Substituting the right-hand side of eq 3 into eq 2 we obtain:

Transcription

1 ph, pk, Solver, nd ll Tht This hndout contins four prts. In Prt I we derive n expression tht reltes the erved sornce of molecule (e.g., dye, cid-se indictor, or fluorescent molecule to the frctionl sornce of oth the cidic nd sic forms of the molecule, the pk of the molecule, nd the ph of the solution. In Prt II we show how one uses the Solver lgorithm of the spredsheet progrm Excel to fit experimentl dt to the nonliner expression derived in Prt I. In Prt III we derive different expression tht llows for liner, lest-squres fit of the dt. Finlly, in Prt IV we show how to fit experimentl dt to n n th order polynomil nd how to use the 2 nd derivtive of the n th order polynomil to help us estimte the prmeter of interest (i.e., pk. Prt I: Derivtion In the following derivtion we will ssume tht we re working with molecule (e.g., n cid-se indictor tht hs n ionizle functionl group. The erved sornce of this molecule or compound ( t some totl concentrtion c equls the sornce of the cidic form ( times the frction of the compound tht is in the cidic or protonted form (f plus the sornce of the sic form ( times the frction of the compound tht is in the sic or unprotonted form (f. f f ( We cn rewrite eq s follows: [In ] [In ] (2 [In ] [In ] [In ] The right-hnd-most side of eq 2 contins the rtio of the unprotonted or sic form of the indictor to the protonted or cidic form. We cn express this rtio in n lterntive form, which is rerrngement of the Henderson-Hssellch eqution, s follows: [ In ] phpk 0 ( (3 Sustituting the right-hnd side of eq 3 into eq 2 we otin: (phpk 0 (4 (phpk 0 Eq 4 is in form tht is convenient to use in spredsheets ecuse we void potentil lgorithmic prolems of estimting rtios of very smll numers. Becuse eq 4 is

2 nonliner expression, we need to use nonliner, lest-squres fitting to fit experimentl dt to eq 4. To ccomplish this nonliner, lest-squres fitting, we will use the Solver lgorithm in the spredsheet progrm Excel, which is the suject of Prt II of this hndout. Prt II: Using the Solver lgorithm in Excel In n experiment one collects dt of the compound of interest (t prticulr wvelength t different ph vlues, s shown columns nd B of the portion of the spredsheet shown in Figure. (Plese note tht the compound used in the following exmple is not the sme s the one you will use in your l so you should not expect the dt for this compound to mtch the dt you otin in l; lso, it is ssumed tht the totl concentrtion of the compound is constnt in ech of the smples, throughout the entire experiment. B C D E F G ph clc ( - clc 2 Prmeter E-05 = = pk = sum = Figure. Portion of n Excel spredsheet used to clculte the chnge in erved sornce ( s function of ph. Before we finish descriing ll of the fetures of the spredsheet shown in Figure, we should plot the dt to get visul impression of how our dt pper. n exmple of plot is shown in Figure 2. 2

3 ph Figure 2. Plot of the vlues s function of ph. To construct this plot, one highlights cells 2 through 8 (the ph vlues nd cells B2 through B8 (the vlues nd then selects: Insert nd then Chrt nd then XY Sctter nd then Next. When one selects Next new screen will pper, nd on this new screen one should see tht the defult of columns hs een selected, which correctly indictes tht dt (in this exmple re rrnged in columns. When one selects Next gin, one then hs the option to lel the x- nd y-xes nd finlly the option to select Finish. n inspection of the plot shown in Figure 2 revels tht the inflection or midpoint of the dt seems to correspond to ph of out 7. The inflection point corresponds to the pk of the indictor, nd we will use this informtion to enter our guess of the vlue of the pk in cell G4 of the portion of the spredsheet shown in Figure. The Solver lgorithm requires tht we enter n initil guess or estimte of the vlue(s of the prmeter(s tht we wnt the lgorithm to estimte through the nonliner, lest squres fitting. This initil guess needs to e resonle so tht the lgorithm will converge upon solution. This requirement for resonle initil guess demonstrtes the importnce of visully inspecting one s experimentl dt. In the present exmple, our initil guess of the vlue of the pk ws 7, nd this vlue ws entered into cell G4 of the spredsheet shown in Figure. Note tht the vlue in cell B2 is used s our initil guess for the vlue of (cell G2 nd the vlue in cell B8 is used s our initil guess for the vlue of (cell G3. To clculte the clc vlue in cell C2 of the spredsheet shown in Figure, one enters the following function sttement: =($G$2+$G$3*0^(2-$G$4/(+0^(2-$G$4. Note how this function sttement corresponds to the right-hnd side of eq 4. The estimtes of,, nd pk, cells G2, G3, nd G4, respectively, hve dollr symols ($ rcketing their column nd row positions. This rcketing of the column nd row positions with dollr symols designtes these cells s solute references such tht the vlues in these cells re used repetedly in the susequent clcultions. Note lso tht the ph (cell 2 does not hve the $ symols entered with it. By leving out the $ symols we indicte tht this cell is reltive reference nd its vlue will not e used repetedly when the remining vlues in column C re clculted. Insted, the ph vlue in the cell elow 3

4 2 (i.e., the ph represented in cell 3 will e used to clculte the vlue tht is entered into cell C3, nd the vlue in the cell elow 3 (i.e., 4 will e used to clculte the vlue tht is entered into cell C4, nd so on. To fill-in the remining vlues in column C, one highlights cell C2 nd then moves the cursor to the lower right-hnd corner, t which time the ppernce of the cursor chnges from n open plus sign to closed plus sign. When you see the chnge in ppernce of the cursor, you cn drg the cursor down the column nd the pproprite vlues will e clculted. For exmple, if one highlights cell C3 (fter filling in column C then one would see: =($G$2+$G$3*0^(3- $G$4/(+0^(3-$G$4. Column D of Figure is used to clculte the squred differences etween the erved nd clculted sornce vlues. For exmple, the function sttement in cell D2 is: =(B2-C2^2. The remining cells in column D (down to cell D8 re filled-in s discussed ove for column C. Cell D0 contins the sum of the squred devitions of cells D2 through D8. The function sttement in cell D0 ws =sum(d2:d8. The vlue of cell D0 (i.e., the sum of the squred devitions of erved nd clculted vlues is wht we wnt the Solver lgorithm to minimize; we wnt the Solver lgorithm to minimize this sum y chnging the vlue(s of the prmeter(s in column G of Figure. To run Solver, we first highlight cell D0 nd from the menu we select Tools nd then Solver. If one does not see Solver then it needs to e dded to the Tools menu. To dd Solver, one selects dd-ins from the Tools menu nd then selects Solver dd In. Plese note tht the Solver ddition is not ville on some versions of Excel run y Mcintosh computers, so you will hve to use PC in one of the computer centers on cmpus or otin the ptch mentioned in the URL tht ws posted on D2L. Once Solver hs successfully een dded to the Tools menu, it should pper ech susequent time the Tools menu is chosen. fter we select Solver new window should open. We should then see $D$0 under the heding Set Trget Cell. This solute reference ($D$0 ws the cell tht we left highlighted prior to selecting Solver. Next to the heding Equl to: we should see the options Mx nd Min, nd we will select Min since we wnt Solver to minimize the sum of the squred devitions. We lso will see By chnging cells: elow which is mini-window. We cn either select the icon on the right of this mini-window, which will llow us to highlight cells G2, G3, nd G4, or we cn type $G$2:$G$4 in the window. Once we hve completed these steps, we will select Solve, nd then we will see new window, which should indicte tht Solver hs found solution. We lso should see tht the defult Keep Solver Solution hs een selected. We will click OK nd then we will see tht the estimtes we entered into cells G2, G3, nd G4 hve chnged, long with the respective clc nd ( clc 2 vlues in columns C nd D. ccordingly, the sum highlighted in cell D0 will hve een minimized, which is the gol of determining the est-fit vlues. The vlues in cells G2, G3, nd G4 re the fitted estimtes of,, nd pk, respectively. (lterntively, we could hve held fixed the nd vlues nd used Solver to minimize the sum in cell D0 y chnging only the pk estimte in cell G4. The ppernce of the spredsheet fter running Solver s descried ove is shown in Figure 3. Note tht our initil guess of 7 for the estimte of the pk ws pretty close to the fitted vlue of 6.99 in cell G4 of Figure 3. 4

5 B C D E F G ph clc ( - clc 2 Prmeter = = pk = E sum = Figure 3. Portion of n Excel spredsheet fter the Solver lgorithm ws run to minimize the sum in cell D0. The lst thing we should do is to show how well our experimentl dt mtch our theoreticl expression. We do this y highlighting cells 2 through 8, B2 through B8, nd C2 through C8 of Figure 3 nd following the steps descried ove for the construction of the plot in Figure 2. When we re done we will hve two sets of dt points. By defult Excel will cll the vlues Series nd the clc vlues Series 2. We wnt to keep the Series dt s discrete points, ut we do not wnt the Series 2 dt to remin s discrete points. Rther, these vlues re our est-fit line; thus, we do the following. We right-click on Series 2 dt point in the figure, nd then we left-click on Formt Dt Series, which ppers in new window. When the next window ppers, we select None under Mrker nd we select utomtic or Custom under Line. When we re done, our figure should pper s shown elow in Figure ph Figure 4. plot of the erved sornce s function of ph with the estfit line tht ws determined using the Solver lgorithm in Excel. 5

6 The est-fit line in Figure 4 looks little choppy. Given the nture of our theoreticl expression, it should e smooth. If we wnt to get Excel to do etter jo of representing the est-fit line, we need to enter the vlue of 4 in cell 9 of our spredsheet (Figure 3. Then, in cell 0 type =9+0.0 nd hit enter. We should see the vlue 4.0 entered in cell 0. We select cell 0 nd move the cursor to the lower right-hnd corner until we see the ppernce of the cursor chnge from n open plus sign to closed plus sign, t which time we cn fill down this column until we rech the vlue of 0.8. Then, we plce our cursor on cell C8 nd continue filling down this column until we rech the lst of the newly filled-in ph vlues in column. Now, we construct the figure similr to wht ws descried ove, ut this time we will hve to specify which dt elong to Series nd which elong to Series 2. When we do this, we get much etter est-fit line, s shown in Figure ph Figure 5. plot of the erved sornce s function of ph with more refined est-fit line s descried in the text. Note tht lmost ll of the dt points in Figure 4 (nd in Figure 5 fll close to the estfit line, which is good sign tht the theory we developed ove ws pproprite to the tsk of estimting the pk of the compound. One of the downsides to using the Solver lgorithm is tht it does not provide (i error estimtes of the fitted prmeters, nd (ii n estimte of the coefficient of determintion or r 2 vlue, which is mesure of the extent to which the fitting ccounted for or explined the vrince in our experimentl dt. In clss nd in l we will get round the first of these drwck y collecting two sets of dt t two different wvelengths. We will use the two sets of dt to clculte the verge pk (t prticulr wvelength for ech set. We will verge the two estimtes of the pk collected t one wvelength, nd then we will verge the two estimtes of the pk collected t the other wvelength. We lso will clculte the stndrd devition for ech verge. The etter wy to pproch this is to fit ll of the dt simultneously with more sophisticted softwre progrm tht provides the error estimtes. We will del with the second of the ove drwcks (i.e., the lck of n estimte of the coefficient of determintion or r 2 vlue in future ssignment. Once gin, more powerful softwre progrm would provide this estimte s well. 6

7 Prt III: Derivtion of Liner Expression. Since the compound in question is either protonted or unprotonted, the sum of the frctionl forms of the compound must equl unity (f + f =, nd we cn rewrite eq s follows: f f (5 ( n lgeric mnipultion of eq 5 yields: ( f ( ( (6 [In ] [In ] ( Some dditionl lgeric rerrngements led to: [In ] ( ( ( ( (7 If we tke the log (se 0 of oth the left- nd right-hnd-most sides of eq 7 we otin: [In ] ( log log ( (8 The left-hnd side of eq 8 is prt of the fmilir Henderson-Hssellch (HH eqution. If we sustitute the right-hnd side of eq 8 into the HH eq nd rerrnge, we hve: ( log ( ph pk (9 Eq 9 is in the form of stright line where the left-hnd side is nlogous to the dependent vrile y, ph is the independent vrile, the slope m is equl to unity, nd the y-intercept is equl to pk. Thus, eq 9 is our sought-fter liner expression. Now tht we hve our desired expression, we look t our exmple dt (Figure nd use the t ph 4 (.9 for nd t ph 0.8 (0.29 for. Since we hve discussed how to use Excel to determine the liner lest-squres est-fit line, I will leve tht up to you. Plese note, however, tht oth the y-intercept nd the x-intercept of eq 9 should provide estimtes of the pk nd these estimtes idelly should gree i.e., the estimte of the slope should e close to unity. ny devitions from unity will cuse the two estimtes not to gree with ech other. There re some other cvets to using eq 9, which you will discover in the course of fitting the dt from Figure or when you fit your ctul experimentl dt to eq 9. I will let you discover wht these cvets re. Tht is prt of the lerning process tht I wnt you to experience. 7

8 Prt IV: Fitting Experimentl Dt to n n th Order Polynomil. n inspection of the dt in Figure 2 shows trnsition or inflection point where the slope of the slope of n nticipted est-fit line chnges s we move from ph 6 to ph 8. In fct, it ecomes more nd more negtive (steeper s we pproch the inflection point, nd then it grdully ecomes less nd less negtive fter we pss the inflection point. t the point of trnsition or inflection the slope of the slope psses through zero, which we erve vi sign chnge in the second derivtive of est-fit, n th order polynomil. Since we nticipte tht the inflection point is etween ph 6.5 nd 7.8 we highlight cells 4 through 7 nd B4 through B7 of the spredsheet shown in Figure nd copy nd pste the vlues in these cells into new worksheet in Excel. We then crete new chrt s shown in Figure y = x x x R 2 = ph Series Poly. (Series Figure 6. Plot of the erved sornce s function of ph for select ph vlues. The est-fit line is to 3 rd order polynomil, the eqution of which is shown, long with the coefficient of determintion (r 2 or R 2. The est-fit line in Figure 6, which Excel clls Trendline, is otined y right-clicking on dt point nd selecting dd Trendline. We then select Polynomil nd specify 4 th order. Finlly, we select Options nd check the oxes next to Disply eqution on chrt nd Disply R-squred vlue on chrt. Note tht the eqution shown is tht of 3 rd order polynomil rther thn the specified 4 th order polynomil. The 3 rd order polynomil is shown ecuse the fit to the dt ws perfect s indicted y the R-squred vlue of unity. To use the resulting eqution to determine where the inflection point occurs, we crete new column of ph vlues tht run from ph 6.5 to 7.8 with 0.0 increments in etween s descried ove in reference to Figure 5. For simplicity, let us sy tht these vlues run from cell B Figure 7. Portion of n Excel sheet in which the second derivtive of the 3 rd order polynomil is used to estimte the pk of the compound. 8

9 through 3. Then, in the cell B we type =6*-0.575* + 2*.304. This eqution is the second derivtive of the eqution shown in Figure 6. We then fill down column B nd note when the sign chnges etween two cells. portion of the spredsheet is shown in Figure 7. The sign chnge in column B occurs etween cells B79 nd B80. The ph vlues in the djcent cells (79 nd 80 re 7.28 nd 7.29, so the pk is etween these two vlues. Since the vlue in cell B79 is much closer to zero thn is the vlue in cell B80, we estimte tht the pk is closer to 7.28, rther thn the verge of 7.28 nd Note tht this estimte of the pk is resonly close to the estimte of 6.99 determined ove, ut I wnt you to think out why these estimtes do not gree more closely. You will follow this procedure (long with the other, ove-mentioned procedures in Prts I III when you perform the nlogous experiment with the fluorescent molecule fluorescein in n upcoming l. 9