Mathematics for Chemistry Workbook

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A workbook for chemists on the underlying mathematics needed to study chemistry at beginning undergraduate level. Videos of worked solutions to many of the problems in this workbook can be also found in JorumOpen. CC License: Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales

Text of Mathematics for Chemistry Workbook

  • 1. ThisworkislicensedunderaCreativeCommonsAttributionNonCommercialShareAlike2.0License. Author PatrickJOMalleyOwnerPatrickJOMalley TitleMathematicsforChemistryWorkbook Keywords ukoer,sfsoer,mathsforchemistry,quantitativechemistry,differentiation,integration,naturallogarithm,simultaneousequations,straightlinegraphs,exponentials,unitconversion.Description Aworkbookcontainingessentialmathematicaltechniquesforchemistrystudents LanguageEnglishFilesize30MbFileFormat Word2003Document1

2. Introduction Much of chemistry is quantitative. In the laboratory you will need to calculate yields, do calculations involvingpHandpKa,plotgraphsandestimateexperimentalerrors.Tofollowthecorechemistrylectures youwillneedamasteryofsimplealgebra,trigonometryandcalculus.Inshorttogetthemostoutofthis course and to get good marks in examinations and practical work, you need a certain level of mathematicalskills.Thiscourseaimstoequipyouwiththeseskills.Thereareseveralimportantpointstonoteaboutthiscourse.Chem 10511 is student centred learning. This means you have the full course material, practice exercises and answers to the exercises plus video screencasts of worked solutions for some of the exercises. You work through this material at your own pace in your own time. In addition a practice exam test is provided for each of the real tests - so that you know the sort of thing to expect in the real test. All up to date information on the course should be accessed through the Chem 10511 Blackboard site. You work though the material in your own time at your own pace, using the workbook,screencastsandonlinepracticetests. Thereisadropin cliniceveryFridayinRoom7.29between1.02.0 pm Specialised PASSsessionsareconductedtoprovideassistance. AssessmentisviathreeBlackboardTests.The drop-in clinic (in my office 7.29) is to go through particular problems that you still don't understand - even with the course material, screencasts and answers. Bring along you working/attempts so we can go through the problem together. This is a clinic it does not contain lectures or formal presentations.The material covered is divided into 3 sections each of which are examined using a Blackboardtest. The3sectionsarepresentedintheworkbookunderheadingsTests1,2and3.Youwillbedividedinto 3 or 4 groups for each test. You will be given 45 minutes to complete each test. Please consult the Blackboardsiteforalluptodateinformationconcerning groupallocationsneartothetestdates.All testswilltakeplaceintheChemistryComputerClusteronthegroundflooroftheChemistrybuilding.Allteststakeplacebetween15pmonthefollowingdates:rd Test123 October2009.th Test213 November2009.th Test311 December2009. For the overall mark the individual tests are weighted in the ratio 20:30:50 for tests 1, 2 and 3 respectively.These tests will be invigilated and University regulations will apply. Library cards must be on view.Nonotesortextbooksmaybeused.2 3. Practice Blackboardtestquizzes areavailable foruseatanytime. We stronglyrecommendthatyoutrythesebeforeattemptingtherealtest.Toaccessthese followthelinksontheBlackboardsite. Bringpaperandpenswithyoutothesetests.Forthefirsttestnocalculatorsareallowed,butthesewillbeneededfortests2and3. Ausefultextbooktoaccompanythecourseis: BeginningMathematicsforChemistry,SKScott,OUP(1995) A final word we believe it extremely important that all our students have a mastery of themathematicalskillscontainedinthiscourse.Pleasenote,however,thatwearenotaskingforanimpossiblyhighstandard!Manyofyouwillhavecoveredmuchofthematerialalready forthoseof you who have not on satisfactory completion of this course, you will have acquired theseskills. 3 4. Test1Material 4 5. AlgebraAnaturalstartingpointinanymathematicseducationisalgebra.Theabilitytoperformbasicalgebraic manipulations,toexpandorfactoriseexpressionsandrearrangeformulae,isfundamentaltohandlingof numericaldataanddevelopingphysicalmodelsofthewaytheuniverseworks.Youshouldalwaysbearinmindthatalgebraisnothingmorethancommonsense!Itissimplyawayofstatingsystematicallywhatweareallowedtodowith numbers.Remember,attheendoftheday,thelettersinalgebraicexpressionssimplystandfornumbersandwearedealingwithnothingmorethan arithmetic.Ourstartingpointistodefinecarefullythepropertiesofnumbers.NegativeNumbers Beforewelookatsomebasicalgebraitisimportantthatyouareclearontheuseofnegativenumbers.Theearlyhistoryofrealnumbershadthreeimportantstages.Positivenumbers1,2,3....Thesedevelopedfromtheneedtocountobjectssheep,coinsetc.e.g. 3+2=57 6=1Zero Theinclusionofzeroasanumberisalater(Arabic)idea.Itsimplifiescalculations.e.g.3 3=0Negativenumbers1,2,3 The idea of negative numbers is later still. They greatly simplify the rules of algebra and are fundamentaltoeverydayscientificexpressions. e.g. 58=3Realnumbers32 1 01 23 4 5 ve numbers + ve numbersAddinganegativenumberisthesameassubtractingapositivenumber. a+b=a b Subtractinganegativenumberisthesameasaddingapositivenumber. ab=a+bMultiplicationrules positive positive=positivea b=ab positive negative=negativea b=ab negative positive=negative a b=ab negative negative=positive a b=ab5 6. ArithmeticbasicrulesAdditiona+b=b+a Theorderdoesn'tmatter e.g. 2+5=5+2 Bothequal7Multiplicationa b=b a Theorderdoesn'tmatter e.g. 2 5=5 2 Bothequal10Subtractiona -b b- a Theordermatters a -b=(b -a) e.g. 2 -5 5- 2 -3 3Divisiona b a 1 Theordermatters= b ab b a 2 5 e.g. 0. 254 . 5 2 CommonFactorsexpandinga(x +y)=ax +aya(x -y)=ax -aye.g. 2(3 +4)=(2 3) + (2 4)2(3 -4)=(2 3) - (2 4)MultiplyingBrackets(a+b)(c +d)=ac +ad+bc+bd 2 (x+a)(x+b)=x +ax+bx+ab 2= x +(b+a)x+abe.g.(4 + 6)(2 + 5) = (4 2) + (4 5) + (6 2) + (6 5) 6 7. Factorising ac +ad+bc+bd=a(c +d)+b(c+d) =(c +d)(a+b) =(a+b)(c +d)e.g.(4 2) + (4 5) + (6 2) + (6 5) = 4(2 + 5) + 6(2 + 5)=(2 +5)(4+6)=(4+6)(2 +5)Factorisingisjustausefulwayoftidyingupexpressions.Innowaydoesitchange theunderlyingvaluewhennumbersaresubstitutedforthevariables.Factorising expressionsissomewhatofanartthatonlycomeswithplentyofpractice.Equationswithasinglevariable10x 2 xe.g. y=5x+3y=y=ax +bx+cy=sin (x+2) aYouwillmeetmanyexpressionsofthistypeinthecourse.Theequationhasasinglevariable(xinthese examples)andanumberofconstants(5,3,a,b,c,...).Itisessential thatyoucanperformbasic manipulationoftheseexpressions.Moreonmultiplyingbrackets y =(5+x)(x+3) =(5 x)+(5 3)+(x x)+(x 3) expand 2=5x+15+x +3xsimplifyeachterm 2=x +5x+3x+15reorder 2=x +(5+3)x+15 collectfactors 2=x +8x+15 simplifyfactorsBothexpressionsareexactlythesameinvalue.Theyarejustdifferentformsofthesamething.GeneralRule (x+a)(x+b)=(x x)+(x b)+(a x)+(a b)2 =x +bx+ax+ab 7 8. 2=x +(b+a)x+ab SpecialCases222 (x+a) =(x+a)(x+a)=x +2ax+a 2 22 2 (x+a)(x a)=x ax+ax a=x a222 (x a) =(x a)(x a)=x 2ax+aMultiplebrackets 2 (x+1)(x+2)(x+3) =(x+1)(x +5x+6) 32 2=x +5x +6x+x +5x+6 32=x +6x +11x+6 8 9. PracticeExercise1Expandthefollowingexpressions(x+1)(x+6)(4x+1)(2x+2)(x+4)(x 6)(2x 1)(3x+2)(2x 1)(3x 2)(x 1)(x+6)(x 3)(6x+4) 3 (x+1) 3 (x+a)Factorisethefollowingexpressions 2 x 1 2 x +2x+1 2 x 2x+19 10. FractionsAdditionandsubtraction 1 1 325+ =? +=putovercommondenominator2 3 6661 11WRONG + CommonError a b a+ b1 1 1 + 2 3 5Thevalueofafractionisnotchangedbymultiplyingtopandbottombythesamenumbere.g.1 2 3 = = 2 4 6Thisprovidesthemethodforaddingandsubtractingfractions 11 1 b 1 ab+a +=+ =ab a b b a ab 11 1 b 1 ab a = =ab a b b a abab a yb x ay+bx+= += xy x yy x xyExample12x +( 2 x+105 )+== 5x5x5xgoingbackwards x+10x 10 12= + =+5x5x5x5x 10 11. Multiplication1 1 1 = a b aba b ab = x y xy 8 6 8 6 e.g. = 2 3 2 3 DivisionInordinaryarithmeticwecanperformdivisionbymultiplyingthenominatorbytheinverseofthe denominator. 1 a 3= a 3 divisionby3multiplicationby1/3Wedothesamethingwithfractions1 1 1 b b = = a b a 1 aa xaba y ay== = bx yxb bx y CancellingcommonfactorsYoucanonlycancelcommonfactorsinthenumeratoranddenominatorifthey multiplyeverythingelse inboth.ax a x x4(x+1) 4 (x+1)4= = = = ay a y y(x+2)(x+1) (x+2)(x+1) (x+2) axcannotcancel asinceaddedindenominator a+y11 12. 4+(x+1) cannotcancel (x+1)sinceaddedinnumerator (x+2)(x+1)12 13. PracticeExercise2Evaluatethefollowingexpressions1 1 + 3 5 2 4 5 9 2 4 3 9 2 4 392 3+ 3x y2 3+ 3x x1 1 + (x+2) (x+3)6 (x+2) (x+3)13 14. RearrangingEquationsThebasicformofanequationisLefthandSide(LHS)=RightHandSide(RHS)Akeyskillistobeabletoperformmanipulationsonsuchequations.Theseareoftendoneinorderto isolateanexpressionforaparticularvariable.Allvalidmanipulationsdonotalterthetruthofthe expression theymerelychangetheform.Therearereallyonly tworulestomanipulatingequations:AnequationisunaffectedifthesamequantityisaddedorsubtractedfromeachsideAnequationisunaffectedifeachsideismultipliedordividedbythesamequantitye.g.x=y x y equallytrueare:x+3=y+3 or 3x=3y or =a aTheserulesallowusto"move"constants,variableorexpressionsbetweensides.x+7=yinitialequationx+77=y 7subtract7fromeachsidex=y 7 7movedtoothersideor9x=5initialequation9x 5= divideeachsideby9 9 95 x= 9movedtootherside 9 14 15. IsolatingavariableWecankeepapplyingrulestoisolateavariableononeside 7x+4=5 initialequation 7x+4 4=5 4 subtract4frombothsides 7x=1 movedthe4 7x 1 =divideeachsideby77 7 1x=7movedtootherside7or 7x+4=10initialequation2+y 7x+4=10(2+y) multiplyby(2+y) 7x=10(2+y)4 subtract4 7x=20+10y 4expand 7x=16+10y 7x=2(8+5y) collectcommonfactor2( +5 ) 2 8 y x== (8+5y) divideby7 7 7Anexaminationoftheaboveexamplesprovidessimplerrulesforgettingthesameresults.Tomoveanexpressionfollowinga+or signtotheothersidewesimplychangeitssign x 9=2ybecomesx=2y+9 4y 4yx+=5becomesx=5 3+y3+y15 16. Tomoveanominatorordenominatoracrosswemultiplybytheinverse 2y 9x=2y becomes x=9 54yx =5 becomes x=4y x =2becomes x=2(y+3)y+3ChemistryRelatedExampleThe2ndorderkineticsexpressionfortheconcentrationofaspecieswithtimetisgivenby:1 1 - =kt c c 0 wherecistheconcentrationattimet,c istheinitialconcentration(attimezero)andkistherate0 constant.Findanexpressionforc. 1 1 =kt+movec overcc0 01 c kt+1 = 0putovercommonfactorc c0 c0=(1+c kt)0movebottomc over 0c c =(1+c kt)c 00 moveco