Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws

Size: px
Start display at page:

Download "Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws"

Transcription

1 Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species ha ake par in he reacion ino an oherwise empy conainer. The reacion rae was hen measured. Wha paerns do you noice abou he reacion raes? Can reacion raes be prediced? Daa Taken a 98 K Daa Taken a 35 K Exp. [] [B] [C] Rae Exp. [] [B] [C] Rae M M M M M M M M M M M M 3 M M M 3 M M M 4 M M M 4 M M M 5 M M M 5 M M M 6 M 3 M M 6 M 3 M M 7 M M M 4 7 M M M M M M M M M 3 9 M M M 4 9 M M M 8.5 M 6 M 3 M.5 M 6 M 3 M Big Idea: The raes of chemical reacions are described by simple expressions ha allow us o predic he composiion of a reacion mixure a anyime. These expressions also sugges he seps in which he reacions akes place. Chaper 5 Chemical Kineics o Thermo Review o Reacion Raes o o Concenraion and o o Explaining Reacion Rae Facors Reacion Raes Reacion Raes Caalys: subsance ha increases he reacion rae wihou being consumed in he reacion. Homogeneous Caalys: caalys ha is in he same phase as he reacans. Heerogeneous Caalys: caalys ha is in a differen phase han he reacans. Reacion Raes: The change in concenraion of one of he reacans or producs divided by he ime inerval over which he change akes place. verage Rae of Consumpion of R: R P verage Rae of Producion of P: Noe: Raes are always posiive, herefore, since he reacans are consumed, a negaive sign mus be added o make he rae posiive. Unique verage Rae (UR) 3 4 Reacion Raes Insananeous Rae of Reacion The bes approximaion o he rae a a single insan is obained by drawing a line angen o he plo of he concenraion agains ime. The slope of he angen line is called he insananeous rae of he reacion. Rae Law: n equaion expressing he insananeous reacion rae in erms of he concenraions, a any insan, of he subsances aking par in he reacion. Noe: k is he rae consan and x,y, are he orders of reacion. Noe: This form of he rae law is called he differenial rae law. Noe: The unis of rae are always, herefore, he unis of k will differ depending on he overall reacion order. 5 6

2 Things o know abou he rae law: How o find he unis of k Sep : Deermine he order of he Rae laws can conain producs, reacans, caalyss bu usually only saring maerial. reacion Rae laws do no conain inermediaes. Examples for rae=k[][b]: Rae laws can ONLY be deermined Overall order experimenally. Sep : Subrac from he overall order Orders do NOT correlae wih coefficiens in Examples for rae=k[][b]: balanced equaion! Orders can be an inegers, zeros, fracions, Sep 3: Find posiives, OR negaives! # Each species has is own individual reacion Examples for rae=k[][b]: order. The overall reacion order is he sum of he individual orders found in he reacion. 7 8 Order h s nd Deermine he rae law: + B + C D ime Iniial Concenraion [] 4 6 Rae Rae Law Experimen [] o (M) [B] o (M) [C] o (M) Iniial Rae ( ) General Rae Law: ) Order wih respec o : 3) Order wih respec o B: 4) Order wih respec o C: 9 Deermine he rae law and k for + B C Experimen [] o (M) [B] o (M) Iniial Rae ( Deermining Order (long way) Sep : Find wo experimens in which he concenraions of everyhing, excep one species, is held consan. Sep : Divide he rae laws for hese wo experimens by each oher. Noe: This will cancel ou k and all oher variables excep for he order ha you are rying o deermine. Mah Noe: Sep 3: Solve for order. Mah Noe: I is someimes useful o ake he log of boh sides of he equaion. The log(x y )=ylog(x).

3 The rae law for he following reacion NO(g) +O (g) NO (g) was experimenally found o be in he form rae=k[no] x [O ] y I was also found ha when he NO concenraion was doubled, he rae of he reacion increases by a facor of 4. In addiion, when boh he O and he NO concenraion were doubled, he rae increases by a facor of 8. Wha is he reacion order of O? a) h b) s c) nd d) 3 rd e) None of he above Deermine he rae law and k for + B C Experimen [] o (M) [B] o (M) Iniial Rae ( Pseudo Order Reacion: reacion in which he rae law can be simplified because all bu one of he species have virually consan concenraions. 3 4 Zero Order Inegraed Rae Law Firs Order Inegraed Rae Law ) Rae k dx x cons d ) k 5) k 3) 4) d k d k k ) Rae k d ) k 3) d k 4) d k dx x 5) ln ln ln x cons ln k k ln 5 6 Second Order Inegraed Rae Law ) Rae k dx cons x x d ) k 5) k 3) 4) d k d k k The rae law for +B C + D was only found o be dependen on. Using he following daa deermine he rae law and k. (s) [] (M)

4 (s) [] [] (M) ln[] - - ln[] []^- 5 5 [] - (M - ) Calculae he concenraion of N O afer he firs order decomposiion: N O(g) N (g) + O (g). The rae of decomposiion of N O = k[n O]. The reacion has coninued a 78ºC for. ms, and he iniial concenraion of N O was. M and k = Half Life: i akes for he concenraion o drop o half he iniial amoun s Order h Order k k k k ln k ln k ln k ln ln ln k ln ln ln k ln k nd Order k k k Reacion Mechanism: The seps by which a reacion akes place. Elemenary Reacion: One of he reacion seps in an overall reacion. Noe: RTE LWS CN BE DETERMINED FROM ELEMENTRY RECTIONS. Noe: In order o be a valid mechanism he sum of he elemenary reacions mus equal he overall reacion

5 Mechanism : Elemenary Reacion(s): Mechanism : Elemenary Reacion(s): Reacion Inermediae: species ha plays a role in he reacion bu does no appear in he reacion. Moleculariy: The number of reacan molecules/aoms aking par in an elemenary reacion. Noe: Differen elemenary reacions usually give differen raes. Noe: In order o be a valid mechanism he rae law derived from he elemenary reacions mus mach he experimenally deermined rae law. Examples: Moleculariy of Moleculariy of 5 6 Use he experimenally found rae laws given below o deermine which reacion is mos likely o occur in a single sep. Experimenally Found a) NO (g)+f (g)no F(g) b) H (g)+br (g)hbr(g) c) NO(g)+O (g)no (g) + O (g) d) NO (g)+co(g) NO(g) + CO (g) Consider he following hypoheical reacion: + B E. The mechanism for his reacion is: () + B C (slow) () B + C D (fas) (3) D E (fas) The rae law consisen wih his mechanism is: a) rae=k[][b] b) rae=k[] [B] c) rae=k[] d) rae=k[][b] e) None of he above 7 8 Relaing he Rae and Equilibrium Consans The reacion NO + Cl NOCl was experimenally found o have he rae law: rae = k[cl ][NO]. Which mechanism could no be he correc mechanism? +B C Equilibrium (Rae forward = Rae reverse ) a) Cl Cl (fas equilibrium) Cl + NO NOCl (slow) b) NO + Cl NOCl (slow) NOCl NOCl 3 +NOCl (fas equilibrium) NOCl 3 NOCl + Cl (fas equilibrium) c) NO + Cl NOCl d) NO N O (fas equilibrium) N O + Cl NOCl (slow) e) ll of he above are possible mechanisms Equilibrium Consan C k K B k k K k Rae Consan 9 3 5

6 Muli Sep Reacions Wih Unknown Speeds Sep : Wrie an expression for he rae of formaion of one of he final producs (someimes he produc of ineres is specified). If possible selec a produc ha is only in sep. Sep : Use he seady sae approximaion o solve for he concenraion of inermediaes. Sep 3: Plug back ino overall rae equaion. The rae consan for he second-order gasphase reacion HO(g) + H (g) H O(g) + H(g) varies wih he emperaure as shown here: Temperaure (K) Rae Consan ( Deermine he acivaion energy. a).4-5 b) 4. 3 c) 4. 4 d) None of he above 3 3 Collision Theory (Gases Only) Rae = Collision Frequency Fracion wih Sufficien Energy How would we ge collision frequency Size of he molecules/aoms verage velociy of he molecules/aoms Concenraion of molecules/aoms Problem: lhough he rae has he righ form (rrhenius) i predics a larger reacion rae han is found. Soluion: Serics need o be aken ino accoun. How would we ge fracion wih sufficien energy? Bolzmann disribuion 4 E a = Minimum energy needed for reacion o occur. = Takes ino accoun number of collisions and serics Take way From Chaper 5 civaed Complex Theory (Soluions) Big Idea: The raes of chemical reacions are described by simple expressions ha allow us o predic he composiion of a reacion mixure a anyime. These expressions also sugges he seps in which he reacions akes place. Thermo Review Know he difference beween hermo and kineics Thermodynamics allows us o predic if a reacion will occur. Kineics allows us o predic how fas a reacion will occur. Be able o draw reacion coordinaes along wih labeling reacans, producs, inermediaes, ransiion saes, and acivaion energy. (8, 8, 8) Reacion Raes Be able o explain how a caalys can increase reacion rae. (84, 88)

7 Take way From Chaper 5 Take way From Chaper 5 Reacion Raes (Coninued) Be able o calculae he average rae of reacion of species given he average rae of reacion of anoher species. Be able o calculae he unique average rae of a reacion. Rae Law Know ha rae laws mus be deermined experimenally. Be able o deermine he order of a reacion and each individual species. Be able o calculae rae law from experimenal daa. (7, 8, 9,, ) Double concenraion and rae says he same: h order Double concenraion and rae doubles: s order Double concenraion and rae quadruples: nd order If none of hese use he mah rick Be able o deermine he unis of he rae consan (8, 4, 5) The unis of concenraion are consan (M) The unis of rae are consans Rae Law (Coninued) Know ha rae problems can be simplified when he concenraion of species is high and essenially unchanging (pseudo order reacions). (34, 37, 5) Be able o use he inegraed rae law o perform calculaions. (4, 45, 46, 48) h order: s order: ln nd order: Be able o idenify he order of he reacion by ploing daa. (8, 9, 3, 3, 33, 36) [] vs. linear h order, slope = -k ln[] vs. linear s order, slope= -k [] - vs. linear nd order, slope = k Be able o calculae half-life/ nd half-life/ec Take way From Chaper 5 Take way From Chaper 5 (5) Be able o wrie a rae law of an elemenary reacion. (55) Be able o deermine he rae law of muli sep reacions. (56,57) Be able o eliminae inermediaes from muli sep reacions. Equilibrium (59,6,64) Seady sae approximaion (65,66) Concenraion Caalys Be able o draw poenial energy diagram of reacions Temperaure Know ha mos reacions follow rrhenius behavior. (68,73,74,65,75,77,78,) ln ln (coninued) Surface rea Know he ideas behind collision heory (gases). Know he ideas behind acivaed complex heory (soluions)

AP Chemistry--Chapter 12: Chemical Kinetics

AP Chemistry--Chapter 12: Chemical Kinetics AP Chemisry--Chaper 12: Chemical Kineics I. Reacion Raes A. The area of chemisry ha deals wih reacion raes, or how fas a reacion occurs, is called chemical kineics. B. The rae of reacion depends on he

More information

2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts?

2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts? AP Chemisry Tes (Chaper 12) Muliple Choice (40%) 1) Which of he following is a kineic quaniy? A) Enhalpy B) Inernal Energy C) Gibb s free energy D) Enropy E) Rae of reacion 2) Of he following quesions,

More information

Chapter 12. Chemical Kinetics

Chapter 12. Chemical Kinetics Chaper. Chemical Kineics Chemisry: The Sudy of Change FCT ORY Processes => Chemical Reacions (Redox, cid-base, Precipiaion in q. Soln) Two facors of chemical reacions How fas? => Chemical Kineics (Mechanism,

More information

Chapter 13 Homework Answers

Chapter 13 Homework Answers Chaper 3 Homework Answers 3.. The answer is c, doubling he [C] o while keeping he [A] o and [B] o consan. 3.2. a. Since he graph is no linear, here is no way o deermine he reacion order by inspecion. A

More information

Chapter 14 Homework Answers

Chapter 14 Homework Answers 4. Suden responses will vary. (a) combusion of gasoline (b) cooking an egg in boiling waer (c) curing of cemen Chaper 4 Homework Answers 4. A collision beween only wo molecules is much more probable han

More information

CHEMICAL KINETICS: 1. Rate Order Rate law Rate constant Half-life Temperature Dependence

CHEMICAL KINETICS: 1. Rate Order Rate law Rate constant Half-life Temperature Dependence CHEMICL KINETICS: Rae Order Rae law Rae consan Half-life Temperaure Dependence Chemical Reacions Kineics Chemical ineics is he sudy of ime dependence of he change in he concenraion of reacans and producs.

More information

Chapter 14 Chemical Kinetics

Chapter 14 Chemical Kinetics /5/4 Chaper 4 Chemical Kineics Chemical Kineics Raes of Reacions Chemical Kineics is he sudy of he rae of reacion. How fas does i ake place? Very Fas Reacions Very Slow Reacions Acid/Base Combusion Rusing

More information

Chapter 15: Phenomena

Chapter 15: Phenomena Chapter 5: Phenomena Phenomena: The reaction A(aq) + B(aq) C(aq) was studied at two different temperatures (298 K and 350 K). For each temperature the reaction was started by putting different concentrations

More information

Chapter 14 Chemical Kinetics

Chapter 14 Chemical Kinetics # of paricles 5/9/4 Chemical Kineics Raes of Reacions Chemical Kineics is he sudy of he rae of reacion. How fas does i ae place? Very Fas Reacions Very Slow Reacions Chaper 4 Chemical Kineics Acid/Base

More information

Northern Arizona University Exam #1. Section 2, Spring 2006 February 17, 2006

Northern Arizona University Exam #1. Section 2, Spring 2006 February 17, 2006 Norhern Arizona Universiy Exam # CHM 52, General Chemisry II Dr. Brandon Cruickshank Secion 2, Spring 2006 February 7, 2006 Name ID # INSTRUCTIONS: Code he answers o he True-False and Muliple-Choice quesions

More information

Math 333 Problem Set #2 Solution 14 February 2003

Math 333 Problem Set #2 Solution 14 February 2003 Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial

More information

Advanced Organic Chemistry

Advanced Organic Chemistry Lalic, G. Chem 53A Chemisry 53A Advanced Organic Chemisry Lecure noes 1 Kineics: A racical Approach Simple Kineics Scenarios Fiing Experimenal Daa Using Kineics o Deermine he Mechanism Doughery, D. A.,

More information

CHEMICAL KINETICS Rate Order Rate law Rate constant Half-life Molecularity Elementary. Complex Temperature dependence, Steady-state Approximation

CHEMICAL KINETICS Rate Order Rate law Rate constant Half-life Molecularity Elementary. Complex Temperature dependence, Steady-state Approximation CHEMICL KINETICS Rae Order Rae law Rae consan Half-life Moleculariy Elemenary Complex Temperaure dependence, Seady-sae pproximaion Chemical Reacions Kineics Chemical ineics is he sudy of ime dependence

More information

A First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18

A First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18 A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly

More information

KINEMATICS IN ONE DIMENSION

KINEMATICS IN ONE DIMENSION KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec

More information

Solutions to Assignment 1

Solutions to Assignment 1 MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we

More information

Math 2214 Solution Test 1A Spring 2016

Math 2214 Solution Test 1A Spring 2016 Mah 14 Soluion Tes 1A Spring 016 sec Problem 1: Wha is he larges -inerval for which ( 4) = has a guaraneed + unique soluion for iniial value (-1) = 3 according o he Exisence Uniqueness Theorem? Soluion

More information

04. Kinetics of a second order reaction

04. Kinetics of a second order reaction 4. Kineics of a second order reacion Imporan conceps Reacion rae, reacion exen, reacion rae equaion, order of a reacion, firs-order reacions, second-order reacions, differenial and inegraed rae laws, Arrhenius

More information

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms

More information

dy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page

dy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,

More information

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities: Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial

More information

CHAPTER 12 DIRECT CURRENT CIRCUITS

CHAPTER 12 DIRECT CURRENT CIRCUITS CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As

More information

5.2. The Natural Logarithm. Solution

5.2. The Natural Logarithm. Solution 5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e 2.718 281 828 59. Like π, e also occurs frequenly in naural phenomena. In fac,

More information

4.5 Constant Acceleration

4.5 Constant Acceleration 4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),

More information

CHAPTER 16 KINETICS: RATES AND MECHANISMS OF CHEMICAL REACTIONS

CHAPTER 16 KINETICS: RATES AND MECHANISMS OF CHEMICAL REACTIONS CHAPTER 6 KINETICS: RATES AND MECHANISMS OF CHEMICAL REACTIONS FOLLOW UP PROBLEMS 6.A Plan: Balance he equaion. The rae in ers of he change in concenraion wih ie for each subsance is [ A] expressed as,

More information

APPM 2360 Homework Solutions, Due June 10

APPM 2360 Homework Solutions, Due June 10 2.2.2: Find general soluions for he equaion APPM 2360 Homework Soluions, Due June 10 Soluion: Finding he inegraing facor, dy + 2y = 3e µ) = e 2) = e 2 Muliplying he differenial equaion by he inegraing

More information

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,

More information

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Chapter 7: Solving Trig Equations

Chapter 7: Solving Trig Equations Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions

More information

Math 2214 Solution Test 1B Fall 2017

Math 2214 Solution Test 1B Fall 2017 Mah 14 Soluion Tes 1B Fall 017 Problem 1: A ank has a capaci for 500 gallons and conains 0 gallons of waer wih lbs of sal iniiall. A soluion conaining of 8 lbsgal of sal is pumped ino he ank a 10 galsmin.

More information

KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow

KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow 1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering

More information

04. Kinetics of a second order reaction

04. Kinetics of a second order reaction 4. Kineics of a second order reacion Imporan conceps Reacion rae, reacion exen, reacion rae equaion, order of a reacion, firs-order reacions, second-order reacions, differenial and inegraed rae laws, rrhenius

More information

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.

dt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3. Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies

More information

1.6. Slopes of Tangents and Instantaneous Rate of Change

1.6. Slopes of Tangents and Instantaneous Rate of Change 1.6 Slopes of Tangens and Insananeous Rae of Change When you hi or kick a ball, he heigh, h, in meres, of he ball can be modelled by he equaion h() 4.9 2 v c. In his equaion, is he ime, in seconds; c represens

More information

Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k

Challenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,

More information

MA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions

MA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by

More information

Math 10B: Mock Mid II. April 13, 2016

Math 10B: Mock Mid II. April 13, 2016 Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.

More information

המחלקה : ביולוגיה מולקולרית הפקולטה למדעי הטבע

המחלקה : ביולוגיה מולקולרית הפקולטה למדעי הטבע טל : 03-9066345 פקס : 03-9374 ראש המחלקה: ד"ר אלברט פנחסוב המחלקה : ביולוגיה מולקולרית הפקולטה למדעי הטבע Course Name: Physical Chemisry - (for Molecular Biology sudens) כימיה פיזיקאלית )לסטודנטים לביולוגיה

More information

Reaction Order Molecularity. Rate laws, Reaction Orders. Determining Reaction Order. Determining Reaction Order. Determining Reaction Order

Reaction Order Molecularity. Rate laws, Reaction Orders. Determining Reaction Order. Determining Reaction Order. Determining Reaction Order Rae laws, Reacion Orders The rae or velociy of a chemical reacion is loss of reacan or appearance of produc in concenraion unis, per uni ime d[p] d[s] The rae law for a reacion is of he form Rae d[p] k[a]

More information

6.003 Homework 1. Problems. Due at the beginning of recitation on Wednesday, February 10, 2010.

6.003 Homework 1. Problems. Due at the beginning of recitation on Wednesday, February 10, 2010. 6.003 Homework Due a he beginning of reciaion on Wednesday, February 0, 200. Problems. Independen and Dependen Variables Assume ha he heigh of a waer wave is given by g(x v) where x is disance, v is velociy,

More information

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010

Simulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid

More information

Vehicle Arrival Models : Headway

Vehicle Arrival Models : Headway Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where

More information

u(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x

u(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x . 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih

More information

Some Basic Information about M-S-D Systems

Some Basic Information about M-S-D Systems Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,

More information

2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance

2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion

More information

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still. Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in

More information

1. VELOCITY AND ACCELERATION

1. VELOCITY AND ACCELERATION 1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under

More information

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile

More information

Homework 2 Solutions

Homework 2 Solutions Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,

More information

5.1 - Logarithms and Their Properties

5.1 - Logarithms and Their Properties Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We

More information

4.6 One Dimensional Kinematics and Integration

4.6 One Dimensional Kinematics and Integration 4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of

More information

d = ½(v o + v f) t distance = ½ (initial velocity + final velocity) time

d = ½(v o + v f) t distance = ½ (initial velocity + final velocity) time BULLSEYE Lab Name: ANSWER KEY Dae: Pre-AP Physics Lab Projecile Moion Weigh = 1 DIRECTIONS: Follow he insrucions below, build he ramp, ake your measuremens, and use your measuremens o make he calculaions

More information

EXERCISES FOR SECTION 1.5

EXERCISES FOR SECTION 1.5 1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler

More information

Physics 20 Lesson 5 Graphical Analysis Acceleration

Physics 20 Lesson 5 Graphical Analysis Acceleration Physics 2 Lesson 5 Graphical Analysis Acceleraion I. Insananeous Velociy From our previous work wih consan speed and consan velociy, we know ha he slope of a posiion-ime graph is equal o he velociy of

More information

Phys1112: DC and RC circuits

Phys1112: DC and RC circuits Name: Group Members: Dae: TA s Name: Phys1112: DC and RC circuis Objecives: 1. To undersand curren and volage characerisics of a DC RC discharging circui. 2. To undersand he effec of he RC ime consan.

More information

Predator - Prey Model Trajectories and the nonlinear conservation law

Predator - Prey Model Trajectories and the nonlinear conservation law Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories

More information

Math 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.

Math 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11. 1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be

More information

Sub Module 2.6. Measurement of transient temperature

Sub Module 2.6. Measurement of transient temperature Mechanical Measuremens Prof. S.P.Venkaeshan Sub Module 2.6 Measuremen of ransien emperaure Many processes of engineering relevance involve variaions wih respec o ime. The sysem properies like emperaure,

More information

2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes

2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion

More information

Chapter 2. First Order Scalar Equations

Chapter 2. First Order Scalar Equations Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.

More information

Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution

Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his

More information

13.3 Term structure models

13.3 Term structure models 13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)

More information

Motion along a Straight Line

Motion along a Straight Line chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4)

More information

Suggested Problem Solutions Associated with Homework #5

Suggested Problem Solutions Associated with Homework #5 Suggesed Problem Soluions Associaed wih Homework #5 431 (a) 8 Si has proons and neurons (b) 85 3 Rb has 3 proons and 48 neurons (c) 5 Tl 81 has 81 proons and neurons 43 IDENTIFY and SET UP: The ex calculaes

More information

Math 116 Second Midterm March 21, 2016

Math 116 Second Midterm March 21, 2016 Mah 6 Second Miderm March, 06 UMID: EXAM SOLUTIONS Iniials: Insrucor: Secion:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam. 3. This exam has pages including

More information

IB Physics Kinematics Worksheet

IB Physics Kinematics Worksheet IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?

More information

= ( ) ) or a system of differential equations with continuous parametrization (T = R

= ( ) ) or a system of differential equations with continuous parametrization (T = R XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of

More information

( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:

( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively: XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of

More information

AP Calculus BC Chapter 10 Part 1 AP Exam Problems

AP Calculus BC Chapter 10 Part 1 AP Exam Problems AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a

More information

Today: Graphing. Note: I hope this joke will be funnier (or at least make you roll your eyes and say ugh ) after class. v (miles per hour ) Time

Today: Graphing. Note: I hope this joke will be funnier (or at least make you roll your eyes and say ugh ) after class. v (miles per hour ) Time +v Today: Graphing v (miles per hour ) 9 8 7 6 5 4 - - Time Noe: I hope his joke will be funnier (or a leas make you roll your eyes and say ugh ) afer class. Do yourself a favor! Prof Sarah s fail-safe

More information

The equation to any straight line can be expressed in the form:

The equation to any straight line can be expressed in the form: Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he

More information

Reading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.

Reading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4. PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence

More information

SMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.

SMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15. SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a

More information

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3 and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or

More information

EE650R: Reliability Physics of Nanoelectronic Devices Lecture 9:

EE650R: Reliability Physics of Nanoelectronic Devices Lecture 9: EE65R: Reliabiliy Physics of anoelecronic Devices Lecure 9: Feaures of Time-Dependen BTI Degradaion Dae: Sep. 9, 6 Classnoe Lufe Siddique Review Animesh Daa 9. Background/Review: BTI is observed when he

More information

Sterilization D Values

Sterilization D Values Seriliaion D Values Seriliaion by seam consis of he simple observaion ha baceria die over ime during exposure o hea. They do no all live for a finie period of hea exposure and hen suddenly die a once,

More information

!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)

!!#$%&#'()!#&'(*%)+,&',-)./0)1-*23) "#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5

More information

Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan

Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure

More information

Displacement ( x) x x x

Displacement ( x) x x x Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh

More information

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures. HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =

More information

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx. . Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.

More information

Exam 1 Solutions. 1 Question 1. February 10, Part (A) 1.2 Part (B) To find equilibrium solutions, set P (t) = C = dp

Exam 1 Solutions. 1 Question 1. February 10, Part (A) 1.2 Part (B) To find equilibrium solutions, set P (t) = C = dp Exam Soluions Februar 0, 05 Quesion. Par (A) To find equilibrium soluions, se P () = C = = 0. This implies: = P ( P ) P = P P P = P P = P ( + P ) = 0 The equilibrium soluion are hus P () = 0 and P () =..

More information

Final Spring 2007

Final Spring 2007 .615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o

More information

where the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP).

where the coordinate X (t) describes the system motion. X has its origin at the system static equilibrium position (SEP). Appendix A: Conservaion of Mechanical Energy = Conservaion of Linear Momenum Consider he moion of a nd order mechanical sysem comprised of he fundamenal mechanical elemens: ineria or mass (M), siffness

More information

Math 115 Final Exam December 14, 2017

Math 115 Final Exam December 14, 2017 On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname):

More information

Math Final Exam Solutions

Math Final Exam Solutions Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,

More information

SPH3U1 Lesson 03 Kinematics

SPH3U1 Lesson 03 Kinematics SPH3U1 Lesson 03 Kinemaics GRAPHICAL ANALYSIS LEARNING GOALS Sudens will Learn how o read values, find slopes and calculae areas on graphs. Learn wha hese values mean on boh posiion-ime and velociy-ime

More information

Math 116 Practice for Exam 2

Math 116 Practice for Exam 2 Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem

More information

Notes 8B Day 1 Doubling Time

Notes 8B Day 1 Doubling Time Noes 8B Day 1 Doubling ime Exponenial growh leads o repeaed doublings (see Graph in Noes 8A) and exponenial decay leads o repeaed halvings. In his uni we ll be convering beween growh (or decay) raes and

More information

4.1 - Logarithms and Their Properties

4.1 - Logarithms and Their Properties Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,

More information

MA Study Guide #1

MA Study Guide #1 MA 66 Su Guide #1 (1) Special Tpes of Firs Order Equaions I. Firs Order Linear Equaion (FOL): + p() = g() Soluion : = 1 µ() [ ] µ()g() + C, where µ() = e p() II. Separable Equaion (SEP): dx = h(x) g()

More information

The average rate of change between two points on a function is d t

The average rate of change between two points on a function is d t SM Dae: Secion: Objecive: The average rae of change beween wo poins on a funcion is d. For example, if he funcion ( ) represens he disance in miles ha a car has raveled afer hours, hen finding he slope

More information

8. Basic RL and RC Circuits

8. Basic RL and RC Circuits 8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics

More information

ENV 6015 Solution to Mixing Problem Set

ENV 6015 Solution to Mixing Problem Set EN 65 Soluion o ixing Problem Se. A slug of dye ( ) is injeced ino a single ank wih coninuous mixing. The flow in and ou of he ank is.5 gpm. The ank volume is 5 gallons. When will he dye concenraion equal

More information

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B) SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) 6 8 4 6 Diameer The able above gives he measuremens of he B(x)

More information

Unsteady Mass- Transfer Models

Unsteady Mass- Transfer Models See T&K Chaper 9 Unseady Mass- Transfer Models ChEn 6603 Wednesday, April 4, Ouline Conex for he discussion Soluion for ransien binary diffusion wih consan c, N. Soluion for mulicomponen diffusion wih

More information

Section 4.4 Logarithmic Properties

Section 4.4 Logarithmic Properties Secion. Logarihmic Properies 59 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies

More information

Y 0.4Y 0.45Y Y to a proper ARMA specification.

Y 0.4Y 0.45Y Y to a proper ARMA specification. HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where

More information

Physics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008

Physics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008 Physics 221 Fall 28 Homework #2 Soluions Ch. 2 Due Tues, Sep 9, 28 2.1 A paricle moving along he x-axis moves direcly from posiion x =. m a ime =. s o posiion x = 1. m by ime = 1. s, and hen moves direcly

More information