orig For example, if we dilute ml of the M stock solution to ml, C new is M and the relative uncertainty in C new is

Transcription

1 hapter 5 May of the problem i thi chapter require a regreio aalyi lthough equatio for thee calculatio are highlighted i the olutio to the firt uch problem, for the remaiig problem, both here ad elewhere i thi tet, the reult of a regreio aalyi imply are provided Be ure you have acce to a cietific calculator, a preadheet program, uch a Ecel, or a tatitical oftware program, uch a R, ad that you kow how to ue it to complete a regreio aalyi 1 For each tep i a dilutio, the cocetratio of the ew olutio, ew, i orig Vorig ew Vew where orig i the cocetratio of the origial olutio, V orig i the volume of the origial olutio take, ad V ew i the volume to which the origial olutio i diluted propagatio of ucertaity for ew how that it relative ucertaity i u u u V V u V V ew orig orig ew a k + a k + ` j ew orig orig ew For eample, if we dilute 100 ml of the 1000 M tock olutio to 100 ml, ew i M ad the relative ucertaity i ew i u ew ` 1000 j + ` 1 00 j + ` 10 0 j 94 # 10 ew The abolute ucertaity i ew, therefore, i uew ( # 10 M) #( 94 # 10 ) 94 # The relative ad the abolute ucertaitie for each olutio cocetratio are gathered together i the table that follow (all cocetratio are give i moll ad all volume are give i ml) The ucertaitie i the volumetric glaware are from Table 4 ad Table 43 For a V orig of 100 ml ad of 0100 ml, the ucertaitie are thoe for a µl digital pipet For a erial dilutio, each tep ue a 100 ml volumetric pipet ad a 100 ml volumetric flak; thu hapter 5 Stadardizig alytical Method M See hapter 4 to review the propagatio of ucertaity 39 ew orig V orig V ew u Vorig u Vew

2 40 Solutio Maual for alytical hemitry 1 u ew orig ew ew u ew For the et of oe-tep dilutio uig the origial tock olutio, each olutio require a differet volumetric pipet; thu ew orig V orig V ew u Vorig u Vew u ew orig ew ew u ew Note that for each ew, the abolute ucertaity whe uig a erial dilutio alway i equal to or better tha the abolute ucertaity whe uig a igle dilutio of the origial tock olutio More pecifically, for a ew of M ad of M, the improvemet i the abolute ucertaity i approimately a factor of, ad for a ew of M, the improvemet i the abolute ucertaity i approimately a factor of 6 Thi i a ditict advatage of a erial dilutio O the other had, for a erial dilutio a determiate error i the preparatio of the M olutio carrie over a a determiate error i each ucceive olutio, which i a ditict diadvatage We begi by determiig the value for k i the equatio Stotal k + Sreag where S total i the average of the three igal for the tadard of cocetratio, ad S reag i the igal for the reaget blak Makig appropriate ubtitutio 1603 k ( 1 0ppm) + 00

3 hapter 5 Stadardizig alytical Method 41 ad olvig for k give it value a ppm 1 Subtitutig i the igal for the ample ( ppm ) + 00 ad olvig for give the aalyte cocetratio a 733 ppm 3 Thi tadard additio follow the format of equatio 59 Samp Spike V o V Vo V Vtd + td f f V f i which both the ample ad the tadard additio are diluted to the ame fial volume Makig appropriate ubtitutio ml ml 5 00 ( ) ml 100 ml # # ml + ppm # 5 00 ml ppm 008 ad olvig give the aalyte cocetratio,, a 800 ppm The cocetratio of aalyte i the origial olid ample i 1 g ( mgl)( 50 L) c m 1000 mg -3 # # 10 % ww 1 00 gample Here we aume that a part per millio i equivalet to mgl 4 Thi tadard additio follow the format of equatio 511 Samp Spike Vo V V Vtd td o+ + td V o+ V td i which the tadard additio i made directly to the olutio that cotai the aalyte Makig appropriate ubtitutio ( )( ) ml 10 0ppm 1 00 ml 5 00 ml ml ml ml ppm ad olvig give the aalyte cocetratio,, a 191 ppm 5 To derive a tadard additio calibratio curve uig equatio 510 S k Vo V V Vtd pike a td o+ + td V o+ V k td we multiply through both ide of the equatio by V o + V td S ( V + V ) k V+ k V pike o td o td td how i Figure SM51, the lope i equal to k ad the y-itercept i equal to k V o The -itercept occur whe S pike (V o + V td ) equal zero; thu 0 kv o+ ktdvtd -itercept V o S pike (V o + V td ) lope k y-itercept k V o td V td Figure SM51 Stadard additio calibratio curve baed o equatio 51

4 4 Solutio Maual for alytical hemitry 1 a remider, for thi problem we will work through the detail of a uweighted liear regreio calculatio uig the equatio from the tet For the remaiig problem, it i aumed you have acce to a calculator, a preadheet, or a tatitical program that ca hadle mot or all of the relevat calculatio for a uweighted liear regreio ad the -itercept i equal to V o We mut plot the calibratio curve thi way becaue if we plot S pike o the y-ai veru td # " Vtd ( Vo+ Vtd), o the -ai, the the term we idetify a y-itercept kv o Vo+ Vtd i ot a cotat becaue it iclude a variable,v td, whoe value chage with each tadard additio 6 Becaue the cocetratio of the iteral tadard i maitaied at a cotat level for both the ample ad the tadard, we ca fold the iteral tadard cocetratio ito the proportioality cotat K i equatio 51; thu, uig S, S IS, ad for the tadard S S k 33 K K( 1 00 mgl) IS kis IS give K a 0665 Lmg Subtitutig i S, S IS, ad K for the ample ( Lmg) give the cocetratio of aalyte i the ample a 8 mgl 7 For each pair of calibratio curve, we eek to fid the calibratio curve that yield the mallet ucertaity a epreed i the tadard deviatio about the regreio, r, the tadard deviatio i the lope, b1, or the tadard deviatio i the y-itercept, b0 (a) The calibratio curve o the right i the better choice becaue it ue more tadard ll ele beig equal, the larger the value of, the maller the value for r i equatio 519, ad for b0 i equatio 51 (b) The calibratio curve o the left i the better choice becaue the tadard are more evely paced, which miimize the term i i equatio 51 for b0 (c) The calibratio curve o the left i the better choice becaue the tadard pa a wider rage of cocetratio, which miimize the term ( i - X) i equatio 50 ad i equatio 51 for b1 ad b0, repectively 8 To determie the lope ad the y-itercept for the calibratio curve at a ph of 46 we firt eed to calculate the ummatio term that appear i equatio 517 ad i equatio 518; thee are: i y i i yi i Subtitutig thee value ito the equatio 517 ( 6 # ) - ( # 131 0) b1 ( 6 # ) - ( 308) 4 477

5 hapter 5 Stadardizig alytical Method 43 give the lope a 477 M, ad ubtitutig ito equatio ( 477 # 308 4) b give the y-itercept a 69 The equatio for the calibratio curve i S total 477M # d -69 Figure SM5 how the calibratio data ad the calibratio curve To fid the cofidece iterval for the lope ad for the y-itercept, we ue equatio 519 to calculate the tadard deviatio about the regreio, r, ad ue equatio 50 ad equatio 51 to calculate the tadard deviatio i the lope, b1, ad the tadard deviatio i the y-itercept, b0, repectively To calculate r we firt calculate the predicted value for the igal, V y i, uig the kow cocetratio of d + ad the regreio equatio, ad the quared reidual error, ( yi -V y) ; the table below ummarize thee reult i i y i V y i ( yi -Vy) i ddig together the lat colum, which equal 798, give the umerator for equatio 519; thu, the tadard deviatio about the regreio i 798 r To calculate the tadard deviatio i the lope ad i the y-itercept, we ue equatio 50 ad equatio 51, repectively, uig the tadard deviatio about the regreio ad the ummatio term outlied earlier; thu 6# ( ) ( 6 # ) - ( 308) 4 b1 ( ) # ( 6 # ) - ( 308) 4 b With four degree of freedom, the cofidece iterval for the lope ad the y-itercept are b 1 b1! tb1 477!( 776)( 0 018) 477! 036 M S total () [d + ] (M) Figure SM5 alibratio curve at ph 46 for the data i Problem 58

6 44 Solutio Maual for alytical hemitry 1 reidual error [d + ] (M) Figure SM53 Plot of the reidual error for the calibratio tadard i Problem 58 at a ph of 46 S total () ph 37 ph [d + ] (M) Figure SM54 alibratio curve for the data i Problem 58 at a ph of 37 ad at a ph of 46 b 1 b0! tb0-69!( 776)( 0 758) -69! 01 (b) The table below how the reidual error for each cocetratio of d + plot of the reidual error (Figure SM53) how o dicerible tred that might caue u to quetio the validity of the calibratio equatio i y i Vy i yi -Vyi (c) regreio aalyi for the data at a ph of 37 give the calibratio curve equatio a S total 143 M # d -5 0 The more eitive the method, the teeper the lope of the calibratio curve, which, a how i Figure SM54, i the cae for the calibratio curve at ph 37 The relative eitivitie for the two ph i the ratio of their repective lope kph kph The eitivity at a ph of 37, therefore, i three time more eitive tha that at a ph of 46 (d) Uig the calibratio curve at a ph of 37, the cocetratio of d + i the ample i + ( ) [ ] Stotal b d M b1 143 M To calculate the 95% cofidece iterval, we firt ue equatio 55 d r 1 1 ^Samp - Stdh b m ( b1) ^tdi - to determie the tadard deviatio i the cocetratio where the umber of ample, m, i oe, the umber of tadard,, i i, the tadard deviatio about the regreio, r, i 86, the lope, b 1, i 143, the average igal for the oe ample, S amp, i 663, ad the average igal for the i tadard, S td, i 687 t firt glace, the term ( tdi - td), where tdi i the cocetratio of the i th tadard ad td i the average cocetratio for the tadard, eem i 1 td h

7 hapter 5 Stadardizig alytical Method 45 cumberome to calculate We ca implify the calculatio, however, by recogizig that ( tdi - td) i the umerator i the equatio that give the tadard deviatio for the cocetratio of the tadard, d Becaue d i eay to determie uig a calculator, a preadheet, or a tatitical oftware program, it i eay to calculate ( tdi - td) ; thu i 1 ( tdi - td) ( - 1)( d) ( 6-1)( 641 ) 3487 Subtitutig all term back ito equatio 55 give the tadard deviatio i the cocetratio a ^ h d ( 143 ) ( 3487) The 95% cofidece iterval for the ample cocetratio, therefore, i d 49 9! ( 776)( 14) 499! 59M 9 The tadard additio for thi problem follow equatio 510, which, a we aw i Problem 55, i bet treated by plottig S pike (V o + V td ) o the y-ai v V o the -ai, the value for which are d ( i - ) - 1 td td V td (ml) S pike (arb uit) S pike (V o + V td ) td V td Figure SM55 how the reultig calibratio curve for which the calibratio equatio i S ( V + V ) # V pike o td td td To fid the aalyte cocetratio,, we ue the abolute value of the -itercept, V o, which i equivalet to the y-itercept divided by the lope; thu V ( ) b o ml k which give a 13 ppb To fid the 95% cofidece iterval for, we ue a modified form of equatio 55 to calculate the tadard deviatio i the -itercept Vo r 1 " Spike( Vo+ Vtd), b + 1 ( b1) ( tdivtdi- tdvtd) where the umber of tadard,, i four, the tadard deviatio about the regreio, r, i 00155, the lope, b 1, i , the i 1 S pike (V o + V td ) td V td Figure SM55 Stadard additio calibratio curve for Problem 59

8 46 Solutio Maual for alytical hemitry 1 average igal for the four tadard, Spike( Vo+ Vtd), i 147, ad the term ( tdivtdi- tdvtd) i Subtitutig back ito thi equatio give the tadard deviatio of the -itercept a " 147, ( )( 18# 10 ) Vo 4 Dividig Vo by V o give the tadard deviatio i the cocetratio,, a Vo 197 V o The 95% cofidece iterval for the ample cocetratio, therefore, i 1 3! ( 4 303)( ) 1 3! ppb S S IS IS Figure SM56 Iteral tadard calibratio curve for the data i Problem 51 epected aborbace meaured aborbace Figure SM57 Plot of the meaured aborbace value for a erie of pectrophotometric tadard veru their epected aborbace value The origial data i from Problem 45 1 (a) For a iteral tadardizatio, the calibratio curve place the igal ratio, S S IS, o the y-ai ad the cocetratio ratio, IS, o the -ai Figure SM56 how the reultig calibratio curve, which i characterized by the followig value lope (b 1 ): 5576 y-itercept (b 0 ): 3037 tadard deviatio for lope ( b1 ): 0314 tadard deviatio for y-itercept ( b0 ): 0781 Baed o thee value, the 95% cofidece iterval for the lope ad the y-itercept are, repectively b 0 b0! tb0 3037!( 3 18)( ) 3037! 484 b 1 b1! tb1 5576!( 3 18)( ) 5576! 1001 (b) The author cocluded that the calibratio model i iappropriate becaue the 95% cofidece iterval for the y-itercept doe ot iclude the epected value of 0 cloe obervatio of Figure SM56 how that the calibratio curve ha a ubtle, but ditict curvature, which ugget that a traight-lie i ot a uitable model for thi data 11 Figure SM57 how a plot of the meaured value o the y-ai ad the epected value o the -ai, alog with the regreio lie, which i characterized by the followig value: lope (b 1 ): 9996 y-itercept (b 0 ): tadard deviatio for lope ( b1 ): tadard deviatio for y-itercept ( b0 ): 0011 For the y-itercept, t ep i

9 hapter 5 Stadardizig alytical Method 47 t ep b0 b0 - b0 ad t ep for the lope i t ep b1 b1 - b For both the y-itercept ad the lope, t ep i le tha the critical value of t(05,3), which i 318; thu, we retai the ull hypothei ad have o evidece at a 05 that the y-itercept or the lope differ igificatly from their epected value of zero, ad, therefore, o evidece at a 05 that there i a differece betwee the meaured aborbace value ad the epected aborbace value 1 (a) Kowig that all three data et have idetical regreio tatitic ugget that the three data et are imilar to each other cloe look at the value of y ugget that all three data et how a geeral icreae i the value of y a the value of become larger, although the tred eem oiy (b) The reult of a regreio aalyi are gathered here parameter Data Set 1 Data Set Data Set 3 b b b b r ad are i agreemet with the value reported i part (a) Figure SM58 how the reidual plot for all three data et For the firt data et, the reidual error are cattered at radom aroud a reidual error of zero ad how o particular tred, uggetig that the regreio model provide a reaoable eplaatio for the data For data et ad for data et 3, the clear patter to the reidual error idicate that either regreio model i appropriate (c) Figure SM59 how each data et with it regreio lie For data et 1, the regreio lie provide a good fit to what i rather oiy data For the ecod data et, we ee that the relatiohip betwee ad y i ot a traight-lie ad that a quadratic model likely i more appropriate With the eceptio of a apparet outlier, data et 3 i a traight-lie; removig the outlier i likely to improve the regreio aalyi (d) The apparet outlier i the third poit i the data et ( 1300, y 174) Figure SM510 how the reultig regreio lie, for which lope (b 1 ): 345 reidual error reidual error reidual error (a) (b) (c) Figure SM58 Reidual plot for (a) data et 1; (b) data et ; ad (c) data et 3 The dahed lie i each plot how the epected tred for the reidual error whe the regreio model provide a good fit to the data

10 48 Solutio Maual for alytical hemitry 1 y y y (a) (b) (c) Figure SM59 Regreio plot for the data from (a) data et 1; (b) data et ; ad (c) data et 3 y Figure SM510 Regreio plot for data et 3 after removig the apparet outlier y-itercept (b 0 ): 401 tadard deviatio for lope ( b1 ): 0031 tadard deviatio for y-itercept ( b0 ): 009 tadard deviatio about the regreio ( r ): Note that r, b0, ad b1 are much maller after we remove the apparet outlier, which i coitet with the better fit of the regreio lie to the data (e) The aalyi of thi data et drive home the importace of eamiig your data i a graphical form uggeted earlier i the awer to part (a), it i difficult to ee the uderlyig patter i a data et whe we look at umber oly 13 To complete a weighted liear regreio we firt mut determie the weightig factor for each cocetratio of thallium; thu y - i y i (avg) yi ( i) w i where y i (avg) i the average of the eve replicate meauremet for each of the i tadard additio, ad yi i the tadard deviatio for thee replicate meauremet; ote that the icreae i yi with larger value of i idicate that the idetermiate error affectig the igal are ot idepedet of the cocetratio of thallium, which i why a weighted liear regreio i ued here The weight i the lat colum are calculated uig equatio 58 ad, a epected, the um of the weight i equal to the umber of tadard To calculate the y-itercept ad the lope, we ue equatio 56 ad equatio 57, repectively, uig the table below to orgaize the variou ummatio i y i (avg) w i i w i y i w i i w i i y i total

11 hapter 5 Stadardizig alytical Method 49 b 1 i 1 i 1 i 1 i 1 wiy i i- w i i wy i i wii - c wiim i 1 ()( ) - ( 665)( ) ()( ) - ( 665) b 0 i 1 i 1 wiyi- b1 w i i ( )( 0 665) 4 The calibratio curve, therefore, i S total µ + ( 1443 µppm) # Figure SM511 how the calibratio data ad the weighted liear regreio lie Tl S total Tl Figure SM511 alibratio data ad calibratio curve for the data i Problem 513 The idividual poit how the average igal for each tadard ad the calibratio curve i from a weighted liear regreio The blue tick mark alog the y-ai how the replicate igal for each tadard; ote that the pacig of thee mark reflect the icreaed magitude of the igal idetermiate error for higher cocetratio of thallium

12 50 Solutio Maual for alytical hemitry 1