In 47–50, find a Taylor series for f(t) about t=0. Assuming the Laplace transform of f(t) can be computed term by term, find an expansion for in powers of 1/s. If possible, sum the series. F(t) = sin t

4/4/16 EXAMAPRIL14 th Ø 4/12NickTreichishostingareviewsession@415pminFulmer125 Ø 4/12Finneganishostingareviewsessioninthepit@7 Ø 4/11@515theChemclubishostingareviewsession AXE • • n=thenumberofbondinggroupsonthecentralatom • m=thenumberoflonepairsonthecentralatom • ANYtypeofbond(single,double,triple)isonebondinggroup • n+m=thenumberofelectrongroupsonthecentralatom ElectronGeometry • Thenumberofelectrongroupsdeterminesthis • Thereareonly5,seetable10.1 • Determinestheidealbondangles-elementstryandarrangethemselvesasfar awayfromeachotheraspossiblebecausetheelectronsrepulsethemselves MolecularGeometry(shape)-electrongeometryandthenumberoflonepairs determinethemoleculargeometry • Themolecularshape,thebonpolarities,andtheformalchargedistribution determinethemolecularpolarity 4/6/16 BondAngles • ElectronGeometrydeterminestheidealbondangles(theyareapproximate) Linear:180degreeangles Triganolplanar:120degreeangles Tetrahedral:109.5degreeangles Trigonalbipyramidal:90,120degreeangles Octahedral:90degreeangles • Lonepairstakeupmorespacethanbondingpairs • Doublebondstakeupmorespacethanasinglebonds • ImportantNote:Whendealingwithresonancestructures,theamountof spacethe“rotating”doublen=bondtakesupisnegligiblebecauseitis technicallyalldoublebonds ▯ • ElectronGeometry:TriganolBipyramidal • ℎ:Linearduetothe180degreeangle HowdowedothebondanglesforthismoleculeByusingtheideathatdouble bondstakeupmorespacethanasingle,andlonepairstakemorespacethanbonds • OctahedralElectronGeometry: ▯ ▯ • Molecularshape:squareplanar,duetothe90degreeangle • BondanglesforF-Xe-Fis90degrees • ElectronGeometry:Tetrahedral • MolecularShape:Triganolpyramidal • BondAnglesforH-N-Hislessthan109.5becausethelonepairwillmakethe anglesmallerthanthatofanormaltetrahedralmolecule NoticehowitisTetrahedralwiththelonepair,fortheshapethelonepairisnot drawn PolarandNon-PolarMolecules • Somethingispolarifithasanegativesideandanonpolarside • Inmoleculesthisisdeterminedbysymmetryaroundthecentralatom • If a molecule is symmetric (by polar bonds)around the central atom, the charges cancel eachother out • If the molecule is not semetric(by polar bonds) than the molecule is polar • A numeric measure of polarity is the Dipole moment(μ) • The more polar a molecule the larger the dipole moment • Is ▯ polar or non-polar Non-polar • Is ▯polar or non polar Non polar • Is ▯olar or non polar Polar • Look at the bonds and the way the bonds are arranged to see if they are polar or not! ValenceBondTheory • Acovalentbondisproducedbytheoverlapoforbitalsintheregion betweenthetwoatoms • Thegreatertheoverlap,thestrongerthebond. • But,theatomicorbitalsdonotalwayspointintherightdirectionsto producetheshapesthatthemoleculesaresupposedtohave Toexplainthis,itisassumedthattheatomicorbitalsaremixedtoproduce‘hybrid’ orbitalsthatpointinthecorrectdirection. • Hybridorbitalsarenamedfortheorbitalsthatcontributetothem.For Example isacombinationofonesorbitalandtwoporbitals • ImportantNote:Allorbitalsthataremixedmustcomefromthesame shell.ForExample3pand3scanbehybrids,but2pand4scannotbe combinedtogether. • The▯totalnumberoforbitalsisunchanged(Therewillbe2orbitals,three orbitalsandsoon) EachhybridhasitsownGeometryassociatedwithit,asshownbelow NoticehowforthegroundstateofCarbonthe2sorbitalisfull,butthe2porbitalis not▯evenhalffull,thismakescarbonunstablesothetwoorbitalscombinetoform ,makingcarbonstable • Singlebondsareformedbyoverlappingorbitalsinbetweenthe atoms.Thesebondsarecalledsigma(σ)bonds. • Doubleandtriplebondsalsocontainaσ-bondbutthesecond(andthird bond)cannotformdirectlybetweentheatoms(becausethereisalreadya pairofelectronsthere) • Theadditionalbondsareformedbyunhybridizedp-orbitalsthat overlapintheareastoeithersideofthesigmabond.Thesebondsare calledpi(π)bonds. • Adoublebondconsistsofaσ-bondandaπ-bond. • Atriplebondconsistsofaσ-bondandtwoπ-bonds. ThisDiagramshouldhelpyouvisualizetheprincipalofsigmaandpibonds