Ring Opening Polymerization of Lactide for The synthesis of Poly (Lactic Acid)

Size: px
Start display at page:

Download "Ring Opening Polymerization of Lactide for The synthesis of Poly (Lactic Acid)"

Transcription

1 ig peig Polymerizatio of Lactide for The sythesis of Poly (Lactic Acid) eported by Keith A. Porter Date 2 March 2006 ITDUCTI Poly (lactic acid) (PLA) shows promise as a valuable alterative to petroleum-based polymers for uses as plastics, fibers, ad coatigs. PLA is produced from reewable resources such as cor ad sugar beets, ad the material is biodegradable thus makig it ideal for idustrial use. 1,2 PLA is derived from lactic acid, which exists as two eatiomers: L- ad D-lactic acid, 1 ad 2. Lactic acid forms PLA C 3 1 C 3 2 upo polycodesatio; however, this is a equilibrium reactio ad difficulties completely removig water ca limit the maximum molecular weight attaied due to hydrolysis of the ester bods. A solutio to this problem is the use of the cyclic dimmer lactide. Lactide udergoes ig peig Polymerizatio (P) to give PLA. ice this is ot a codesatio polymerizatio, there is o eed for the removal of water. Cargill Dow LLC has capitalized o the P of lactide to produce about 300 x 10 6 lbs/year of PLA. 3 PLA belogs to a group of polymers kow as polyesters, which are amog the more versatile of orgaic polymers. The P of lactide is thermodyamically drive by the relief of agle strai ad switchig from (E) to (Z) ester coformatio upo rig opeig. 4 There are three stereoisomers of lactide, D-lactide (3), L-lactide (4), ad meso-lactide (5). This semiar will focus o the stereoselective sythesis of the differet PLA microstructures ad the properties that arise from stereohomogeeity BACKGUD IFMATI PLA is a aliphatic, hydrolyzable polyester (6). Upo exposure to heat ad water, the polymer breaks dow to low molecular weight oligomers. 5 Further degradatio ca occur by the actio of microorgaisms i the body ad i the eviromet to produce lactic acid i the C 3 former ad carbo dioxide ad water i the latter. PLA has may properties similar to its 6 petro-chemical based couterparts. PLA is glossy, trasparet, ad flavor-resistat, ad it ca be elastic or tough. The polymer has may uses ragig from bottles to high temperature plastic cookig dishes. 6 25

2 TEE-MGEEITY F PLA Four differet stereoisomers of PLA are show i Figure 1. Isotactic PLA is formed from either pure D- or L-lactide, ad the sequetial stereogeic carbos have the same absolute cofiguratio. I cotrast, Isotactic ydiotactic eterotactic Isotactic tereoblock m Figure 1. Differet microstructures of PLA sydiotactic PLA has alteratig cofiguratios of the sequetial stereoceters. Atactic PLA has a radom distributio of cofiguratios about the stereoceters while its heterotactic couterpart has regios of stereo-homogeeity. 7 Isotactic stereoblock PLA is similar to isotactic PLA but differs i that rac-lactide is used istead of pure L- or D-lactide. Thakur, Muso ad co-workers 8 showed that the tacticity ca be determied directly by 1 M aalysis of the methie hydroge. The most commo method for determiig the degree of crystalliity is by Differetial caig Calorimetry (DC). 9 As stereoregularity icreases, so does the overlap betwee PLA helices, thus icreasig the umber of stabilizig dipole-dipole iteractios. 10 The stroger the itermolecular forces, the greater the mechaical ad thermal properties of the polymer. 11 ice the properties arise from the stereochemistry 7, the developmet of ew types of stereocomplexes are vital to make stroger, more durable PLA. YTEI F PLA WIT PTICALLY ACTIVE LACTIDE Isotactic PLA is formed by the polymerizatio of either pure L or D-lactide. everal differet types of metal catalysts such as zic (II) ad titaium(iv) have bee evaluated, but ti (II) 2- ethylhexaoate ((ct) 2 ) (7) has bee most widely used because of high reactio rates, the solubility i the moomer melt, ad the ability to produce high molecular weights. 1,12 The coordiatio-isertio 26

3 mechaism is show i Figure 2. Molecular modelig suggests that two alcohols (these alcohols ca be iitiators such as Me or ipr or the propagatig hydrolyzed lactide) exchage with the octoate ligads (8a) followed by the coordiatio of lactide to the metal ceter (8b). Isertio of the alcohol (8c) followed by rig-opeig (8d) geerates a liear moomer (8e) ad starts propagatio. The P of eat lactide with (ct) 2 gives PLA havig molecular weights up to 10 6 g/mol at o C with catalyst cocetratio of ppm i 2-5 hours. 5 A major drawback of the ti catalyst is the icorporatio of the toxic metal o the polymer chai ed ad the resultig toxicity risk i biomedical applicatios. Aother is the broad polydispersity idex ragig from The large rage was attributed to the ti complex reactig with impurities i L-lactide to form ew iitiator species leadig to a iitiatio rate that is slower tha the propagatio rate equiv 8a Lactide 8b 8e Lactide 8d 8c Figure 2. Coordiatio-isertio mechaism of (ct) 2 catalyzed polymerizatio of L-lactide. YTEI F PLA WIT rac-lactide Kietic esolutio of rac-lactide ad the Formatio of Isotactic tereoblocks The kietic resolutio of rac-lactide was accomplished with two differet chiral alumium complexes passky ad co-workers 14a,b reported oe of the most importat discoveries i the stereocotrol of PLA; the use chiral Al(iPr)[()-(alBiap)] (()-9) catalyst promoted high selectivity i the kietic resolutio of rac-lactide. ()-9 showed a 20:1 preferece for the polymerizatio of D- 27

4 lactide over L-lactide (Figure 3). This was a major advace i the stereocotrolled polymerizatio of PLA because the separatio of rac-lactide to yield eatiopure lactide is expesive ad tedious. ()-9 D L 20:1 ratio L Figure 3. Kietic resolutio of rac-lactide. At 60 % coversio, the product was idetified by circular dichroism (CD) as poly(d-lactic acid) with a T m of 170 o C. At 100% coversio, the T m rose to 187 o C, idicatig that evetually, isotactic stereoblock PLA was sythesized. Feije ad co-worers 16 reported the first eat P of rac-lactide usig a chiral catalyst, (, ) sale-al catalyst (10) i the sythesis of isotactic stereoblock PLA. Alumium catalyst (, )-10 was prefered the P of L-lactide over D-lactide by 14:1. The polymer had a T m of which is lower tha that of the reported stereoblock complex reported by passky. The lower T m could be attributed to lower stereoselectivity. Catalysts ()-9 ad (, )-10 could be used i the sythesis of eat L-lactide; if the polymer reactio were moitored carefully by CD, after polymerizatio of the reactive lactide (Dlactide is active with 9 ad L-lactide is active with 10), the o-polymerized left over could be distilled off uder reduced pressure. owever o such study has yet bee reported. Al ipr But Al tbu ()-9 tbu But (, )-10 ythesis of eterotactic PLA from rac-lactide Achiral catalysts are easier to sythesize ad geerally less expesive tha their chiral couterparts; cosequetly a research priority is to achieve stereocotrolled Z polymerizatios without chiral catalysts. I the polymerizatio of rac-lactide, the bulky ature of achiral catalyst 11 ad the ature of the polymer chai-ed stereoceter were used to ifluece the stereochemistry of the last iserted moomer, which determies the eatiomer echaied. If (, ) cofiguratio is echaied, the isotactic PLA is 11 formed, whereas if the chai ed attacks (, ), the heterotactic PLA forms. Complex = ipr 28

5 11 was foud to be highly active i the polymerizatio of rac-lactide with 90% of the likages formed betwee alteratig L ad D-lactide. After 20 mi at 20 o C, 11 polymerized rac-lactide to 95% coversio to PLA with a molecular weight of g/mol ad a polydispersity idex of This material was also a o-crystallie, amorphous complex with a T g of 49 o C; however sice PLA is mostly used i low temperature eviromets, heterotactic PLA seems like a cheap alterative to the use of optically pure L or D-lactide YTEI F PLA FM meso-lactide ythesis of ydiotactic PLA ydiotactic polymers ofte have better properties tha their isotactic couterparts, such as icreased crystallizatio rates ad higher glass trasitio temperatures ad meltig poits. ecet studies have thus targeted the sythesis of sydiotactic PLA with efficiet seterochemical cotrol ad high yields. 15 Previous results by Thakur 21 ad Kricheldork 22 have demostrated low coversio (up to 75 %) of meso-lactide to sydiotactic PLA. The two pathways for the stereoselectivity of this polymerizatio are show i Figure 4. Cosistet attack at the eatiotopic A or B acyl- bod leads to sydiotactic PLA while alteratig attack o B ad A leads to heterotactic PLA. Coates ad co-workers reported that ()-9 exhibits a strog preferece for opeig of meso-lactide at the carboyl group adjacet to the stereogeic ceter, (Figure 5). eterotactic A k BA >>k BB B meso-lactide k B >>k A ydiotactic Figure 4. Differet types of meso-lactide polymerizatio L Al L Al L Al L Al =AlL +1 Figure 5. AlL -catalyzed polymerizatio of meso-lactide 29

6 1 M aalysis of the polymer proved high sydiotactic cotet (96%). The polymer had a T g ad T m of 50.7 ad 149 o C, respectively, which were lower tha the values of isotactic PLA at 60 ad 180 o C. This deviatio was attributed to a lower degree of crystalliity sice 96% of moomer was coverted to sydiotacic PLA. YTEI F ETETACTIC PLA Coates ad co-workers were iterested i determiig if sydiotactic PLA could be sythesized with a optically impure catalyst, rac-9. They foud that the reactio gave amorphous heterotactic polymer with a T g = 43.2 o C (Figure 6). The 1 M spectrum revealed heterotactic PLA. The authors have previously reported that ()-9 prefers to attack the carboyl closest to the stereoceter. After iitiatio, the polymer chais switch eatiomeric alumium catalyst before isertio ito the ext moomer site. Molecular modelig suggests that ()-9 prefers the ceter ad ()-9 prefers the L s Al L r Al L Al L Al ipr ipr Polymer exchage A L Al L Al ipr ipr meso-lactide B epeat A, B L Al L Al ipr ipr Figure 6. ythesis of heterotactic PLA from meso-lactide usig rac-8 ceter. GAIC CATALYT F P F LACTIDE A importat cosideratio i the polymerizatio of lactide is the removal of metal cotamiats, boud to the chai ed before applicatio i resorbable biomaterials. 23 The applicatio of orgaocatalysts to cotrolled lactide polymerizatio would be a highly viable alterative to 30

7 orgaometallic approaches. everal orgaic compouds have demostrated high activity ad eatioselectivity i a umber of commo orgaic trasformatios. edrick ad co-workers were the first to report the use of first orgaic catalyst 4-(dimethylamio)pyridie (DMAP), 12, i the P of lactide. 23 The catalytic cycle for 12 (cheme 1a) ad -eterocyclic carbee (C) 13 (cheme 1b) are show. Iitiatio occurs whe a alcohol reacts with the lactide-orgaic catalyst complex, leavig a CF 3 termial -hydroxyl group to act as a ucleophile to F 3 C react with additioal lactide moomer. igh coversios of up to 99% i two hours were obtaied with C 13. Thiourea-amie catalyst 14 has also bee used but the P occurs by a differet catalytic mechaism. Thiourea-amie 14 -bods to the carboyl oxyge through 25 the thiourea ad the alcohol iitiator is directed to attack the acyl-oxyge bod by the tertiary amie. cheme 1. Proposed mechaism of P of Lactide with 12 ad 13. (a) DMAP DMAP (b) Mes Mes Mes Mes C These orgaic catalysts are relatively iexpesive ad highly active, ad they yield PLA with low PDI because they are livig polymerizatios. 23 rgaocatalysts provide attractive substitutes for the P of lactide for biomedical ad evirometal applicatios. 26 CCLUI AD FUTUE WK PLA has recetly attracted much attetio as a biodegradable polymer for the replacemet of oilbased material. everal differet types of PLA have bee sythesized with high stereoregularity ad i high yield. More recet work has bee devoted to the sythesis of orgaocatalysts because they are less 31

8 toxic tha metal catalysts. edrick has prepared a series of orgaocatalysts that have bee highly reactive for the P of lactide. owever, so far they have ot show as high stereocotrol or high molecular weights as the metal catalysts have. o doubt, ew orgaocatalysts will be sythesized that will withstad the high temperature of the lactide melt while maitaiig stereocotrol ability. EFEECE (1) wift, G. Acc. Chem. es. 1993, 26, 105. (2) Itavaara, M.; Karjomaa,.; eli, J.F.; Chemosphere 2002, 46, 879. (3) otes from the Packagig Laboratory: Polylactic Acid A Excitig Uiversity of Florida. < ew Packagig Material. (4) teves, M. P. Polymer Chemistry a Itroductio 3 rd ed. xford Uiversity Press, Y, 1999, Chapter 3. (5) Drumright,. E.; Gruber, P..; Adv. Mater. 2000, 12, (6) Cargill Dow LLC website < (7) Chada, M. Advaced Polymer Chemistry A Problem olvig Guide. Marcel Dekker Ic. Y 2000, Chapter 1 (8) Thakur, K. A. M.; Muso, E. J. et al. Macromolecules 1997, 30, 2422 (9) Thermal Characterizatio of Polymeric Materials Vol 2. Turi, E. A., Ed.; Academic Press: a Diego 1997 (10) Ikada, Y.; Jamshidi K., Tsuji,., yo,.. Macromolecules 1987, 20, 904. (11) Du, Y. D.; Lemstra, P. J.; ijehuis, A. J.; Va Aert,. A. M.; Bastiaase, C. Macomolecules (12) (a) chwach, G.; Coudae, J.; Egel,.; Vert, M. J. Polym. ci. A, Polym. Chem. 1994, 32, 2965 (b) Bourissou, D.; Marti-Vaca, B.; Dechy-Cabaret,. Chem. ev. 2004, 104, 6147 (13) yer, M.; tridsberg, K.; Albertsso, A.; vo check,.; vesso, M. Macromolecules , 3877 (14) passy,.; Wisiewski, M.; Plutta, C.; Le Borge, A. Macromol. Chem. Phys. 1996, 197, (15) vitt, T. M.; Coates, G. W.; J. Am. Chem. oc. 2002, 124, (16) (a) Zhog, Z.; Dijkstra, P. J.; Feije, J. J. Am. Chem. oc. 2003, 125, (b) Ma,.; kuda, J. Macromolecules 2005, 38, 2665 (17) vitt, T. M.; Coates, G. W. J. Am. Chem. oc. 1999, 121, 4072 (18) Cheg, M.; Attygalle, A. B.; Lobkovsky, E. B.; Coates, G. W. J. Am. Chem. oc. 1999, 121, (19) Chamberlai, B. M.; et al. J. Am. Chem. oc. 2001, 123, 3229 (20) Wu, J-C.; uag, B-.; sueh, M-L.; Lai, -L.; Li, C-C. Polymer 2005, 46, 9784 (21) Thakur, K. A. M.; et al. Macromolecules 1997, 30, 2422 (22) Kricheldorf,..; Lee,.. Polymer 1995, 36, 2995 (23) Coer, E. F.; yce, G. W.; Myers, M.; Mock, A.; edrick, J. L. J. Am. Chem. oc. 2002, 124, 914 (24) (a) ederberg, F.; Coer, E. F.; Moller, M.; Thierry, G.; edrick, J. L. Agew. Chem. It. Ed. 2001, 40, (b) yce, G. W.; Glauser, T.; Coer, E. F.; Mock, A.; Waymouth,. M.; edrick, J. L. J. Am. Chem. oc. 2003, 125, 3046 (25) Dove, A. P.; Pratt,. C.; Lohmeijer, B. G.; Waymouth,. M.; edrick, J. L. J. Am. Chem. oc. 2005, 127, (26) Amgoue, A.; Thomas, C. M.; oisel, T.; Carpetier, J-F. Chem. Eur. J. 2006, 12,

Chapter 2. Aromatic Polyketones from α-aminonitriles

Chapter 2. Aromatic Polyketones from α-aminonitriles hapter 2 Aromatic Polyketoes from α-amioitriles 2.1 ythesis of α-amioitriles α-amioitriles 1 are a class of compouds which cotai the amie ad itrile fuctioal groups o the same carbo. The geeral structure

More information

Polymerization Lab p. 1 Polymerization Lab

Polymerization Lab p. 1 Polymerization Lab Polymerizatio Lab p. 1 Polymerizatio Lab Itroductio: Polymers (Greek-PLY...may ad MES...parts) have existed sice the begiig of life. Both "atural" ad "sythetic" polymers are a itegral part of our life.

More information

Macromolecules 2003, 36,

Macromolecules 2003, 36, Cyclic AFT/MADIX CT Agets by obert B. Logi rlogicosultig.com Cyclic AFT CTA's ca polymerize viyl moomers ito cyclic polymers. Cyclic polymers because of their uique properties compared to their liear couterparts

More information

Free Radical Polymerization

Free Radical Polymerization Free Radical Polymerizatio Referece: Aspe Polymers: Uit Operatios ad Reactio Models, Aspe Techology, Ic., 2013. 1. Itroductio The free-radical bulk/solutio polymerizatio model is applicable to bulk ad

More information

Enzymes in Organic Chemistry. Enzymes!

Enzymes in Organic Chemistry. Enzymes! Ezymes i rgaic Chemistry March 17 th, 2015 Ezymes! Ezymes are biological catalysts. They icrease the rate at which equilibrium is reached, but they do ot affect the equilibrium. Ezymes differ from ordiary

More information

POLYMERS

POLYMERS POLYMERS Short Aswer Questios: **1.What is PHBV? How is it useful to ma? As. PHBV is Poly β-hydroxy butyrate-co- β-hydroxy valerate.it is a biodegradable polymer. It is used i speciality packig, orthopaedic

More information

Polymerization of Glycidyl Nitrate via Catalysis with BF3 THF : Compatibility with the Activated Monomer (AM) Mechanism

Polymerization of Glycidyl Nitrate via Catalysis with BF3 THF : Compatibility with the Activated Monomer (AM) Mechanism ATK THIKL INC Polymerizatio of Glycidyl Nitrate via Catalysis with BF3 THF : Compatibility with the Activated Moomer (AM) Mechaism Prepared For 2004 IM/EM Techical Symposium November 15-17, 2004 Sa Fracisco,

More information

Structure. Extremely interesting BPs (very high Tm ~ C) Excellent thermic and mechanical properties

Structure. Extremely interesting BPs (very high Tm ~ C) Excellent thermic and mechanical properties tructure Extremely interesting BPs (very high Tm ~ 220-230 C) Excellent thermic and mechanical properties Limitations: no straigthforward synthetic pathways available (Expensive material) Applications:

More information

POLYMER SCIENCE and TECHNOLOGY I

POLYMER SCIENCE and TECHNOLOGY I POLYMER SCIENCE ad TECHNOLOGY I Dr. Işıl Gerçek Beşkardeş 2018 What is the greatest cotributio of chemistry to sciece ad society? Developmet of POLYMERIZATION is oe of the biggest ivetios chemistry has

More information

MOTION. The easy stuff; The gaseous state - Translational motion. The crystalline state - Oscillations about a Mean position.

MOTION. The easy stuff; The gaseous state - Translational motion. The crystalline state - Oscillations about a Mean position. The easy stuff; MOTION The gaseous state - Traslatioal motio The crystallie state - Oscillatios about a Mea positio The hard stuff; The liquid state - Coupled motios MOTION IN POLYMERS LARGE SCALE MOTION

More information

Structural Confirmation of Polymeric Material Using MS/MS and Fragmentation Mechanisms

Structural Confirmation of Polymeric Material Using MS/MS and Fragmentation Mechanisms Structural Cofirmatio of Polymeric Material Usig MS/MS ad Fragmetatio Mechaisms Kirste Crave, 1 Michael McCullagh, 1 Pascal Gerbaux, 2 ad Julie De Witer 2 1 Waters Corporatio, Machester, UK 2 Uiversity

More information

Synthesis of Polymers Prof. Paula Hammond Lecture 21: Living Anionic Polymerization, Effects of Initiator and Solvent

Synthesis of Polymers Prof. Paula Hammond Lecture 21: Living Anionic Polymerization, Effects of Initiator and Solvent 10.569 Sythesis of Polymers Prof. Paula Hammod Lecture 21: Livig Aioic Polymerizatio, Effects of Iitiator ad Solvet Livig Polymerizatio d M ] R p = = k p M ]M ] dt M ] ] I o (assumes all iitiator is active

More information

Different FRP Resin Chemistries for Different Chemical Environments

Different FRP Resin Chemistries for Different Chemical Environments Differet FRP Resi Chemistries for Differet Chemical Eviromets Do Kelley, Jim Graham, Thom Johso Ashlad Performace Materials ABSTRACT The corrosio challeges posed by differet chemical eviromets madate the

More information

Functional Polymers From Renewable Resources

Functional Polymers From Renewable Resources Fuctioal Polymers From Reewable Resources S. Shag, A. Ro, S. J. Huag ad R. A. Weiss Polymer Program Uiversity of Coecticut Storrs, CT New Eglad Gree Chemistry Cosortium Aual Meetig Uiversity of Maie ro,

More information

Biorenewable Polymers 1: The Isotactic Polymerisation of Lactide

Biorenewable Polymers 1: The Isotactic Polymerisation of Lactide 4A3 Advanced Polymer Synthesis Biorenewable Polymers 1: The Isotactic Polymerisation of Lactide Dr. Ed Marshall Rm: M220, RCS 1 e.marshall@imperial.ac.uk www.ch.ic.ac.uk/marshall/4a3.html 4A3 - Slide 1

More information

Polymers. One of the most remarkable and useful polymers known is Teflon, formed via polymerization of tetrafluoroethylene as shown below:

Polymers. One of the most remarkable and useful polymers known is Teflon, formed via polymerization of tetrafluoroethylene as shown below: Experimet 8 Chem 276 Fall Semester 2010 Page 1 of 5 Polymers Itroductio A polymer is a gigatic chailike molecule made by coectig may small molecules, kow as moomers. The process by which polymers are formed

More information

Asymmetric Diels Alder Reactions

Asymmetric Diels Alder Reactions Asymmetric Diels Alder eactios Chiral Auxiliaries (-)-8-eylmethol favorable!-stackig 3 C 3 C C 3 AlCl 3 3 C 3 C C 3 s-tras Lewis-acid complex AlCl 3 Cl JACS, 1975, 1610 frot-face approach of diee B 3 C

More information

All Excuses must be taken to 233 Loomis before 4:15, Monday, April 30.

All Excuses must be taken to 233 Loomis before 4:15, Monday, April 30. Miscellaeous Notes The ed is ear do t get behid. All Excuses must be take to 233 Loomis before 4:15, Moday, April 30. The PYS 213 fial exam times are * 8-10 AM, Moday, May 7 * 8-10 AM, Tuesday, May 8 ad

More information

( ) ( ), (S3) ( ). (S4)

( ) ( ), (S3) ( ). (S4) Ultrasesitivity i phosphorylatio-dephosphorylatio cycles with little substrate: Supportig Iformatio Bruo M.C. Martis, eter S. Swai 1. Derivatio of the equatios associated with the mai model From the differetial

More information

CHAPTER 11. Practice Questions (a) OH (b) I. (h) NH 3 CH 3 CO 2 (j) C 6 H 5 O - (k) (CH 3 ) 3 N conjugate pair

CHAPTER 11. Practice Questions (a) OH (b) I. (h) NH 3 CH 3 CO 2 (j) C 6 H 5 O - (k) (CH 3 ) 3 N conjugate pair CAPTER 11 Practice Questios 11.1 (a) O (b) I (c) NO 2 (d) 2 PO 4 (e) 2 PO 4 (f) 3 PO 4 (g) SO 4 (h) N 3 (i) C 3 CO 2 (j) C 6 5 O - (k) (C 3 ) 3 N 11.3 cojugate pair PO 3 4 (aq) C 3 COO(aq) PO 2 4 (aq)

More information

Towards the Development of a High-spin Organic Polymer

Towards the Development of a High-spin Organic Polymer Appedix II: Towards the Developmet of a High-spi rgaic Polymer 194 II.1 Itroductio The developmet of pure orgaic materials with ferromagetic properties is of log stadig theoretical ad experimetal iterest.

More information

Miscellaneous Notes. Lecture 19, p 1

Miscellaneous Notes. Lecture 19, p 1 Miscellaeous Notes The ed is ear do t get behid. All Excuses must be take to 233 Loomis before oo, Thur, Apr. 25. The PHYS 213 fial exam times are * 8-10 AM, Moday, May 6 * 1:30-3:30 PM, Wed, May 8 The

More information

Chemical Kinetics CHAPTER 14. Chemistry: The Molecular Nature of Matter, 6 th edition By Jesperson, Brady, & Hyslop. CHAPTER 14 Chemical Kinetics

Chemical Kinetics CHAPTER 14. Chemistry: The Molecular Nature of Matter, 6 th edition By Jesperson, Brady, & Hyslop. CHAPTER 14 Chemical Kinetics Chemical Kietics CHAPTER 14 Chemistry: The Molecular Nature of Matter, 6 th editio By Jesperso, Brady, & Hyslop CHAPTER 14 Chemical Kietics Learig Objectives: Factors Affectig Reactio Rate: o Cocetratio

More information

Samples questions on Polymer Chemistry

Samples questions on Polymer Chemistry Samples questios o Polymer Chemistry Symbols carry usual meaigs Module 1: Lecture 1-3 1. For a polydisperse polymer sample rak the M, M w, M z, ad M v accordig to their values. I 10 g of a polystyree sample

More information

European Polymer Journal

European Polymer Journal Europea Polymer Joural 45 (2009) 155 164 Cotets lists available at ScieceDirect Europea Polymer Joural joural homepage: www.elsevier.com/locate/europolj Experimetal studies o the rig opeig polymerizatio

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.

The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c. 5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =

More information

Influence of reaction conditions on the ozonation process

Influence of reaction conditions on the ozonation process Ifluece of reactio coditios o the ozoatio process Leka Sumegová, Já Derco, Michal Melicher Faculty of Chemical ad Food Techology, Slovak Uiversity of Techology, Radliského 9, 81237 Bratislava, Slovakia

More information

Chapter 1 Introduction

Chapter 1 Introduction hapter 1 Itroductio Sythetic polymers [1] are vital materials used i moder daily life from packagig, electroics, medical devices, clothig, vehicles, buildigs, etc., due to their ease of processig ad light

More information

Statistics 511 Additional Materials

Statistics 511 Additional Materials Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability

More information

CHEE 221: Chemical Processes and Systems

CHEE 221: Chemical Processes and Systems CHEE 221: Chemical Processes ad Systems Module 3. Material Balaces with Reactio Part a: Stoichiometry ad Methodologies (Felder & Rousseau Ch 4.6 4.8 ot 4.6c ) Material Balaces o Reactive Processes What

More information

Announcements, Nov. 19 th

Announcements, Nov. 19 th Aoucemets, Nov. 9 th Lecture PRS Quiz topic: results Chemical through Kietics July 9 are posted o the course website. Chec agaist Kietics LabChec agaist Kietics Lab O Fial Exam, NOT 3 Review Exam 3 ad

More information

OPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES

OPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES OPTIMAL ALGORITHMS -- SUPPLEMENTAL NOTES Peter M. Maurer Why Hashig is θ(). As i biary search, hashig assumes that keys are stored i a array which is idexed by a iteger. However, hashig attempts to bypass

More information

University of California, Los Angeles Department of Statistics. Hypothesis testing

University of California, Los Angeles Department of Statistics. Hypothesis testing Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Elemets of a hypothesis test: Hypothesis testig Istructor: Nicolas Christou 1. Null hypothesis, H 0 (claim about µ, p, σ 2, µ

More information

Stability Evaluation of Surfactant-Grafted Polyacrylamide Used in Oilfields by Evolution of Ammonia

Stability Evaluation of Surfactant-Grafted Polyacrylamide Used in Oilfields by Evolution of Ammonia Joural of Materials Sciece ad hemical Egieerig, 1,, 15-19 Published lie May 1 i SciRes. http://www.scirp.org/joural/msce http://dx.doi.org/1.3/msce.1.53 Stability Evaluatio of Surfactat-Grafted Polyacrylamide

More information

A B A B. 2. Homolytic Bond Dissociation Energies (DH ) 1. Introduction: How Radicals Form and How They React Heterolysis

A B A B. 2. Homolytic Bond Dissociation Energies (DH ) 1. Introduction: How Radicals Form and How They React Heterolysis hapter 10 Radical Reactios reated by Professor William Tam & Dr. Phillis hag h. 10-1 About The Authors These Powerpoit Lecture Slides were created ad prepared by Professor William Tam ad his wife Dr. Phillis

More information

Lecture III-2: Light propagation in nonmagnetic

Lecture III-2: Light propagation in nonmagnetic A. La Rosa Lecture Notes ALIED OTIC Lecture III2: Light propagatio i omagetic materials 2.1 urface ( ), volume ( ), ad curret ( j ) desities produced by arizatio charges The objective i this sectio is

More information

Rémy Catteaux Laboratoire des Matériaux, Céramiques et Procédés Associés Maubeuge. SiO 2 -CaO-Na 2 O-P 2 O 5 bioglasses prepared via sol gel method

Rémy Catteaux Laboratoire des Matériaux, Céramiques et Procédés Associés Maubeuge. SiO 2 -CaO-Na 2 O-P 2 O 5 bioglasses prepared via sol gel method émy Catteaux Laboratoire des Matériaux, Céramiques et Procédés Associés Maubeuge Si 2 -Ca-Na 2 -P 2 5 bioglasses prepared via sol gel method Fields of use : - Boes Subtitutes - Surgery BCerS Mos 2011 1

More information

There is no straightforward approach for choosing the warmup period l.

There is no straightforward approach for choosing the warmup period l. B. Maddah INDE 504 Discrete-Evet Simulatio Output Aalysis () Statistical Aalysis for Steady-State Parameters I a otermiatig simulatio, the iterest is i estimatig the log ru steady state measures of performace.

More information

A statistical method to determine sample size to estimate characteristic value of soil parameters

A statistical method to determine sample size to estimate characteristic value of soil parameters A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig

More information

On the convergence rates of Gladyshev s Hurst index estimator

On the convergence rates of Gladyshev s Hurst index estimator Noliear Aalysis: Modellig ad Cotrol, 2010, Vol 15, No 4, 445 450 O the covergece rates of Gladyshev s Hurst idex estimator K Kubilius 1, D Melichov 2 1 Istitute of Mathematics ad Iformatics, Vilius Uiversity

More information

ALDEHYDES AND KETONES

ALDEHYDES AND KETONES ALDEHYDES AND KETONES Itroductio The group C=O is kow as carboyl group. This group is foud i aldehydes, ketoes, carboxylic acids, esters ad acid derivatives. I this group carbo atom makes sp hybridizatio.

More information

4. Amines, Amides and Amino Acids

4. Amines, Amides and Amino Acids 4. Amies, Amides ad Amio Acids amig Amies These ed i amie. There is, however, rather cofusigly two ways of usig this suffix. The exam board ted to use the propylamie commo versio where the ame r propa-1-amie

More information

Efficient GMM LECTURE 12 GMM II

Efficient GMM LECTURE 12 GMM II DECEMBER 1 010 LECTURE 1 II Efficiet The estimator depeds o the choice of the weight matrix A. The efficiet estimator is the oe that has the smallest asymptotic variace amog all estimators defied by differet

More information

N Goalby chemrevise.org

N Goalby chemrevise.org 21. olymers There are two types of polymerisatio additio ad codesatio Additio olymerisatio A additio polymer forms whe usaturated moomers react to form a polymer Moomers cotai = bods oly(alkees) are chemically

More information

Recurrence Relations

Recurrence Relations Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The

More information

10.6 ALTERNATING SERIES

10.6 ALTERNATING SERIES 0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose

More information

Frequentist Inference

Frequentist Inference Frequetist Iferece The topics of the ext three sectios are useful applicatios of the Cetral Limit Theorem. Without kowig aythig about the uderlyig distributio of a sequece of radom variables {X i }, for

More information

4.3 Growth Rates of Solutions to Recurrences

4.3 Growth Rates of Solutions to Recurrences 4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.

More information

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation

II. Descriptive Statistics D. Linear Correlation and Regression. 1. Linear Correlation II. Descriptive Statistics D. Liear Correlatio ad Regressio I this sectio Liear Correlatio Cause ad Effect Liear Regressio 1. Liear Correlatio Quatifyig Liear Correlatio The Pearso product-momet correlatio

More information

SNAP Centre Workshop. Basic Algebraic Manipulation

SNAP Centre Workshop. Basic Algebraic Manipulation SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test.

Because it tests for differences between multiple pairs of means in one test, it is called an omnibus test. Math 308 Sprig 018 Classes 19 ad 0: Aalysis of Variace (ANOVA) Page 1 of 6 Itroductio ANOVA is a statistical procedure for determiig whether three or more sample meas were draw from populatios with equal

More information

Power and Type II Error

Power and Type II Error Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error

More information

NUCLEATION 7.1 INTRODUCTION 7.2 HOMOGENEOUS NUCLEATION Embryos and nuclei CHAPTER 7

NUCLEATION 7.1 INTRODUCTION 7.2 HOMOGENEOUS NUCLEATION Embryos and nuclei CHAPTER 7 CHAPER 7 NUCLEAION 7.1 INRODUCION I this text, we focus our attetio o crystallie solids that form from the melt. he process begis with the creatio of a cluster of atoms of crystallie structure, which may

More information

Kinetics of Complex Reactions

Kinetics of Complex Reactions Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet

More information

Grey Correlation Analysis of China's Electricity Imports and Its Influence Factors Hongjing Zhang 1, a, Feng Wang 1, b and Zhenkun Tian 1, c

Grey Correlation Analysis of China's Electricity Imports and Its Influence Factors Hongjing Zhang 1, a, Feng Wang 1, b and Zhenkun Tian 1, c Applied Mechaics ad Materials Olie: 203-0-3 ISSN: 662-7482, Vols. 448-453, pp 258-262 doi:0.4028/www.scietific.et/amm.448-453.258 204 Tras Tech Publicatios, Switzerlad Grey Correlatio Aalysis of Chia's

More information

DETERMINATION OF MECHANICAL PROPERTIES OF A NON- UNIFORM BEAM USING THE MEASUREMENT OF THE EXCITED LONGITUDINAL ELASTIC VIBRATIONS.

DETERMINATION OF MECHANICAL PROPERTIES OF A NON- UNIFORM BEAM USING THE MEASUREMENT OF THE EXCITED LONGITUDINAL ELASTIC VIBRATIONS. ICSV4 Cairs Australia 9- July 7 DTRMINATION OF MCHANICAL PROPRTIS OF A NON- UNIFORM BAM USING TH MASURMNT OF TH XCITD LONGITUDINAL LASTIC VIBRATIONS Pavel Aokhi ad Vladimir Gordo Departmet of the mathematics

More information

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity

More information

Series: Infinite Sums

Series: Infinite Sums Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree

More information

New Structure of Monocyclic Fluoropolymer for 157nm Photoresist

New Structure of Monocyclic Fluoropolymer for 157nm Photoresist New Structure of Moocyclic luoropolymer for 157m Photoresist Masataka Eda, Yoko Takebe, Osamu Yokokoji Research eter, Asahi Glass o., Ltd. Akihiko Otoguro, Shigeo Irie, Toshiro Itai, Kiyoshi ujii Semicoductor

More information

ENGI Series Page 6-01

ENGI Series Page 6-01 ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,

More information

Data Analysis and Statistical Methods Statistics 651

Data Analysis and Statistical Methods Statistics 651 Data Aalysis ad Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasii/teachig.html Suhasii Subba Rao Review of testig: Example The admistrator of a ursig home wats to do a time ad motio

More information

Statistical Fundamentals and Control Charts

Statistical Fundamentals and Control Charts Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,

More information

6 Sample Size Calculations

6 Sample Size Calculations 6 Sample Size Calculatios Oe of the major resposibilities of a cliical trial statisticia is to aid the ivestigators i determiig the sample size required to coduct a study The most commo procedure for determiig

More information

Frequency Domain Filtering

Frequency Domain Filtering Frequecy Domai Filterig Raga Rodrigo October 19, 2010 Outlie Cotets 1 Itroductio 1 2 Fourier Represetatio of Fiite-Duratio Sequeces: The Discrete Fourier Trasform 1 3 The 2-D Discrete Fourier Trasform

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

Chromatography-Free Esterification Reactions Using a Bifunctional Polymer

Chromatography-Free Esterification Reactions Using a Bifunctional Polymer Title Chromatography-Free Esterificatio Reactios Usig a Bifuctioal Polymer Author(s) Ma, S; Toy, PH Citatio Sylett, 2016, v. 27. 08, p. 1207-1210 Issued Date 2016 URL http://hdl.hadle.et/10722/234028 Rights

More information

Chapter 8: Estimating with Confidence

Chapter 8: Estimating with Confidence Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig

More information

Course Outline. Designing Control Systems. Proportional Controller. Amme 3500 : System Dynamics and Control. Root Locus. Dr. Stefan B.

Course Outline. Designing Control Systems. Proportional Controller. Amme 3500 : System Dynamics and Control. Root Locus. Dr. Stefan B. Amme 3500 : System Dyamics ad Cotrol Root Locus Course Outlie Week Date Cotet Assigmet Notes Mar Itroductio 8 Mar Frequecy Domai Modellig 3 5 Mar Trasiet Performace ad the s-plae 4 Mar Block Diagrams Assig

More information

Lecture 3: August 31

Lecture 3: August 31 36-705: Itermediate Statistics Fall 018 Lecturer: Siva Balakrisha Lecture 3: August 31 This lecture will be mostly a summary of other useful expoetial tail bouds We will ot prove ay of these i lecture,

More information

Expectation and Variance of a random variable

Expectation and Variance of a random variable Chapter 11 Expectatio ad Variace of a radom variable The aim of this lecture is to defie ad itroduce mathematical Expectatio ad variace of a fuctio of discrete & cotiuous radom variables ad the distributio

More information

Chapter 14: Chemical Equilibrium

Chapter 14: Chemical Equilibrium hapter 14: hemical Equilibrium 46 hapter 14: hemical Equilibrium Sectio 14.1: Itroductio to hemical Equilibrium hemical equilibrium is the state where the cocetratios of all reactats ad products remai

More information

9.4.3 Fundamental Parameters. Concentration Factor. Not recommended. See Extraction factor. Decontamination Factor

9.4.3 Fundamental Parameters. Concentration Factor. Not recommended. See Extraction factor. Decontamination Factor 9.4.3 Fudametal Parameters Cocetratio Factor Not recommeded. See Extractio factor. Decotamiatio Factor The ratio of the proportio of cotamiat to product before treatmet to the proportio after treatmet.

More information

LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES

LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES J Lodo Math Soc (2 50, (1994, 465 476 LONG SNAKES IN POWERS OF THE COMPLETE GRAPH WITH AN ODD NUMBER OF VERTICES Jerzy Wojciechowski Abstract I [5] Abbott ad Katchalski ask if there exists a costat c >

More information

1 Introduction to reducing variance in Monte Carlo simulations

1 Introduction to reducing variance in Monte Carlo simulations Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by

More information

MA131 - Analysis 1. Workbook 9 Series III

MA131 - Analysis 1. Workbook 9 Series III MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................

More information

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3

(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3 MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special

More information

EE 505. Lecture 28. ADC Design SAR

EE 505. Lecture 28. ADC Design SAR EE 505 Lecture 28 ADC Desig SAR Review from Last Lecture Elimiatio of Iput S/H C LK X IN S/H Stage 1 r 1 Stage 2 r 2 Stage k r k Stage m r m 1 2 k m Pipelied Assembler (Shift Register

More information

Filter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and

Filter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and Filter bas Separately, the lowpass ad highpass filters are ot ivertible T removes the highest frequecy / ad removes the lowest frequecy Together these filters separate the sigal ito low-frequecy ad high-frequecy

More information

Finally, we show how to determine the moments of an impulse response based on the example of the dispersion model.

Finally, we show how to determine the moments of an impulse response based on the example of the dispersion model. 5.3 Determiatio of Momets Fially, we show how to determie the momets of a impulse respose based o the example of the dispersio model. For the dispersio model we have that E θ (θ ) curve is give by eq (4).

More information

4.7 Organic Chemistry

4.7 Organic Chemistry 4.7 rgaic hemistry rude oil rude oil is a fiite resource foud i rocks. rude oil is the remais of a aciet biomass cosistig maily of plakto that was buried i mud. Alkaes Most of the hydrocarbos i crude oil

More information

A proposed discrete distribution for the statistical modeling of

A proposed discrete distribution for the statistical modeling of It. Statistical Ist.: Proc. 58th World Statistical Cogress, 0, Dubli (Sessio CPS047) p.5059 A proposed discrete distributio for the statistical modelig of Likert data Kidd, Marti Cetre for Statistical

More information

Example 2. Find the upper bound for the remainder for the approximation from Example 1.

Example 2. Find the upper bound for the remainder for the approximation from Example 1. Lesso 8- Error Approimatios 0 Alteratig Series Remaider: For a coverget alteratig series whe approimatig the sum of a series by usig oly the first terms, the error will be less tha or equal to the absolute

More information

Bertrand s Postulate

Bertrand s Postulate Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a

More information

Research on real time compensation of thermal errors of CNC lathe based on linear regression theory Qiu Yongliang

Research on real time compensation of thermal errors of CNC lathe based on linear regression theory Qiu Yongliang d Iteratioal Coferece o Machiery, Materials Egieerig, Chemical Egieerig ad Biotechology (MMECEB 015) Research o real time compesatio of thermal errors of CNC lathe based o liear regressio theory Qiu Yogliag

More information

Advanced Course of Algorithm Design and Analysis

Advanced Course of Algorithm Design and Analysis Differet complexity measures Advaced Course of Algorithm Desig ad Aalysis Asymptotic complexity Big-Oh otatio Properties of O otatio Aalysis of simple algorithms A algorithm may may have differet executio

More information

Similarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall

Similarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig

More information

A Simple Optimum-Time FSSP Algorithm for Multi-Dimensional Cellular Automata

A Simple Optimum-Time FSSP Algorithm for Multi-Dimensional Cellular Automata A Simple Optimum-Time FSSP Algorithm for Multi-Dimesioal Cellular Automata HIROSHI UMEO Uiversity of Osaka Electro-Commuicatio umeo@cyt.osakac.ac.jp KINUO NISHIDE Uiversity of Osaka Electro-Commuicatio

More information

Upper bound for ropelength of pretzel knots

Upper bound for ropelength of pretzel knots Upper boud for ropelegth of pretzel kots Safiya Mora August 25, 2006 Abstract A model of the pretzel kot is described. A method for predictig the ropelegth of pretzel kots is give. A upper boud for the

More information

Principle Of Superposition

Principle Of Superposition ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give

More information

577. Estimation of surface roughness using high frequency vibrations

577. Estimation of surface roughness using high frequency vibrations 577. Estimatio of surface roughess usig high frequecy vibratios V. Augutis, M. Sauoris, Kauas Uiversity of Techology Electroics ad Measuremets Systems Departmet Studetu str. 5-443, LT-5368 Kauas, Lithuaia

More information

Revision Topic 1: Number and algebra

Revision Topic 1: Number and algebra Revisio Topic : Number ad algebra Chapter : Number Differet types of umbers You eed to kow that there are differet types of umbers ad recogise which group a particular umber belogs to: Type of umber Symbol

More information

Solutions to Equilibrium Practice Problems

Solutions to Equilibrium Practice Problems Solutios to Equilibrium Practice Problems Chem09 Fial Booklet Problem 1. Solutio: PO 4 10 eq The expressio for K 3 5 P O 4 eq eq PO 4 10 iit 1 M I (a) Q 1 3, the reactio proceeds to the right. 5 5 P O

More information

Molecular Mechanisms of Gas Diffusion in CO 2 Hydrates

Molecular Mechanisms of Gas Diffusion in CO 2 Hydrates Supportig Iformatio Molecular Mechaisms of Gas Diffusio i CO Hydrates Shuai Liag, * Deqig Liag, Negyou Wu,,3 Lizhi Yi, ad Gaowei Hu,3 Key Laboratory of Gas Hydrate, Guagzhou Istitute of Eergy Coversio,

More information

Abstracts. p43. p107. M. Tomµs-Gamasa. U. Scholz, W. Dong, J. Feng, and W. Shi. M. B. Hay and J. D. Hicks

Abstracts. p43. p107. M. Tomµs-Gamasa. U. Scholz, W. Dong, J. Feng, and W. Shi. M. B. Hay and J. D. Hicks I 2.1.1.1 Alkylamies with Aryl ad Alkeyl Electrophiles M. Tomµs-Gamasa p9 This review covers selected efficiet sythetic methods for metal-catalyzed cross-couplig reactios that ca directly covert simple

More information

Chapter 22 Developing Efficient Algorithms

Chapter 22 Developing Efficient Algorithms Chapter Developig Efficiet Algorithms 1 Executig Time Suppose two algorithms perform the same task such as search (liear search vs. biary search). Which oe is better? Oe possible approach to aswer this

More information

Supporting Information

Supporting Information Supportig Iformatio Rig-peig Copolymerizatio of Maleic Ahydride with Epoxides: A Chai-Growth Approach to Usaturated Polyesters Agela M. DiCiccio ad Geoffrey W. Coates * Departmet of Chemistry ad Chemical

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

Supporting Information

Supporting Information Supportig Iformatio Cotrolled Radical Polymerizatio of Acrylates Regulated by Visible Light Nicolas J. Treat, ett P. Fors, Joh W. Kramer, Matthew Christiaso, Chie- Yag Chiu, Javier Read de Alaiz, ad Craig

More information