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Th€ sca e, on two pieces of measu' .9 gla$wa.e The whne nlmbeu (left) be onq ro a medsur n9 cylndetwh le rhe black nlmbe6 (cenre) malkour mlch smalervolumes on ther de ofa graduared piperre.A qEate' deqee of measurhg accuacycan be obtained by us n9 the plperc rathe.rhan the

sci€nt sts need to be p nciped andactwirh nt€gr ty and honesry.

'One _{a m of the physlcal sciences} has b@. io qivean exact pictur.o ofthematerla wond One achievehent... has been ro p@ve rhatrhis aim k unattainab ei

Whara@theimp i.ation, of thi5 claim for ihe aspnaionsoiscience?

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Measurernent

and

data processing

1 1 . 1 . 2 1 1 . 1 . 3 1 1 . 1 . 4 1 1 . 1 . 5

Science isa communal activity and it is important tbat

information is shared openly and honestly. An essential

part ofthis pro€ess is the way the international scientific

community subje€ts the indings ofscientists to intense

critical scrutiny through the repetition ofexPeriments and

r h e p e c r r e v i e w o f r e \ u l r { i n i o u r n a l s a n d a t c o n f e r e n c e s

All measurements have uncertainties and it is important

these are reportedwhen data is exchanged, as these limit

the conclusions that can be legitimat€ly drawn Scien€e ha5

progres.ed and is one ofrhe mo"t successtu I en t erpriser

i n o u r ': u l l u r e b e c a u s e l h e ' e i n h e r e n t u n c e r l d i n t i e s

are re€ognized. Chemistry provides us t{ith a deeP

understanding ofthe material world but it does Dot offer

Data collected from investigations are often presented

in sraphical form. A sraph is a useful tool as it sbows

relationships between variables and identifies data Points

which do not fit the general trend and so gives another

measure ofthe reliability ofthe data,

ASSessment statements

! 1.1 Uncerlainty and error in measurement

1 1.1.1 Describe and give examples of random uncertainties and systematic error5,

Distinguish between preq3ion and accuftcy.

Describe howthe effects of random uncertainties may be reduced. State random uncertainty as an uncertainty range (t).

Slate the results of calculations to the appropriate number of signiflcant figures.

'11,2 _{uncertainties in calculated results}

1 1.2.1 State uncertainties a5 absolute and percentage uncertalntles. 11.2.2 Detemine the uncertainties in results.

| 1.3 G ra ph ical techniq ues

'11.3.1 _{Sketch graphs to reprcsent dependences and interpret graph} behaviout

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Uncertainty and error in

measurement

Uncertainty

in measurement

\l..rsurement is an inrportant part ofchemislry.In the laboratory,vou will use

::il.rcnt mcasuring apparatus and there willbe tines when you havc to select the

:nrumeDt that is Drost appropriatc for your task from a range ol possibilities.

..rppose, _{for example, you wanted 25 cmr ofwntcr, you could choose} from

r(.rsuring cylindcrs, pipettcs, burettes, volumetri. fhsks ofdiffcrcnt sizes, or

...n an anilyticnl balnncc ifyou knor{ the deDsity. All of thcsc could bc uscd to

-,.rsurc r volumc of25crnr, but\vith difierert levels ofunccrtainly.

Uncertainty

in analogue instruments

\. unce(ainty rrngc lpplies to any expe.nnenhl va lu c. Some pi€ces oiipparaius . : . , r r t h c d e g r e e o f u n c c r h i n t y , i n o l h e r c a s e s y o u w i l l h n v e t o m a k c r i u d g c m e n t . ..:pposc yrtr are rskcd to mcasrrc thc volunrc ofwrter in the nrcasuriiii cylindcr . : r t r v n n r F i g u r e I l . l . T h c b o t ( o m o i l h e r D e n i s c u s o f a l i q u i d u s u a l l y l i c s b c t w c c n ' i o gr.rdurtions u n d s o th e f i D : l l i i g o r e o i t h c r c i r d i n g h a s to b e e s t i n r a t e d . T h e . : r i l l c s t divisiur in thc mersurlDg c y i i n d c r i s a c n r s o i { e s h o u l d r c p o r t rh e

' l ! c rs 62 12 cnlr. lhc sinrc coDsidctllions u p p l y t o o t h e r c q u i p n r c n t s u c h d s .:r.rtcs rrd rlcohol thcflnonrcters thrt hNvc !n.1logrc scalcs. The tncettainty of . -.nelogue scale is ! half the !mall.!t divisionl

Lhir srheun.e(a nrtrange n themeasur.! cy .dor I the c ose !p phoro be ow?

V

An analyrica ba ance , oneot dre mon pre. se innrumenls n. s.hoo

a b o r a t o r y _{l h s r a d 9la nnrume.l}

^.

F l g u r . i l l T h e v o l u m e B a d n q shou d be taken lrom the botrom ofrhe nre.ii.ls You.ou d rcport rh.vo !nre a s 6 2 ( n r b l r th s E n o r a n e x a . r v a u e

< ^i a.oho thennometer wirh a

5 m a e 5 r d v s o n o f l ' C T h e u . c e r r a l . r y s05'C so rhe remperaiur€ shoLld be r e . o r d . d a s 2 5 0 1 0 s ' c

'Ite

uncerl.inty of .n.n.lo9ue r<ale ir i hallthesm.llelt

-Ei

lE

rrad(ltt

gw€

Th€ mas oI the water c recoded at 100.00 t0.01 _9.

nf, uno.nlnty of. dlgh.l sl. lr + rh. rm.ll.3t 3(|1.

MearuE you rcactlon time, l.{ow _{90 to M,heinemann.couk/} hotlinkr in5e( the expcss code 4259P and cl ck on lhis .ctivity.

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Uncertainty in digital instruments

A top pan balance has a digital scale. The mass ofthe sample ofwater shown here

is I 00.00 g but the last digit is uncertain. The d€g.e€ of uncertainty is 10.0 i g: the

smallest scale division. T

Other sources of uncertainty

Chemists are inter€sted in measuringhow properties change during a rea€tion

and this can lead to additional sources of uncertainty. Wlen time m€asurements

ar€ taken for example, the rea€tion time ofthe experimenter should be

Similarlythere a.e uncertainties in judging, forexample the point that an

indi€ator changes colour when measuring the end-point of a titration, or what is

the t€mp€rature at a particular tim€ duringan exothermic reaction, orwhat is the

voltage of an electrochemi€al cell. These extra uncertainties should be noted even

if they are not actually quantified when data are collected in experimental work.

2 A .ewad _{is 9 len for a missing} diamond, which has a rcpo(ed masr of 9.92 _{l0O5 9. You} frnd a dlamond and measufe l$ massa510.1 10.29. qould !h s be the missing diar,rondT

Slgnlflcant figuros in measunomonis

Th€ digits in th€ measurement up to and includingth€ first unc€rtain digitare

the signifcant figures ofthe measurement. There are two significant figures,

for example, in 62 crnr and five in l00.00g.The z€ros are significant hereas they

signiry that th€ uncertainty range is :r 0.01g. The number ofsigniFcant figures

may not always be clear. Ifa tim€ measurement is 1000s, for example, are th€re

one, two, three or four signincant fi8u.es? As it is not clear, it is us€tuIto use

scientiic notation to avoid any confusion with one non-zero digit on th€ leftof

th€ decimal Doint. 0.98 for €xamDle is written as 9.8 X I 0-r.

3 Express the following in itandard notation:

{e) 0.04s (b) 222 cmi {.1 0030 s

Experir

The experir the general

Random

being too hi

is why ir is t( l a t h e s m e p repeatabl., il

Suppose rhc

rhc followint

0 . 1 2 3 4 I ,

q/$emat

0 . l l 1 6 g , 0

. IeasurinS lrcltom sil

Lcuracy a

T}l. smaller rh

4 whatisthenumberof slqnificantf gures Ineach of the follMing?

td) 30"c

{d) l50O g

Measurements

Signlfi(ant figures

Measurements

signifi.ant figures

1 0 0 0 s 0 . 4 5 m o l d m 3 2

1 x 1 0 3 s 4 . 5 x l 0 j m o l d m I 2

1 . 0 x 1 0 3 s 2 4 . 5 0 x 1 0 j m o l d m I 3

1 . 0 0 x 103 s 3 4 . 5 0 0 x 10 'mol dm 3

1 . 0 0 0 x 10r s 4 4 . 5 0 0 0 x 1 0 'm o l d m 3 5

,-t

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Experimental

errors

The experimental error in a result is the ditrerence between the recorded value and

the generally accepted or literature value. Errors can be categorized as random or

Random €rors

wlen an experim€nter approximates a reading,there n an equal probability of

b e i n g t o o h i g h o r r o o l o w T h i s i " J r a n d o m e r r o r '

Random errors are caused by:

. the readability of the measuring instrument

. the effects ofchanges in the surroundings such as temperature variations and

. insufficient data

. the observer misinterpreting the reading.

A5theyare random, the er.ors can be reduced through rePeated measurements This

is why it is Sood practice to duplicate experiments when designing experiments

Ifthe same person duplicates the experiment with the same result the results are

repeatable, if several experim€nters duplicate the results they are reproducible.

suppose the mass of a piece ofmagn€sium ribbon is measured several times and

the following results obtainedl

0 . 1 2 3 4 _{E , 0 . 1 2 3 2 9 , 0 . 1 2 3 3 8 , 0 . 1 2 3 4 9 , 0 . 1 2 3 5 9 , 0 . 1 2 3 6 9} ( 0 . 1 2 1 4 - 0 . 1 2 3 2 - 0 . l 2 J t - 0 . 1 2 3 4 - 0 . 1 2 J 5 0 . 1 2 J 6 ) _

I h e a v c r r g e v d l u e - - - _o S

_ o.t234E

The mass is reported as 0.1234 a 0.0002 g as it is in the range 0.1232-0.12368

Systematlc errorg

systematic €rrors occur as a result of poor €xPerimental design orProcedure. They

cannot be reduc€d by repeating theexperiments. SuPPose the top pan balanc€

was iDcorrectly zeroed iD the previousexample and th€ following results were

0 . 1 2 3 6 g , 0 . 1 2 3 4 g , 0 . 1 2 3 5 g , 0 , 1 2 3 6 8 , 0 . 1 2 3 7 g , O \2388

-{l thevalues are too high by 0.00029.

. (0-1236 + 0.1234 + 0.1235 + 0.\236 + 0.1237 + 0.\238) _{_}

6

= 0.12369

Eramples of systematic errors:

. Measuring the volume of water from the toP of the meniscus rather than the

bottom will lead to volumes which aretoo high.

. Overshooting the volume of a liquid detiv€red in a titration will lead to volumes

which are too high.

. Heat losses in an exothermic reaction will iead to smaller temperatures changes.

Svstemaii€ errors can be reduced by careful erTerirnental design

lccuracy and precision

The smaler the systernatic error, the greater will be the accuracy. The smaller the

r.ndom uncertainties, the greater will be the pfecision The masses of magnesium

m the earlier example are neasured to the same precision but the frist set ofvalues

. Ex.ml6er'3 hlnt You sholld compareyolr resu $ to irerarlre va ues

. Ex.Dln.?3hlnt whe. eva lar n! inve(lgar ons d st ngu 5h between systemat c and random erc6

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Meas!rement and data processing

Precise measurements have small random errors and are reproducible in repeated

trials.Accurate measurements have small systemaric errors and give a result close

to the accepted value.

F l g u E rl . 2 T h e s e t o f . e a d i . g s o n > the eft are for hiqh acc!.acyand low precis on The read ngs on the.iqhra@

tor low ac.uacyand h 9h pccisio.. Eg

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5 Fepeated mea3uremenBof a quantiry can reduc€ rhe effecrsot

C l a n d I

Additlo

Wo.tod

tbktlbtr

Percen

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Uncertainties in calculated results

nificant

figures in calculations

Uncertainties in the raw data lead to uncertainties in processed data and it is

important that these are propagated in a consistent way.

Multiplicaiion and division

C o n s i d e r a s a n p l e o f s o d i u m c h l o r i d e w i t h a mass o f 5 . 0 0 1 0 . 0 1 g a n d a v o l u m e

of 2.3 1 0.1 cmr. What is its density?

Using a calculator:

d e n . i r y p . . 1 " * - - 5 ! Q = 2 . l 7 r 9 l J g + . t g c m

Can we claim to know the deDsity to such pre€ision when the value is based on les

The value is rnisleading as the mass lies in the range 4.99-5.01 g and the volume i5

between 2.2-2.4cmr. Th€ best we can do is to give a rang€ ofvalues for the densil\

The maximum value is obtained when the maxirnum value for the mass is

combined with the mininum value of the volumc.

5 . 0 1 ^ ^ - - ^ - .

P,"' -;-2 -2 27727}8c.ti' '

and the mjnjmum value is obtaiDed bycornbining the minimum rnass with a

maximum valu€ for the volune.

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The densityfalls in the range berween the maximum andminimum value.

The second significant figure is uncertain and the repo(ed value must be reported

to this precision as 2.2 gcm j. The precision of the density is limited by the

lolume measurement as this is tbe least precise.

This leads to a simple rule. wheneveryou multiply or divide data, the answer

should be quot€d to the same number ofsignincant figures as the least precise data.

Addition and subtraction

\\4ren values are added or subtra€ted, the number ofdecimal places deternine the

precision ofthe calculated value.

suppose we need the total mass of h{o pieces ofzinc ofmass l.2l g and 0.56 g.

The total mass = 1.77 g can be given to two decimal places as the balance was

precise to 1 0.01 in both cases.

Similarly$'hen calculating a temperature increase from 25.2'C to 35.2'C.

Temperature increase = 35.2-25.2'C = )0.0'C

Worked erample

Report the total mass of solution prepared by adding 50 g of water to 1.00 g oi

sugar. Would the use of a more precise balance for the mass ofsugar result in a

more precise total mass?

&tudon

Total mass = 50 + 1.00 g - 5l g

The precision of the total is limited by the prechion ofthe mass of the water.

Lsing a more precise balance for the mass ofsugar would bave not improved the

Percentage

uncertainties

and errors

An uncertaintyof 1s is more significant for time measurements of l0s than it

b for 100 s. It is helpful to express the un€ertainty using absolute, fractional or

Frcentage values,

The iractional uncertainty = absolute uncertainty/measured value.

This can be expressed as a percentage (see Key fact box below).

I

/ abrolul€ un..rtllntv \ r,.E .r.d. un..rr.lnrv = | ::::i i::i r r lx|o0%

uncertainty should not be contused with p€rcentage €rror. Percentage

is a measure of how close the €xpedmental v.lue is to the literature or

ted value.

@

Wh.w.r you moltlply or dlvlde d.t!. lhe.nseer sholld b.quotedtothesrnenunb.r ofslgn|fontfgorc3sth.

Wh6n.v.r yoll .dd or 3ubtr..t d.r., the rBe.r tholld b. quot.d to th. 3.me nunber of d..lmd pl.<.! I th. le.3t

. Ex.nln.r'3hlhit when evJ lar n! po.edu.esyou shou d d 5cu$ the precis on a rd accuracyol rhc measuremenl5 You 5hou d 5pe. h.aly

o o k i t t r e p r o . e d u r e i n d u l e o f

a i s sity.

J ra@Dted v.lue - exD.rln.nr.lv49] x roo* P.r.enr.s..mr=

Ivleasurement and data processing

whd .ddlng .r robn dlng |t!.q|m6l9 the lnc.n.lnty k th. {m of th. .biolut

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Propagation

of uncertainties

AddlUon and subtraction

Conside. two bnrette readings:

Initial readingi a 0.05 cmr : 1 5.05

Iinal reading/ :! 0.05 cmr: 37.20

wlat value should be reported for the volume delivered?

Tbe initial reading is in the range: 15.00 15.10

r n c n n a l r e i d l n S r s In r n e r a n q e : J / , t r - t / , 2 5

The maximum volume is formed by cornbininsthe maximum final reading Irith

the minimum initial reading: v o l , r " r = 37.25-15.00 = 22.2scnl

The minimom vohrne is fonned bycombining the minimum linalvolumewith

the maximum initial readingl

v o l r r = 3 7 . 1 5 1 5 . 1 0 = 22.05cmr

therefore vol = 22.15 :t0.1cmr

The volume depends on two measurements and the uncertainty is the sum ofthe

two absohrte uncertainties.

This result crn be generalized.

When ndding or subtracting meisurcments, the uncertainty G the sum ofthe

absolutc uncertnintics.

Multlpllcatlon and dlvlslon

Workingout the uDcertairrty in calculated values can be a time-coDsuming process. C o n s i d e r t h e d e n s i t y c a l c u l a t i o n :

As discussed earlier the densityshould oDlybe given to rwo significant figures given

the uncertaintyin the nass and volumevalues. The uncertrintyin thecalculated

value of the density is 7% (given to one significant figure). This is equal ro the

sum ofthe uncertainties in the mass and volume values: (5 + 2olo to rhe same

level ofaccurary). This approximate resuit provides us with a simple treaiment of

propagating uncertainties when rnultiplying ard dividing measurements

Wlen multiplyingor dividing measurements, the totalpercentage uncertaintyis

the sum ofthe individual percentage uncertainties. The absolute uncertainty can

then be calculated from the percentage uncertainty.

lvorto

T h . l e n !

Sorudot

Discut

(-rn th('dil

.nrh!lpr.h :rg€lened n

'rnis is dis.1

Wh.n hultlplyhg or dlvldlng n...urem.nq th. totll Per.ent.ge un..n.hty 13 rh. rum of rh. lndlvldo.l P.r..nt geun.ert lntl.r. lh..b$lut [email protected].!r th.nbe..l.!l.t d from the p.E.nr.ge unc€rtalnty.

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Absolute uncertainty % Uncertainty

N,laximum value Minlmum value

:1f,= noo

23.5

2 )

Absolute uncertalnty

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Worked example

The lengths of the sides ofa wooden block are measured and the diagram shoivs ihe measured vilues with their uncertainti€s.

What is the perceDtage uncertainty in the calculated area of the block?

Solution

Area = 40.0 x 20.0 nrni : 800mm: (area is given to three significlnt fisures)

( o l , u n c e r t a i D t y o f a r e a ) = ( % u n c e r t a n r t y o f l e n g t h ) + ( o / o u n c e r t a i r t y o f b r e a d t h )

'/o unccrr inryof leDgdr = (0.5/40.0) X 10001, - r 25olo

9/o uncertrinty oibrerdrb - (0.s/20.0) x 100oi = 2.so/o

o/o _{uncertrinty oi rea = 1.25 + 2.5 = 3.75 - 4%'}

A b s o l u t e u n c e t r i n t y = (3.75l100) x 800mmr = 30Dnn' Arca = 800 a30 nrmj

6 The.on.envatlon oia 50 wion ofhydoch o c ac d = 1.00 a0.05 mo dm-rand the vo lne = l0.010.lcmr crl.llate the number ot mol€sand 9 ve the arrsolte unce.lalniy

Discussing

errors and uncedainties

AD exper ilt]cnh I conclusioD nrust lrkc into rccount aDysyste'natic e|rors rDd

rnndom uDcertiintics. Yo u should recogDizc whcn thc uDcertuintyol one of thc

mcrsurcnients is nruch grertcr thrn thc odrers is dris ivill thcn hrvc thc nrijor

. f f e c l o n th e u n c e r t i i n t y o i t h e 6 n n l r e s u l t . T h e r p p . o x i n r r t c u n c c r t a i n t y c u r

be trkcn as being due to tb.rt qurntity uloDe. ID thernonctric cxperinrents, for

exrDrple, thc thcrmometer often produccs the |rrost uDcertnii resrlls, purticularly

ior reactions which prod ce snrill temperaturc differences

Cln the difterence bctween the expernrent^l dnd literature valu€ be explaincd in

trrms ofthe uncertairties oi the measureNents or wcrc other systenratic erus

involved? This questions Deeds to be answered when evillurtinS an experinental

procedurc. Heat loss to the surrcundinSs, for example, accounts for experirnental

enthalpy chnnges lor exothermic reactions beiDg lower dran literrture values.

Suggested modilications, s ch as inrproved insulation to reduce heat exchange

beFveen the systcm and the srrorndings, should atteDrpt to reduce thcse errors

Thh k discussed iD nrorc detailin Chapter 5.

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To find theabolute un.ertalnty In.Galculated value ror db or;: I Fitrd the pe(entage

un.en.lntylnu and b.

un.ert.lntler of4.nd 6 to fnd the per<entag. u..Enalnty In the .al.ul!t.d

3 <onverr thk per.entag. [email protected] to an .biolute

. E r . n l n . r ! h l n l : , , e c i r u r t e n n . r . r h i , . n . ! o f l t r r l r r r L n l r r s gr-Fare, o, eq!r r. l

t o n o r n r o r e u r r n i , ! ! f h . r n i i a r r . 5 f n ! e $ r h a , , : i ) r r i n . . l . r i ! . s ncacu al o.s nrou l .!l b! ,olndcd o l i l o r v o d u n n e . e j i a r y r r p r r t 5 ! r i

t"-l

1l

r g i v e n

ntv is

7 what is the ma n source of eror n experimenls ca( ed out toder€rm neenthalpy.ha.qes . a

A !.cenanvolumemeasuremenrs B hearexchangewth the surou.ding5

C u.cenanue5 n the.oncenlrations ofthe solut o.s D mpurues . the reaqents

. Exlmln€r'i hlnir Th.'. nr.L:.. n o v a r i a r i o n i n r r r e p r . . t . r l r r . : , - _ i

5 h o u d b e u ; . i l . i r i : - i . . p r o . i 5 ! . q f . r . : r ; - ' : r ' : i v . r i l e s l r r : . . : ' :

-+ 4 0 . 0 1 0 , 5 m m

1-=>-:-= :-

-Measurement and data processing

lhe IndeFrd.nt v.rl.bl. ls the .d!5e .nd ls plotied on rhe horlrontrl.xl5,Th. d.p.nd.ni vtrl.blehth6.trer.nd k plott do.th.v.nbl.xk.

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CraCrrlcat

techniques

A graph is often the best method ofpresenting and analysing data.It shows the

relationship between the hdep€nd€nt variable plotted on rhe horizonral axis

and the d€pendent variable on the vertical axis and gives an indication ofthe

reliability of the measurements.

Plotting graphs

Wlen you draw a gfaph you should:

. cive the graph a title.

Label the axes with both quantities and units.

Use the available space as effectively as possible,

Use sensible linear scales there should Lre no uneven jumps.

Plot all the points correctly.

A line ofbest fit should be drawn srnoothly and clearly. It do€s not have to go

through allthe points but should showthe overalltrend.

Identiryrny points nhich do notagree with the general trend.

Think carefully about dre inclusion ofthc origin. The point (0,0) can be the

most trccurate data point or it car be irrelevant.

Th. g.

ideil 8: T}le p.! iDrcrpd

Erron

Srsr.mari

t graph c

t ,

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Flturu 1i.3 Anraighr inegraph > wh ch pa$es through th€ orlg n 5hows rhal thedependent vaiab e

s proportional torhe ndependent

The 'best-fit' straight line

In many cases the b€st procedure is to find a way of plotting the data to produce a

straight line. The'best-fif line passes as near to ns many ofthe points as possible.

For example,a straight Iine through the origin is the nost appropriate way to join

t h e s e t o f p o i n t s i n F i g u r e I l . 3 l

The best-6t line does not necesarily pas t-hrough any ofthe points plotted.

Two properties of a straight line are particularly usetul: the gradi€nt and the

intercept.

Finding the gradient and the Intercept

The equation for a straightline _{is f = m' + c.}

r is the independent variable, _{), is the dependent} variable, 7n is the gradient and ris

the intercept on the v€rtical r,ris.

The gradient ofa straight line is the increase in the dependent vaiable divided

bythe increase iD the independent variable. The triangle used to calculate the

gradient shouldbe as large as possible.

< F i s u r e l l . 4

n = ^ r

u _independent

The gradient of a straight line has unitsi the units of the vertical ards divid€d by

the units of the horizontal ards. Sometimes a line has to be extended beyond the

range of measurements of the graph. This is call€d extrspolalion. Absolute zero,

for example, can be found by e{trapolating the volume/temperature graph for an

ideal gas.

The process ofassuming that the trend line applies between two points is called

interpolstion. The gradient of a curve at any point is the gradient ofthe tangent

to the curve at that point.

The slopeofthe cu.ve ls the gradienr of the tangent arthlr point.

< Fbur.1r.6 Th sg€ph shows how the conceniradon of a Gactant decreases wthtime.Th€ g.adlent ofa s ope ls 9ven bythe glad ent of the rangent al lhat poinr _{llle "aqLranon} of rhe tangenr was calcllated by comput$ sofrwa€ The rareatrhe po m shown is -0.1I moldm I m n_r.Thenegativevalle shosrharih Eaciantconcentmtion

sdecrcasing whh ndedi.g time. - 2 7 3 ' C ₀

lemperatu.efc

Flgsru1l.5

< Fbur.tl.7 A sy*ematlc e(or prcduces a dlsplac€d straight ine. Random uncenahti€s lead to poht5 on borh s des ofthe ped€.t stlaight line. E

E E

z

E

4 . 7 0,6 0.5 0,4 0,3 0,2

- 0 . 1 1 0 9 x _{+ 0 . 3 8 1 8}

2

Errors and graphs

Systematic errors and random uncertainties can often be recognized liom a graph.

A graph combines the results ofmany measurements and so minimiz€s the effects

I ngut rr.r

'".i

- 2 7 3 ' C ₀ t€mperat!re/'C

< F b u r e rr . 7 A s y n e n J t . . r . p r o d ! . € s a d s p .. e d n . 3 9 1 i ' : B a n d o m u n c e ( i i n l e s .i d : . . , - , borh s des 01 Ihe Derr..r !i,: r" = < Flgure rr,5 f hir qraph show5 hor' rF.

w t h r i m e T h e q r a d e n t o l r s o p e s 9 ven by the 9rad ent ot the tinqenl il r h a r p o . t T h e e q u a ( o n o l t h € t J n ! ! N i wds cr.ulircd lr!.ompuLer soltfl e ' l h e

r a L e a h - " p o i n t s h o w . s - 0 l l m o d o r m i n . T h e n e q a t v € v a u € shows rhat lhat reacLant.on.enlrat o.

s decreasing wirh .(ea5.! I'm.

o independent

l he grrdient ofa stluight line has units; the units of the vcrtical axis divided bv the units of the horizontal axis sometnnes a line has to be cxtended beyond dre ringe oimelsurencnls oftbe grrph lhis is callcd extrapolation. Absokte zero, for exanrplc, crn be Ll1nd by extrapolating fte volumre/teDrperrture Sraph tor un

' l h e

p L o c c s s o f r s s u i n g t h r t t h c t r e n d li n e a p p l i e s b e t w e e n t w o p o i n t s i s c i l l e d i n t e r p o l a t i o n . l h e g r r d i c D t o f a c u r v e _{d t r n y Poirrl is the grrdicnt ofthe tangent} l o d r e c u r v c u l t h . t p o i n t .

0.6

0.2 0 , 1 0

2

Errors

and graphs

S,ysterratic c|lo$ und rNndom unccrltrirries caD olierr be r€coSnized tionr ' gnPh

,\ grrph conrbincs thc results ol'nr.uy nrersuremeDts rnd so n]iDini,cs the efiects

of .undonr unccrtanrties irr the tnersurenrents

E 6

E

E E

The slope ofthe curve 15 the grad ent ofthe tangent at this poini

Measurement and data process ng

g

g

I

I

1

Choosing

what to plot to produce

a straight line

ID manycases,the best way to analyse neasucments is to find a wayofpiotting the data to produce a straight linc.

For exanple thc idenl gxs equation:

Pt/ = ,RT

crn b€ rcarrrnged to gile a straight line graph:

P = , ' 1 1 7 - l t ]

'lhe

prcssurc is inverselyproportional to dre vohunc. This relitionsbiP is clcarly seeD ivhcn a gr.rph of l/ vagainsi P gives N straight line passing through thc origin

l constrnt tenrPcraiure.

Using spreadsheets

to plot graphs

'lhcrc

rrc nrany sofiwdrc prckrgcs which .llow grrphs lo be Plottcd und trnilyscdi t h c e q u l l i o n o l r h c b c n l l t l i n c c r n b c g i v e n d n d o t h c r ProPerties c a l c l l l a t c d F o r c \ i m p l c , th c lu | g c n t ro t h c . u r v c il r l r i s u r c l l . 6 h d s l h c c q u a t i o n :

l ' = 0 l l 0 9 r + 0 1 8 1 8

n r t h e g r n d i c n l o l t h c lu n g e n l u l t h i t p o i n t - - 0 l l n n n d l n ' m i n (lrrc shoukl, howcvcf, bc trkrn whcn using lhcsc Pxckrgcs.

' l h c s c t

o l d a t i p o i n l s c i D c i t h c L b e j o i n e d b y r b c s t fi l s t r i i g h t li n e w h i c h d o c s n o l p a s s $ r o u g h r n r p o i n t e x c e p r t h c o t i g i i r

) ' = l 6 2 5 5 r i ( R ' = 0 9 5 1 7 )

or r polyno Dr iil rvhich givcs a perfect 6t.s indicatcd bythe llr vilue of I

/ = - 0 . 0 1 8 3 r ' ' + 0 . 2 6 6 7 ! - 1 . 2 0 8 3 - f r + 1 . 7 3 3 3 x r + 1 . 4 2 6 7 i r ( 1 ? ' : t)

The polynonrialequriion is unlikely, hoi{ever, to bc physically sigDilicant as any series ofrandom points c.D lit a polynom ial of sufllcient lcngth,iust as any tlvo ponlts dcline a strright line.

a

l

"

_{; # k . '}

F i g u r e l l . a T h s n r a i q h t n . 9 r r F h j 0!!1l,dr theDiesrrc ! nvP6ely , F p o r l o r l l n I o r f e v o [ n c

F i g u r e 1 1 . 9 A r . q r a l n n w h i { h ) . r i r n 6 J p r r r c . r l i l ! m r r . . . $ i ' y 1 r b . r ( c a ' D r o 0 l i h c r . i r . f n r i f l

l : i :

: ' : :

! !

5 . : . 1

: ?

: : . :

a - : - .

The vo ume 14 pressure Pand temperatLrre /and numb€r of rno es of an ideal ga5 ae

related by rhe idea gas equallon:Py= ,RL Ithe r€latonship between press!re and

vo ume at constanl temp€raiure ola lix€d dmount of gas B nvesligated experimenta y,

whkh one ofthe fo ow ng plots wou d produce a near graphT

A P a q a i n s t Y 8 P a q a i n n l

-L

7 a g a n n 7

D No plotcan produce a stra ght lif€

T h e m a s 5 o l a n o b i € c t i s m e d s u r e d a s 1 . 6 5 2 g and l s v o u m e 1 . 1 c m r . i t h e d € n s i q

(mass p€r unitvolume)is caku ated from these va uet to how many sign icdnt figures

s t o rd b p e p ' e s s € d )

8 2 C 3 D 4

T h e t i n r e i o r a 2 . 0 0 c n r s a m p € o f rn a q n e s i u m r i b b o n t o re a c l c o m p l e t e y w t h 2 0 . 0 c m r

ol 1.00 mo dm r hydroch oric ac d s mealur€d four limes by a student Th€ read ngs €

between 48 8 and 49 2s Th s measurement s best recorded as:

A 4 8 . 8 a 0 2 s 8 4 8 . 8 1 0.4 s C 4 9 . 0 1 0 2 ! D 4 9 . 0 a 0 4 5

A student measures the vo ume ofwater incorrecty by reading the lop lnst€ad of the

bottoan ol men scus,This errorw allect:

A neilh€r the precision nor the accuracy ollhe readings

I o n y the a c c u r a c y o l t h € r e i d i n g s

C on y the precision ol the read ngs

D both th€ pr€cson and the accurary ofthe readifgs

A k n o w n v o u m e o f l o d i u m h y d r o x d € s o l u t o n i s a d d € d t o a c o n l c a f a s k u s n g a

p p€tte.A burette is Lrsed to measurs thevolume oi hydroch oric acd n€ed€d to

n€utra ze the sod um hydroxide.Whkh oflhe fo ow nq woLr d ead to a systematc€rror

i n the r e s u t s 7

I lhe use 01a wel burette

I lhe lse oid wet pipelt€ l l the ! s € o f a w e t c o n c a ta s k A a n d L l o f y

B aid Ll only C l a n d l l o n l y D , l l a n d L l

fte numb€r ofsqnificaitfigures that should be repofted fol th€ mass increase whlch s

c b t a i n e d b y i a k i n g t h € d i l l e r e n c e b e t w e e n r € a d i n g s o f I1 . 6 2 : 5 g and 1 0 . 5 8 0 5 9 5 : . l

V

Measurernent

and

data

processing

l

A 0 . 2 6 6 g s a m p l e o f z i n c a d d e d t o h y d r o c h l o c a c i d . 0 . 1 8 6 9 o f z i n c l s l a l e r r e c o v e r e d

fom the ac d, What is the percentag€ mass loss oi the zinc to the coffe( number oJ

siqnificant fgures?

A 3 0 % B 3 0 . 1 0 / o c 34.41% D 3 0 . 0 8 %

Whkh type oferols can cancelwhen differences ln quantiti€s €€ ca culated?

ll syslematrc e|r0r5

A l o n y B l l o f y C l a n d l l

D neither or ll

The enthalpy change of the rea(on:

Cuso4(aq) + Zn(s)+ Zfsoi(aq) + CLls)

was del€rmifed using the proced!re outlined on page 103.

A S 9 U m t n g :

. al the heat of reaction passes inlo the water

The molarenlhalpy chanqe can be cakuated frcrn the t€mperature change of the

50lut on using the €xpressionl

l r - r \

L H = + a ! _ : : ! , ^ , , ' 2 , /. r r o _ .. LL!JU!]

where .B?o s th€ specific heat capacity of watet _{1,," h the} tempeGture ol the copper

sulfale before z ncwas add€d _{and Ini sthe naximum} temperature ofthe coppef

sulfate soluton afterthe z ncwas added,

The following r€sults were r€corded:

the mr

ibilitl

u s i n g . l h e ri s i

ass€agi

Al Ar a t . 1 a l . 2

A.2 hin

a 2 . l

A2.3

Ai Intn

A 3 r (

13.2 t

I

A . ] 3 T n J 4 '

l c L r s o a l = 0.s00moldm 3

(a) Ca culate the tenrperatrre change du ng the reaction and givethe absolute

uncenainty,

(b) cak! atethe peKentage urcertaintyofth s tempe€ture change.

(c) Calculat€ the molar enthalpychange ofrca(ion.

(d) Assuming the uncertainties in any other measurcmefts are negligibe, determ nethe

percentage uncelrainly in the experimenta valle ofthe enthalpychange.

21.2

Calcu ate th€ absolute uncenainty.

lle iterature valueforthe standard enthalpy change of reaction = -2l7kl mol I.

Comment on any differences between theexperimenta and liierature va ues,

teJ

(0

10 The litelature value for the enthalpy change of reaction bexaeen copper sullate and zifc is

217klmol '.An 18 stldent followed the procedure outlined on page 103 to obtain an