H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3.


 Isabel Butler
 2 years ago
 Views:
Transcription
1 . Spetrosopy Mss spetrosopy igh resolution mss spetrometry n e used to determine the moleulr formul of ompound from the urte mss of the moleulr ion For exmple, the following moleulr formuls ll hve rough M r of 60, ut more preise M r n give the moleulr formul. e.g. M r = 60.0 moleulr formul = 4 M r M r = moleulr formul = 8 = moleulr formul = 4 N igh resolution mss spetrosopy n mesure the mss to 5 d.p. This n help differentite etween ompounds tht pper to hve similr Mr (to the nerest whole numer) Aurte msses of toms: =.0078 =.0000 = N = 4.00 Exmple A ompound is found to hve n urte reltive formul mss of It is thought to e either or N N. lulte the M r of eh ompound to 4 deiml ples to work out whih one it is. = (.0000 x ) + ( x ) + (.0078 x6) = N N. = (.0000 x ) + (4.00 x ) + (.0078 x6) = Frgmenttion When orgni moleules re pssed through mss spetrometer, it detets oth the whole moleule nd frgments of the moleule. Moleulr ion formed: M [M] +. + e The moleule loses n eletron nd eomes oth n ion nd free rdil Severl peks in the mss spetrum our due to frgmenttion. The Moleulr ion frgments due to ovlent onds reking: [M] +. X + + Y. Reltively stle ions suh s rotions R + suh s + nd ylium ions [R=] + re ommon. The more stle the ion, the greter the pek intensity. The pek with the highest mss/hrge rtio will e normlly due to the originl moleule tht hsn t frgmented (lled the moleulr ion). As the hrge of the ion is + the mss/ hrge rtio is equl to Mr. This proess produes n ion nd free rdil. The ion is responsile for the pek The yl group present in ronyls, esters id derivtives is R ommon stle ion + Mss spetrum for utne = 58 Eqution for formtion moleulr ion 4 0 [ 4 0 ] +. + e m/z 58 Equtions for formtion of frgment ions from moleulr ions [ 4 0 ] +. [ ] + +. m/z 4 [ 4 0 ] +. [ ] + +. m/z 9 Mss spetrum for utnone The high pek t 4 due to stility of yl group 4 [ ] + Eqution for formtion moleulr ion [ ] +. + e m/z 7 Equtions for formtion of frgment ions from moleulr ions [ ] +. [ ] + +. m/z 57 9 [ ] + [ ] + 57 [ ] +. 7 [ ] +. [ ] + +. m/z 4 [ ] +. [ ] + +. m/z 9 N Goly hemrevise.org
2 Infrred spetrosopy ertin groups in moleule sor infrred rdition t hrteristi frequenies omplited spetr n e otined thn provide informtion out the types of onds present in moleule ABVE 500 m  Funtionl group identifition BELW 500 m  Fingerprinting omplited nd ontins mny signls piking out funtionl group signls diffiult. This prt of the spetrum is unique for every ompound, nd so n e used s "fingerprint". e.g. = m   (id) m  Use n IR sorption tle provided in exm to dedue presene or sene of prtiulr onds or funtionl groups A omputer will ompre the IR spetr ginst dtse of known pure ompounds to identify the ompound use spetr to identify prtiulr funtionl groups nd to identify impurities, limited to dt presented in wvenumer form Spetr for utnl = Asorption or trough in etween m  rnge indites presene of = ond Alwys quote the wve numer rnge from the dt sheet  sorptions tend to e rod Asorption or trough in etween m  rnge indites presene of  ond in n id Spetr for ethnoi id = rogue sorptions n lso our nd re inditors of impurities N Goly hemrevise.org
3 NMR spetrosopy Different types of NMR There re two min types of NMR. NMR. (proton) NMR There is only round % in orgni moleules ut modern NMR mhines re sensitive enough to give full spetr for The spetr is simpler spetrum thn the NMR Equivlent ron toms. In NMR spetrum, there is one signl (pek) for eh set of equivlent toms. d peks 4 peks, dinitroenzene, dinitroenzene,4 dinitroenzene N peks N 4 peks peks d 4 peks peks l d peks d e N 5 peks Equivlent ydrogen toms. In n NMR spetrum, there is one signl for eh set of equivlent toms. Ethnol hs groups of different hydrogen toms sets of equivlent s: rtio ::9 sets of equivlent s: rtio :: In ddition the intensity (integrtion vlue) of eh signl is proportionl to the numer of equivlent toms it represents. Br d d d 4sets of equivlent s: rtio 6::: signl sets of equivlent s: rtio :: 4 sets of equivlent s: rtio ::: d N Goly hemrevise.org
4 Solvents Smples re dissolved in solvents without ny toms, e.g. l 4, Dl. This mens tht in the NMR the solvent will not give ny peks The sme solvent is used in NMR nd in this se there will e one pek due to the solvent tht will pper on the spetrum. owever, it is known where this pek is so it n e ignored. In the exm it is likely this pek will not our on the spetr. lirtion nd shift A smll mount of TMS (tetrmethylsilne) is dded to the smple to lirte the spetrum TMS is used euse: its signl is wy from ll the others it only gives one signl it is nontoxi it is inert it hs low oiling point nd so n e removed from smple esily Si tetrmethylsilne The sme lirtion ompound is used for oth nd NMR The spetr re reorded on sle known s the hemil shift (δ), whih is how muh the field hs shifted wy from the field for TMS.. The δ is mesure in prts per million (ppm) is reltive sle of how fr the frequeny of the proton signl hs shifted wy from tht for TMS δ hemil shift (ppm) 0 NMR shift The δ depends on wht other toms/groups re ner the more eletronegtive groups gives greter shift. δ ppm N Goly hemrevise.org δ ppm 4
5 NMR shift SpinSpin oupling in Nmr In high resolution NMR eh signl in the spetrum n e split into further lines due to inequivlent s on neighouring toms. ppm Nulei in identil hemil environments do not show oupling mongst themselves! Splitting of pek = numer of inequivlent s on neighouring toms + signl singlet doulet triplet qurtet pperne Split numer of peks numer of neighouring inequivlent toms 4 0 reltive size : :: ::: The pek due to group will e triplet s it is next to ( ron with s) The pek due to group will e qurtet s it is next to ( ron with s) The pek due to group will e singlet s it is next to ron with no s) The pek due to group will e triplet s it is next to ron with s Shift Integrtion tre The pek due to group will e singlet s it is next to ron with 0 s Shift..6 Integrtion tre The pek due to group will e qurtet s it is next to ron with s Shift.74. Integrtion tre ppm N Goly hemrevise.org 5
6 hromtogrphy hromtogrphy is n nlytil tehnique tht seprtes omponents in mixture etween moile phse nd sttionry phse. Seprtion y olumn hromtogrphy depends on the lne etween soluility in the moving phse nd retention in the sttionry phse. A solid sttionry phse seprtes y dsorption, A liquid sttionry phse seprtes y reltive soluility The moile phse my e liquid or gs. The sttionry phse my e solid (s in thinlyer hromtogrphy, TL) or either liquid or solid on solid support (s in gs hromtogrphy, G) If the sttionry phse ws polr nd the moving phse ws non polr e.g. exne. Then nonpolr ompounds would pss through the olumn more quikly thn polr ompounds s they would hve greter soluility in the nonpolr moving phse. (Think out intermoleulr fores) PL stnds for high performne liquid hromtogrphy. PL: sttionry phse is solid sili PL: moile phse liquid In gsliquid hromtogrphy G the moile phse is inert gs suh s nitrogen, helium, rgon. The Sttionry phse is liquid on n inert solid. GsLiquid hromtogrphy Gsliquid hromtogrphy n e used to seprte mixtures of voltile liquids. The time tken for prtiulr ompound to trvel from the injetion of the smple to where it leves the olumn to the detetor is known s its retention time. This n e used to identify sustne. Flow ontrol In gsliquid hromtogrphy, the moile phse is gs suh s helium nd the sttionry phse is high oiling point liquid sored onto solid. Smple in oven Some ompounds hve similr retention times so will not e distinguished. disply Bsi gsliquid hromtogrphy will tell us how mny omponents there re in the mixture y the numer of peks. It will lso tell us the undne of eh sustne. The re under eh pek will e proportionl to the undne of tht omponent. rrier gs olumn detetor Wste outlet It is lso possile for gsliquid hromtogrphy mhine to e onneted to mss spetrometer, IR or NMR mhine, enling ll the omponents in mixture to e identified. GMS is used in nlysis, in forensis, environmentl nlysis, irport seurity nd spe proes. Most ommonly mss spetrometer is omined with G to generte mss spetr whih n e nlysed or ompred with spetrl dtse y omputer for positive identifition of eh omponent in the mixture. N Goly hemrevise.org 6
7 Bringing it ll together. Work out empiril formul Elementl nlysis 66.6%.8%.9%. Using moleulr ion pek m/z vlue from mss spetrum lulte Moleulr formul moleulr ion pek m/z vlue= /.8/.9/6 =5.555 =.8 = =4 =8 = Mr empiril formul 4 8 = 7 If Mr moleulr formul 44 then ompound is 8 6. Use IR spetr to identify min onds/funtionl group 8 6 ould e n ester, roxyli id or omintion of lohol nd ronyl. Look for IR spetr for = nd  onds There is = ut no  sorptions, so must e n ester.  = 4. Use NMR spetr to give detils of ron hin 4 peks only 4 different environments. singlet of re 9 At δ =0.9 Mens groups 9 Pek t δ 4 shows Are suggests Qurtet mens next to Pek t δ. shows = Are suggests Singlet mens djent to with no hydrogens Pek t δ. shows R Are mens Triplet mens next to 5 4 δ ppm Put ll together to give finl struture N Goly hemrevise.org 7
6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR
6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht
More information6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR
6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht
More information3.15 NMR spectroscopy Different types of NMR There are two main types of NMR 1. C 13 NMR 2. H (proton) NMR
.5 NMR spetrosopy Different types of NMR There re two min types of NMR. NMR. (proton) NMR There is only round % in orgni moleules ut modern NMR mhines re sensitive enough to give full spetr for The spetr
More informationAnalytical Techniques Chromatography
Anlytil Tehniques hromtogrphy hromtogrphy is n nlytil tehnique tht seprtes omponents in mixture etween moile phse nd sttionry phse. Types of hromtogrphy inlude: thinlyer hromtogrphy (TL) plte is oted
More information22.Analytical Techniques Chromatography
.Anlytil Tehniques hromtogrphy hromtogrphy is n nlytil tehnique tht seprtes omponents in mixture etween moile phse nd sttionry phse. Types of hromtogrphy inlude: thinlyer hromtogrphy (TL) plte is oted
More information1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes.
1 This question is out men ond enthlpies nd their use in the lultion of enthlpy hnges. Define men ond enthlpy s pplied to hlorine. Explin why the enthlpy of tomistion of hlorine is extly hlf the men ond
More informationChapter 4rth LIQUIDS AND SOLIDS MCQs
Chpter 4rth LIQUIDS AND SOLIDS MCQs Q.1 Ioni solis re hrterize y () low melting points () goo onutivity in soli stte () high vpour pressure () soluility in polr solvents Q.2 Amorphous solis. () hve shrp
More informationGeneralization of 2Corner Frequency Source Models Used in SMSIM
Generliztion o 2Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville
More informationCALCULATING REACTING QUANTITIES
MODULE 2 14 WORKSHEET WORKSHEET For multiplehoie questions 1 5 irle the letter orresponding to the most orret nswer. 1 The lned eqution for the urning of utnol (C 4 H 9 OH) is given elow: C 4 H 9 OH(l)
More information1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light.
1 This igrm represents the energy hnge tht ours when eletron in trnsition metl ion is exite y visile light. Give the eqution tht reltes the energy hnge ΔE to the Plnk onstnt, h, n the frequeny, v, of the
More informationTrigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis  The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
More informationLogarithms LOGARITHMS.
Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the
More informationActivities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions
MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd
More informationReview Topic 14: Relationships between two numerical variables
Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationNumbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point
GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply
More information4cyanopentanoic acid dithiobenzoate (CPADB) was synthesized as reported by Y.
Eletroni upplementry Mteril (EI) for Journl of Mterils Chemistry B This journl is The Royl oiety of Chemistry 2012 ynthesis of 4ynopentnoi id dithioenzote (CPADB). 4ynopentnoi id dithioenzote (CPADB)
More informationProject 6: Minigoals Towards Simplifying and Rewriting Expressions
MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy
More informationThermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R
/10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?
More informationCS311 Computational Structures Regular Languages and Regular Grammars. Lecture 6
CS311 Computtionl Strutures Regulr Lnguges nd Regulr Grmmrs Leture 6 1 Wht we know so fr: RLs re losed under produt, union nd * Every RL n e written s RE, nd every RE represents RL Every RL n e reognized
More informationNondeterministic Automata vs Deterministic Automata
Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n
More informationLesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.
27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
More information8 THREE PHASE A.C. CIRCUITS
8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the solled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),
More informationPAIR OF LINEAR EQUATIONS IN TWO VARIABLES
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,
More informationInstructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting.
ID: B CSE 2021 Computer Orgniztion Midterm Test (Fll 2009) Instrutions This is losed ook, 80 minutes exm. The MIPS referene sheet my e used s n id for this test. An 8.5 x 11 Chet Sheet my lso e used s
More informationChem Homework 11 due Monday, Apr. 28, 2014, 2 PM
Chem 44  Homework due ondy, pr. 8, 4, P.. . Put this in eq 8.4 terms: E m = m h /m e L for L=d The degenery in the ring system nd the inresed sping per level (4x bigger) mkes the sping between the HOO
More informationPart 4. Integration (with Proofs)
Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1
More informationSpacetime and the Quantum World Questions Fall 2010
Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36  out 3% () P (x 1
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More informationMaintaining Mathematical Proficiency
Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationA Study on the Properties of Rational Triangles
Interntionl Journl of Mthemtis Reserh. ISSN 09765840 Volume 6, Numer (04), pp. 89 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn
More informationFirst compression (06.3 GPa) First decompression ( GPa) Second compression ( GPa) Second decompression (35.
0.9 First ompression (06.3 GP) First deompression (6.32.7 GP) Seond ompression (2.735.5 GP) Seond deompression (35.50 GP) V/V 0 0.7 0.5 0 5 10 15 20 25 30 35 P (GP) Supplementry Figure 1 Compression
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2 elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most wellknown nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationNon Right Angled Triangles
Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in nonright ngled tringles. This unit
More informationComparing the Preimage and Image of a Dilation
hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) nglengle Similrity Theorem (.2) SideSideSide Similrity
More informationMAT 403 NOTES 4. f + f =
MAT 403 NOTES 4 1. Fundmentl Theorem o Clulus We will proo more generl version o the FTC thn the textook. But just like the textook, we strt with the ollowing proposition. Let R[, ] e the set o Riemnn
More informationDorf, R.C., Wan, Z. T Equivalent Networks The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
orf, R.C., Wn,. T Equivlent Networks The Eletril Engineering Hndook Ed. Rihrd C. orf Bo Rton: CRC Press LLC, 000 9 T P Equivlent Networks hen Wn University of Cliforni, vis Rihrd C. orf University of
More informationDiscrete Structures Lecture 11
Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.
More informationSomething found at a salad bar
Nme PP Something found t sld r 4.7 Notes RIGHT TRINGLE hs extly one right ngle. To solve right tringle, you n use things like SOHHTO nd the Pythgoren Theorem. n OLIQUE TRINGLE hs no right ngles. To solve
More informationSECTION A STUDENT MATERIAL. Part 1. What and Why.?
SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are
More informationFinite State Automata and Determinisation
Finite Stte Automt nd Deterministion Tim Dworn Jnury, 2016 Lnguges fs nf re df Deterministion 2 Outline 1 Lnguges 2 Finite Stte Automt (fs) 3 Nondeterministi Finite Stte Automt (nf) 4 Regulr Expressions
More informationIowa Training Systems Trial Snus Hill Winery Madrid, IA
Iow Trining Systems Tril Snus Hill Winery Mdrid, IA Din R. Cohrn nd Gil R. Nonneke Deprtment of Hortiulture, Iow Stte University Bkground nd Rtionle: Over the lst severl yers, five sttes hve een evluting
More informationwhere the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b
CS 2942 9/11/04 Quntum Ciruit Model, SolovyKitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits  Universl Gte Sets A lssil iruit implements multioutput oolen funtion f : {0,1}
More information1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
More informationLecture Notes No. 10
2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite
More informationLecture 6: Coding theory
Leture 6: Coing theory Biology 429 Crl Bergstrom Ferury 4, 2008 Soures: This leture loosely follows Cover n Thoms Chpter 5 n Yeung Chpter 3. As usul, some of the text n equtions re tken iretly from those
More information1 Which of the following summarises the change in wave characteristics on going from infrared to ultraviolet in the electromagnetic spectrum?
Which of the following summrises the chnge in wve chrcteristics on going from infrred to ultrviolet in the electromgnetic spectrum? frequency speed (in vcuum) decreses decreses decreses remins constnt
More informationPYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:
PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for rightngled tringles
More informationAppendix C Partial discharges. 1. Relationship Between Measured and Actual Discharge Quantities
Appendi Prtil dishrges. Reltionship Between Mesured nd Atul Dishrge Quntities A dishrging smple my e simply represented y the euilent iruit in Figure. The pplied lternting oltge V is inresed until the
More information12.4 Similarity in Right Triangles
Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right
More informationEigenvectors and Eigenvalues
MTB 050 1 ORIGIN 1 Eigenvets n Eigenvlues This wksheet esries the lger use to lulte "prinipl" "hrteristi" iretions lle Eigenvets n the "prinipl" "hrteristi" vlues lle Eigenvlues ssoite with these iretions.
More informationSolving Radical Equations
Solving dil Equtions Equtions with dils: A rdil eqution is n eqution in whih vrible ppers in one or more rdinds. Some emples o rdil equtions re: Solution o dil Eqution: The solution o rdil eqution is the
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationNONDETERMINISTIC FSA
Tw o types of nondeterminism: NONDETERMINISTIC FS () Multiple strtsttes; strtsttes S Q. The lnguge L(M) ={x:x tkes M from some strtstte to some finlstte nd ll of x is proessed}. The string x = is
More information22: Union Find. CS 473u  Algorithms  Spring April 14, We want to maintain a collection of sets, under the operations of:
22: Union Fin CS 473u  Algorithms  Spring 2005 April 14, 2005 1 UnionFin We wnt to mintin olletion of sets, uner the opertions of: 1. MkeSet(x)  rete set tht ontins the single element x. 2. Fin(x)
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More informationMath 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)
Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte singlevrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion
More information18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2106
8. Problem Set Due Wenesy, Ot., t : p.m. in  Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the oneelement, linerly epenent sets forme from these. (b) Wht re the twoelement, linerly
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationThis chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2
1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion
More informationAcidBase Equilibria
Tdeusz Górecki Ionic Equiliri AcidBse Equiliri BrønstedLory: n cid is proton, se is. Acid Bse ( 3 PO 4, O), ( N 4 ) nd ( PO  4 ) cn ll ehve s cids. Exmple: 4 N N3 Sustnces hich cn ehve oth s cids nd
More information6.5 Improper integrals
Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =
More information6. Photoionization of acridine through singlet and triplet channels
Chpter 6: Photoioniztion of cridine through singlet nd triplet chnnels 59 6. Photoioniztion of cridine through singlet nd triplet chnnels Photoioinztion of cridine (Ac) in queous micelles hs not yet een
More informationSurds and Indices. Surds and Indices. Curriculum Ready ACMNA: 233,
Surs n Inies Surs n Inies Curriulum Rey ACMNA:, 6 www.mthletis.om Surs SURDS & & Inies INDICES Inies n surs re very losely relte. A numer uner (squre root sign) is lle sur if the squre root n t e simplifie.
More informationCHAPTER 20: Second Law of Thermodynamics
CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het
More informationLogic Synthesis and Verification
Logi Synthesis nd Verifition SOPs nd Inompletely Speified Funtions JieHong Rolnd Jing 江介宏 Deprtment of Eletril Engineering Ntionl Tiwn University Fll 2010 Reding: Logi Synthesis in Nutshell Setion 2 most
More informationTOPPER SAMPLE PAPER  5 CLASS XI MATHEMATICS. Questions. Time Allowed : 3 Hrs Maximum Marks: 100
TOPPER SAMPLE PAPER  5 CLASS XI MATHEMATICS Questions Time Allowed : 3 Hrs Mximum Mrks: 100 1. All questions re compulsory.. The question pper consist of 9 questions divided into three sections A, B nd
More informationCEM143 MWF 8:00 8:50 am. October 5, 2018
CEM43, Fll 208 st Miterm CEM43 MWF 8:00 8:50 m st Miterm toer 5, 208 Nme: Setion: PID: TA: This is lose ook n note exmintion. This exm hs 35 questions. Answer ll questions on the seprte nswer sheet (ule
More informationCEM143 MWF 8:00 8:50 am. October 5, 2018
CEM43, Fll 208 st Miterm CEM43 MWF 8:00 8:50 m st Miterm toer 5, 208 Nme: Setion: PID: TA: This is lose ook n note exmintion. This exm hs 35 questions. Answer ll questions on the seprte nswer sheet (ule
More information( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More informationThe Properties of Stars
10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product
More informationSOLUTIONS TO ASSIGNMENT NO The given nonrecursive signal processing structure is shown as
SOLUTIONS TO ASSIGNMENT NO.1 3. The given nonreursive signl proessing struture is shown s X 1 1 2 3 4 5 Y 1 2 3 4 5 X 2 There re two ritil pths, one from X 1 to Y nd the other from X 2 to Y. The itertion
More informationTable of Content. c 1 / 5
Tehnil Informtion  t nd t Temperture for Controlger 032018 en Tble of Content Introdution....................................................................... 2 Definitions for t nd t..............................................................
More informationAP CALCULUS Test #6: Unit #6 Basic Integration and Applications
AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret
More informationChapter E  Problems
Chpter E  Prolems Blinn College  Physics 2426  Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current
More informationCalculus Cheat Sheet. Integrals Definitions. where F( x ) is an antiderivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx
Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous AntiDerivtive : An ntiderivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.
More informationGreen s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e
Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let
More informationCS 573 Automata Theory and Formal Languages
Nondeterminism Automt Theory nd Forml Lnguges Professor Leslie Lnder Leture # 3 Septemer 6, 2 To hieve our gol, we need the onept of Nondeterministi Finite Automton with moves (NFA) An NFA is tuple
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationElectromagnetism Notes, NYU Spring 2018
Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
More informationCHENG Chun Chor Litwin The Hong Kong Institute of Education
PEhing Mi terntionl onferene IV: novtion of Mthemtis Tehing nd Lerning through Lesson Study onnetion etween ssessment nd Sujet Mtter HENG hun hor Litwin The Hong Kong stitute of Edution Report on using
More informationChapter Gauss Quadrature Rule of Integration
Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to
More informationChapter 8 Roots and Radicals
Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationSection 4.4. Green s Theorem
The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higherdimensionl nlogues) with the definite integrls
More informationAP Calculus AB Unit 4 Assessment
Clss: Dte: 004 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd AmGm inequlity 2. Elementry inequlities......................
More information5. Every rational number have either terminating or repeating (recurring) decimal representation.
CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd
More informationSECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS
SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS INSITU PROBING OF DOMAIN POLING IN Bi 4 Ti 3 O 12 THIN FILMS BY OPTICAL SECOND HARMONIC GENERATION YANIV BARAD, VENKATRAMAN GOPALAN Mterils Reserh Lortory
More informationCore 2 Logarithms and exponentials. Section 1: Introduction to logarithms
Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI
More information