Rates of chemical reactions
|
|
- Mavis Kelley
- 6 years ago
- Views:
Transcription
1 Rtes of chemicl rections
2 Mesuring rtes of chemicl rections Experimentl mesuring progress of the rection Monitoring pressure in the rection involving gses 2 NO( g) 4 NO ( g) + O ( g) n(1 α) 2αn 1 αn 2 3 p= (1 + α) p0 2 Asorption t prticulr wvelength (e.g. Br 2 elow) H ( g) + Br ( g) 2 HBr( g) 2 2 Conductnce of the ionic solution ( CH ) CCl( q) + H O( q) ( CH ) COH ( q) + H + ( q) + Cl ( q)
3 Mesuring rtes of chemicl rections Flow method Stopped-flow method Flsh photolysis Chemicl quench flow Freeze quench method
4 Rtes of chemicl rections A + 2B 3C+ D Instntneous rte of consumption of rectnt: d[ R]/ Instntneous rte of formtion of product: From stoichiometry dp [ ]/ dd [ ] 1 [ ] [ ] 1 [ ] = dc = da = db 3 2 Rte of the rection: 1 1 dni 1 dξ v = = V ν V i where ξ = In cse of heterogeneous rection the rte will e defined s mol/m 2 s n n i i0 ν i the extent of the rection
5 Rection order Rection rte is generlly proportionl to the concentrtion of the regents to some power v= k[ A] [ B]... Depends only on T nd not on the concentrtion The power is generlly not equl to the stoichiometric coefficients! Order of rection: v= k[ A][ B] First order in A, first order in B, overll second order. v = k Zero order. v k A B 1/2 = [ ] [ ] Hlforder in A, first order in B, overll tree-hlves order.
6 Experimentl determintion of the rte lw Isoltion method A + B P where ll the components except one re present in lrge mounts (therefore their concentrtion is constnt) experiment repeted t severl concentrtion v= k[ A] [ B ] = k [ A] 0 Advntges: esy interprettion of dt non-integrl orders cn e esily determined Disdvntges needs to e repeted for every rectnt physiologicl conditions cn not lwys e used if high concentrtion re required high concentrtions might ffect rection mechnism
7 Experimentl determintion of the rte lw Method of initil rtes A + B P severl initil concentrtion of A mesured (gin ssuming tht the concentrtions re constnt) v= k[ A ] [ B ] = k [ A ] v = k [ A ] logv = logk+ log[ A ] Advntges: esy interprettion of dt non-integrl orders cn e esily determined not necessry to use lrge ccess of ny regent Disdvntges needs to e repeted for mny times for every rectnt rection processes involving e.g. products cn e missed
8 Rection order Exmple 2 2 I( g) + Ar( g) I ( g) + Ar( g) [I 0 ] x mol dm -3 [Ar] x mmol dm
9 Integrted rte lws First order rection. Let s find concentrtion of regent A fter time t Slope -> k da [ ] = ka [ ] A A da [ ] = k [ A ] 0 0 t [ A] ln = kt [ A0 ] [ A] = [ A ] e kt 0
10 Integrted rte lws First order rection [ A] = [ A ] e kt 0 Hlf-life time required for concentrtion to drop y ½. kt 1 [ A0 ] 2 ln 2 1/2 = ln = ln 2 t1/2 = [ A0 ] k Time constnt time required for concentrtion to drop y 1/e: τ = 1 k
11 Integrted rte lws Second-order rections da [ ] = ka [ ] 2 A A 0 da [ ] 2 [ A ] = k t = [ A] [ A ] 0 kt [ A] [ A ] 0 = + kt A 0 1 [ ] Second order Hlf-life depends on initil concentrtion, long when concentrtion is low First order 1 [ A ] 1 [ A ] = t = 2 1 [ ] [ ] 0 0 1/2 + kt1/2 A0 k A0
12 Integrted rte lws
13 Rection pproching equilirium Generlly, most kinetics re mde fr from equilirium where reverse rections re not importnt. Close to equilirium the mount of products is significnt nd reverse rection should e considered. A B v= k[ A] B A v= k'[ B] da = ka [ ] + k [ B] [ A] + [ B] = [ A] 0 da = ka [ ] + k ([ A] 0 [ A]) ( k+ k ) t k + ke [ A] = [ A] 0 k+ k k k [ B] k [ A] eq = [ A], [ B] = [ A], K = = k+ k k+ k A k eq 0 eq 0 [ ] eq k k K k k Generlly, for multistge rection: =...
14 Rection pproching equilirium Relxtion denotes return of the system to equilirium fter externl influence, e.g. temperture jump (vi electricl dischrge, or microwve, or lser pulse) 1 τ t / τ x= x0e = k + k Indeed t the new condition: da = k( x+ [ A] eq) + k( x+ [ B] eq) = = ( k + k ) x x k nd k cn e found from the relxtion experiment (s K is known)
15 The temperture dependence of rection rte The rte constnts of most rection increse with the temperture (usully 2-4 times for every T rise 25 ->35 C) Experimentlly for mny rections k follows Arrhenius eqution E ln k = ln A RT A pre-exponentil (frequency) fctor, E ctivtion energy High ctivtion energy mens tht rte constnts depend strongly on the temperture, zero would men rection independent on temperture
16 The temperture dependence of rection rte Stges of rection: Regents Activtion complex Product Trnsition stte k = Ae E / RT Frction of collision with required energy Activtion energy is the minimum kinetic energy rectnts must hve to form the products. Pre-exponentil is rte of collisions Arrhenius eqution gives the rte of successful collisions.
17 Elementry rections Most rections occur in sequence of steps clled elementry rections. Moleculrity of n elementry rection is the numer of molecules coming together to rect (e.g. uni-moleculr, imoleculr) Uni-moleculr: first order in the rectnt da [ ] A P = k[ A] Bimoleculr: first order in the rectnt da [ ] A + B P = k[ A][ B] Proportionl to collision rte H + Br2 HBr + Br
18 Consecutive elementry rections Rection cn proceed through formtion of n intermedite: k k A I P da [ ] = k[ A] di [ ] = k[ A] k[ I] dp [ ] = k[ I] Solution for A should e in form: [ A] = [ A] 0e kt di [ ] k + k[ I] = k[ A] e [ I] = ( e e )[ A] k k kt kt kt 0 0 [ A] + [ I] + [ P] = [ A] ke [ P] = 1 + ke [ A] kt kt 0 0 k k
19 Consecutive elementry rections The stedy-stte pproximtion Assumption: fter initil induction period the concentrtion of intermedite stys constnt di [ ] 0 Then nd di [ ] dp [ ] = k [ A] k [ I] 0 = k [ I] k [ A] ( kt ) 0 [ P] 1 e [ A]
20 Consecutive elementry rections The rte-limiting step Suppose k >>k thn if intermedite is formed it decys rpidly into P, so forming intermedite is rte-limiting step. kt kt ke ke kt [ P] = 1 + [ A] [ P] = ( 1 e )[ A] k k 0 0 Rte of formtion of P doesn t depend on decy rte of I.
21 Consecutive elementry rections Kinetic nd thermodynmic control of rections kinetic control: fr from equilirium A+ B P v P = k [ A][ B] ( ) A+ B P v P = k [ A][ B] ( ) [ P ] [ P] = k k thermodynmic control : concentrtion t equilirium re defined y equilirium constnts (Gis energies)
22 Consecutive elementry rections Pre-equiliri k k A+ B I P k This condition rises when k >>k. Then nd K dp [ ] = [ I] k [ A][ B] = k = k [ I] = k K[ A][ B] k Second order form with composite rte constnt
23 The kinetic isotope effect Oservtion of slowing down the rection upon replcement n tom with hevier isotope fcilitte determining the rte limiting step Primry kinetic isotope effect: rte limiting requires reking ond involving the isotope Secondry kinetic isotope effect: reduction of the rection rte even when the rte limiting step doesn t require reking ond involving the isotope
24 Unimoleculr rections As the molecule cquires the energy s result of collision why the rection is still first order? Lindemnn-Hinshelwood mechnism * k * da [ ] 2 A+ A A + A = k[ A] * * k da [ ] * A + A A+ A = k [ A ][ A] * * k da [ ] * A P = k[ A ] If the lst step is rte-limiting the overll rection will hve first order kinetics
25 Lindemnn-Hinshelwood mechnism * k * da [ ] 2 A+ A A + A = k[ A] * * k da [ ] * A + A A+ A = k [ A ][ A] * * k da [ ] * A P = k[ A ] da * [ ] * [ ] A = k A k A A k A 2 * * [ ] [ ][ ] [ ] 2 k[ A] = k + k [ A] 2 dp * kk [ A] = k [ A ] = k + k [ A] If the rte of dectivtion is much higher tht unimoleculr decy thn: dp kk A kk A = k + k [ A] k 2 [ ] [ ] The Lindemnn-Hinshelwood mechnism cn e tested y reducing the pressure (slowing down the ctivtion step) so the rection will switch to the second order.
26 Lindemnn-Hinshelwood mechnism testing for the theory 2 dp kk [ A] = k + k [ A] 1 k 1 = + k k k k [ A] isomeriztion of CHD=CHD
27 RRK-model RRK (Rice-Rmsperger-Kssel) model is correction to Lindemnn-Hinshelwood mechnism tht tkes into ccount energy distriution over the modes of motion s 1 E * k( E) = 1 k E s numer of modes E* - energy required to rek the ond
28 The ctivtion energy of the composite rection Let s consider Lindemnn-Hinshelwood mechnism nd pply Arrhenius-like temperture dependence to ech rte constnt k ( E ( )/ )( ( )/ ) RT E RT Ae Ae ( E ( )/ ) RT Ae = kk AA k = = A { E ( ) + E ( ) E ( )}/ RT e Overll ctivtion energy cn e positive or negtive E ( ) + E ( ) > E ( ) E ( ) + E ( ) < E ( )
29 Prolems E22.6 At 518 C, the hlf-life for the decomposition of smple of gseous cetldehyde (ethnl) initilly t 363 Torr ws 410 s. When the pressure ws 169 Torr, the hlf-life ws 880 s. Determine the order of the rection. E22.10 The second-order rte constnt for the rection CH 3 COOC 2 H 5 (q) + OH - (q) CH 3 CO 2- (q) + CH 3 CH 2 OH(q) is 0.11 dm 3 mol -1 s -1. Wht is the concentrtion of ester fter () 10 s, () 10 min when ethyl cette is dded to sodium hydroxide so tht the initil concentrtions re [NOH] = mol dm -3 nd [CH 3 COOC 2 H 5 ] = mol dm -3? E22.16 The effective rte constnt for gseous rection tht hs Lindemnn Hinshelwood mechnism is s -1 t 1.30 kp nd s -1 t 12 P. Clculte the rte constnt for the ctivtion step in the mechnism.
Lecture 6: Diffusion and Reaction kinetics
Lecture 6: Diffusion nd Rection kinetics 1-1-1 Lecture pln: diffusion thermodynmic forces evolution of concentrtion distribution rection rtes nd methods to determine them rection mechnism in terms of the
More informationUNIVERSITY OF MALTA DEPARTMENT OF CHEMISTRY. CH237 - Chemical Thermodynamics and Kinetics. Tutorial Sheet VIII
UNIVERSITY OF MALTA DEPARTMENT OF CHEMISTRY CH237 - Chemicl Thermodynmics nd Kinetics Tutoril Sheet VIII 1 () (i) The rte of the rection A + 2B 3C + D ws reported s 1.0 mol L -1 s -1. Stte the rtes of
More informationCHEMICAL KINETICS
CHEMICAL KINETICS Long Answer Questions: 1. Explin the following terms with suitble exmples ) Averge rte of Rection b) Slow nd Fst Rections c) Order of Rection d) Moleculrity of Rection e) Activtion Energy
More information4. CHEMICAL KINETICS
4. CHEMICAL KINETICS Synopsis: The study of rtes of chemicl rections mechnisms nd fctors ffecting rtes of rections is clled chemicl kinetics. Spontneous chemicl rection mens, the rection which occurs on
More informationModule 2: Rate Law & Stoichiomtery (Chapter 3, Fogler)
CHE 309: Chemicl Rection Engineering Lecture-8 Module 2: Rte Lw & Stoichiomtery (Chpter 3, Fogler) Topics to be covered in tody s lecture Thermodynmics nd Kinetics Rection rtes for reversible rections
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationAcid-Base Equilibria
Tdeusz Górecki Ionic Equiliri Acid-Bse Equiliri Brønsted-Lory: n cid is proton, se is. Acid Bse ( 3 PO 4, O), ( N 4 ) nd ( PO - 4 ) cn ll ehve s cids. Exmple: 4 N N3 Sustnces hich cn ehve oth s cids nd
More informationWhich of the following describes the net ionic reaction for the hydrolysis. Which of the following salts will produce a solution with the highest ph?
95. Which of the following descries the net ionic rection for the hydrolysis of NH4Cl( s)? A. NH4 ( q) Cl & ( q) NH4Cl( s) B. NH Cl & 4 ( s) NH4 ( q) Cl ( q) C. Cl ( q) H O & 2 ( l) HCl( q) OH ( q) D.
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory 2. Diffusion-Controlled Reaction
Ch. 4 Moleculr Rection Dynmics 1. Collision Theory. Diffusion-Controlle Rection Lecture 17 3. The Mteril Blnce Eqution 4. Trnsition Stte Theory: The Eyring Eqution 5. Trnsition Stte Theory: Thermoynmic
More informationChemistry Department. The Islamic University of Gaza. General Chemistry B.(CHEMB 1301) Time:2 hours الرقم الجامعي... اسم المدرس...
The Islmic University of Gz Chemistry Deprtment Generl Chemistry B.(CHEMB 1301) Time:2 hours 60 اسم الطالب... الرقم الجامعي... اسم المدرس... R = 8.314 J/mol.K, or = 0.0821 L.tm/mol.K Q1- True ( ) or flse(
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose nti-derivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationCHAPTER 20: Second Law of Thermodynamics
CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationCHM Physical Chemistry I Chapter 1 - Supplementary Material
CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information3.2.2 Kinetics. Maxwell Boltzmann distribution. 128 minutes. 128 marks. Page 1 of 12
3.. Kinetics Mxwell Boltzmnn distribution 8 minutes 8 mrks Pge of M. () M On the energy xis E mp t the mximum of the originl pek M The limits for the horizontl position of E mp re defined s bove the word
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationDIRECT 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 informationMath 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More informationINTRODUCTION. The three general approaches to the solution of kinetics problems are:
INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The
More informationLECTURE 14. Dr. Teresa D. Golden University of North Texas Department of Chemistry
LECTURE 14 Dr. Teres D. Golden University of North Texs Deprtment of Chemistry Quntittive Methods A. Quntittive Phse Anlysis Qulittive D phses by comprison with stndrd ptterns. Estimte of proportions of
More informationdy ky, dt where proportionality constant k may be positive or negative
Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.
More informationZ b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...
Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.
More informationProblem 22: Buffer solutions 1. The equilibrium, which governs the concentration of H + within the solution is HCOOH! HCOO + H + + Hence K
Problem : Buffer solutions. The equilibrium, hich governs the concentrtion of H ithin the solution is HCOOH! HCOO H [HCOO ] 4 Hence. [HCOOH] nd since [HCOOH] 0.00 M nd [HCOO ] 0.50 M -4 0.00 4..8 M 0.50
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More information6. Photoionization of acridine through singlet and triplet channels
Chpter 6: Photoioniztion of cridine through singlet nd triplet chnnels 59 6. Photoioniztion of cridine through singlet nd triplet chnnels Photoioinztion of cridine (Ac) in queous micelles hs not yet een
More information7/19/2011. Models of Solution Chemistry- III Acids and Bases
Models of Solution Chemistry- III Acids nd Bses Ionic Atmosphere Model : Revisiting Ionic Strength Ionic strength - mesure of totl concentrtion of ions in the solution Chpter 8 1 2 i μ ( ) 2 c i z c concentrtion
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationMATH 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 informationDynamic equilibrium occurs when the forward and reverse reactions occur at the same rate.
MODULE 2 WORKSHEET8 EQUILIBRIUM Syllus reference 9.3.2 1 Clssify ech of the following sttements s true or flse. For the flse sttements rewrite them so they re true. For chemicl equilirium to e estlished
More informationProperties 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 informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationCHAPTER 08: MONOPROTIC ACID-BASE EQUILIBRIA
Hrris: Quntittive Chemicl Anlysis, Eight Edition CHAPTER 08: MONOPROTIC ACIDBASE EQUILIBRIA CHAPTER 08: Opener A CHAPTER 08: Opener B CHAPTER 08: Opener C CHAPTER 08: Opener D CHAPTER 08: Opener E Chpter
More informationPsychrometric Applications
Psychrometric Applictions The reminder of this presenttion centers on systems involving moist ir. A condensed wter phse my lso be present in such systems. The term moist irrefers to mixture of dry ir nd
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationAcid Base Equilibrium Review
Acid Bse Equilirium Review Proof of true understnding of cid se equilirium culmintes in the ility to find ph of ny solution or comintion of solutions. The ility to determine ph of multitude of solutions
More informationReview 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 informationA. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationFig. 1. Open-Loop and Closed-Loop Systems with Plant Variations
ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses
More informationMATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs
MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching
More informationHydronium or hydroxide ions can also be produced by a reaction of certain substances with water:
Chpter 14 1 ACIDS/BASES Acids hve tste, rect with most metls to produce, rect with most crbontes to produce, turn litmus nd phenolphthlein. Bses hve tste rect very well well with most metls or crbontes,
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More information1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation
1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationChapter 17: Additional Aspects of Aqueous Equilibria
1 Chpter 17: Additionl Aspects of Aqueous Equilibri Khoot! 1. Adding Br to sturted queous solution of decreses its solubility in wter. BSO 4, Li CO 3, PbS, AgBr. Which of the following mitures could be
More information1. Weak acids. For a weak acid HA, there is less than 100% dissociation to ions. The B-L equilibrium is:
th 9 Homework: Reding, M&F, ch. 15, pp. 584-598, 602-605 (clcultions of ph, etc., for wek cids, wek bses, polyprotic cids, nd slts; fctors ffecting cid strength). Problems: Nkon, ch. 18, #1-10, 16-18,
More informationStrong acids and bases. Strong acids and bases. Systematic Treatment of Equilibrium & Monoprotic Acid-base Equilibrium.
Strong cids nd bses Systemtic Tretment of Equilibrium & Monoprotic cid-bse Equilibrium onc. (M) 0.0.00 -.00-5.00-8 p Strong cids nd bses onc. (M) p 0.0.0.00 -.0.00-5 5.0.00-8 8.0? We hve to consider utoprotolysis
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationThe Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases.
Section 1 Section 2 The module clcultes nd plots isotherml 1-, 2- nd 3-metl predominnce re digrms. ccesses only compound dtbses. Tble of Contents Tble of Contents Opening the module Section 3 Stoichiometric
More information4 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 informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationMATH 3795 Lecture 18. Numerical Solution of Ordinary Differential Equations.
MATH 3795 Lecture 18. Numericl Solution of Ordinry Differentil Equtions. Dmitriy Leykekhmn Fll 2008 Gols Introduce ordinry differentil equtions (ODEs) nd initil vlue problems (IVPs). Exmples of IVPs. Existence
More informationMAT187H1F Lec0101 Burbulla
Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is
More informationWhen a force f(t) is applied to a mass in a system, we recall that Newton s law says that. f(t) = ma = m d dt v,
Impulse Functions In mny ppliction problems, n externl force f(t) is pplied over very short period of time. For exmple, if mss in spring nd dshpot system is struck by hmmer, the ppliction of the force
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationPart I: Basic Concepts of Thermodynamics
Prt I: Bsic Concepts o Thermodynmics Lecture 4: Kinetic Theory o Gses Kinetic Theory or rel gses 4-1 Kinetic Theory or rel gses Recll tht or rel gses: (i The volume occupied by the molecules under ordinry
More informationThe 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 informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationMath 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 informationSummarizing Remarks λ λ λ. : equilibrium geometry
112 Summrizing Remrks... 112 Summrizing Remrks The theory underlying chemicl processes, in prticulr chemicl equilibrium is mture science. The bsis of the edifice is Quntum Mechnics! For prticulr volume
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationState 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 informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationFundamentals of Analytical Chemistry
Homework Fundmentls of nlyticl hemistry hpter 9 0, 1, 5, 7, 9 cids, Bses, nd hpter 9(b) Definitions cid Releses H ions in wter (rrhenius) Proton donor (Bronsted( Lowry) Electron-pir cceptor (Lewis) hrcteristic
More informationChapter 16 Acid Base Equilibria
Chpter 16 Acid Bse Equilibri 16.1 Acids & Bses: A Brief Review Arrhenius cids nd bses: cid: n H + donor HA(q) H(q) A(q) bse: n OH donor OH(q) (q) OH(q) Brønsted Lowry cids nd bses: cid: n H + donor HA(q)
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationPHYS 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 information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationthan 1. It means in particular that the function is decreasing and approaching the x-
6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the
More informationGreen s functions. f(t) =
Consider the 2nd order liner inhomogeneous ODE Green s functions d 2 u 2 + k(t)du + p(t)u(t) = f(t). Of course, in prctice we ll only del with the two prticulr types of 2nd order ODEs we discussed lst
More informationScientific notation is a way of expressing really big numbers or really small numbers.
Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific
More informationDesigning Information Devices and Systems I Discussion 8B
Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More information4.1. Probability Density Functions
STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
More informationA. Limits - L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. -1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.
A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by
More informationIdeal Gas behaviour: summary
Lecture 4 Rel Gses Idel Gs ehviour: sury We recll the conditions under which the idel gs eqution of stte Pn is vlid: olue of individul gs olecules is neglected No interctions (either ttrctive or repulsive)
More information221B Lecture Notes WKB Method
Clssicl Limit B Lecture Notes WKB Method Hmilton Jcobi Eqution We strt from the Schrödinger eqution for single prticle in potentil i h t ψ x, t = [ ] h m + V x ψ x, t. We cn rewrite this eqution by using
More informationMath 42 Chapter 7 Practice Problems Set B
Mth 42 Chpter 7 Prctice Problems Set B 1. Which of the following functions is solution of the differentil eqution dy dx = 4xy? () y = e 4x (c) y = e 2x2 (e) y = e 2x (g) y = 4e2x2 (b) y = 4x (d) y = 4x
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationOrganic Acids - Carboxylic Acids
Orgnic Acids - rboxylic Acids Orgnic cids - crboxylic cid functionl group rboxylic cids re redily deprotonted by bses such s NO eg 3 O O - + O - + O 3 O O Acid Bse onjugte Bse onjugte Acid This rection
More information