Fitting a Polynomial to Heat Capacity as a Function of Temperature for Ag. by

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1 Fitting a Polynomial to Heat Capacity as a Function of Temperature for Ag. by Theresa Julia Zielinski Department of Chemistry, Medical Technology, and Physics Monmouth University West Long Branch, J tzielins@monmouth.edu Copyright Theresa Julia Zielinski All rights reserved. You are welcome to use this document in your own classes but commercial use is not allowed without the permission of the author. Place in the Curriculum: This document fits into the thermochemistry section of a standard physical chemistry course. It might also be used as part of the material covered when discussing the 3rd law of thermodynamics. This document could easily be extended by the user to include integration techniques and evaluation of 3rd law entropies. Prerequisites: 1. Basic functional skill with the Mathcad program. Mathcad 2001i 04 higher. 2. Understanding of mathematical concepts: polynomials, matrix methods for solving a set of simultaneous equations. The background information describing the mathematics and matrix approach are contained in the fitting.pdf file that accompanies this Mathcad module. A full deep understanding of the matrix method is not a necessary requirement for being able to use this document. 3. Introduction to the concepts of the heat capacity and heat capacity as a function of temperature. Goal: This document demonstrates the method of fitting a polynomial of any reasonable power to a set of data. It provides you with an introduction to non-linear curve fitting and the opportunity to develop skills for doing curve fitting with Mathcad. It also permits exploration of the concepts of measuring the goodness of fit and creates the situation where you are asked to ustify a choice by using quantitative statistical arguments. The document shows an example of how to use matrix methods to solve simultaneous equations in physical chemistry. Student Tasks: 1. Interactively work through the material presented here. Examine each part of the document and relate the Mathcad implementation to the equations described in the text file, fitting.doc that accompanies this document. 3. Complete the exercises at the end of this document. 4. Present your results in the form of an annotated computer generated Mathcad document. Page 1

2 Performance Obectives: When you are finished with this document you will be able to: 1. Import a file of comma delimited data into a variable in a Mathcad document. 2. Use matrix methods for manipulation of data in a curve fitting procedure. 3. Choose an appropriate polynomial fit for the data and ustify that choice on the basis of defensible criteria for goodness of fit. 4. Compute the residuals and variance for a curve fitting procedure. 5. Plot the fitted curve and experimental data with suitable scaling and detail. 6. Write the polynomial for the fit using the coefficients provided by the fitting process. Mathcad otes: In Mathcad you may insert comma delimited files into arrays. There are two ways to do this. First, the comma delimited file can be associated with a variable name. This is accomplished by pulling down the File menu and choosing Associate. There you will see that space is provided for filling in the Mathcad variable name and for identifying the comma delimited file. Only one file may be associated with any variable name. The alternative is to use the READPR function and have the file containing the data stored in the same directory or on the same diskette as the Mathcad document that requires the data. After the file name and variable are associated or the data file placed into the appropriate directory, Mathcad can be started and the file then read into the array with the READPR(filename) command equated to the array name. Filename is the name of a filename.prn file on the diskette or hard disk. If you use the Associate option then the extension of the file called filename may be anything you wish. In the example below AG is the variable name. CP is the array name. The linked file is a comma delimited file called silver.csv. If you wish to use this feature for other substances you must prepare your comma delimited file separately. This is easy to do. Many spreadsheet programs will allow you to export files as comma delimited, CSV, files. Alternatively these can be created by using a text editor and saving the data as an ASCII file or typing the data directly into a Mathcad document. You should practice these methods in order to increase their range of computer skills within the context of chemistry rich topics. Page 2

3 The polynomial fit procedure starts here. P := 19 P = the number of data points. If you create a cvs file with more data points then you need to change this number for this document to work. If there are too few data elements in your cvs file the program will not work. Always check compatibility of file and array sizes. := P 1 i := 0, 1.. i = the index used to label each data point. In this example there are 19 data points. = the number of degrees of freedom. CP := C:\..\silver.csv Here the csv file is loaded into the document. The file silver.csv was created using Excel. It was then saved from Excel as a comma delimited text file. One gets the cvs file into Mathcad by Insert/Component/FileReadOrWrite/Browse. Then chose the cvs file you created with Excel. ote the form of the expression that reads in the silver data file. 0 1 ote the data in the array CP CP = T := i C := i CP i0, CP i1, C = Each line in CP is a data set. The first number is the temperature and the second is the heat capacity at that temperature. These are assigned to the one-dimensional arrays (vectors) T and C. T i and C i are individual elements of these arrays. T is the temperature and C is the heat capacity at that temperature. Only some of the points are shown to the left. All are present and each one can be viewed by changing the subscript on the C 11 term ust to the left. ote: Mathcad starts arrays with the first index equal to zero. This notation is maintained in this document but not in the accompanying text document. Be careful not to confuse this with the subscript used for the polynomial fitting parameters Page 3

4 The task before us is to examine the data and to determine the best polynomial expression that represents the data. We can start by fitting the data to a cubic equation. (Refer to class notes in the file fitting.doc or fitting.rtf where least squares fitting of polynomials to data is outlined. The variable names and matrix names are not the same here as in the notes.) m:= 2 The m=2 equation here is toggled off. The equation that actively defines m is given below near the graph of the polynomial. This permits easy experimentation later in the document without the need to scroll through the document many times. The global variable assignment m=2 was used for m below. Later you will experiment with this value to get a good polynomial fit for the data. := 0.. m k := 0.. m and k run from 0 to m where m is the order of the polynomial to be fit to the data. For example a quadratic equation has m=2 but will have 3 fitting parameters. By changing m below you can examine the effect of the order of the polynomial on the goodness of fit. The graph and fitting parameters are shown at the end of this document. := A k, ( T i ) ( T i) b := i = 0 i = 0 a := A 1 b C T i ( i) These define the elements of the matrix used in the polynomial fit procedure. The column vector containing the parameter values is called 'a.' The matrix equation that solves for 'a' is written ust as you see it written on paper in the notes for this exercise. Page 4

5 f( T, a) := m = 0 a ( T) This is the general expression for the polynomial that is fitted to the data. f(t,a) is the heat capacity at temperature T. By varying T the appropriate heat capacity is calculated. (Caution: if you set T to a specific value here in the document several operations below will be damaged.) ext it is necessary to determine the goodness of fit of the the polynomial to the data and the variance in the fitting parameters. ote how the the equations from the notes are used. Ra ( ) := i = 0 ( C f T, a i ( i )) 2 Calculating the Residual (R). otice that R is a function of the fitting parameters. Ra ( ) ssq := This is the variance of the fit. (ssq) m ssq = This is the numerical value of ssq. Every time you change the order of the polynomial you will get a new value of ssq. The important point here is that you can choose the best polynomial. What criteria will you use? There are at least two important ones. ( ), SSQ := ssq A 1 The variance for individual parameters are calculated too. (ote that Mathcad recognizes that ssq and SSQ are two different variable names.) SSQ = The variance in a parameter, SSQ 1 is shown to the left. This is the variance for the a 1 parameter. Students should vary the subscript to find the variance of the other parameters. Results must be recorded in a note book. Page 5

6 Here is the graph showing the fit of the function, solid line, and the data as points. To the lower left is the parameter column vector. You may change the value of m here and explore the fitting as the order of the polynomial increases. m 2 30 a = C i k a k ( ) k T i ssq = ssq is repeated here for convenience T i Closure Activities: 1. Write the full expression for the polynomial with the best fit. 2. Explain why you chose this polynomial. 3. Tabulate the variances for each parameter used to fit the polynomial. 4. Tabulate the goodness of fit for each polynomial tried during the exercise. 5. Be prepared to explain the process used in this exercise during an exam. Reference: Data Reduction and Error Analysis for the Physical Sciences by Philip R. Bevington McGraw-Hill Book Company Y 1969 Acknowledgment: Partial support for this work was provided to TJZ by the ational Science Foundation's Division of Undergraduate Education through grant DUE # and by the ew Traditions proect at the University of Wisconsin - Madison through the ational Science Foundation's Division of Undergraduate Education through grant DUE # Page 6

Non-Linear Least Squares Analysis with Excel

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