The rate constant of a chemical reaction k (units are dependent on what order the reaction is) is dependent on temperature, where typically increasing the temperature increases the reaction rate, meaning that the reaction goes faster upon increasing the temperature. At a company X, the Research \& Development department, which does not have a member with a Chemical Engineering background, has done experiments to measure the rate constant of a first order reaction at different temperatures. The data they found from their experiments is as follows: \begin{tabular}{|l|l|} \hline Temperature [K] & k[1/s] \\ \hline 298 & 0.01575 \\ \hline 303 & 0.02281 \\ \hline 308 & 0.01866 \\ \hline 313 & 0.02155 \\ \hline 318 & 0.03372 \\ \hline 328 & 0.03765 \\ \hline 338 & 0.05201 \\ \hline 348 & 0.06416 \\ \hline 358 & 0.09250 \\ \hline 368 & 0.1184 \\ \hline 378 & 0.1283 \\ \hline 388 & 0.1736 \\ \hline 398 & 0.1988 \\ \hline \end{tabular} This data has been plotted in the cell below for your convenience. The ultimate goal of the company is to develop a model by which they can estimate the yield of their process at a variety of temperatures, and so they would like to find a description of the rate constant as a function of temperature. Part 1 - Interpolation of the data (35 points) The first approach of the R\&D team is to do an interpolation of the data as they are reasonably confident in the measurements they took and think that interpolation will yield the most accurate results describing the data. In this first part of the assignment, you will use interpolation to investigate if this approach is advisable. You can decide on using Newton or Lagrange polynomials. - Based on the data available, what would be the order of the polynomial? (2 points) - In the cells below, write a code for a polynomial fit (you can choose which one you use) of the above data. Using this fit, plot the data and your fit. (10 points) - Comment on the applicability of the global polynomial interpolation. (5 points) - With this fit, calculate at what temperature the reaction rate constant equals 0.100s1. Also calculate at what temperature the reaction rate is 0.150 and 0.175s1 (8 points) - NOTE: For this, you may use root-finding methods we have discussed in Section 2. - Comment on your results for these calculations and potential ways to prevent the problems that you are encountering in the previous question. (10 points) \# Write your interpolation code here Python \# Use this cell to plot the data and your polynomial and to the required calculations Python Answer cell for part 1 - The order of the polynomial is [YOUR ANSWER HERE] - When using this type of curve-fitting for the data available, we observe [COMMENT ON THE APPLICABILITY OF THE USED METHOD] - Base on the current fit, fill the following table - Comment on your ability to fill the table with the current curve fitting approach, the reliability of the answers and ways to increase the reliability or solve some of the encountered problems. After evaluation of the above data, the R&D engineers question the applicability of interpolation on the data they have collected. They move to a least squares regression method to get a better description of the measured data. They want to do both linear regression and polynomial regression to compare the suitability of the methods. - Write a method for using least squares polynomial regression on the given data. (7 points) - Test different orders of polynomial fit and determine the goodness of the fit by calculating the sum of residuals and r2 for the different order polynomials. ( 5 points) - Tip: write a for loop to test different orders Remember that the sum of residuals is equal to: Sr=i=1nei2=i=1n(yi,datayi,fit)2 - Plot the second order polynomial and find the temperatures for which the reaction rates are 0.100,0.150,0.17 : and 0.210s1 respectively based on this second order polynomial. (8 points) - NOTE: For this, you may use root-finding methods we have discussed in Section 2. - Comment on your results for these calculations and potential ways to prevent the problems that you are encountering in the previous question. ( 10 points) \# Write your regression code here \# Use this cell to plot your data and do the required calculations Python - For a polynomial fit of order n, the sum of residuals Sr equals: - Comment on your ability to fill the table with the current curve fitting approach, the reliability of the answers and ways to increase the reliability or solve some of the encountered problems. The rate constant of a chemical reaction k (units are dependent on what order the reaction is) is dependent on temperature, where typically increasing the temperature increases the reaction rate, meaning that the reaction goes faster upon increasing the temperature. At a company X, the Research \& Development department, which does not have a member with a Chemical Engineering background, has done experiments to measure the rate constant of a first order reaction at different temperatures. The data they found from their experiments is as follows: \begin{tabular}{|l|l|} \hline Temperature [K] & k[1/s] \\ \hline 298 & 0.01575 \\ \hline 303 & 0.02281 \\ \hline 308 & 0.01866 \\ \hline 313 & 0.02155 \\ \hline 318 & 0.03372 \\ \hline 328 & 0.03765 \\ \hline 338 & 0.05201 \\ \hline 348 & 0.06416 \\ \hline 358 & 0.09250 \\ \hline 368 & 0.1184 \\ \hline 378 & 0.1283 \\ \hline 388 & 0.1736 \\ \hline 398 & 0.1988 \\ \hline \end{tabular} This data has been plotted in the cell below for your convenience. The ultimate goal of the company is to develop a model by which they can estimate the yield of their process at a variety of temperatures, and so they would like to find a description of the rate constant as a function of temperature. Part 1 - Interpolation of the data (35 points) The first approach of the R\&D team is to do an interpolation of the data as they are reasonably confident in the measurements they took and think that interpolation will yield the most accurate results describing the data. In this first part of the assignment, you will use interpolation to investigate if this approach is advisable. You can decide on using Newton or Lagrange polynomials. - Based on the data available, what would be the order of the polynomial? (2 points) - In the cells below, write a code for a polynomial fit (you can choose which one you use) of the above data. Using this fit, plot the data and your fit. (10 points) - Comment on the applicability of the global polynomial interpolation. (5 points) - With this fit, calculate at what temperature the reaction rate constant equals 0.100s1. Also calculate at what temperature the reaction rate is 0.150 and 0.175s1 (8 points) - NOTE: For this, you may use root-finding methods we have discussed in Section 2. - Comment on your results for these calculations and potential ways to prevent the problems that you are encountering in the previous question. (10 points) \# Write your interpolation code here Python \# Use this cell to plot the data and your polynomial and to the required calculations Python Answer cell for part 1 - The order of the polynomial is [YOUR ANSWER HERE] - When using this type of curve-fitting for the data available, we observe [COMMENT ON THE APPLICABILITY OF THE USED METHOD] - Base on the current fit, fill the following table - Comment on your ability to fill the table with the current curve fitting approach, the reliability of the answers and ways to increase the reliability or solve some of the encountered problems. After evaluation of the above data, the R&D engineers question the applicability of interpolation on the data they have collected. They move to a least squares regression method to get a better description of the measured data. They want to do both linear regression and polynomial regression to compare the suitability of the methods. - Write a method for using least squares polynomial regression on the given data. (7 points) - Test different orders of polynomial fit and determine the goodness of the fit by calculating the sum of residuals and r2 for the different order polynomials. ( 5 points) - Tip: write a for loop to test different orders Remember that the sum of residuals is equal to: Sr=i=1nei2=i=1n(yi,datayi,fit)2 - Plot the second order polynomial and find the temperatures for which the reaction rates are 0.100,0.150,0.17 : and 0.210s1 respectively based on this second order polynomial. (8 points) - NOTE: For this, you may use root-finding methods we have discussed in Section 2. - Comment on your results for these calculations and potential ways to prevent the problems that you are encountering in the previous question. ( 10 points) \# Write your regression code here \# Use this cell to plot your data and do the required calculations Python - For a polynomial fit of order n, the sum of residuals Sr equals: - Comment on your ability to fill the table with the current curve fitting approach, the reliability of the answers and ways to increase the reliability or solve some of the encountered problems