Fitting a Polynomial to Heat Capacity as a Function of Temperature for Ag. Mathematical Background Document

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1 Fttng Polynol to Het Cpcty s Functon of Teperture for Ag. thetcl Bckground Docuent by Theres Jul Zelnsk Deprtent of Chestry, edcl Technology, nd Physcs onouth Unversty West ong Brnch, J tzelns@onouth.edu Theres Jul Zelnsk, 996, 997, 998. You re welcoe to use ths docuent n your own clsses but coercl use s not llowed wthout the persson of the uthor. Fttng thetcl functon to experentl dt s useful technque for representng dt wth n nlytcl functon. To do ths we need to dentfy the type of functon nd procedure for optzng the fttng preters n the functon. Polynol expressons y be sutble choce n ny cses. Polynols lso provde n esy begnnng for exnng the bsc procedure for fttng functon to set of dt. The obectve s to choose polynol expresson such tht the su of the squres of the devtons between the functon nd the dt re nzed. Here we wll use the flr lest squres procedure tht ost students lern bout n the physcl chestry lbortory. The thetcs for ths s reltvely strght forwrd nd cn be found n ost physcl chestry texts or lbortory nuls. Consder polynol f ( x) x x x x () where,,, up to re the fttng preters, nd s the order of the polynol. If you hve set of dt consstng of x nd y vlues, then you cn clculte n f(x) for ech x vlue,.e. f(x ). The f(x ) cn then be copred to the y vlues. If the f(x ) nd y gree for every x then you hve functon f(x) tht fts the dt. We use sttstcs to deterne the degree of goodnes of ths ft. A frst pproch s to use the ethod of lest squres. Ths s the ethod tht wll be developed n these notes. Exercse : For qudrtc functon the coeffcents re.5,., nd -4.5x -4. Copute f(x ) for the followng prs of x, y dt: 3, 4.8; 5,.7; 7, 6.3; 9, 9.. The ethod of lest squres requres tht the su of the squres of the devtons,.e. squre of the resduls, be nzed. Ths s defned s: [ ] R y f( x ) () where f(x ) s the vlue of f(x) for dt pont nd there re dt ponts n the experentl set. The experentl dt re tbulted n prs of x nd y vlues. f(x ) s the coputed vlue of the property y nd t s copred to the experentl vlue y to obtn the resdul. R s the resdul functon. It s the su of the squres of the devtons nd s soetes clled SSD n thcd docuents. It doesn t tke Creted: Februry 996 fttng.doc Author: Theres Jul Zelnsk odfed: July, 998 Pge

2 long to get used to the for of ths functon so tht you wll be ble to recognze t no tter wht ne s hs. The dscusson tht follows does not consder the experentl weght of the dt ponts y. Exercse : Copute R for the dt n Exercse.. nzton of the resduls s ccoplshed by stndrd pplcton of clculus. We wnt to fnd the best set of fttng preters, nely those whch nze R. Therefore we tke the dervtve wth respect to ech fttng preter nd set t equl to zero s shown n equton 3.,,, etc. (3) The result s set of sultneous equtons, one for ech coeffcent. Soe of these equtons re: ( y x x ) ( ) x x ( ) x x (4) where there s one equton for ech fttng preter,.e. equtons becuse the powers of x rnge fro to n the polynol. The suton s tken over the,, 3, dt ponts. In thcd ths y pper s,,, 3. - dependng on how the rry nuberng s set n the docuent you re usng. Exercse 3: Show tht the equtons n (4) coe fro the equtons n (3). Rerrngng (4) gves: ( x x ) 3 ( x x x ) 3 ( ) y y x x x x y x 4 (5) Systes of lner equtons lke ths re best represented by trx notton. Creted: Februry 996 fttng.doc Author: Theres Jul Zelnsk odfed: July, 998 Pge

3 Creted: Februry 996 fttng.doc Author: Theres Jul Zelnsk odfed: July, 998 Pge 3 A b (6) otce the bold font on the coponents of equton (6). Ths ndctes tht these re trces. trx A s squre syetrc rel trx. b nd re colun vectors. We see ther explct for here n equtons 7 nd 8. Proble: Check your skll wth trx npultons by showng tht the trx equton (6) corresponds to representton of the set of sultneous equtons shown n equton (5). Hnt: use the trx nd vectors n equton (7) nd (8). A 3 (7) nd b y (8) The eleents n trx A re esly wrtten by usng the followng expresson:

4 A,k ( x ) ( x ) k (9) where s the row nd k s the colun. nd k correspond to the subscrpt of the coeffcent used n the polynol expresson. Do not confuse ths wth the rnge nteger,, tht dentfes the dt ponts. nd k rnge fro to,.e. over the totl nuber of fttng preters. The defult ntl ndex for vectors nd rrys s zero n thcd. If our dt were represented n ths defult then would lso run fro to - nd the trx n equton (7) nd the vectors n equton (8) would need to be rewrtten. It s portnt to be flexble but ccurte wth our use of the rnge vrble. In lke fshon we wrte the vector b n equton () ( ) () b y x where b s row eleent n the colun vector b. There re eleents n the vector b nd eleents n the vector. The trx A s by squre. Through trx lgebr we cn solve for the vector A. Be creful bout how you ultply trces. A - A A b A b () The vector contns ll of the fttng preters n the polynol expnson. We obtn the vrnce of the ft nd the vrnce of the fttng preters by: R σ σ σ A () where,,... nd R s the su of the squres of the devtons t the end of the fttng process. Queston : Wht crter would you use to deterne whch polynol you would choose to represent set of dt? The docuent Slver.cd s thcd pleentton of polynol curve fttng. Ths docuent, the thcd docuent, nd sple dt re t the thcd WWW ste or dskette provded by your nstructor. The thcd docuent shows vrous opertons but you should frst experent wth vryng the order of the polynol nd observng the ft of the functon to the dt. Record your observtons n your note book. After you hve deonstrted copetence wth the requred obectves of ths lesson you Creted: Februry 996 fttng.doc Author: Theres Jul Zelnsk odfed: July, 998 Pge 4

5 ght lke to try other sets of dt or functons to lern ore bout curve fttng. Other thcd docuents on the WWW ste deonstrte lterntve curve fttng pproches. Queston : How does the goodness of ft prove wth ncresng order of the fttng polynol? Queston 3: How does one choose the best polynol to represent the dt? Queston 4: Use your textbook or nother text to copre the functons used to represent the het cpcty of terls s functon of teperture. Queston 5: Wht s the crter would you use for choosng to use one functon or nother to use when fttng polynol to het cpcty dt? References: Johnson, K. Jeffrey, uercl ethods n Chestry ; ew York: rcel Dekker, Inc., 98. Bevngton, Phlp R., Dt Reducton nd Error Anlyss for the Physcl Scences ; ew York: cgrw Hll Book Copny, 969. Hng, R. W., uercl ethods for Scentsts nd Engneers, nd Ed.;ew York: Dover Publctons, 986. Acknowledgent: Prtl support for ths work ws provded to TJZ by the tonl Scence Foundton's Dvson of Undergrdute Educton through grnt DUE # nd by the ew Trdtons proect t the Unversty of Wsconsn - dson through the tonl Scence Foundton's Dvson of Undergrdute Educton through grnt DUE # Creted: Februry 996 fttng.doc Author: Theres Jul Zelnsk odfed: July, 998 Pge 5