8A, Chapter 1. Review of Atomic Structure: Chloroform, CHCl 3. Hydrogen, H Carbon, C Chlorine, Cl. ** Where are the electrons and what are they doing?

Size: px
Start display at page:

Download "8A, Chapter 1. Review of Atomic Structure: Chloroform, CHCl 3. Hydrogen, H Carbon, C Chlorine, Cl. ** Where are the electrons and what are they doing?"

Transcription

1 EM 8A, Lecture 1 Structure & Bondg - rbitls & Electron onfigurtion - Lewis Structures - Vlence Bond ory & ybrid rbitls - ondensed & Skeletl Structures Review Amic Structure: hloro, l ydrogen, rbon, hlore, l ** Where re nd wht re dog? Bond à rbitls à Use eriodic tble ssign electron configurtion (e- config) L1-1

2 Electron onfigurtion (e-config) olumn Re* Totl #e- B N F Full e- config # Vlence e- Vlence e- config rbitl Digrm Lewis dot (m) Lewis dot (s) * Reresenttive m column on eriodic tble. Lecture 1 L1-

3 Vlence Bond ory - ovlent s ed shrg e- through terctions () orbitls Ex. ydrogen, Sigm (σ) Direct orbitl overl Loclized e- shrg AKA Sgle Ex. hloro ybridiztion = combg s & orbitls llow n m mke desired numbers nd tye s Pi (π) Bond Deloclized e- shrg orbitls L1-

4 1.9 ybrid rbitls nd Structure Vlence Bond ory 14 Acetylene In ddition g sgle nd s shrg nd trons, reectively, crbon ybridiztion14 APTER lso cn trile shrg six 1 Structure nd Bondg To ccount trile such s cetylene, Exmle 1 Structure nd Bondg orbitl, n. Imge tht, i we need third kd APTER orbitls result, nd one orbitl rems orbitls. Three combg or three orbitls, crbon s orbitl izes unchnged. Like s, sgle orbitls Two re unsymmetricl boutresult, nd orbit orbitl. orbitls nucleus nd re lyoriented ecific direction so cn unchnged. orbitls re oriented rt on x-xis, s. three orbitls lieremg lne t ngles 10 one n orbitls on y-xis nd # chrge 4 re erendiculr remg orbitl erendiculr lne, orbitls. Three orbitls result, nd one orbitl rems shown 1.15.s shown ybrid rbitls nd Structure 1.9 ybrid rbitls nd Structure Acetylene 1.8 ybrid rbitls nd Structure clouds* unchnged. Like s, orbitls re unsymmetricl bout In ddition g sgle nd s shrg nd elec Acetylene 1.15 ybridiztion. 1.1 ybridiztion. ecific direction so cn nucleus nd re ly oriented rbitls trons, reectively, crbon lso cn trile shrg six. 1 orbitls re oriented equivlent nd 1.1 ethne. s. three three orbitls lieorbitls In lne t ngles 10 sgle one n s shrg ddition g nd 10 electo ccount trile such s cetylene, q is, wy ech eren lie lne t ngles 10 one. crbon crbon ed remg orbitl erendiculr lne, s shown 1.1. trile shrg six trons, reectively, crbon lso cn need third kd orbitl, n. Imge tht, sted For diculrwe remg orbitls n nd sgle unized overl orbitls. To ccount trile such s cetylene, q, combg or three orbitls, crbon s orbitlclrity, izes orbitl (red/blue) is erendiculr (red/blue). smller lobesonly 1.1 ybridiztion. we need third.imge tht, sted kd orbitl, n 14 lne. sgle orbitl. Two orbitls result, nd orbitls orbitls rerem not shown. three equivlent orbitls combg or10 three orbitls, crbon s orbitl izes only unchnged. orbitls re oriented rt on x-xis, while lie lne t ngles 10 one sgle orbitl. Two orbitls result, nd orbitls rem s n nd sgle unized one orbitls re erendiculr on y-xis nd z-xis, orbitls. Three orbitlsremg result, nd orbitl rems unchnged. orbitls re oriented rt on x-xis, while orbitl (red/blue) is erendiculr re unchnged. Like s, shown orbitls unsymmetricl bout ne Anor orbitls re erendiculr on y-xis nd remg z-xis, s lne. nucleus nd re ly oriented ecific direction so cn crbon shown crbon ybridiztion. crbon ms roch ech orbitl 1.1 ybridiztion. three equivlent orbitls lie lne t ngles 10 one n nd sgle unized orbitl (red/blue) is erendiculr lne. s. three orbitls lie lne t ngles 10 one n orbitls re oriented crbon1.1. overl hed-on remg orbitl s shown. At sme 1.15 ybridiztion. erendiculr eren lne, wy ech 111. orbitls ech crbon z z " sidewys overl z orbitls reoriented Side view To view diculr remg orbitls y orbitls overl similrly y y ". net effect is th wy ech eren(red/blue). six roch nd tion crbon crbon diculr remg orbitls crbons 10 iztion ech trile. tw g At orbitls echtime, hydrogen comlete (red/blue). hed-on 154 m overl. sme unized Side view To view lene ( is 1.). orbitls terct sidewys overl wht clled i (). ne Anor " results shrg combtion n Ethne ech crbons iztion roch nd orbitl 1. tion nd hed-on overl. At sme time, unized crbon crbon ( ne Anor cetylene. crbon ms crbon ms roch ech orbitls on ech 1.14). Note tht occuy region centered orbitls terct sidewys overl wht is clled i (). re joed oneoverl crbon hed-on. At nd sme nuclei, whileresults " occuy belowtime, le combtion n nd " shrg bove orbitls " s. crbon ms roch ech orbitls on ech nd Problem 1.8 z orbitls ech drwn nuclei. z z " sidewys overl, nd tion crbon crbon ( nd crbon crbon overl hed-on. Atngle, sme time, is shrg Drw le- rone,. Predict vlue ech centered similrly overl ethylene, y. net effect 1.14). Note tht occuy region To comlete ms s y orbitls y " hydrogen zoverll orbitls ech crbon z sidewys overl, nd dicte she. z " nd orbitl orbitls. six nd tion crbon crbon trile. remnuclei, while " occuy bove nd below le remg thus hs lnr, similrly ". net effect is shrg y orbitlsg overl y y hydrogen comlete cety nd ech drwn nuclei.problem 1.9 orbitls ngles roximtely 10. ( ctul vlues re six nd tion 1.). trile. remside view To view To comlete ethylene, hydrogen ms crbon crbon s lene ( onvert followg moleculr model hexne, comonent gsole, le- ngle.) Ech ngle nd 11. g orbitls ech hydrogen comlete cety remg orbitls. thus hs lnr, (gry, ivory ). hs length orbitls m nd strength 464 kj/mol (111 kcl/mol). " ( 1.). lene e-config nd crbons ngles roximtely 10. ( ctul vlues re iztion roch ech orbitl 1. orbitls 1.14 ngle nd 11. orbitls Ech cetylene. hed-on overl. At sme time,ngle.) unized ms " crbon orbitl 1. hs joed length nd strength Structure 464 kj/mol 1.8 ybrid nd (111 1 ethylene. ne rt orbitls terct sidewys overl wht is clled kcl/mol). i (). re one m rbitls " 1.14 orbitls nd " results cetylene. nd crbon ms ethylene results " s. combtion n shrg orbitls " re joed one overl (hed-on) tion orbitls, nd crbon crbon ( rbitl orbitls 1.1 ethne. nd " s. or occuy region centered 1.14).nd Note thtrt results crbon crbon is ed orbitl Digrm " (sidewys) overl unexne ethylene. ne rt ethylene results (hed-on) overl orbitls, nd or rt results " (sidewys) overl unized orbitls (red/blue). " hs electron density bove nd below le drwn nuclei. crbon crbon rbon crbon tri " 106 m nuclei, while " occuy nd below le overl bove orbitls. For ized orbitls (red/blue). orbitl drwn nuclei. clrity, smller lobes " hs electron orbitls re not shown. To density comlete ethylene, hydrogen ms s bove nd below le orbitls " 10 m remg orbitls. thus hs lnr, drwn nuclei. orbitls orbitls " nd ngles roximtely 10. ( ctul vlues re orbitls " ngle nd 11. ngle.) Ech orbitls crbon crbon rbon crbon orbitls rbon crbon trile " hs length m nd strength 464 kj/mol (111 kcl/mol). 1.8 ybrid rbitls nd Structure crbon m oyright Lerng. All Rightsethne Reserved. My not be coied, scnned, or dulicted, becuse whole or rt.trile Due electronic rbon crbon crbon rbon crbon s we ve seen010engge methne nd re clled sgle s rights, some third rty content my be suressed orbitls " 11. lerng Ediril review hs deemed tht ny suressed content does not mterilly ffect overll exerience. engge Lerng reserves right remove dditionl content t ny time if su result shrg one electron ir ed ms. It ws m m recognized nerly 150 yers go, however, tht crbon ms cn lso ethylene. ne rt She Tetrhedrl Trigonl Plnr Ler ethylene results s shrg electron irs ms or trile s shr (hed-on) overl orbitls, Bond Angles m irs., hs m g three electron stnce, P nd nd or rt results 10 m 154 mmore " (sidewys) overl unconts crbon crbon, while cetylene hs 14 m conts crbon crbon trile. ized orbitls (red/blue). exmles q nd 10 m Ethne " hs electron ow re multile s described vlence ory? we disdensity bove nd below le oyright 010 engge Lerng. Rights Reserved. My not be coied, or dulicted, whole rt. Due electronic third rty content my be suressed ebook nd/or ehter(s). cussedall orbitls scnned, Section 1.6, weor sid tht rights, some vlence-shell Ediril review hs deemed tht ny suressed content does not mterilly ffect overll lerng exerience. engge Lerng reserves right remove dditionl content t ny time if subsequent rights restrictions require it. drwn nuclei. mic orbitls crbon combe equivlent s. Imge oyright 010 engge Lerng. AllDue Rights Reserved. Mysome not be coied, or dulicted, whole or ebook " rt. Due ehter(s). electronic rights, some third rty content my be suressed ebook nd/or ehter(s). orbitls oyright 010 engge Lerng. All Rights Reserved. My not be coied, scnned, or dulicted, whole or rt. electronic rights, third rtyscnned, content my be suressed nd/or Ediril review hs deemed tht ny suressed content does mterilly overll lerng exerience. engge Lerng reserves it.right remove dditionl content t ny time if subsequent rights restrictions require it. oril review hs deemed tht ny suressed content does not mterilly ffect overll lerng exerience. engge Lerng reserves not right removeffect dditionl content t ny time if subsequent rights restrictions require Problem 1.8 sted tht s orbitl combes only three vilble crbon not be crbon rbon crbon oyright 010 engge Lerng. All Rights Reserved. My coied, scnned, or dulicted, whole or rt. Due electronic rights, some third rty content my be suressed ebook nd/or ehter(s). Drw le- rone, hs Predict content vlue notech ngle, Ediril review deemed suressed does mterilly ffect overll lerng exerience. engge Lerng reserves right remove dditionl content t ny time if subsequent rights restrictions require it. tht. ny nd dicte overll she. 11. Problem 1.9 * hrge cloud =010m or108.7 lone ir round centrl m; number m Reserved. oyright engge All Rights notbe coied, scnned, or dulicted, whole ornt rt. Due electronic rights, some third rty s content my be suressed ebook nd/or ehter(s). onvert followg moleculr model hexne, comonent Lerng. gsole, le My (gry, ivory ). " bo Ediril review hs deemed tht ny suressed content does not mterilly ffect overll lerng exerience. engge Lerng reserves right remove dditionl content t ny time if subsequent rights restrictions require it. 14 m oyright 010 engge Lerng. All Rights Reserved. My not be coied, scnned, or dulicted, whole or rt. Due electronic rights, some third rty content my be suressed ebook nd/or ehter(s). Ediril review hs deemed tht ny suressed content does not mterilly ffect overll lerng exerience. engge Lerng reserves right remove dditionl content t ny time if subsequent rights restrictions require it. exne L1-4

5 Reresenttions rgnic Molecules Le-Bond (Lewis) ondensed Skeletl (zig-zg) ( ) Indicte iztion (,, or ) every,, nd N m 17 6 N N Dero-Prover (deot-jected contrcetive) "Fictitious " ( trg uroses only) Next time Polrity, Forml hrge, Resonnce ** Tke ~0 m skim hter.1-.6 bee lecture, use Redg Questions L1-5

Key for Chem 130 Second Exam

Key for Chem 130 Second Exam Nme Key for Chem 130 Second Exm On the following pges you will find questions tht cover the structure of molecules, ions, nd solids, nd the different models we use to explin the nture of chemicl bonding.

More information

Chem 130 Second Exam

Chem 130 Second Exam Nme Chem 130 Second Exm On the following pges you will find questions tht cover the structure of molecules, ions, nd solids, nd the different models we use to explin the nture of chemicl bonding. Red ech

More information

BOND ORDER (BO): Single bond Þ BO = 1; Double bond Þ BO = 2; Triple bond Þ BO = 3 Bond Order Þ bond strength and bond length

BOND ORDER (BO): Single bond Þ BO = 1; Double bond Þ BO = 2; Triple bond Þ BO = 3 Bond Order Þ bond strength and bond length EMISTRY 104 elp Sheet #1 hem 103 Review (Text: h 6, h 7) Do topics pproprite for your lecture Prepred y Dr. Tony Jco http://www.chem.wisc.edu/res/clc (Resource pge) Nuggets: Electronegtivity (6.7), Bond

More information

Please write neatly!

Please write neatly! Nme Chem 130 Second Exm On the following pges re eight questions tht consider the structure of molecules, ions, nd solids, nd the different models we use to explin the nture of chemicl bonding. Red ech

More information

Chem 130 Second Exam

Chem 130 Second Exam Nme Chem 130 Second Exm On the following pges you will find questions tht cover the structure of molecules, ions, nd solids, nd the different models we use to explin the nture of chemicl bonding. Red ech

More information

Chem 130 Second Exam

Chem 130 Second Exam Nme Chem 130 Second Exm On the following pges you will find seven questions covering vries topics rnging from the structure of molecules, ions, nd solids to different models for explining bonding. Red

More information

3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression.

3 x x x 1 3 x a a a 2 7 a Ba 1 NOW TRY EXERCISES 89 AND a 2/ Evaluate each expression. SECTION. Eponents nd Rdicls 7 B 7 7 7 7 7 7 7 NOW TRY EXERCISES 89 AND 9 7. EXERCISES CONCEPTS. () Using eponentil nottion, we cn write the product s. In the epression 3 4,the numer 3 is clled the, nd

More information

USA Mathematical Talent Search Round 1 Solutions Year 25 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 25 Academic Year 1/1/5. Alex is trying to oen lock whose code is sequence tht is three letters long, with ech of the letters being one of A, B or C, ossibly reeted. The lock hs three buttons, lbeled A, B nd C. When the

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

Chapter 3 Structures of Coordination Compounds

Chapter 3 Structures of Coordination Compounds hpter 3 Structures of oordintion ompounds Problem Solutions: 3.1. Ethnol nd dimethylether re isomers becuse they hve the sme number nd types of toms but different properties. urthermore, they re structurl

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil

More information

Lesson 1.6 Exercises, pages 68 73

Lesson 1.6 Exercises, pages 68 73 Lesson.6 Exercises, pges 68 7 A. Determine whether ech infinite geometric series hs finite sum. How do you know? ) + +.5 + 6.75 +... r is:.5, so the sum is not finite. b) 0.5 0.05 0.005 0.0005... r is:

More information

k and v = v 1 j + u 3 i + v 2

k and v = v 1 j + u 3 i + v 2 ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended

More information

fractions Let s Learn to

fractions Let s Learn to 5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

More information

QUANTUM CHEMISTRY. Hückel Molecular orbital Theory Application PART I PAPER:2, PHYSICAL CHEMISTRY-I

QUANTUM CHEMISTRY. Hückel Molecular orbital Theory Application PART I PAPER:2, PHYSICAL CHEMISTRY-I Subject PHYSICAL Pper No nd Title TOPIC Sub-Topic (if ny) Module No., PHYSICAL -II QUANTUM Hückel Moleculr orbitl Theory CHE_P_M3 PAPER:, PHYSICAL -I MODULE: 3, Hückel Moleculr orbitl Theory TABLE OF CONTENTS.

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

A B= ( ) because from A to B is 3 right, 2 down.

A B= ( ) because from A to B is 3 right, 2 down. 8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.

More information

The Regulated and Riemann Integrals

The Regulated and Riemann Integrals Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue

More information

Definition :- A shape has a line of symmetry if, when folded over the line. 1 line of symmetry 2 lines of symmetry

Definition :- A shape has a line of symmetry if, when folded over the line. 1 line of symmetry 2 lines of symmetry Symmetry Lines of Symmetry Definition :- A shpe hs line of symmetry if, when folded over the line the hlves of the shpe mtch up exctly. Some shpes hve more thn one line of symmetry : line of symmetry lines

More information

Infinite Geometric Series

Infinite Geometric Series Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to

More information

Problem Set 3 Solutions

Problem Set 3 Solutions Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,

More information

On the diagram below the displacement is represented by the directed line segment OA.

On the diagram below the displacement is represented by the directed line segment OA. Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

More information

7.2 The Definite Integral

7.2 The Definite Integral 7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where

More information

Consider a potential problem in the half-space dened by z 0, with Dirichlet boundary conditions on the plane z = 0 (and at innity).

Consider a potential problem in the half-space dened by z 0, with Dirichlet boundary conditions on the plane z = 0 (and at innity). Problem.7 Consier otentil roblem in the hlf-sce ene by z 0, with Dirichlet bounry conitions on the lne z 0 (n t innity)..7.. Write own the rorite Green function G(~x; ~x 0 ). G D (~x; ~x 0 ) (x x 0 ) (x

More information

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Families of Solutions to Bernoulli ODEs

Families of Solutions to Bernoulli ODEs In the fmily of solutions to the differentil eqution y ry dx + = it is shown tht vrition of the initil condition y( 0 = cuses horizontl shift in the solution curve y = f ( x, rther thn the verticl shift

More information

Mathematics Extension 1

Mathematics Extension 1 04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

Solutions to Physics: Principles with Applications, 5/E, Giancoli Chapter 16 CHAPTER 16

Solutions to Physics: Principles with Applications, 5/E, Giancoli Chapter 16 CHAPTER 16 CHAPTER 16 1. The number of electrons is N = Q/e = ( 30.0 10 6 C)/( 1.60 10 19 C/electrons) = 1.88 10 14 electrons.. The mgnitude of the Coulomb force is Q /r. If we divide the epressions for the two forces,

More information

The Properties of Stars

The 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 information

Riemann is the Mann! (But Lebesgue may besgue to differ.)

Riemann is the Mann! (But Lebesgue may besgue to differ.) Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >

More information

NOT TO SCALE. We can make use of the small angle approximations: if θ á 1 (and is expressed in RADIANS), then

NOT TO SCALE. We can make use of the small angle approximations: if θ á 1 (and is expressed in RADIANS), then 3. Stellr Prllx y terrestril stndrds, the strs re extremely distnt: the nerest, Proxim Centuri, is 4.24 light yers (~ 10 13 km) wy. This mens tht their prllx is extremely smll. Prllx is the pprent shifting

More information

2008 Mathematical Methods (CAS) GA 3: Examination 2

2008 Mathematical Methods (CAS) GA 3: Examination 2 Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed

More information

Andrew Ryba Math Intel Research Final Paper 6/7/09 (revision 6/17/09)

Andrew Ryba Math Intel Research Final Paper 6/7/09 (revision 6/17/09) Andrew Ryb Mth ntel Reserch Finl Pper 6/7/09 (revision 6/17/09) Euler's formul tells us tht for every tringle, the squre of the distnce between its circumcenter nd incenter is R 2-2rR, where R is the circumrdius

More information

Quadratic Residues. Chapter Quadratic residues

Quadratic Residues. Chapter Quadratic residues Chter 8 Qudrtic Residues 8. Qudrtic residues Let n>be given ositive integer, nd gcd, n. We sy tht Z n is qudrtic residue mod n if the congruence x mod n is solvble. Otherwise, is clled qudrtic nonresidue

More information

Chapter 16. Molecular Symmetry

Chapter 16. Molecular Symmetry I. Smmetr Chpter 6. Moleculr Smmetr Elements xis mirror plne inversion center... Opertions rottion bout n xis reflection thru plne inversion thru center Five smmetr elements nd corresponding opertions:

More information

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of

More information

Crystalline Structures The Basics

Crystalline Structures The Basics Crystlline Structures The sics Crystl structure of mteril is wy in which toms, ions, molecules re sptilly rrnged in 3-D spce. Crystl structure = lttice (unit cell geometry) + bsis (tom, ion, or molecule

More information

11.1 Exponential Functions

11.1 Exponential Functions . Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

More information

5.2 Exponent Properties Involving Quotients

5.2 Exponent Properties Involving Quotients 5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9.

5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9. Regulr Expressions, Pumping Lemm, Right Liner Grmmrs Ling 106 Mrch 25, 2002 1 Regulr Expressions A regulr expression descries or genertes lnguge: it is kind of shorthnd for listing the memers of lnguge.

More information

Equations, expressions and formulae

Equations, expressions and formulae Get strted 2 Equtions, epressions nd formule This unit will help you to work with equtions, epressions nd formule. AO1 Fluency check 1 Work out 2 b 2 c 7 2 d 7 2 2 Simplify by collecting like terms. b

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble)

More information

Theoretical foundations of Gaussian quadrature

Theoretical foundations of Gaussian quadrature Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

More information

4.1 One-to-One Functions; Inverse Functions. EX) Find the inverse of the following functions. State if the inverse also forms a function or not.

4.1 One-to-One Functions; Inverse Functions. EX) Find the inverse of the following functions. State if the inverse also forms a function or not. 4.1 One-to-One Functions; Inverse Functions Finding Inverses of Functions To find the inverse of function simply switch nd y vlues. Input becomes Output nd Output becomes Input. EX) Find the inverse of

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

Natural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring

Natural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring More generlly, we define ring to be non-empty set R hving two binry opertions (we ll think of these s ddition nd multipliction) which is n Abelin group under + (we ll denote the dditive identity by 0),

More information

CHEMISTRY. 31 (b) The term acid rain was coined by Robert Augus 32 (c)

CHEMISTRY. 31 (b) The term acid rain was coined by Robert Augus 32 (c) CHEMISTRY 31 (b) The term cid rin ws coined by Robert Augus 32 (c) Initil At (4-3 moles equilibrium mole mole Hence, mole Hence, number of moles of t equilibrium =2-1=1 mole Number of moles of t equilibrium

More information

Dynamics: Newton s Laws of Motion

Dynamics: Newton s Laws of Motion Lecture 7 Chpter 4 Physics I 09.25.2013 Dynmics: Newton s Lws of Motion Solving Problems using Newton s lws Course website: http://fculty.uml.edu/andriy_dnylov/teching/physicsi Lecture Cpture: http://echo360.uml.edu/dnylov2013/physics1fll.html

More information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

(6.5) Length and area in polar coordinates

(6.5) Length and area in polar coordinates 86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx + 6 6 x x dx Centre of mss +

More information

7.3 Problem 7.3. ~B(~x) = ~ k ~ E(~x)=! but we also have a reected wave. ~E(~x) = ~ E 2 e i~ k 2 ~x i!t. ~B R (~x) = ~ k R ~ E R (~x)=!

7.3 Problem 7.3. ~B(~x) = ~ k ~ E(~x)=! but we also have a reected wave. ~E(~x) = ~ E 2 e i~ k 2 ~x i!t. ~B R (~x) = ~ k R ~ E R (~x)=! 7. Problem 7. We hve two semi-innite slbs of dielectric mteril with nd equl indices of refrction n >, with n ir g (n ) of thickness d between them. Let the surfces be in the x; y lne, with the g being

More information

Abstract inner product spaces

Abstract inner product spaces WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the

More information

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 Review of Sine, Cosine and Tangent for Right Angles Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences

UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Department of Electrical Engineering and Computer Sciences UNIVERSITY OF CALIFORNIA, BERKELEY College of Engineering Deprtment of Electricl Engineering nd Computer Sciences Eld Alon Homework #3 Solutions EECS4 PROBLEM : CMOS Logic ) Implement the logic function

More information

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q

More information

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?

Log1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1? 008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing

More information

Intro to Nuclear and Particle Physics (5110)

Intro to Nuclear and Particle Physics (5110) Intro to Nucler nd Prticle Physics (5110) Feb, 009 The Nucler Mss Spectrum The Liquid Drop Model //009 1 E(MeV) n n(n-1)/ E/[ n(n-1)/] (MeV/pir) 1 C 16 O 0 Ne 4 Mg 7.7 14.44 19.17 8.48 4 5 6 6 10 15.4.41

More information

n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1 The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

More information

W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

More information

GRADE 4. Division WORKSHEETS

GRADE 4. Division WORKSHEETS GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.

More information

DIRECT CURRENT CIRCUITS

DIRECT CURRENT CIRCUITS DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

More information

3. Vectors. Home Page. Title Page. Page 2 of 37. Go Back. Full Screen. Close. Quit

3. Vectors. Home Page. Title Page. Page 2 of 37. Go Back. Full Screen. Close. Quit Rutgers University Deprtment of Physics & Astronomy 01:750:271 Honors Physics I Lecture 3 Pge 1 of 37 3. Vectors Gols: To define vector components nd dd vectors. To introduce nd mnipulte unit vectors.

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER 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 information

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year

USA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year 1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.

More information

r 0 ( ) cos( ) r( )sin( ). 1. Last time, we calculated that for the cardioid r( ) =1+sin( ),

r 0 ( ) cos( ) r( )sin( ). 1. Last time, we calculated that for the cardioid r( ) =1+sin( ), Wrm up Recll from lst time, given polr curve r = r( ),, dx dy dx = dy d = (r( )sin( )) d (r( ) cos( )) = r0 ( )sin( )+r( ) cos( ) r 0 ( ) cos( ) r( )sin( ).. Lst time, we clculted tht for crdioid r( )

More information

Section 7.2 Velocity. Solution

Section 7.2 Velocity. Solution Section 7.2 Velocity In the previous chpter, we showed tht velocity is vector becuse it hd both mgnitude (speed) nd direction. In this section, we will demonstrte how two velocities cn be combined to determine

More information

Lecture 08: Feb. 08, 2019

Lecture 08: Feb. 08, 2019 4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny

More information

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:

Physics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions: Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You

More information

Identify graphs of linear inequalities on a number line.

Identify graphs of linear inequalities on a number line. COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point

More information

Recitation 3: More Applications of the Derivative

Recitation 3: More Applications of the Derivative Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

More information

Lesson Notes: Week 40-Vectors

Lesson Notes: Week 40-Vectors Lesson Notes: Week 40-Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples

More information

Chapter 14. Matrix Representations of Linear Transformations

Chapter 14. Matrix Representations of Linear Transformations Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

HQPD - ALGEBRA I TEST Record your answers on the answer sheet.

HQPD - ALGEBRA I TEST Record your answers on the answer sheet. HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property

More information

Riemann Sums and Riemann Integrals

Riemann 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, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

More information

Student Activity 3: Single Factor ANOVA

Student Activity 3: Single Factor ANOVA MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether

More information

Mathematics Extension 2

Mathematics Extension 2 00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd

More information

ex. Line Bond Structure like Lewis Dot but using lines not dots (and no lone pairs shown)

ex. Line Bond Structure like Lewis Dot but using lines not dots (and no lone pairs shown) hemical Formulas: 1. Empirical smallest whole number ratio (ex. 2 ) 2. Molecular actual number of atoms in the specified ratio (ex. 6 12 ) 3. Structural shows order of attachment of all atoms ex. Lewis

More information

This lecture covers Chapter 8 of HMU: Properties of CFLs

This lecture covers Chapter 8 of HMU: Properties of CFLs This lecture covers Chpter 8 of HMU: Properties of CFLs Turing Mchine Extensions of Turing Mchines Restrictions of Turing Mchines Additionl Reding: Chpter 8 of HMU. Turing Mchine: Informl Definition B

More information

Boolean Algebra. Boolean Algebras

Boolean Algebra. Boolean Algebras Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with: - two binry opertions, commonly denoted by + nd, - unry opertion, usully denoted by or ~ or, - two elements usully clled zero

More information

PHYS Summer Professor Caillault Homework Solutions. Chapter 2

PHYS Summer Professor Caillault Homework Solutions. Chapter 2 PHYS 1111 - Summer 2007 - Professor Cillult Homework Solutions Chpter 2 5. Picture the Problem: The runner moves long the ovl trck. Strtegy: The distnce is the totl length of trvel, nd the displcement

More information

In-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill

In-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 In-Clss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the

More information

Riemann Sums and Riemann Integrals

Riemann 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 information

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present? University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the

More information

Section 8.1 The Covalent Bond

Section 8.1 The Covalent Bond Section 8.1 The Covalent Bond Apply the octet rule to atoms that form covalent bonds. Describe the formation of single, double, and triple covalent bonds. Contrast sigma and pi bonds. Relate the strength

More information

MATH SS124 Sec 39 Concepts summary with examples

MATH SS124 Sec 39 Concepts summary with examples This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples

More information

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

More information

Equations and Inequalities

Equations and Inequalities Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in

More information