CHEMISTRY TEXT Uncertainty and Error in Measurement

8/16/2019 CHEMISTRY TEXT Uncertainty and Error in Measurement

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MeasureMent and data ProcessingChapter 11

070822 Chem Chap 11-3 for correc289 289 7/12/2007 8:55:11 AM070822 Chem Chap 11-3 for correc290 290 7/12/2007 8:55:12 AM

MeasureMent anddata processing

11.1 Uncertainty and error in mea!r ement

11.2 Uncertaintie in ca"c!"ated r e!"t

11.3 #raphica" techni$!e

11.1 uncertainty & error in MeasureMent

11.1.1 %ecr i&e and 'i(e e)amp"e of r andom !ncertaintie and ytematic er r or.

11.1.2 %itin'!ih &et*een pr eciion and acc!racy.

11.1.3 %ecr i&e ho* the eff ect of r andom !ncertaintie may &e r ed!ced.

11.1.+ ,tate r andom !ncertainty a an !ncertainty ran'e .

11.1.5 ,tate the r e!"t of ca"c!"ation to the appropriate n!m&er of i'nificant fi'!r e.

2007

(a"!e 8.31+ 4 mo" 1 6 1 i o&(io!"y more acc!rate thanone that 'i(e a (a"!e of 8.103 4 mo" 1 6 1. heacc!racy i !!a""y mea!red a the percenta'e de(iationfrom the accepted (a"!e !in' the e)preion:

ercenta'e de(iation

: ;) p e r im e n t a " ( a "!e A cce p te d ( a "!e t can &e een that the percenta'e de(iation of the t*o(a"!e $!oted for the idea" 'a contant are 0.0+< and2.5+< r e pecti(e"y:

ercenta'e de(iation 8.317 - 8.31+ =100 8.31+

0.03>08 0.0+ to 1 sig. fig. precision of top line

8.103 - 8.31+n practica" cience? the re!"t of e)periment are ne(er comp"ete"y re"ia&"e a there are a"*ay e)per imenta"

error and !ncertaintie in(o"(ed. t i ther ef or eim portant?

ercenta'e de(iation : 2.53788

8.31+ : = 100

epecia""y in $!antitati(e *or@? to &e a&"e to ae thema'nit!de of thee and their effect on the re"ia&i"ity of the fina" re!"t. t i important to differentiate &et*eenthe acc!racy of a re!"t and the preciion of a re!"t. heacc!racy of the re!"t i a mea!re of ho* c"oe the re!"ti to ome accepted? or "iterat!re (a"!e for the $!antity

&ein' determined. or e)amp"e? an e)periment that 'i(ea (a"!e of 8.317 4 mo" 1 6 1 for the idea" 'a contantaccepted

2.5+ to 3 sig. fig. precision of top line

he preciion of the re!"t i a mea!re of the certaintyof the (a"!e determined? !!a""y $!oted a a (a"!e.A'ain? *ith re'ard to the 'a contant? one 'ro!pmi'ht $!ote a re!"t of 8.3+ 0.03 4 mo"-1 6 -1? *hi"tanother 'r o! p 'i(e a (a"!e of 8.513 0.00> 4 mo" 1 6 1.Bhi"t the r e!"t of the former 'ro!p i the moreacc!rate i.e. c"oer to

11 c o r

e

Accepted (a"!e : =

: :


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the accepted (a"!e? that of the "atter 'ro!p i the mor e precie i.e. ha the ma""et !ncertainty. he *ay that!ncertaintie are ca"c!"ated i dic!ed e)teni(e"y"ater in the cha pter .

he re"ationhip &et*een acc!racy and preciion ii""!trated in i'!re 1101.

Acc!rate and precie theidea"

1nacc!r ate? &!t pr ecie

1nacc!r ate andimprecie

Figure 1101 The r elationship between accuracyand pr ecision

Random uncertainties ma@e a mea!rement "e precie? &!t not in any partic!"ar direction? in other *ord the act!a" (a"!e may &e either 'reater or ma""er

than the (a"!e yo! record. andom !ncertaintie or err or arie mot"y from inade$!acie or "imitation inthe intr!ment. hi may &e a re!"t of ho* precie"y yo! can read a meter or ca"e of? +3for e)amp"e? the &!rette in i'!re 1102.Be *o!"d pro&a&"y ta@e the readin' a+3.>? &!t in o doin' *e are ayin' thatit i nearer to +3.> than it i to +3.5 or +3.7? in other *ord it i 'reater than+3.55 had it &een "e *e *o!"d ha(er ecor ded it a +3.5 and ma""er than+3.>5 had it

&een 'reater *e *o!"d ha(e recorded it a

+++3.7? hence *e ho!"d record thi (a"!ea +3.> 0.05. ,ee i'!re 1102.

Figure 1102 A sample r eading

,imi"ar"y if *e are readin' a di'ita" intr!ment? !ch a a &a"ance? then *e ho!"d record the !ncertainty a &ein'ha"f of the "at di'it. or e)amp"e the readin' 37.3>1on a di'ita" reado!t ho!"d &e recorded a 37.3>1 0.0005? a!min' the reado!t i teady. f the readin' if"!ct!atin'? then yo! ha(e to try to etimate the de'ree of certainty yo!

fee" yo! ha(e in the (a"!e yo! record. f the "at di'it@eep 'oin' !p or do*n &y one? yo! may fee" yo! canay the (a"!e i definite"y 'reater than 37.359 and "ethan 37.3>3? o yo! record it a 37.3>1 0.001.

n ome cae? !ch a many thermometer? it i on"y poi&"e to read a ca"e to the nearet 0.2 that i? one

*o!"d record 23.0? 23.2? 23.+ etc.? &!t ne(er an oddfina" di'it !ch a 23.3. n thi cae the !ncertainty*o!"d &e0.1? &eca!e a readin' of 23.2 mean it i 'reater than23.0? &!t "e than 23.+.

D!antitati(e apparat! !!a""y ha a nomina" random!ncertainty *hich ref"ect the to"erance !ed in itman!fact!re. Bith (o"!metric apparat! it i poi&"eto p!rchae &oth A-'rade and -'rade apparat!. or e)amp"e? *ith a 25 cm3 &!"& pipette the man!fact!r er E!ncertainty for A-'rade i 0.03 cm3? *herea for -

'rade it i 0.0> cm

3

.Far'e random !ncertaintie o&(io!"y decreae the

preciion of the (a"!e o&tained and can a"o "ead toinconitent re!"t if the proced!re i repeated. f on"yone readin' i ta@en? then a "ar'e random !ncertaintycan "ead to an inacc!rate re!"t. epeatin' e)per imenta"determination ho!"d? ho*e(er? increae the preciionof the fina" re!"t a the random (ariation tatitica""ycance" o!t.

Systematic errors a"*ay affect a re!"t in a partic!"ar direction and hence the acc!racy of the e)periment. hey

arie from f"a* or defect in the intr!ment or f r omerror in the *ay that the mea!rement *a ta@en. f?f or e)amp"e? a t!dent ta@e the ma of an empty*ei'hin' &ott"e to ca"c!"ate the ma of o"id !ed?rather than re-*ei'hin' it after tippin' the contentinto a &ea@er? then thi *o!"d &e a ytematic error

&eca!e the ma of o"id in the &ea@er *i"" a"*ay &e"i'ht"y "e than the (a"!e the t!dent !e &eca!eome o"id may ha(e &een "eft in the *ei'hin' &ott"e. tco!"d ne(er &e hi'her. t i often diffic!"t to a""o* for !ch error $!antitati(e"y? &!t the direction in *hich it*o!"d affect the fina" re!"t can a"*ay &e determined.or e)amp"e? in thi cae? if the o"id *ere !ed to prepare

a o"!tion that *a then titrated *ith a tandard o"!tionfrom a &!rette? then it mi'ht e)p"ain *hy the &!rettereadin' *a "e than the e)pected (a"!e? &!t it co!"d note)p"ain a hi'her readin'. ften the or der of ma'nit!de ietimated. n thi e)amp"e? a!min' the ma ta@en *a10.000 ' Go"id "eft in the &ott"eE *o!"d not &e a&"e toe)p"ain *hy the titre *a 10< "e than e)pected

yo! *o!"d !re"y ha(e noticed 1 ' of remainin'o"id..

a@in' the initia" readin' of a &!rette *hen it *a *e""a&o(e head hei'ht *o!"d a"o 'i(e rie to a ytematic

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r. ara""a) error? ca!ed &y not readin' thee

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perpendic!"ar"y *o!"d? in thi cae? 'i(e a (a"!e for theinitia" readin' that *a 'r eater than the tr!e r eadin'.Hence the (a"!e o&tained f or the titre *o!"d &econitent"y "o*er than the tr!e (a"!e. Another e)amp"eof a ytematic err or *o!"d &e to read the top rather than the &ottom of the menic! in mea!rin' cy"inder and pi pette? *hich *o!"d conitent"y de"i(er a ma""er (o"!me than that r ecor ded. ;rror in the ca"i&ration of intr!ment? for e)amp"e pH meter? a"o "ead toytematic error. he mot fami"iar ytematic error iheat e)chan'e to the !r r o!ndin' *hich? inca"orimetric e)periment? ine(ita&"y "ead to ma""er temperat!re chan'e than *o!"d &e o&er(ed in a perfect"y in!"ated ytem.

A "ar'e ytematic error can "ead to the re!"t &ein'inacc!rate and? if it i a (aria&"e ytematic error i.e. ita"*ay ha an effect in a partic!"ar direction? &!t the iIeof the error (arie? it can "ead to inconitent re!"t.

f yo! are in(o"(ed in ca"c!"ation in(o"(in'e)per imenta" data then thee ho!"d &e com&ined o ato ref"ect a"" of the !ncertaintie in(o"(ed and that ico(ered in the ne)t ection. f? ho*e(er? yo! are doin'ca"c!"ation in(o"(in' data that yo! ha(e &een 'i(en?then it i important to $!ote the fina" re!"t to a preciionthat ref"ect that of thee data. he 'enera" r!"e to app"yare:

J or m!"tip"ication and di(iion? the re!"t ho!"dha(e the ame n!m&er of significant figures athe "eat precie piece of data.

J or addition and !&traction the re!"t ho!"dnot ha(e more decimal places than the "eat precie piece of data.

,!ppoe yo! *ih to ca"c!"ate the form!"a of a f"!orideof !rani!m 'i(en that 5.8+ ' of !rani!m prod!ced8.>37 ' of the f"!oride and that the mo"ar mae of !rani!m and f"!orine are 238.03 and 19.00 ' mo" 1

repecti(e"y. f yo! *ere initia""y a@ed for the ma of f"!orine preent in the compo!nd? thi ho!"d &e $!oteda 2.80 ' rather than2.797 ' a there are on"y 2 d.p. in the ma data f or !rani!mK note that the fina" Iero in the ma of f"!orine

ho!"d &e *r itten a it i i'nificant/. he mo"e of !rani!m 5.8+

0.02+53+7 ho!"d &e $!oted a 0.02+5 a the ma of !rani!m i on"y @no*n to 3 i'nificant fi'!re.Fi@e*ie? the mo"e of f"!orine 2.797 0.1+72105 ho!"d

&e $!oted a 0.1+7 and the :U ratio - 0.1+7... >.0000937 a >.00. t i 'ood practice epecia""y *ithmodern ca"c!"ator to @eep intermediate re!"t to thema)im!m preciion in the memory of the ca"c!"atorand then to ro!nd off the fina" re!"t to the appropriate

preciion.

eXtensionAnother factor re"ated to e)perimenta" data iconitency. he conitency of a re!"t i a mea!re of ho* r epr od!ci &"e the re!"t i *hen the e)periment irepeated. or e)amp"e? the re!"t of t*o 'ro!p for determinin' the (a"!e of the 'a contant "iterat!re(a"!e 8.31+ 4 mo"-1 6 -1 are 'i(en:

#ro!p A 8.537K 8.+87K 8.598K 8.+92K 8.+72Mean 8.517 4 mo"-1 6 -1 an'e 0.12>

#ro!p 8.13K 7.9+K 8.++K 8.5+K 8.22Mean 8.25 4 mo"-1 6 -1 an'e 0.>0

t can &e een that thoe o&tained &y #ro!p A are far more conitent and more precie than thoe o&tained &y

#ro!p ? e(en tho!'h thoe of #ro!p are moreacc!rate. he conitency i indicated &y the ran'ema)im!m (a"!e minim!m (a"!e? tho!'h moreo phiticated indicator? !ch a tandard de(iation?co!"d &e !ed to compare conitency. he !ncertaintyof a et of r epeated mea!rement ho!"d ref"ect theconitency of the re!"t. here are many tatitica"*ay of etimatin' the !ncertainty of repeatedmea!rement? &!t pro&a&"y one of the imp"et i todi(ide the ran'e of the re!"t X

max

X min

&y t*ice the $!are root of the n!m&er of r eadin'ta@en L:

X X X ma) min2N

App"yin' thi? the re!"t of #ro!p A ho!"d &e $!oteda8.517 0.028 4 mo"-1 6 -1 or perhap more correct"y 8.520.03 4 mo"-1 6 -1 and that of 'ro!p a 8.25 0.13 4 mo"-1

6 -1. Lote that the re!"t o&tained &y #ro!p confir mthe "iterat!re (a"!e it i *ithin the !ncertainty *hereathoe of #ro!p A do not. Bhich 'ro!p performed the

&etter e)per imentO

i'!re 1103 e)tend the re"ationhip &et*een acc!racyand preciion ho*n in i'!re 1101 to inc"!de theconcept of conitency.

238.03

19.00

0.02+5...

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Fiter at!r e (a"!e

Acc!r ate? precie and conitent the idea"

1nacc!r ate? &!t pr ecie and conitent

Fe pr ecie? &!t acc!rate and conitent

Fe conitent? &!t acc!rate and pr ecie

1nacc!r ate? im pr ecie and inconitentPP

Figure 1103 The r elationship between accuracy! precision and consistency

11.2 uncertainties in calculatedresults

11.2.1 ,tate !ncertaintie a a&o"!te and percenta'e !ncertaintie.

11.2.2 %etermine the !ncer taintie in r e!"t.

2007

he !ncertaintie in indi(id!a" mea!rement can &ecom&ined to ca"c!"ate the !ncertainty in the fina" (a"!eof the $!antity &ein' determined. ne *ay to etimatethi *o!"d &e to a!me a"" the !ncertaintie *ere in thedirection that *o!"d 'i(e the "ar'et (a"!e of the fina"$!antity that i ta@in' the "ar'et poi&"e n!m&er if the(a"!e i added or !ed to m!"tip"y and the ma""et (a"!eif it i !&tracted or !ed to di(ide and reca"c!"ate ther e!"t *ith thee data.

An a"ternati(e that i often eaier to app"y i to !e thefo""o*in' imp"e r!"e:

J add absolute !ncertaintie *hen addin' or !&tractin' n!m &er

J add percentage !ncertaintie *hen m!"ti p"yin'or di(idin' n!m &er

J Qm!"tip"y percentage !ncertaintie &y thee)ponentia" *hen raiin' to a po*erR

he a&o"!te !ncertainty i the act!a" !ncertainty in the(a"!e? for e)amp"e 0.05 for a $!antity that ha the(a"!e28.5 0.05. he percenta'e !ncertainty i the a&o"!te!ncertainty e)preed a a percenta'e of the (a"!e. or e)amp"e the percenta'e !ncertainty of 28.5 0.05 i0.18< 100 = 0.05 . he third point i p!t in par enthee &eca!e it i a cone$!ence of the econd a X 2 X.X .,o app"yin' the econd r!"e the !ncertainty of X 2 *i""

&e do!&"e the percenta'e !ncertainty of X . ,imi"ar"y the percenta'e !ncertainty of N X *i"" &e ha"f the per centa'e!ncertainty of X .

,!ppoe yo! *ant to ca"c!"ate the (a"!e and !ncertaintyof X ? *here

X " A # $ %

'i(en the

(a"!e:

A 123 0.5K # 12.7 0.2K % +.3 0.1

X 123 = 12.7 +.3 1033.2 note that thi ha not yet &een r o!nded

to an appropriate preciion a the

!ncertainty ha not &een ca"c!"ated

Act!a" !ncertainty in # $ % 0.3 add act!a"!ncertaintie? 0.2 S 0.1 0.3

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< !ncertainty in A 0.+07< 100 = 0.5

/

< !ncertainty in # $ % 3.571< 100 =0.3

< !ncertainty in X 3.978<

add percenta'e !ncertaintie? 0.+07 S 3.571

Act!a" !ncertainty in X +1.1 1033.2 = 3.978

herefore X 1033.2 +1.1

he !!a" practice i to on"y 'i(e the !ncertainty to onei'nificant fi'!re and then to ro!nd off the (a"!e to aimi"ar n!m&er of decima" p"aceK hence the fina"K r e!"tho!"d &e $!oted a

X = 1030 +0

Lote that? e(en tho!'h # and % *ere mea!red to'r eater preciion? &eca!e they are !&tracted theycontr i &!te m!ch more to the fina" !ncertainty than doeA.

Bhen !in' a "iterat!re (a"!e in ca"c!"ation? a!me it preciion i "imited to the n!m&er of di'it 'i(en. or e)amp"e? if the re"ati(e atomic ma of ch"orine i 'i(ena35.+5 it ho!"d &e ta@en a 0.005. n ome cae?ho*e(er? the !ncertainty of one $!antity i m!ch 'reater than that of a"" of the other? o the !ncertainty of the

fina" (a"!e can &e conidered to &e d!e to on"y that factor and hence ha(e the ame percenta'e !ncertainty. n theecae it i impor tant to tate that yo! are i'norin' theminor !ncertaintie in other data. Conider aca"orimetry e)periment that 'a(e the fo""o*in' re!"t:

nitia" temperat!re 21.> 0.1TC

T 2.> 0.2TC in addition and !&traction? the!ncertaintie are added/

< !ncertainty 100 = 0.2

7.7<

& " m c UT

200 ' = +.18 4 ' 1 = 2.> TC

2173.> 4

' &

2173.>0.0500

+3+72 4 mo" 1

A the !ncertainty in the temperat!re chan'e i o m!ch'reater than that of the other $!antitie? the per centa'e!ncertainty in the fina" re!"t *i"" &e ta@en a 7.7 @4 mo" 1? the percenta'e dicrepancy *o!"d &e:

ina" temperat!re 2+.2 0.1TC +5.> +3 +5.>

5.7<

Ma of *ater heated 200 0.5 '

Amo!nt of "imitin' rea'ent 0.0500 0.00005 mo"

Com&ined *ith the (a"!e &e"o* from a data &oo@:

,pecific heat capacity of *ater +.183 0.0005 4 ' 1 6 1

t *o!"d &e accepta&"e to tate that the per centa'e!ncertainty in the heat a&or&ed *a &ein' ta@en a

&ein' the ame a the percenta'e !ncertainty of thetem perat!r e

n thi cae the percenta'e dicrepancy i ma""er thanthe !ncertainty in the e)perimenta" (a"!e 7.7

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chan'e 100 =0.2

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f the "iterat!re (a"!e for the entha"py chan'e *a 5>.2@4 mo" 1? the percenta'e dicrepancy *o!"d &e:

5>.2= 100 23.5<

n thi cae the percenta'e dicrepancy i m!ch 'r eater than co!"d &e acco!nted for &y the !ncertaintie in the

e)per imenta" (a"!e 7.7


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11. graphicaltechni!ues

11.3.1 ,@etch 'r aph to r epr eent dependence and interpret 'raph &eha(io!r .

HA HS S A

nitia" Concentration a 0 0

;$!i"i&ri!m concentration a(x x x

,!&tit!tin' in the e$!i"i&ri!m e)preion:

Q HS R.Q A R x.x x2

x2

11.3.2 Contr!ct 'raph from e)perimenta" data. ) a

Q HA R a $ x " a $ x * a

11.3.3 %r a* &et-fit "ine thr o!'h data point on a 'r aph.

) .a * x2 o x * + ) a . Na

11.3.+ %etermine the (a"!e of phyica" $!antitie from 'r aph.

2007

Hence? if thi appro)imation i (a"id? the QHSR x *o!"d &e proportiona" to NQHAR Na and thi co!"d &e teted &y dra*in' a 'raph of QHSR a'aint NQHAR and eein' if it *a "inear.

f manip!"ation cannot prod!ce a "inear f !nction?#raph are one of the mot !ef!" *ay for inter pr etin'

cientific data &eca!e they a""o* for direct (i!a"corre"ation &et*een the data o&tained and a partic!"ar cientific mode" or hypothei. n it imp"et f or mthi in(o"(e or'aniin' data o that a "inear 'raph ie)pected.

i'!re 110+ and 1105.

ometime there i an in(ere proportiona"ity or in(ere

"inear re"ationhip? *hich can &e !ed to p"ot a 'raph thatho!"d 'i(e a trai'ht "ine. ,ee i'!re 110> and 1107.

y

1/ x

Figure 110, -nerse proportionality

x

Figure 110/ irect proportionality y

1/ x

x

Figure 110 A linear r elationship

Figure 110 -nerse linear r elationship

f a &!ffer o"!tion i made &y ta@in' a o"!tion of a

*ea@ acid and dio"(in' a o"id a"t of that *ea@ acid init? then the dependence of QHSR on the amo!nt of a o"ida"t added *o!"d &e an in(ere "inear r e"ationhip:

,ometime it i poi&"e to rearran'e an e$!ation that Q HS R.Q A R S R Q HA R

i not a "inear re"ationhip to 'i(e a modified e$!ationthat doe ha(e !ch a re"ationhip. An e)amp"e *o!"d &e

) a

Q HA Ro Q

H

) a.

Q A R

tetin' to ee if the appro)imation of ne'"ectin' theeffect of diociation on the concentration of a *ea@acid i

hi co!"d &e teted &y eein' if a 'raph of QHSRa'aint 1 a thi *i"" &e proportiona" to 1 *a

ma of a"t Q A R(a"id: "inear? a!min' the (o"!me and concentration of the acid

are @ept contant.

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Bhen dra*in' 'raph it i !!a" to chooe the a)e othat the independent (aria&"e fre$!ent"y time i p"otteda"on' the horiIonta" a)i and the dependent (aria&"eon the (ertica" a)i. he ca"e ho!"d &e choen o a toma)imie the !e of the 'raph area? ta@in' into acco!ntany e)trapo"ation of the data that may &e re$!ired. Afair"y imp"e ca"e ho!"d a"o &e choen 2? +? 5? 8 or

10 $!are e$!a" to one !nit to faci"itate the eay p"ottin' of data. More comp"e) ca"e for e)amp"e? 7$!are e$!a" to one !nit may ma)imie the !e of the'raph area? &!t fre$!ent"y "ead to error in the p"ottin' of point.

Care m!t a"o &e ta@en to ha(e eno!'h data point toen!re that the 'raph rea""y i "inear. or e)amp"e it i$!etiona&"e *hether the data in i'!re 1108 a rea""ydo repreent a trai'ht "ine rather than a c!r(e? &!t theaddition of e)tra data point? a ho*n in i'!re 1108&? confirm that they do indeed decri&e a "inear r e"ationhi p? tho!'h if thee point had &een e"e*here? ain i'!re 1108 c? they co!"d ha(e indicated that there"ationhip i non- "inear.

S

#raph can &e !ed to chec@ the (a"idity of ar e"ationhi p? a in the e)amp"e a&o(e? or to determineome (a"!e f r om either the intercept in *hich cae the!nit are the ame a thoe of the a)i or the 'radient of the "ine in *hich cae the !nit are thoe of the (ertica"a)i di(ided &y thoe of the horiIonta" a)i. ,ome of the more common 'raph that are enco!ntered in

chemitry and their !e are conidered:

dea" 'a "a* .4 n. 5.T

o 4 " n. 5.T

or " n. 5.T 4 Bhere i the pre!re? 4 the (o"!me? n the n!m&er of mo"e and T the a&o"!te temperat!re of the 'a amp"e.

5 i the idea" 'a contant.

Mot common"y i p"otted a'aint - 1 / atcontant temperat!re oy"e-Mariotte Fa*. A "inear 'raph ho* the (a"idity of thi re"ationhip and? if then!m&er of mo"eof 'a can &e determined for e)amp"e &y @no*in' itma and the temperat!re i mea!red? then the (a"!e of ythe idea" 'a contant 5 can &e fo!nd from the 'radient.

S ,ee i'!re 1109.S

14

x

Figure 1106 7a8S S

y S SS

SS S

x

Figure 1106 7b8S

3

Figure 1109 #o yle: s ;aw

A"ternati(e"y 4 can &e p"otted a'aint T at contant pre!re Char"eE Fa*. A'ain a "inear 'raph ho*the (a"idity of thi re"ationhip and? if the n!m&er of mo"e of 'a i @no*n and the pre!re i mea!red?then the (a"!e of the idea" 'a contant 5 can &e fo!ndfrom the 'radient. he data can a"o &e e)trapo"ated

&ac@ to 4 0 to find the (a"!e of a&o"!te Iero. ,ee

i'!re 1110.SS

y S4

S SS S

x

Figure 1106 7 c8

T

Figure 1110 %harles: s ;aw

c o r e

S

4

/


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29>


17/22


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Figure 111/ An actiation energy gr aph

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Chapter 11

;)ercie 11.1 ;)ercie 11.2

o!r 'ro!p ha(e done a ca"orimetric e)periment todetermine the entha"py chan'e of a reaction in a$!eo!o"!tion? for *hich the "iterat!re (a"!e i 38.73 @4 mo"-1.he (a"!e fo!nd &y the 'ro!p in @4 mo"-1 *ere:

A 35.1 0.3 3>.5 0.5C 33.2 0.1 % 3+.7 0.2

1. Bhich re!"t ha the "o*et a&o"!te !ncertaintyO

A 3+.875 0.017 7+9 5C 0.000+ 0.0001% 87500 200

2. Bhich re!"t ha the "o*et percenta'e !ncer taintyO1. Bhich re!"t i the mot precieO

A 3+.875 0.0172. Bhich re!"t i the mot acc!rateO 7+9 5

C 0.000+ 0.00013. %o yo! thin@ the maVor pro&"em i % 87500 200

A andom err or ,ytematic err or C oth are e$!a""y im portant% t i not poi&"e to te"" from the data

'i(en

+. f yo! *ere to repeat the e)periment *hich of thefo""o*in' i mot "i@e"y to &e the impro(ementthat *o!"d mot impro(e the re!"tO

A Uin' a more precie ther mometer . Lot a!min' that the pecific heat of

the o"!tion e$!a""ed that of *ater.C Uin' pipette rather than mea!r in'

cy"inder.% mpro(in' the in!"ation of the

ca"or imeter .

5. Bhich of the fo""o*in' i the &et *ay to report themean re!"t of the 'ro!pO

A 3+.875 0.100 @4 mo"-1. 3+.9 0.5 @4 mo"-1.C 3+.9 0.8 @4 mo"-1.% 35 2 @4 mo"-1.

>. epeatin' an e)periment a n!m&er of time *i"""ead to a decreae in:

A the random err or . the ytematic err or .C &oth the random and the ytematicerr or . % neither the random and theytematic err or .

3. f ) 7+9 5 and y 3+.8 0.7? *hich one of the fo""o*in' *i"" ha(e the 'reatet per centa'e!ncer taintyO

A ) S y ) - yC )/y% ).Ny

+. #i(e the re!"t and the a&o"!te !ncertainty? to thecorrect preciion? of the fo""o*in' ca"c!"ation:

a 20.1 0.1 S 2.75 0.05 & 115.+ 0.2 = 8.137 0.001c 0.572 0.001 17.> 0.2 - 11+ 1d 1 - 0.27> 0.002/0.02+ 0.001e 52800 100/ N17.2 0.2 - 2.37 0.01

;)ercie 11.3

1. "ot a 'raph of the fo""o*in' data. Contr!ct the &ettrai'ht "ine and !e it to determine the (a"!e of theintercept and the 'radient of the "ine.

) 1 2 3 + 5 > 7 8 9 10y 7.> 10.7 13.2 17.1 19.8 23.0 25.7 28.7 32.9 35.+

2. *o (aria&"e? ) and y? are re"ated &y the

e$!ation: , )-y 5

*here , and are contant. %ra* an appropriate 'raphto in(eti'ate *hether the data &e"o* !pport thi and!e it to find (a"!e for , and .

) 1 2 3 + 5 > 7 8 9 10y 2.52 3.78 +.11 +.35 +.52 +.59 +.>2 +.71 +.73 +.7+

298

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CHAPTER 11 MEASUREMENT AND DATA PROCESSING

(IB TOPIC 11) SUMMARY

Uncertainties and errrs

Random uncertainties or error arie mot"y from inade$!acy or "imitation in theintr!ment or the *ay a mea!rement i made. andom error ma@e a mea!rement "e

precie? &!t not in any partic!"ar direction. hee are *ritten a an !ncertainty ran'e? !cha ++.20 0.05 cm3.

Systematic errors are d!e to identifia&"e ca!e? and arie from f"o* in the intr!ment or error made in ta@in' a mea!rement !ch a an incorrect ca"i&ration of a pH meter orreadin' the top rather than the &ottom of the menic!. ,ytematic error a"*ay affect are!"t in a partic!"ar direction a"*ay ma""er or "ar'er !n"i@e random error. andom!ncertaintie can &e red!ced &y repeatin' readin'K ytematic error can not red!ced &yrepeatin' readin'.

Precisinf an e)periment i repeated many time? the preciion i a mea!re of ho* c"oe therepetition *i"" &e to each other. he preciion or re"ia&i"ity of an e)periment i amea!re of the random error. f the preciion i hi'h then the random error i ma"".

Acc!rac"

he acc!racy of a re!"t i a mea!re of ho* c"oe the re!"t i to ome accepted or"iterat!re (a"!e Acc!racy i a mea!re of the ytematic error. f an e)periment iacc!rate then the ytematic error i (ery ma"".

A mea!rement can ha(e a 'reat de'ree of preciion? yet &e inacc!rate !ch a if the topof a menic! i read in (o"!me readin' !in' a pipette or a mea!rin' cy"inder inteadof he &ottom of the menic!.

Si#ni$icant $i#!res

he n!m&er of i'nificant fi'!re in any ca"c!"ation ho!"d &e &aed on the n!m&er of decima" p"ace/i'nificant fi'!re in the data &aed on the fo""o*in' imp"e treatment:

J n addition and !&traction: Add a&o"!te !ncertaintie

J n m!"tip"ication? di(iion and po*er: Add percenta'e !ncertaintie

J f one !ncertainty i m!ch "ar'er than the other? i'nore the other !ncertaintieand etimate the !ncertainty &aed on the "ar'er one !in' the r!"e a&o(e.

% re 2007 1


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Page 298 Ex 11.1

1. C

2.

3.

+. %

5. C

>. A

Page 298 Ex 11.2

1. C

2. A

3.

+. a 22.85 0.15

& 939.0 0.2

c 55.1 0.8

d 30.2 1.3 or may&e 30 1O

e 13700 200

"eae refer to the re"e(ant chapter for more detai". % re 2007


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Page 298 Ex 11.3

1. rom the 'raph? the intercept i + and the 'radient i 3.1

40

35

30

25

20

15

10

5

0

0 2 4 6 8 10

%

2 earran'e to 'i(e y - 5/, 1/).

hi i an in(ere re"ationhip and o re$!ire a 'raph of 1/) a'aint y.

Bhen 1/) 0? y K 'radient -5/,:

5.50

5.00

4.50

4.00

3.50

3.00

2.50

2.00

0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

1&%

rom the intercept *hen 1/) 0 5 and from the 'radient -2.5 , 2

7ote how important the 1 > x " 1.0 point is?8

" f t th " t h t f d t i" % 2007

"

CHEMISTRY TEXT Uncertainty and Error in Measurement

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