterion indicates that we obtained a good fit of the simulated curve to the data set However the confidence limits were not as good as might be desired especially for K4 An MSC of 13 or more is very good but the confidence limits for the parameters were not very well determined We feel however that the fit is good enough for this example so we plot the results This plot is shown in Figure 9 1 below Model 9 Chart m KOBS vs PH KOBS Calc vs PH
10. KR AO PO KF AO KR PO EXP KF KR T KF KR P KF AO PO KF AO KR PO EXP KF KR T KF KR In order to perform a curve fitting we need some measurements of A and P over a time interval Instead of experimentally determining these values we will do a simulation of the model with some initial parameter values and round the data to two places after the decimal This is reasonable for demonstration purposes since it will produce small errors Experimental data might not be so consistently close to the actual answer but it should not be too different from this data set The data points generated by this method are as follows T A P 0 1 6 0 4 4 1 37 0 63 8 1 2 0 8 12 1 08 0 92 16 0 99 1 01 20 0 92 1 08 24 0 87 1 13 28 0 84 1 16 32 0 82 1 18 36 0 8 1 20 40 0 78 1 22 The values of the parameters that were used to obtain this set of data should be good enough starting points for a least squares curve fitting This is true only for this demonstration because a simplex search is a good means of being assured that the answer that is found is the best answer in the local region of parameter space The initial values are Page 32 of 75 Model 6 First Order Reversible Reaction Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization AO 1 6 0 INF Y N PO 0 4 0 INF Y N KF 0 05 0 INF N N KR 0 03 0 INF N N The least squares fitting is done with KF and KR se
11. Sum of squared deviations 1 5979E 005 6 9853E 018 Standard deviation of data 0 0014133 9 3443E 010 R squared 1 1 Coefficient of determination 1 1 Correlation 1 1 Model Selection Criterion 13 794 12 448 Confidence Intervals Parameter Name A Estimated Value 21 974 Standard Deviation 0 11093 95 Range Univariate 21 718 22 23 95 Range Support Plane 21 643 22 305 Parameter Name EA Estimated Value 11999 Standard Deviation 3 2321 95 Range Univariate 11992 12007 95 Range Support Plane 11990 12009 Variance Covariance Matrix 0 012306 0 35713 10 447 Correlation Matrix 1 0 99607 1 Page 58 of 75 Model 11 Arrhenius Equation Nonlinear Form Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 0 80002 is probably not significant Skewness 0 66904 is probably not significant Kurtosis 0 58966 is probably not significant Weighting Factor 2 Heteroscedacticity 1 3922E 008 Optimal Weighting Factor 2 We find that the fit for the weighted case is better than that for the unweighted case Although the Model Selection Criterion is greater than twelve for the unweighted fit the MSC for the weighted fit is almost fourteen which is excellent The confidence limits for these parameters are also good but they could have been better Since
12. 1 56 0 64 4 0 8 1 3 0 9 6 0 63 1 13 1 07 8 0 51 1 01 1 19 10 0 42 0 92 1 28 12 0 35 0 85 1 35 14 0 3 0 8 1 4 16 0 26 0 76 1 44 18 0 22 0 72 1 48 20 0 19 0 69 1 5 The above data set was generated using some initial parameter values Since we are not trying to prove that the answer obtained from a least squares curve fitting is the best that can be found we will skip the simplex search which would normally be done at this time Instead we will start the curve fitting from the following initial parameter values Model 5 Second Order Irreversible Reaction Page 27 of 75 Parameters Name AO BO PO K2 Value Lower Limit Upper Limit Fixed Linear Factorization 1 5 0 INF Y N 2 0 INF Y N 0 2 0 INF Y N 0 1 0 INF N N For this fitting we select only K2 to be varied The values of the other parameters should not change since they are physically measured constants rather than data we are trying to fit The result of the least squares fitting is K2 0 099043 The sum of squared deviations at this point is 0 00014571 which is reasonably small but not overly much so We now check to see how good the fit was by examining the statistical output which is shown below Data Set Name Model 5 Weighted Sum of squared observations 35 169 Sum of squared deviations 0 00014571 Standard deviation of data 0 0021338 R squared 1 Coefficient of determination 0 99998 Correlation 0 99999 Model Selection Criterion 10 735 Confidence I
13. 8 pH Rate Profile Monoprotic Acid IndVars PH DepVars KOBS Params K1 K2 K3 K4 KA KW H 10 PH FHA H H KA FA KA H KA KOBS K1 H FHA K2 FHA K3 FA K4 KW FA H Model 8 pH Rate Profile Monoprotic Acid Page 41 of 75 In order to perform the least squares curve fitting to determine the rate constants k1 k2 k3 and k4 we need to have a set of measurements of kobs over a range of pH The data set which is obtained by performing a simulation with set values of the parameters is shown below PH KOBS 0 0 6 49 0 5 219 1 0 0 825 1 5 0 394 20 0 258 3 0 0 201 4 0 0 196 5 0 0 195 6 0 0 197 7 0 0 217 8 0 0 415 9 0 2 21 10 0 11 3 11 0 20 4 12 0 22 5 12 5 23 4 13 0 25 6 13 5 32 7 14 0 55 1 Because the data set was generated from given parameter values we will use these figures to begin the least squares fitting The simplex search is omitted because it will not make much difference in finding better starting values The parameters used to generate the data set are Page 42 of 75 Model 8 pH Rate Profile Monoprotic Acid Parameters Name Kl K2 K3 K4 KA KW Value Lower Limit 6 3 0 0 195 0 224 0 32 7 0 1E 010 0 1E 014 0 Upper Limit Fixed Linear Factorization INF INF INF INF INF INF KKZZZZ ZZZZZZ The curve fitting will be performed with a weighting factor of 2 0 since the data is rounded to t
14. 811 95 Range Support Plane 32 485 32 89 Variance Covariance Matrix 1 9795E 005 8 0418E 008 8 7488E 009 7 2514E 007 7 8892E 008 0 00011618 1 4106E 006 1 5347E 007 0 00022604 Correlation Matrix 1 0 19324 1 0 015121 0 078253 1 0 005476 0 028338 0 3622 Page 44 of 75 Model 8 pH Rate Profile Monoprotic Acid Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 2 1054 is probably not significant Skewness 3 0168 indicates the likelihood of a few large negative residuals having an unduly large effect on the fit Kurtosis 0 29276 is probably not significant Weighting Factor 2 Heteroscedacticity 0 078134 Optimal Weighting Factor 1 9219 The above statistics indicate that we obtained an excellent fit of the simulated curve to the data points In particular the Model Selection Criterion is greater than 13 and the confidence limits on the parameter values are very good The variance covariance and correlation matrices do not indicate as much independence of parameters as was found for Model 7 but we are confident that the simulated curve fits the data so we plot the results This plot is shown in Figure 8 1 Model 8 Chart k Observed A m KOBS vs PH KOBS_Calc vs PH Figure 8 1 Plot for pH Rate Profile M
15. a few large negative residuals having an unduly large effect on the fit Kurtosis 1 8923 indicates the presence of a few large residuals of either sign Weighting Factor 0 Heteroscedacticity 1 106 Optimal Weighting 1 106 Factor We see from the above output that we obtained a rather good fit of the curve to the data In particular the confidence limits of the parameters vary by around 1 at the most Considering that the data set may be in error by as much as about 1 5 these results are quite good The Model Selection Criterion for this fit is greater than ten which also indicates that the curve fits the data quite well We therefore conclude that we have obtained reasonably good values of the reaction rates K1 K2 and K3 The plot of the fitted curve and the data set is shown in Figure 15 below Page 74 of 75 Model 14 Parallel First Order Irreversible Reactions Model 14 Chart m AvsT m PivsT m P2vsT m P3vsT A Calc vs T P1 Calc vs T P2 Calc vs T P3 Calc vs T 8 9 10 11 12 Time Figure 14 1 Model 14 Parallel First Order Irreversible Reactions Model 14 Parallel First Order Irreversible Reactions Page 75 of 75
16. data was slightly perturbed We now check to see how good the fit was according to other statistics The summary of these statistics is the following Data Set Name Model 3 Weighted Unweighted Sum of squared observations 57 59 57 59 Sum of squared deviations 0 0001237 0 0001237 Standard deviation of data 0 002427 0 002427 R squared 1 1 Coefficient of determination 0 99999 0 99999 Correlation 1 1 Model Selection Criterion 12 052 12 052 Confidence Intervals Parameter Name K2 Estimated Value 0 30117 Standard Deviation 0 00063318 95 Range Univariate 0 29986 0 30249 95 Range Support Plane 0 29986 0 30249 Variance Covariance Matrix 4 0091E 007 Correlation Matrix 1 Page 20 of 75 Model 3 Second Order Irreversible Reaction Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 0 14459 Is probably not significant Skewness 4 5459E 014 Is probably not significant Kurtosis 1 3242 lindicates the presence of a few large residuals of either sign Weighting Factor 0 Heteroscedacticity 1 381E 014 Optimal Weighting Factor 1 3767E 014 It is instructive to note that the goodness of fit statistics and the confidence limits on the parameters are both quite good We might expect that the data errors would not allow such a good fit but t
17. from state to state Micromath s entire potential liability and the Purchaser s exclusive remedy shall be as follows If Micromath is for any reason unable to deliver a repaired or replacement program which complies with the Limited Warranty the Purchaser may obtain a refund of the purchase price by returning the defective diskette including the instruction manual to Micromath along with a request for a refund In no event will Micromath be liable to the Purchaser for any damages including but not limited to lost profits lost savings or other incidental or consequential damages arising out of the use or inability to use the program even if Micromath is advised of the possibility of such damages or any claim by any other party Some states do not allow the limitation or exclusion of liability or consequential damages so the above limitation or exclusion may not apply to the purchaser Page 5 of 75 Introduction The models in this library are intended to aid those users of Scientist who are working on chemical kinetic problems It is not intended to be a comprehensive resource for information on chemical kinetic models It is assumed throughout this manual that the user is familiar with the types of problems that are used here and of the appropriate units for each of the variables or parameters It is also assumed that the user is familiar with the use of Scientist Please refer to the Scientist User Handbook if you have questions regarding
18. numbers obtained by this method were then rounded to two decimal places after the decimal in order to obtain reasonable errors These sorts of errors could have been produced by experimental measurements but for this demonstration they are more easily generated by simulation The data set used for this model is T A P 0 2 5 0 3 0 77 1 73 6 0 45 2 05 9 0 32 2 18 12 0 25 2 25 15 0 20 2 3 18 0 17 2 33 21 0 15 2 35 24 0 13 2 37 21 0 12 2 38 30 0 11 2 39 The parameter values used to generate this data set are shown below These values will also be the initial values for the least squares curve fitting For this example the usual simplex search will not be done since we are not attempting to show that our answer is the best that we can find Instead we just want to demonstrate the general method for working with the model and produce some sample output to show what sort of curves this model can generate Model 3 Second Order Irreversible Reaction Page 19 of 75 Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization AO 2 5 0 INF Y N AO 2 3 0 INF Y N PO 0 1 INF Y N K2 0 3 0 INF N N We perform a least squares curve fit for the reaction rate K2 by selecting only this parameter and deselecting A0 and PO The result of this fitting is as follows K2 0 30117 The sum of squared deviations for this value of K2 is 0 00012370 which is reasonably good considering that the
19. search Page 10 of 75 Model 1 Zero Order Irreversible Reaction to show that no better solutions exist close to the one found by the least squares fit We will only attempt to find one solution to this problem Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization AO 1 0 INF Y N PO 0 2 0 INF Y N KO 0 02 0 INF N N We now proceed with a least squares fit holding AO and PO fixed We find that the best fit value of KO is KO 0 019935 Which is very close to our initial value of 0 02 The sum of squared deviations at this point is 0 00087078 which is good considering the perturbations in the data If we had not modified our data set by such a large factor we could have obtained a better fit but it is noteworthy that the model produces reasonable results even if the data is somewhat inaccurate To get further information on how well the calculated curve fits our data set we need to look at the statistical output This output is as follows Data Set Name Model 1 Weighted Unweighted Sum of squared observations 8 9349 8 9349 Sum of squared deviations 0 00087078 0 00087078 Standard deviation of data 0 0064394 0 0064394 R squared 0 9999 0 9999 Coefficient of determination 0 99913 0 99913 Correlation 0 99958 0 99958 Model Selection Criterion 6 9581 6 9581 Confidence Intervals Parameter Name KO Estimated Value 0 019935 Standard Deviation 7 7353E 005 95 Range Univariate 0 019774 0 020096 9
20. some time interval The model can also simulate the concentration of A given the initial concentrations of A P1 P2 and P3 and some values of the concentrations of the products measured over a period of time This model can further be used to perform functions similar to the ones listed above for the case of two products by setting K3 and P30 to zero and deselecting them from all calculations For this example we will find the reaction rates for the three product case since this is probably the most common use of the model The model that can be used for the above mentioned procedures is as follows Page 70 of 75 Model 14 Parallel First Order Irreversible Reactions Model 14 Parallel First Order Irreversible Reactions IndVars T DepVars A P1 P2 P3 Params P10 P20 P30 AO K1 K2 K3 T1 EXP K1 K2 K3 T A AO T1 Pl P1O K1 AO 1 T1 K1 K2 K3 P2 P20 K2 AO 1 T1 K1 K2 K3 P3 P30 K3 AO 1 T1 K1 K2 K3 The model shown above requires a data set for least squares curve fitting We obtain this model by performing a simulation with some initial parameter values and rounding the results to two places after the decimal in order to produce errors comparable to those from experimental measurements Since we are attempting to find the reaction rates we need measurements of each of the dependent variables in order to obtain the best fit possible The data set that is generated for this purpose is as follows
21. 02035 95 Range Univariate 2 298 2 3067 95 Range Support Plane 2 296 2 3087 Parameter Name K2 Estimated Value 0 097498 Standard Deviation 4 3858E 005 95 Range Univariate 0 097405 0 097591 95 Range Support Plane 0 097362 0 097635 Page 38 of 75 Model 7 pH Rate Profile Nonelectrolyte Parameter Name K3 Estimated Value 53 749 Standard Deviation 0 033708 95 Range Univariate 53 678 53 82 95 Range Support Plane 53 644 53 854 Variance Covariance Matrix 4 1413E 006 1 4614E 008 1 9235E 009 8 4427E 007 1 1112E 007 0 0011362 Correlation Matrix 1 0 16374 1 0 012308 0 075167 1 Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 1 729 is probably not significant Skewness 5 9646 indicates the likelihood of a few large negative residuals having an unduly large effect on the fit Kurtosis 4 2516 is probably not significant Weighting Factor 2 Heteroscedacticity 0 063766 Optimal Weighting Factor 1 9362 These figures show us that we did obtain a good fit The Model Selection Criterion is larger than ten and the confidence limits do not deviate very much from the calculated values Also the relatively small off diagonal terms in the variance covariance matrix and the correlation matrix show that the parameter values are i
22. 45 0 122 55 0 272 65 0 580 75 0 1180 85 0 2330 95 0 4410 The parameter values used to generate the above data set are as follows Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization S 1 2 0 INF N N H 16000 0 INF N N The above figures will also be used as the starting parameter values for the least squares curve fitting We will not perform a simplex search for this parameter values since the data was generated from them and we are only attempting to demonstrate the use of this model and not to confirm results with it We use a weighting factor of 2 0 for the same reasons that it was used in Model 12 For this example we also deselect S as a linear parameter in the hope of obtaining better results The least squares fitting produces the following results S 1 2008 H 16001 The sum of squared deviations for this fit is 2 0239E 5 which is very good In order to see just how good this fit is we must look at the statistical output which is shown below Page 66 of 75 Model 13 Eyring Equation Nonlinear Form Data Set Name Model 13 Sum of squared observations Sum of squared deviations Standard deviation of data R squared Coefficient of determination Correlation Model Selection Criterion Confidence Intervals Parameter Name S Estimated Value 1 2008 Standard Deviation 0 011323 95 Range Univariate 1 1747 95 Range Support Plane 1 167 Parameter Name H Estimated Value
23. 5 Range Support Plane 0 019774 0 020096 Model 1 Zero Order Irreversible Reaction Page 11 of 75 Variance Covariance Matrix 5 9835E 009 Correlation Matrix 1 Residual Analysis The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 1 1155 Is probably not significant Skewness 0 50302 Is probably not significant Kurtosis 0 38038 Is probably not significant Weighting Factor 0 Heteroscedacticity 0 060377 Optimal Weighting Factor 0 060377 We find that several things are worth looking at in these statistics First the confidence limits for KO are identical to the range initially calculated which implies that there are no solutions close to the one that we found Also the standard deviation of these limits is quite small which is very desirable And lastly the goodness of fit statistics indicate that we obtained a reasonably good fit which is perhaps as good as we can expect for this data set A plot of the simulated curve and the data set is shown in the following Figure 1 1 Page 12 of 75 Model 1 Zero Order Irreversible Reaction Model 1 Chart m AvsT mPvsT A Calc vs T P Calc vs T Concentration Time
24. Model 3 Second Order Irreversible Reaction Zi Figure 4 1 Model 4 Second Order Irreversible Reaction 29 Figure 5 1 Model 5 Second Order Irreversible Reaction 30 Figure 6 1 Model 6 First Order Reversible Reaction ii 35 Figure 7 1 Model 7 pH Rate Profile Nonelectolyte 40 Figure 8 1 Plot for pH Rate Profile Monoprotic Acid i 45 Figure 9 1 Model 9 pH Rate Profile Diprotic ACiId i 51 Figure 10 1 Model 10 Arrhenius Equation Linearized Form 55 Figure 11 1 Model 11 Arrhenius Equation Nonlinear Form 59 Figure 12 1 Model 12 Eyring Equation Linearized Form i 64 Figure 13 1 Model 13 Eyring Equation Nonlinear Form i 69 Figure 14 1 Model 14 Parallel First Order Irreversible Reactions 75 Page 8 of 75 Model 1 Zero Order Irreversible Reaction This model may be used in several different ways First it can be used to find the reaction rate KO given the initial concentration of A AO the initial concentration of P PO and a number of measurements of the reactant A and the product P over a period of time Second it can be used to model the concentration of the reactant A given the initial concentration of P the initial concentration o
25. action IndVars T DepVars A P Params AO PO K1 A AO EXP K1 T P PO AO 1 EXP K1 T The data set used to find the concentration of P over a time interval was generated by selecting some initial parameter values doing a simulation for A and introducing small errors into the data We proceed in this manner in order to produce data which approximates experimental measurements The data set is as follows T A 0 0 5 3 0 43 6 0 38 9 0 31 12 0 27 15 0 24 18 0 2 21 0 18 24 0 15 27 0 13 30 0 11 The parameter values that were used to generate this data will also be used as the starting values of the least squares fitting These values are used instead of the values obtained from a simplex search for demonstration Any other application of this model should be preceded by a simplex search unless other conditions apply These initial parameter values are Model 2 First Order Irreversible Reaction Page 15 of 75 Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization AO 0 5 0 INF Y N PO 0 1 0 INF Y N K1 0 05 0 INF N N We now make sure that P is deselected and A is selected for fitting We fix A0 and PO since they are known and do a fitting only for K1 The values of K1 that best fits the data for A is K1 0 050049 The sum of squared deviations for this fit is 0 00024258 which is not too bad considering the size of the errors in the data for A We
26. ares fit and find that the best fit values of the activation entropy and enthalpy are S 1 0014 H 10000 We also find a sum of squared deviations of 1 1802E 5 which is fairly good To see whether the fit of the calculated curve to the data is good enough we look at the statistical summary that Scientist calculates These statistics are shown below Model 12 Eyring Equation Linearized Form Page 61 of 75 Data Set Name Model 12 Weighted Sum of squared observations 10 Sum of squared deviations 1 1802E 005 Standard deviation of data 0 0012146 R squared 1 Coefficient of determination 1 Correlation 1 Model Selection Criterion 13 653 Confidence Intervals Parameter Name S Estimated Value 1 0014 Standard Deviation 0 0084717 95 Range Univariate 0 98186 1 0209 95 Range Support Plane 0 9761 1 0267 Parameter Name H Estimated Value 10000 Standard Deviation 2 7006 95 Range Univariate 9993 9 10006 95 Range Support Plane 9992 1 10008 Variance Covariance Matrix 7 177E 005 0 022786 7 2935 Correlation Matrix 1 0 99594 1 Page 62 of 75 Unweighted 2 9549E009 2079 1 16 121 1 1 1 13 256 Model 12 Eyring Equation Linearized Form Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 0 4249 is probably n
27. ated Value 0 029988 Standard Deviation 4 9312E 005 95 Range Univariate 0 029886 0 030091 95 Range Support Plane 0 029886 0 030091 Variance Covariance Matrix 2 4317E 009 Correlation Matrix 1 Page 24 of 75 Model 4 Second Order Irreversible Reaction Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 1 4506 Is probably not significant Skewness 3 8415E 013 Is probably not significant Kurtosis 0 32348 Is probably not significant Weighting Factor 0 Heteroscedacticity 8 626E 015 Optimal Weighting Factor 8 6597E 015 These numbers indicate that the fit of the simulated curve to the data was quite good The confidence limits for the K2 are very well determined and the Model Selection Criterion is relatively high indicating a good fit We conclude from this that the model is capable of producing quite good results from experimental data Model 4 Chart AvsT m PvsT A Calc vs T P Calc vs T 0 2 4 6 8 9 2 4 46 46 20 2 24 26 28 32 34 38 38 40 Time Figure 4 1 Model 4 Second Order Irreversible Reaction Model 4 Second Order Irreversible Reaction Page 25 of 75 Model 5 Second Order Irreversible Reaction k A B gt P A B 0 This model has several possible uses First it can determine the second order reac
28. by which they may be obtained we would find a more accurate data set but we will not do so here The plot for this fit is obtained by plotting K logarithmically The plot is shown in Figure 10 1 below Model 10 Chart m K ys TINY K Calc vs TINY 2 8 x10 E 3 3 0 x10 E 3 3 2 x10 E 3 3 4 x10 E 3 3 6 x10 E 3 1 Temperature Kelvin Figure 10 1 Model 10 Arrhenius Equation Linearized Form Model 10 Arrhenius Equation Linearized Form Page 55 of 75 Model 11 Arrhenius Equation Nonlinear Form As with Model 10 this model may be used to find the parameters A and Ea for the following equation where Ea is given in units of calories mole Ea k Ax e R T These parameters can be found given a number of measurements of the temperature in degrees Celsius and the reaction rate This model could also be used to simulate the reaction rate given known values of the parameters but finding the values of A and Ea is more common so we will find them as a demonstration of this model The form that the above equation takes in Scientist is as follows Model 11 Arrhenius Equation Non Linear Form IndVars T DepVars K Params A EA K A EXP EA 1 987 T 273 The data set used for this fitting was found by doing a simulation with some initial parameter values and rounding the results to three decimal places By doing this we create errors which are roughly proportional to the square of the inverse of t
29. e need some measurements of KOBS over a range of PH We obtain data of this sort by performing a simulation of the model over a range of PH given set values for the parameters This data set is as follows Model 9 pH Rate Profile Diprotic Acid Page 47 of 75 PH KOBS 0 0 53 7 0 5 39 1 1 0 34 5 1 5 33 1 2 0 32 6 3 0 32 4 4 0 32 4 5 0 32 4 6 0 32 4 7 0 32 3 8 0 32 1 9 0 29 8 10 0 18 2 11 0 6 61 12 0 4 08 12 5 3 96 13 0 7 04 13 5 24 8 14 0 90 1 The parameter values used to generate this data set will also be used as the initial guesses to begin the least squares fitting We will not do a simplex search since the values should be close enough to the least squares solution for demonstration purposes The initial parameter values are Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization K1 21 3 0 INF N N K2 32 4 0 INF N N K3 4 1 0 INF N N K4 0 1 0 INF N N K5 98 6 0 INF N N KAI 1E 010 0 INF Y N KA2 1E 013 0 INF Y N KW 1E 014 0 INF Y N Page 48 of 75 Model 9 pH Rate Profile Diprotic Acid We now fix KA2 and KW for fitting since we do not want them to vary for this problem A weighting factor of 2 0 will be used in fitting this data since the errors are roughly proportional to the inverse of the squares of the values Problems where the data values varied over several orders of magnitude are more accurately fitted wit
30. f A and a number of measurements of P over a period of time Third it can be used to model the concentration of the product P given the initial concentration of A the initial concentration of P and a number of measurements of A over a given time interval For the example below we have chosen the first of these options that is to find the reaction rate constant KO Model 1 Zero Order Irreversible Reaction Page 9 of 75 The model is as follows Model 1 Zero Order Irreversible Reaction IndVars T DepVars A P Params AO PO KO A AO KO T P PO KO T For this example we need a number of measurements of the concentration of A and the concentration of P We generate an example data set by choosing some initial values for the parameters A0 PO and K0 We then do a simulation with these parameter values and randomly add or subtract 0 01 to provide some uncertainty in the data This data set is as follows T A P 0 1 0 2 3 0 93 0 26 6 0 88 0 33 9 0 82 0 38 12 0 77 0 43 15 0 7 0 5 18 0 63 0 55 21 0 59 0 63 24 0 52 0 68 21 0 46 0 74 30 0 41 0 8 The parameter values which were used to obtain this data set are shown below These values will also serve as our initial estimates for a least squares fitting for K0 We will not perform a simplex search because these values should be close enough to the final solution A more rigorous approach to this problem would include a simplex
31. grade The model form of this equation may be used to find the rate constants ki kz and k3 given a number of measurements of the pH and of Kops typically the observed first order reaction rate It could also be used to simulate the observed reaction rate Kops given values for the reaction rate constants ki kz and ks The model used for these purposes is as follows Model 7 pH Rate Profile Nonelectrolyte IndVars PH DepVars KOBS Params K1 K2 K3 KW H 10 PH KOBS K1 H K2 K3 KW H We will now proceed with an example showing how to find the rate constants ki k and ks since this will be the most typical use of this model To do this we need to construct a data set We perform a simulation with some assumed parameter values and round the results to three significant digits The data set constructed in the above manner for this example is Page 36 of 75 Model 7 pH Rate Profile Nonelectrolyte PH KOBS 0 0 2 4 0 5 0 825 1 0 0 328 1 5 0 17 2 0 0 121 3 0 0 0998 4 0 0 0977 5 0 0 0975 6 0 0 0975 7 0 0 0974 8 0 0 0975 9 0 0 098 10 0 0 103 11 0 0 151 12 0 0 635 12 5 1 80 13 0 5 47 13 5 17 1 14 0 53 8 The parameter values that were used to generate this data set will be used as the initial conditions for the least squares curve fitting We will not refine the values with a simplex search since they should already be close enough to
32. h a weighting factor of 2 0 We start the least squares fitting and find that the best fit values are K1 21 313 K2 32 379 K3 4 0968 K4 0 10885 K5 98 650 The sum of squared deviations at this point is 1 2894E 5 which is good We now examine the statistical summary shown below to see if the fit is as good as the sum of squared deviations indicates Data Set Name Model 9 Weighted Unweighted Sum of squared observations 19 24112 Sum of squared deviations 1 2894E 005 0 0115 Standard deviation of data 0 0009597 0 028661 R squared 1 1 Coefficient of determination 1 1 Correlation 1 1 Model Selection Criterion 13 871 12 781 Confidence Intervals Parameter Name KI Estimated Value 21 313 Standard Deviation 0 049208 95 Range Univariate 21 207 21 418 95 Range Support Plane 21 123 21 502 Parameter Name K2 Estimated Value 32 379 Standard Deviation 0 0095833 95 Range Univariate 32 358 32 399 95 Range Support Plane 32 342 32 415 Model 9 pH Rate Profile Diprotic Acid Page 49 of 75 Parameter Name K3 Estimated Value 4 0968 Standard Deviation 0 0041957 95 Range Univariate 4 0878 95 Range Support Plane 4 0807 Parameter Name K4 Estimated Value 0 10885 Standard Deviation 0 017461 95 Range Univariate 0 071404 95 Range Support Plane 0 0417 Parameter Name K5 Estimated Value 98 65 Standard Deviation 0 085807 95 Range Univariate 98 466 95 Range Support Plane
33. hat these values are acceptable and plot the calculated curve and data points This plot is shown in Figure 13 1 below One additional item that is useful to note is that this model produced results that were approximately as accurate as the results of Model 12 Since both models used data sets with the same number of significant digits either of them could be used with to obtain the best fit solution for this problem Page 68 of 75 Model 13 Eyring Equation Nonlinear Form Model 13 Chart mKvsT K Calc vs T 4 083 SES EE dina 2 063 e ug EE rm 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 Temperature Celsius Figure 13 1 Model 13 Eyring Equation Nonlinear Form Model 13 Eyring Equation Nonlinear Form Page 69 of 75 Model 14 Parallel First Order Irreversible Reactions This model has many possible uses It may be used to find the reaction rates K1 K2 and K3 given the initial concentration of the reagent A the initial concentrations of the products P1 P2 and P3 and a number of measurements of the concentrations of the reagent and the products over a period of time It may also be used to simulated the concentration of any one of the products given the initial concentrations of each of the products P10 P20 and P30 the initial concentration of the reagent AO and a number of measurements of the concentrations of the reagent and the products other than the one being simulated over
34. he curve which was fit to it are shown in Figure 5 1 below Model 5 Second Order Irreversible Reaction Page 29 of 75 Model 5 Chart 3 4 5 6 7 8 9 10 Time m AvsT BysT G PysT Calc vs T B Calc vs T P Calc vs T 11 12 13 14 15 16 17 18 19 20 Figure 5 1 Model 5 Second Order Irreversible Reaction Page 30 of 75 Model 5 Second Order Irreversible Reaction Model 6 First Order Reversible Reaction There are several uses to which this model can be put First it can be employed to find the forward and reverse reaction rates KF and KR given the initial concentration of the reagent A AO the initial concentration of the product P PO and a number of measurements of the concentrations of A and P over a time interval The second use for this model is to simulated the concentration of P given the initial concentrations of A and P AO and PO and a number of measurements of the concentration of A over time The third possible use for this model is to simulate the concentration of A given the initial concentrations of A and P and a number of measurements of the concentration of P over a period of time Since the first option would be the most used we will demonstrate how to work with it in this example The form of this model is as follows Model 6 First Order Reversible Reaction Page 31 of 75 Model 6 First Order Reversible Reaction IndVars T DepVars A P Params AO PO KF KR A
35. he magnitude of the number We will use this fact later when we fit the data The data set for this case is Page 56 of 75 Model 11 Arrhenius Equation Nonlinear Form T K 5 0 8 09E 009 15 0 1 72E 008 25 0 3 48E 008 35 0 6 71E 008 45 0 1 24E 007 55 0 2 22E 007 65 0 3 82E 007 75 0 6 39E 007 85 0 1 04E 006 95 0 1 64E 006 The parameter values that were used to construct this data set are as follows Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization A 22 0 INF N N EA 12000 0 INF N N We will not perform a simplex search to find better starting parameters since the data was generated from these values and we are not attempting to prove that the results we get are the best that can be found We set the weighting factor to 2 0 because the data set was constructed to have errors inversely proportional to the square of the magnitude of each value We then do the least squares curve fitting with both A and EA selected for fitting The results of this fit are as follows A 21 974 EA 11999 The sum of squared deviations for this fit is 1 5979E 5 which is quite good We now examine the statistical output for this fit looking particularly at the difference between the weighted and unweighted statistical values Model 11 Arrhenius Equation Nonlinear Form Page 57 of 75 Data Set Name Model 11 Weighted Unweighted Sum of squared observations 10 4 3962E 012
36. he model is not too complicated to provide us with good limits on the parameters Model 3 Chart m AvsT m PvsT A Calc vs T P Calc vs T Time Figure 3 1 Model 3 Second Order Irreversible Reaction Model 3 Second Order Irreversible Reaction Page 21 of 75 Model 4 Second Order Irreversible Reaction k 2A gt P This model is useful for several different calculations It may be used to compute the reaction rate K2 given the initial concentration of A AO the initial concentration of P PO and a number of measurements of the concentrations of the reagent A and the product P over a period of time It can also be used to simulate either the concentration of A or the concentration of P given a number of measurements of the concentration of the other variable over time and the initial concentrations of both variables In this example we will compute the reaction rate The model used for these calculations is Page 22 of 75 Model 4 Second Order Irreversible Reaction Model 4 Second Order Irreversible Reaction IndVars T DepVars A P Params AO PO K2 A AO 1 2 K2 AO T P PO 2 K2 AO 2 T 1 2 K2 AO T The measurements of the concentrations of A and P were generated for this example by performing a simulation with initial parameter values For any other application the concentrations would have been measured experimentally The data set is as follows
37. how to run Scientist The models in this library are documented in roughly the same manner as the example problems at the end of the Scientist User Manual The equations defining the model are given followed by the form they will take in Scientist A sample data set and initial parameter values are given for each model and the results of the least squares fitting for the models are shown The method used in obtaining the results for these models should not be taken as the ideal method of finding the solution to any particular problem The examples are given only to demonstrate what may be done with each model and how the output might appear A Note on Fitting with Multiple Parameters The examples worked out in this manual generally involve fitting more than one parameter to the data set used in each problem Often there are parameters that could be used to fit the data which are held constant such as the initial concentrations of the reactants or products These parameters can be selected for fitting but some care should be taken in doing so primarily because increasing the number of parameters to be fitted causes the ability to accurately determine the parameters to decrease In these cases it is often necessary to fit some of the parameters while holding the others constant then fit the others while holding the parameters that were originally fit constant and then fitting all of them together This method tends to decrease the difficulty of con
38. hree decimal places corresponding to an error roughly proportional to the inverse of the square of the value We fix KA and KW for fitting since they should not vary for this fit The least squares fitting yields the following results K1 6 3012 K2 0 19489 K3 22 393 K4 32 688 The sum of squared deviations for the fit is 1 9223E 5 which is quite good We now look at the statistical output to determine just how good the fit was This output is as follows Data Set Name Model 8 Sum of squared observations Sum of squared deviations Standard deviation of data R squared Coefficient of determination Correlation Model Selection Criterion Confidence Intervals Parameter Name KI Estimated Value 6 3012 Standard Deviation 0 0044491 95 Range Univariate 6 2917 95 Range Support Plane 6 2856 Model 8 pH Rate Profile Monoprotic Acid Weighted 19 1 9223E 005 0 0011321 l l l 13 573 6 3106 6 3167 Unweighted 6411 4 0 0046027 0 017517 1 1 1 13 304 Page 43 of 75 Parameter Name K2 Estimated Value 0 19489 Standard Deviation 9 3535E 005 95 Range Univariate 0 19469 0 19509 95 Range Support Plane 0 19457 0 19522 Parameter Name K3 Estimated Value 22 393 Standard Deviation 0 010779 95 Range Univariate 22 37 22 416 95 Range Support Plane 22 355 22 43 Parameter Name K4 Estimated Value 32 688 Standard Deviation 0 057899 95 Range Univariate 32 564 32
39. lar demonstration they are probably good enough Model 2 Chart o 5 Calc vs T P Calc vs T O AvsT Time Figure 2 1 Model 2 First Order Irreversible Reaction Model 2 First Order Irreversible Reaction Page 17 of 75 Model 3 Second Order Irreversible Reaction This model has several possible uses First it may be employed to find the second order reaction rate K2 given the initial concentration of the reagent A AO the initial concentration of the product P PO and a number of measurements of the concentration of A and P over time Second it can be used to simulate the concentration of P given the initial concentration of P PO the initial concentration of A AO and a number of observations of A over a period of time Third it can simulate the concentration of A given the initial concentration of P the initial concentration of A and a number of measurements of the concentration of P over a time interval We choose to employ the first option finding the reaction rate for this example The model used for this purpose is as follows Page 18 of 75 Model 3 Second Order Irreversible Reaction Model 3 Second Order Irreversible Reaction IndVars T DepVars A P Params AO PO K2 A AO 1 K2 AO T P PO K2 SQR AO T 1 K2 AO T A data set containing observations of A and P over a period of time was generated by performing a simulation with an initial set of parameter values The
40. lected for fitting since we wish to know both of these values The best fit values that Scientist finds are KF 0 049443 KR 0 029466 The sum of squared deviations for the last step in the fitting is 0 00018026 which is reasonably good We cannot say more about the fit of the simulated curve to the data without looking at the statistical output that Scientist provides This output is shown below Data Set Name Model 6 Weighted Unweighted Sum of squared observations 23 426 23 426 Sum of squared deviations 0 00018026 0 00018026 Standard deviation of data 0 0030022 0 0030022 R squared 0 99999 0 99999 Coefficient of determination 0 99987 0 99987 Correlation 0 99994 0 99994 Model Selection Criterion 8 7943 8 7943 Model 6 First Order Reversible Reaction Page 33 of 75 Confidence Intervals Parameter Name KF Estimated Value 0 049443 Standard Deviation 0 00024065 95 Range Univariate 0 048941 0 049945 95 Range Support Plane 0 048807 0 050079 Parameter Name KR Estimated Value 0 029466 Standard Deviation 0 00024846 95 Range Univariate 0 028948 0 029984 95 Range Support Plane 0 028809 0 030123 Variance Covariance Matrix 5 7913E 008 5 7442E 008 6 1734E 008 Correlation Matrix n 1 Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation
41. n Model Selection Criterion Confidence Intervals Parameter Name A Estimated Value 24 989 Standard Deviation 0 088018 95 Range Univariate 24 786 95 Range Support Plane 24 727 Parameter Name EA Estimated Value 11000 Standard Deviation 2 2323 95 Range Univariate 10995 95 Range Support Plane 10993 Variance Covariance Matrix 0 0077471 0 19568 4 9831 Correlation Matrix 1 0 99594 1 Page 54 of 75 Weighted Unweighted 10 9 7067E 011 8 041E 006 6 0172E 017 0 0010026 2 7425E 009 1 1 1 1 1 1 14 176 13 436 25 192 25 232 11005 11007 Model 10 Arrhenius Equation Linearized Form Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 0 92828 is probably not significant Skewness 0 6995 is probably not significant Kurtosis 0 41513 1s probably not significant Weighting Factor 2 Heteroscedacticity 4 7156E 008 Optimal Weighting Factor 2 It is reassuring to note that the fit for the weighted data is much better than the unweighted fit The Model Selection Criterion is quite high indicating a rather good fit of the calculated curve to the data even though the confidence limits for the parameters were somewhat wider than is desirable If we were attempting to find accurate results instead of demonstrating the method
42. ndependently Model 7 pH Rate Profile Nonelectrolyte Page 39 of 75 determined as we would hope Although some of the statistics are better for the unweighted case we accept the weighted values because they better represent the errors in the data We decide that the fit is good enough for this demonstration and draw the plot of the pH versus the log of the observed reaction rate This plot is shown in Figure 7 1 below Model 7 Chart 7 m KOBS vs PH KOBS Calc vs PH K Observed Figure 7 1 Model 7 pH Rate Profile Nonelectolyte Page 40 of 75 Model 7 pH Rate Profile Nonelectrolyte Model 8 pH Rate Profile Monoprotic Acid The equation that describes the pH rate profile for a monoprotic acid is as follows kos ki H7 fha ko faa ks fa ka OH fa where faa H H K fa Ka H Ka OH K y H In the above equations Kw is the ion product of water 1 0E 14 at 25 degrees Centigrade and K is the acid ionization constant This set of equations in model form may be used to find the reaction rate constants k1 k2 k3 and k4 given a number of measurements of kobs typically the first order observed reaction rate over a set of values of pH This model can also be used to find the acid ionization constant Ka given the reaction rate constants k1 k2 k3 and k4 and the measurements of kobs versus pH The model form of the above equations is as follows Model
43. now take a look at the statistics for this fit to assure ourselves that the fit is good enough for simulating P These statistics are shown below Data Set Name Model 2 Weighted Unweighted Sum of squared observations 0 9298 0 9298 Sum of squared deviations 0 00024258 0 00024258 Standard deviation of data 0 0049253 0 0049253 R squared 0 99974 0 99974 Coefficient of determination 0 99853 0 99853 Correlation 0 99927 0 99927 Model Selection Criterion 6 3421 6 3421 Confidence Intervals Parameter Name K1 Estimated Value 0 050049 Standard Deviation 0 00048924 95 Range Univariate 0 048959 0 051139 95 Range Support Plane 0 048959 0 051139 Variance Covariance 2 3935E 007 Page 16 of 75 Model 2 First Order Irreversible Reaction Correlation Matrix 1 Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 1 2885 Is probably not significant Skewness 0 81981 Is probably not significant Kurtosis 0 48551 Is probably not significant Weighting Factor 0 Heteroscedacticity 0 87949 Optimal Weighting Factor 0 87949 We can see that these figures are not quite as good as we would like them to be In particular the goodness of fit statistics are rather average and the confidence limits are probably a bit wider than we would like However for this particu
44. ntervals Parameter Name K2 Estimated Value 0 099043 Standard Deviation 0 00010902 95 Range Univariate 0 098821 95 Range Support Plane 0 098821 Variance Covariance 1 1886E 008 Page 28 of 75 Unweighted 35 169 0 00014571 0 0021338 1 0 99998 0 99999 10 735 0 099265 0 099265 Model 5 Second Order Irreversible Reaction Correlation Matrix 1 Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 0 31634 is probably not significant Skewness 8 3562 indicates the likelihood of a few large positive residuals having an unduly large effect on the fit Kurtosis 3 6291 is probably not significant Weighting Factor 0 Heteroscedacticity 0 39193 Optimal Weighting Factor 0 39193 The above output is probably a little better than we had expected given a sum of squared deviations as large as we have for this problem The Model Selection Criterion is greater than ten which is quite good and the confidence limits on K2 are within 0 5 of each other which is also good considering the size of the errors in the data set We conclude that this model is able to fit data well and obtain an error of no more than the size of the perturbations of the data We could not ask a model to produce output that was much better The plot of the data set and t
45. odel 14 Weighted Unweighted Sum of squared observations 101 64 101 64 Sum of squared deviations 0 00026357 0 00026357 Standard deviation of data 0 0025355 0 0025355 R squared 1 1 Coefficient of determination 0 99998 0 99998 Correlation 0 99999 0 99999 Model Selection Criterion 10 604 10 604 Page 72 of 75 Model 14 Parallel First Order Irreversible Reactions Confidence Intervals Parameter Name K1 Estimated Value 0 030026 Standard Deviation 4 0953E 005 95 Range Univariate 0 029943 0 030108 95 Range Support Plane 0 029906 0 030145 Parameter Name K2 Estimated Value 0 019959 Standard Deviation 3 8846E 005 95 Range Univariate 0 019881 0 020038 95 Range Support Plane 0 019846 0 020072 Parameter Name K3 Estimated Value 0 050007 Standard Deviation 4 5867E 005 95 Range Univariate 0 049915 0 0501 95 Range Support Plane 0 049874 0 050141 Variance Covariance Matrix 1 6772E 009 1 2173E 010 1 509E 009 1 5236E 010 2 9478E 011 2 1038E 009 Correlation Matrix 1 0 076517 1 0 081112 0 016545 1 Model 14 Parallel First Order Irreversible Reactions Page 73 of 75 Residual Analysis Expected Value The following are normalized parameters with an expected value of 0 0 Values are in units of standard deviations from the expected value Serial Correlation 1 4678 indicates a systematic non random trend in the residuals Skewness 4 5827 indicates the likelihood of
46. on contained in the Scientist Chemical Kinetic Library are copyrighted or trademarked by Micromath and constitute proprietary information that remains the property of Micromath Research Page 4 of 75 LIMITED WARRANTY Micromath warrants that the Scientist Chemical Kinetic Library Handbook and the Scientist Chemical Kinetic Library diskette will be free from defects in materials and in good working order when delivered and will for 90 days after delivery properly perform the functions contained in the program when and only when Scientist is used without material alteration and in accordance with the instructions set forth in the instruction manual Scientist is intended only for nonlinear least squares parameter estimation and Micromath takes no responsibility for subsequent use of those estimates Micromath does not warrant that the functions contained in the program will meet the purchaser s requirements Except for the above limited warranty Scientist is provided as is without any additional warranties of any kind either express or implied By means of example only Scientist specifically is not covered by an implied warranty of merchantability of fitness for a particular purpose Some states do not allow the exclusion of implied warranties and the above exclusion of implied warranties may not apply to the purchaser The Limited Warranty gives the purchaser specific legal rights and the purchaser may also have other rights which vary
47. onoprotic Acid Model 8 pH Rate Profile Monoprotic Acid Page 45 of 75 Model 9 pH Rate Profile Diprotic Acid The equation describing the pH rate profile for a diprotic acid is as follows kobs kl H fH2A k2 fH2A k3 fHA k4 fA KS OH fA Where fH2A H 2 H 7 2 Kal H Kal Ka2 fHA Kal H H 4 2 Kal H Kal Ka2 fA Kal Ka2 H 2 Kal H Kal Ka2 OH Kw H In the above equations Kw is the ion product of water 1 0E 14 at 25 degrees Centigrade and Kal and Ka2 are the acid ionization constants The model form of these equations is normally used to find the rate constants kl k2 k3 k4 and k5 given measurements of kobs typically the first order observed reaction rate over a range of pH It may also be used to find the acid ionization constants given values for the rate constants kl k2 k3 k4 and k5 and the measurements of pH versus kobs Since the first use of the model is more typical we will perform that calculation in this example The model form of the equations is Page 46 of 75 Model 9 pH Rate Profile Diprotic Acid Model 9 pH Rate Profile Diprotic Acid IndVars PH DepVars KOBS Params K1 K2 K3 K4 K5 KAI KA2 KW H 10 PH FH2A H 2 H 2 KA1 H KA1 A2 FHA KA1 H H 2 KA1 H KA1 KA2 FA KAI KA2 HN2 KA1 H KA1 KA2 KOBS K1 H FH2A K2 FH2A K3 FHA K4 FA K5 KW FA H To begin the curve fitting process w
48. ot significant Skewness 1 2081 indicates the likelihood of a few large negative residuals having an unduly large effect on the fit Kurtosis 0 11959 is probably not significant Weighting Factor 2 Heteroscedacticity 184 71 Optimal Weighting Factor 186 71 While studying these statistics we find two things which are noteworthy First the confidence limits for S are not as good as they could be And second the Model Selection Criterion for the weighted case is marginally better than that for the unweighted case This would suggest that by using a weighting factor of 0 0 we could produce roughly the same results However a weighting factor of 0 0 means that only the first few points of this data set is significant since the data following it is one to two magnitudes smaller Weighting the data in this manner means that we essential ignore all but the first two or three points This is not what we would like to have Therefore we find that the results for the weighted case are much more meaningful In order to obtain a linear graphics plot of the calculated curve and the data set it is necessary to specify a logarithmic axis for the dependent variable This plot is shown in Figure 12 1 below Model 12 Eyring Equation Linearized Form Page 63 of 75 Model 12 Chart KDIVT vs TINY KDIVT Calc vs TINY K Temperature 2 7e 3 2 8e 3 2 9e 3 3 0e 3 3 1e 3 32e 3 3 3e 3 3 4e 3 3 5e 3 3 5e 3 1 Temperatu
49. re Kelvin Figure 12 1 Model 12 Eyring Equation Linearized Form Page 64 of 75 Model 12 Eyring Equation Linearized Form Model 13 Eyring Equation Nonlinear Form As in Model 12 this model is represented by the following equation AS AH Kx L LO R xe R T h Where K Boltzmann s Constant h Plank s Constant It may be used to compute the best fit values of the activation entropy AS and the activation enthalpy AH for the case of nonlinear graphics given a number of measurements of the temperature in degrees Celsius and the reaction rate As in the discussion of the previous model this model can be use to find the value of either the activation entropy or enthalpy given the value of the other parameter and the measurements listed above The units for the activation entropy and enthalpy are calories degree mole and calories mole respectively The above equation takes on the following form in Scientist Model 13 Eyring Equation Nonlinear Form IndVars T DepVars K Params S H K 1 3805E 16 T 273 EXP S 1 987 EXP H 1 987 273 T 6 6255E 27 The data set used for this fitting is produced by doing a simulation with some initial parameter values and rounding the resulting figures to three decimal places This data set is as follows Model 13 Eyring Equation Nonlinear Form Page 65 of 75 T K 5 0 2 79 15 0 7 90 25 0 20 9 35 0 51 9
50. roduce small errors The data that was obtained by this method is as follows Page 52 of 75 Model 10 Arrhenius Equation Linearized Form TINV K 0 0027 8 06E 006 0 0028 4 64E 006 0 0029 2 66E 006 0 003 1 53E 006 0 0031 8 81E 007 0 0032 5 06E 007 0 0033 2 91E 007 0 0034 1 67E 007 0 0035 9 62E 008 0 0036 5 53E 008 The initial parameters will be close enough to the solution for this demonstration so we will not perform a simplex search This is not the ideal method for finding the best solution but it is adequate for this example The starting values of the parameters are Parameters Name Value Lower Limit Upper Limit Fixed Linear Factorization A 25 0 INF N N EA 11000 0 INF N N The least squares fitting is done with both parameters selected to be fit and the weighting factor set to 2 0 The weighting factor is set in this manner because the errors in the data set calculated are roughly proportional to the square of the inverse of the magnitude of the data point The results of this calculation are as follows A 24 989 EA 11000 The sum of squared deviations for this fit was 8 0410E 6 which is good The statistical output for this model is shown below Model 10 Arrhenius Equation Linearized Form Page 53 of 75 Data Set Name Model 10 Sum of squared observations Sum of squared deviations Standard deviation of data R squared Coefficient of determination Correlatio
51. the final solution The initial parameter values are Parameters Name Value Lower Limit KI 2 3 0 K2 0 0975 0 K3 53 7 0 KW 1E 014 0 Model 7 pH Rate Profile Nonelectrolyte Upper Limit Fixed Linear Factorization INF INF INF INF ZZZ ZZZZ Page 37 of 75 The least squares fitting with a weighting factor of 2 0 for this problem since the values in this data set vary over a number of the orders of magnitude and therefore the errors for each point are roughly proportional to the square of the inverse of its value We fix KW for fitting since it is a constant depending on temperature and therefore should not vary for this problem We now perform the least squares fitting and obtain the following results K1 2 3024 K2 0 097498 K3 53 749 The sum of squared deviation for this fit is 2 9342E 5 which is quite good We now check the rest of the statistical output that Scientist provides in order to see if they indicate they we obtained as good a fit as the sum of squared deviations implies The statistics for this model are shown below Data Set Name Model 7 Weighted Unweighted Sum of squared observations 19 3227 1 Sum of squared deviations 2 9342E 005 0 002205 Standard deviation of data 0 0013542 0 011739 R squared 1 1 Coefficient of determination 1 1 Correlation 1 1 Model Selection Criterion 12 51 13 76 Confidence Intervals Parameter Name KI Estimated Value 2 3024 Standard Deviation 0 0
52. the fit is so good we accept the resulting values of A and EA The plot of the calculated curve and the data points is shown in Figure 11 1 below Model 11 Chart 2e 6 eden gemene a RE nederste ip deal o nasale ETE A Rio pa SES re m KvsT i A Kale vs T 5 om 5 2 2 3 3 4 s so 55 6 65 70 75 80 6 980 965 Temperature Celsius Figure 11 1 Model 11 Arrhenius Equation Nonlinear Form Model 11 Arrhenius Equation Nonlinear Form Page 59 of 75 Model 12 Eyring Equation Linearized Form The manipulations done with this model are based on the following equation AS AH x L T xe R ker h Where K Boltzmann s Constant h Plank s Constant The model may be use to find the best fit values of the activation entropy AS and the activation enthalpy AH for the linear graphics case given a number of measurements of the inverse of the temperature in degrees Kelvin and the reaction rate divided by the temperature It could also be used to find the entropy or enthalpy given a set value for the other parameter but we will not perform this calculation for this example The activation entropy is reported in units of calories degree mole and the activation enthalpy is in units of calories mole To find the values of these parameters for the nonlinear graphics case use Model 13 The form that the above equation takes in Scientist is as follows Model
53. tion rate K2 given the initial concentrations of the two reagents AO and BO the initial concentration of the product PO and a number of measurements of the reagents A and B and the product P over a time interval It could also be used to simulate the concentration of the product P given the initial concentrations of A and B AO and BO the initial concentration of P PO and a number of measurements of A and B over a period of time Two other uses for this model are to simulate the concentration of A or B given the initial concentrations of each reagent and the product and a number of measurements of the concentration of the other reagent and the product over a time interval This example will demonstrate the first of these options The model for these possible calculations is as follows Page 26 of 75 Model 5 Second Order Irreversible Reaction Il Model 5 Second Order Irreversible Reaction Il A0 Not Equal to BO IndVars T DepVars A B P Params AO BO PO K2 A AO AO BO 1 EXP K2 T BO AO AO BO EXP K2 T BO AO B BO AO BO 1 EXP K2 T BO AO AO BO EXP K2 T BO AO P PO AO BO 1 EXP K2 T BO AO AO BO EXP K2 T BO AO Instead of obtaining experimental measurements for the data we perform a simulation and round the resulting numbers to two places after the decimal to produce small errors The results of this simulation are T A B P 0 1 5 2 0 2 2 1 06
54. verging to the final solution but it may not increase the accuracy of the parameter values We leave it to the users of this library to determine what method is appropriate for the problems being solved Page 6 of 75 Table of Contents Model 1 Zero Order Irreversible Reaction ua 9 Model 2 First Order Irreversible Reaction 14 Model 3 Second Order Irreversible Reaction 18 Model 4 Second Order Irreversible Reaction 22 Model 5 Second Order Irreversible Reaction 26 Model 6 First Order Reversible Reaction i 31 Model 7 pH Rate Profile Nonelectrolyte 36 Model 8 pH Rate Profile Monoprotic ACId i 41 Model 9 pH Rate Profile Diprotic ACId ii 46 Model 10 Arrhenius Equation Linearized FOrm cnc conncncnnnnoss 32 Model 11 Arrhenius Equation Nonlinear Form 56 Model 12 Eyring Equation Linearized Form i 60 Model 13 Eyring Equation Nonlinear Form cnc cana ncnconanoss 65 Model 14 Parallel First Order Irreversible Reactions 70 Page 7 of 75 Table of Figures Figure 1 1 Model 1 Zero Order Irreversible Reaction 13 Figure 2 1 Model 2 First Order Irreversible Reaction 17 Figure 3 1
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