SF Chemical Kinetics.
|
|
- Jacob Long
- 5 years ago
- Views:
Transcription
1 SF Cheical Kinetics. Lecture 5. Microscopic theory of cheical reaction inetics. Microscopic theories of cheical reaction inetics. basic ai is to calculate the rate constant for a cheical reaction fro first principles using fundaental physics. ny icroscopic leel theory of cheical reaction inetics ust result in the deriation of an ression for the rate constant that is consistent with the epirical rrhenius equation. icroscopic odel should furtherore proide a reasonable interpretation of the pre-onential factor and the actiation energy E in the rrhenius equation. We will exaine two icroscopic odels for cheical reactions : The collision theory. The actiated coplex theory. The ain ephasis will be on gas phase biolecular reactions since reactions in the gas phase are the ost siple reaction types.
2 References for Microscopic Theory of Reaction Rates. Effect of teperature on reaction rate. urrows et al Cheistry 3, Section 8.7, pp Collision Theory/ ctiated Coplex Theory. urrows et al Cheistry 3, Section 8.8, pp tins, de Paula, Physical Cheistry 9 th edition, Chapter, Reaction Dynaics. Section.., pp tins, de Paula, Physical Cheistry 9 th edition, Chapter, Section..4-.5, pp Collision theory of biolecular gas phase reactions. We focus attention on gas phase reactions and assue that cheical reactiity is due to collisions between olecules. The theoretical approach is based on the inetic theory of gases. Molecules are assued to be hard structureless spheres. Hence the odel neglects the discrete cheical structure of an indiidual olecule. This assuption is unrealistic. We also assue that no interaction between olecules until contact. Molecular spheres aintain size and shape on collision. Hence the centres cannot coe closer than a distance d gien by the su of the olecular radii. The reaction rate will depend on two factors : the nuber of collisions per unit tie (the collision frequency) the fraction of collisions haing an energy greater than a certain threshold energy E*.
3 Siple collision theory : quantitatie aspects. ( g) ( g) Products Hard sphere reactants Molecular structure and details of internal otion such as ibrations and rotations ignored. Two basic requireents dictate a collision eent. One ust hae an, encounter oer a sufficiently short distance to allow reaction to occur. Colliding olecules ust hae sufficient energy of the correct type to oercoe the energy barrier for reaction. threshold energy E* is required. Two basic quantities are ealuated using the Kinetic Theory of gases : the collision frequency and the fraction of collisions that actiate olecules for reaction. To ealuate the collision frequency we need a atheatical way to define whether or not a collision occurs. The collision cross section s defines when a collision occurs. s d r r d r r Effectie collision diaeter d r r rea = s The collision cross section for two olecules can be regarded to be the area within which the center of the projectile olecule ust enter around the target olecule in order for a collision to occur. 3
4 Maxwell-oltzann elocity distribution function J.C. Maxwell F( ) 4 Gas olecules exhibit a spread or distribution of speeds. F() T 3/ T The elocity distribution cure has a ery characteristic shape. sall fraction of olecules oe with ery low speeds, a sall fraction oe with ery high speeds, and the ast ajority of olecules oe at interediate speeds. The bell shaped cure is called a Gaussian cure and the olecular speeds in an ideal gas saple are Gaussian distributed. The shape of the Gaussian distribution cure changes as the teperature is raised. The axiu of the cure shifts to higher speeds with increasing teperature, and the cure becoes broader as the teperature increases. greater proportion of the gas olecules hae high speeds at high teperature than at low teperature. The collision frequency is coputed ia the inetic theory of gases. We define a collision nuber (units: -3 s - ) Z. Z d n n r Mean relatie elocity Units: s - n j = nuber density of olecule j (units : -3 ) Mean relatie elocity ealuated ia inetic theory. d r r erage elocity of a gas olecule F F( ) 4 d T 3/ Maxwell-oltzann elocity Distribution function T M distribution of elocities enables us to statistically estiate the spread of olecular elocities in a gas Soe aths! 8T Mass of olecule 4
5 We now relate the aerage elocity to the ean relatie elocity. If and are different olecules then The aerage relatie elocity is gien by The ression across. r Hence the collision nuber between unlie olecules can be ealuated. r 8T j 8T j Z d n n r Reduced ass 8T Z nns Zn n Collision frequency factor / For collisions between lie olecules The nuber of collisions per unit tie between a single olecule and other Molecules. Total nuber of collisions between lie olecules. We diide by to ensure That each, encounter Is not counted twice. Z Z n s r 8 T Z n 8T n s / / E * Molecular collision is effectie only if translational energy of reactants is greater than soe threshold alue. Fraction of olecules with inetic energy greater Than soe iniu Threshold alue e* F e * e e * T 5
6 The siple collision theory ression for the reaction rate R between unlie olecules. dn e * R Znn dt T 8 T Z s The rate ression for a biolecular reaction between and. dc R cc dt We introduce olar ariables ogadro constant E* N e * n c N dn dt dc N dt n c N The rate constant for biolecular collisions between lie olecules. oth of these ressions are siilar to the rrhenius equation. Hence the SCT rate ression. dc E * R ZN cc dt RT / T E * N s RT E * z RT The biolecular rate constant for collisions between unlie olecules. Collision Frequency factor / / 8 T E * N s RT E * z RT 6
7 We copare the results of SCT with the epirical rrhenius eqn. in order to obtain an interpretation of the actiation energy and pre-onential factor. / T E * N s RT E * z RT obs E RT, encounters / 8 T E * N s RT E * z RT, encounters obs z '' '' N s 8 SCT predicts that the pre-onential factor should depend on teperature. The threshold energy and the actiation energy can also be copared. ctiation energy exhibits a wea T dependence. T d ln dt rrhenius E RT obs z ' 8 ' N s T d ln E * RT dt RT E E * RT Pre-onential factor SCT E E * SCT : a suary. The ajor proble with SCT is that the threshold energy E* is ery difficult to ealuate fro first principles. The predictions of the collision theory can be critically ealuated by coparing the eriental pre-onential factor with that coputed using SCT. We define the steric factor P as the ratio between P the eriental and calculated factors. calc We can incorporate P into the SCT ression for the rate constant. E * For any gas phase reactions Pz RT P is considerably less than unity. Typically SCT will predict that calc will be in E * Pz the region - Lol - s - regardless of RT the cheical nature of the reactants and products. What has gone wrong? The SCT assuption of hard sphere collision neglects the iportant fact that olecules possess an internal structure. It also neglects the fact that the relatie orientation of the colliding olecules will be iportant in deterining whether a collision will lead to reaction. We need a better theory that taes olecular structure into account. The actiated coplex theory does just that. 7
8 Suary of SCT., encounters / 8 T E * N s RT E * z RT, encounters / T E * N s RT E * z RT Transport property Pz Pz Steric factor (Orientation requireent) Energy criterion E * RT E * RT Weanesses: No way to copute P fro olecular paraeters No way to copute E* fro first principles. Theory not quantitatie or predictie. Strengths: Qualitatiely consistent with obseration (rrhenius equation). Proides plausible connection between icroscopic olecular properties and acroscopic reaction rates. Proides useful guide to upper liits for rate constant. Henry Eyring 9-98 Deeloped (in 935) the Transition State Theory (TST) or ctiated Coplex Theory (CT) of Cheical Kinetics. 8
9 Potential energy surface Can be constructed fro eriental easureents or fro Molecular Orbital calculations, sei-epirical ethods, Various trajectories through the potential energy surface 9
10 Potential energy hypersurface for cheical reaction between ato and diatoic olecule. C * C C Reading reaction progress on PE hypersurface. E E U
11 Energy Transition state theory (TST) or actiated coplex theory (CT). In a reaction step as the reactant olecules and coe together they distort and begin to share, exchange or discard atos. They for a loose structure of high potential energy called the actiated coplex that is poised to pass on to products or collapse bac to reactants C + D. The pea energy occurs at the transition state. The energy difference fro the ground state is the actiation energy E a of the reaction step. The potential energy falls as the atos rearrange in the cluster and finally reaches the alue for the products Note that the reerse reaction step also has an actiation energy, in this case higher than for the forward step. ctiated coplex Transition state E a Ea + Reaction coordinate C + D Transition state theory The theory attepts to lain the size of the rate constant r and its teperature dependence fro the actual progress of the reaction (reaction coordinate). The progress along the reaction coordinate can be considered in ters of the approach and then reaction of an H ato to an F olecule When far apart the potential energy is the su of the alues for H and F When close enough their orbitals start to oerlap bond starts to for between H and the closer F ato H F F The F F bond starts to lengthen s H becoes closer still the H F bond becoes shorter and stronger and the F F bond becoes longer and weaer The atos enter the region of the actiated coplex When the three atos reach the point of axiu potential energy (the transition state) a further infinitesial copression of the H F bond and stretch of the F F bond taes the coplex through the transition state.
12 Therodynaic approach Suppose that the actiated coplex is in equilibriu with the reactants with an equilibriu constant designated K and decoposes to products with rate constant K + actiated coplex products where K Therefore rate of foration of products = [ ] = K [][] [ ] = [][] Copare this ression to the rate law: rate of foration of products = r [][] Hence the rate constant r = K The Gibbs energy for the process is gien by Δ G = RTln (K ) and so K = ( Δ G/RT) Hence rate constant r = ( (Δ H TΔ S)/RT). Hence r = (Δ S/R) ( Δ H/RT) This ression has the sae for as the rrhenius ression. The actiation energy E a relates to Δ H Pre-onential factor = (Δ S/R) The steric factor P can be related to the change in disorder at the transition state Statistical therodynaic approach The actiated coplex can for products if it passes though the transition state The equilibriu constant K can be deried fro statistical echanics q is the partition function for each species ΔE (J ol - )is the difference in internal energy between, and at T= Suppose that a ery loose ibration-lie otion of the actiated coplex with frequency along the reaction coordinate tips it through the transition state. The reaction rate is depends on the frequency of that otion. Rate = [ ] It can be shown that the rate constant r is gien by the Eyring equation the contribution fro the critical ibrational otion has been resoled out fro quantities K and q cancels out fro the equation = oltzann constant h = Planc s constant r K T = h Hence r q = q K = q T q h q ΔE (- ) RT q - ΔE ( RT )
13 Statistical therodynaic approach Can deterine partition functions q and q fro spectroscopic easureents but transition state has only a transient existence (picoseconds) and so cannot be studied by noral techniques (into the area of fetocheistry) Need to postulate a structure for the actiated coplex and deterine a theoretical alue for q. Coplete calculations are only possible for siple cases, e.g., H + H H + H In ore coplex cases ay use ixture of calculated and eriental paraeters Potential energy surface: 3-D plot of the energy of all possible arrangeents of the atos in an actiated coplex. Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state. For the siplest case of the reaction of two structureless particles (e.g., atos) with no ibrational energy reacting to for a siple diatoic cluster the ression for r deried fro statistical therodynaics resebles that deried fro collision theory. Collision theory wors.for spherical olecules with no structure Exaple of a potential energy surface Hydrogen ato exchange reaction H + H H C H H + H C tos constrained to be in a straight line (collinear) H H H C Path C goes up along the alley and oer the col (pass or saddle point) between regions (ountains) of higher energy and descends down along the other alley. Paths and go oer uch ore difficult routes through regions of high energy Can inestigate this type of reaction by collision of olecular/ atoic beas with defined energy state. Deterine which energy states (translational and ibrational) lead to the ost rapid reaction. Mol H H Mol H H C Diagra: 3
14 dantages of transition state theory Proides a coplete description of the nature of the reaction including the changes in structure and the distribution of energy through the transition state the origin of the pre-onential factor with units t - that derie fro frequency or elocity the eaning of the actiation energy E a Rather coplex fundaental theory can be ressed in an easily understood pictorial diagra of the transition state - plot of energy s the reaction coordinate The pre-onential factor can be deried a priori fro statistical echanics in siple cases The steric factor P can be understood as related to the change in order of the syste and hence the entropy change at the transition state Can be applied to reactions in gases or liquids llows for the influence of other properties of the syste on the transition state (e.g., solent effects). Disadantage Not easy to estiate fundaental properties of the transition state except for ery siple reactions theoretical estiates of and E a ay be in the right ball-par but still need eriental alues Relating CT paraeters and rrhenius Paraeters. E E H RT U U condensed phases d ln dt RT PV internal energy of actiation olue of actiation T S H hc R RT pre-onential factor = biolecular reaction T S hc R E RT biolecular gas phase reaction V PV n RT RT RT H E RT ideal gases reaction olecularity =, condensed phases, uniolecular gas phase reactions =, biolecular gas phase reactions 4
15 5 RT E R S c h T R S c h T R E S ln T Pre-onential factor related to entropy of actiation (difference in entropy between reactants and actiated coplex R S c h T PZ collision theory positie P negatie S S P S P steric factor TS less ordered than reactants TS ore ordered than reactants S * lained in ters of changes in translational, rotational and ibrational degrees of freedo on going fro reactants to TS. = olecularity ln CT interpretation of rrhenius Equation.
SF Chemical Kinetics.
SF Chemical Kinetics. Lecture 5. Microscopic theory of chemical reaction kinetics. Microscopic theories of chemical reaction kinetics. basic aim is to calculate the rate constant for a chemical reaction
More informationKinetic Molecular Theory of Ideal Gases
Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions.
More informationKinetic Molecular Theory of. IGL is a purely empirical law - solely the
Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions.
More informationMolecular interactions in beams
Molecular interactions in beas notable advanceent in the experiental study of interolecular forces has coe fro the developent of olecular beas, which consist of a narrow bea of particles, all having the
More informationPHY 171. Lecture 14. (February 16, 2012)
PHY 171 Lecture 14 (February 16, 212) In the last lecture, we looked at a quantitative connection between acroscopic and icroscopic quantities by deriving an expression for pressure based on the assuptions
More informationKINETIC THEORY. Contents
KINETIC THEORY This brief paper on inetic theory deals with three topics: the hypotheses on which the theory is founded, the calculation of pressure and absolute teperature of an ideal gas and the principal
More informationKinetic Theory of Gases. Chapter 33 1/6/2017. Kinetic Theory of Gases
1/6/017 Kinetic Theory of Gases Kinetic Theory of Gases Chapter 33 Kinetic theory of gases envisions gases as a collection of atos or olecules in otion. Atos or olecules are considered as particles. This
More informationChemical Kinetics : time course of chemical transformation
Chemistry 3 Chapter 8, pp.339-43 Kotz, Chapter 5, pp.67-73. Lecture - Chemical Kinetics: Integrated rate equations Chemical Kinetics : time course of chemical transformation Thermodynamics tells us whether
More informationKinetic Theory of Gases: Elementary Ideas
Kinetic Theory of Gases: Eleentary Ideas 17th February 2010 1 Kinetic Theory: A Discussion Based on a Siplified iew of the Motion of Gases 1.1 Pressure: Consul Engel and Reid Ch. 33.1) for a discussion
More informationChapter 1 Introduction and Kinetics of Particles
Chapter 1 Introduction and Kinetics of Particles 1.1 Introduction There are two ain approaches in siulating the transport equations (heat, ass, and oentu), continuu and discrete. In continuu approach,
More informationLecture 2: Differential-Delay equations.
Lecture : Differential-Delay equations. D. Gurarie A differential equation, or syste:, ; of the syste:, 0 0 0 0 y f y t y t y, predicts a (near) future state 0 0 y t dt y f y t dt, fro its current state,
More informationKinetic Theory of Gases: Elementary Ideas
Kinetic Theory of Gases: Eleentary Ideas 9th February 011 1 Kinetic Theory: A Discussion Based on a Siplified iew of the Motion of Gases 1.1 Pressure: Consul Engel and Reid Ch. 33.1) for a discussion of
More informationHORIZONTAL MOTION WITH RESISTANCE
DOING PHYSICS WITH MATLAB MECHANICS HORIZONTAL MOTION WITH RESISTANCE Ian Cooper School of Physics, Uniersity of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS ec_fr_b. This script
More informationRecommended Reading. Entropy/Second law Thermodynamics
Lecture 7. Entropy and the second law of therodynaics. Recoended Reading Entropy/econd law herodynaics http://en wikipedia http://en.wikipedia.org/wiki/entropy http://2ndlaw.oxy.edu/index.htl. his site
More informationGAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME. PHYSICAL SCIENCE Grade 11 SESSION 11 (LEARNER NOTES)
PYSICAL SCIENCE Grade 11 SESSION 11 (LEARNER NOTES) MOLE CONCEPT, STOICIOMETRIC CALCULATIONS Learner Note: The ole concept is carried forward to calculations in the acid and base section, as well as in
More informationln P 1 saturation = T ln P 2 saturation = T
More Tutorial at www.littledubdoctor.co Physical Cheistry Answer each question in the space provided; use back of page if extra space is needed. Answer questions so the grader can READILY understand your
More information1 (40) Gravitational Systems Two heavy spherical (radius 0.05R) objects are located at fixed positions along
(40) Gravitational Systes Two heavy spherical (radius 0.05) objects are located at fixed positions along 2M 2M 0 an axis in space. The first ass is centered at r = 0 and has a ass of 2M. The second ass
More informationClassical systems in equilibrium
35 Classical systes in equilibriu Ideal gas Distinguishable particles Here we assue that every particle can be labeled by an index i... and distinguished fro any other particle by its label if not by any
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015
Lecture : Transition State Teory. tkins & DePaula: 7.6-7.7 University o Wasinton Departent o Ceistry Ceistry 453 Winter Quarter 05. ctivated Kinetics Kinetic rate uations are overned by several principles.
More informationChapter 4: Hypothesis of Diffusion-Limited Growth
Suary This section derives a useful equation to predict quantu dot size evolution under typical organoetallic synthesis conditions that are used to achieve narrow size distributions. Assuing diffusion-controlled
More informationm potential kinetic forms of energy.
Spring, Chapter : A. near the surface of the earth. The forces of gravity and an ideal spring are conservative forces. With only the forces of an ideal spring and gravity acting on a ass, energy F F will
More informationScattering and bound states
Chapter Scattering and bound states In this chapter we give a review of quantu-echanical scattering theory. We focus on the relation between the scattering aplitude of a potential and its bound states
More informationExample A1: Preparation of a Calibration Standard
Suary Goal A calibration standard is prepared fro a high purity etal (cadiu) with a concentration of ca.1000 g l -1. Measureent procedure The surface of the high purity etal is cleaned to reove any etal-oxide
More informationXI PHYSICS M. AFFAN KHAN LECTURER PHYSICS, AKHSS, K. https://promotephysics.wordpress.com
XI PHYSICS M. AFFAN KHAN LECTURER PHYSICS, AKHSS, K affan_414@live.co https://prootephysics.wordpress.co [MOTION] CHAPTER NO. 3 In this chapter we are going to discuss otion in one diension in which we
More informationNewton's Laws. Lecture 2 Key Concepts. Newtonian mechanics and relation to Kepler's laws The Virial Theorem Tidal forces Collision physics
Lecture 2 Key Concepts Newtonian echanics and relation to Kepler's laws The Virial Theore Tidal forces Collision physics Newton's Laws 1) An object at rest will reain at rest and an object in otion will
More informationKey Terms Electric Potential electrical potential energy per unit charge (JC -1 )
Chapter Seenteen: Electric Potential and Electric Energy Key Ter Electric Potential electrical potential energy per unit charge (JC -1 ) Page 1 of Electrical Potential Difference between two points is
More informationReading from Young & Freedman: For this topic, read the introduction to chapter 25 and sections 25.1 to 25.3 & 25.6.
PHY10 Electricity Topic 6 (Lectures 9 & 10) Electric Current and Resistance n this topic, we will cover: 1) Current in a conductor ) Resistivity 3) Resistance 4) Oh s Law 5) The Drude Model of conduction
More informationMolecular Speeds. Real Gasses. Ideal Gas Law. Reasonable. Why the breakdown? P-V Diagram. Using moles. Using molecules
Kinetic Theory of Gases Connect icroscopic properties (kinetic energy and oentu) of olecules to acroscopic state properties of a gas (teperature and pressure). P v v 3 3 3 But K v and P kt K v kt Teperature
More informationAn Approximate Model for the Theoretical Prediction of the Velocity Increase in the Intermediate Ballistics Period
An Approxiate Model for the Theoretical Prediction of the Velocity... 77 Central European Journal of Energetic Materials, 205, 2(), 77-88 ISSN 2353-843 An Approxiate Model for the Theoretical Prediction
More informationAll Excuses must be taken to 233 Loomis before 4:15, Monday, April 30.
Miscellaneous Notes he end is near don t get behind. All Excuses ust be taken to 233 Loois before 4:15, Monday, April 30. he PHYS 213 final exa ties are * 8-10 AM, Monday, May 7 * 8-10 AM, uesday, May
More informatione = n 1 ( ) 3 [ m 3] = n [ m 3] n
Magnetospheric Physics - Hoework Solutions, /7/4 7. Plasa definition Can a plasa be aintained at teperatures of T e K Hint: Calculate the density liit using the plasa paraeter and explain your result).
More information1 The properties of gases The perfect gas
1 The properties of gases 1A The perfect gas Answers to discussion questions 1A. The partial pressure of a gas in a ixture of gases is the pressure the gas would exert if it occupied alone the sae container
More informationNonuniqueness of canonical ensemble theory. arising from microcanonical basis
onuniueness of canonical enseble theory arising fro icrocanonical basis arxiv:uant-ph/99097 v2 25 Oct 2000 Suiyoshi Abe and A. K. Rajagopal 2 College of Science and Technology, ihon University, Funabashi,
More informationOptical Properties of Plasmas of High-Z Elements
Forschungszentru Karlsruhe Techni und Uwelt Wissenschaftlishe Berichte FZK Optical Properties of Plasas of High-Z Eleents V.Tolach 1, G.Miloshevsy 1, H.Würz Project Kernfusion 1 Heat and Mass Transfer
More informationLecture 6. Announcements. Conservation Laws: The Most Powerful Laws of Physics. Conservation Laws Why they are so powerful
Conseration Laws: The Most Powerful Laws of Physics Potential Energy gh Moentu p = + +. Energy E = PE + KE +. Kinetic Energy / Announceents Mon., Sept. : Second Law of Therodynaics Gie out Hoework 4 Wed.,
More informationLecture #8-3 Oscillations, Simple Harmonic Motion
Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life.
More informationOBJECTIVES INTRODUCTION
M7 Chapter 3 Section 1 OBJECTIVES Suarize data using easures of central tendency, such as the ean, edian, ode, and idrange. Describe data using the easures of variation, such as the range, variance, and
More informationForce and dynamics with a spring, analytic approach
Force and dynaics with a spring, analytic approach It ay strie you as strange that the first force we will discuss will be that of a spring. It is not one of the four Universal forces and we don t use
More informationQuestion 1. [14 Marks]
6 Question 1. [14 Marks] R r T! A string is attached to the dru (radius r) of a spool (radius R) as shown in side and end views here. (A spool is device for storing string, thread etc.) A tension T is
More informationPhysics 2107 Oscillations using Springs Experiment 2
PY07 Oscillations using Springs Experient Physics 07 Oscillations using Springs Experient Prelab Read the following bacground/setup and ensure you are failiar with the concepts and theory required for
More informationI affirm that I have never given nor received aid on this examination. I understand that cheating in the exam will result in a grade F for the class.
Che340 hysical Cheistry for Biocheists Exa 3 Apr 5, 0 Your Nae _ I affir that I have never given nor received aid on this exaination. I understand that cheating in the exa will result in a grade F for
More informationChem/Biochem 471 Exam 3 12/18/08 Page 1 of 7 Name:
Che/Bioche 47 Exa /8/08 Pae of 7 Please leave the exa paes stapled toether. The forulas are on a separate sheet. This exa has 5 questions. You ust answer at least 4 of the questions. You ay answer ore
More informationBALLISTIC PENDULUM. EXPERIMENT: Measuring the Projectile Speed Consider a steel ball of mass
BALLISTIC PENDULUM INTRODUCTION: In this experient you will use the principles of conservation of oentu and energy to deterine the speed of a horizontally projected ball and use this speed to predict the
More informationNational 5 Summary Notes
North Berwick High School Departent of Physics National 5 Suary Notes Unit 3 Energy National 5 Physics: Electricity and Energy 1 Throughout the Course, appropriate attention should be given to units, prefixes
More informationThe product of force and displacement ( in the direction of force ), during which the force is acting, is defined as work.
5 WORK, ENERGY ND POWER Page 5. Work The product of force and displaceent ( in the direction of force ), during which the force is acting, is defined as work. When N force is applied on a particle and
More informationChapter 2: Introduction to Damping in Free and Forced Vibrations
Chapter 2: Introduction to Daping in Free and Forced Vibrations This chapter ainly deals with the effect of daping in two conditions like free and forced excitation of echanical systes. Daping plays an
More informationBlock designs and statistics
Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent
More informationA Possible Solution to the Cosmological Constant Problem By Discrete Space-time Hypothesis
A Possible Solution to te Cosological Constant Proble By Discrete Space-tie Hypotesis H.M.Mok Radiation Healt Unit, 3/F., Saiwano Healt Centre, Hong Kong SAR Got, 28 Tai Hong St., Saiwano, Hong Kong, Cina.
More information2 Q 10. Likewise, in case of multiple particles, the corresponding density in 2 must be averaged over all
Lecture 6 Introduction to kinetic theory of plasa waves Introduction to kinetic theory So far we have been odeling plasa dynaics using fluid equations. The assuption has been that the pressure can be either
More informationCHAPTER 16 KINETICS: RATES AND MECHANISMS OF CHEMICAL REACTIONS
CHAPTER 6 KINETICS: RATES AND MECHANISMS OF CHEMICAL REACTIONS 6. Changes in concentrations of reactants (or products) as functions of tie are easured to deterine the reaction rate. 6. Rate is proportional
More informationImproved Hidden Clique Detection by Optimal Linear Fusion of Multiple Adjacency Matrices
Iproed Hidden Clique Detection by Optial Linear Fusion of Multiple Adjacency Matrices Hianshu Nayar, Benjain A. Miller, Kelly Geyer, Rajonda S. Caceres, Steen T. Sith, and Raj Rao Nadakuditi Departent
More informationAP Physics Thermodynamics Wrap-up
AP Physics herodynaics Wrap-up Here are your basic equations for therodynaics. here s a bunch of the. 3 his equation converts teperature fro Fahrenheit to Celsius. his is the rate of heat transfer for
More information13 Harmonic oscillator revisited: Dirac s approach and introduction to Second Quantization
3 Haronic oscillator revisited: Dirac s approach and introduction to Second Quantization. Dirac cae up with a ore elegant way to solve the haronic oscillator proble. We will now study this approach. The
More informationChemistry 432 Problem Set 11 Spring 2018 Solutions
1. Show that for an ideal gas Cheistry 432 Proble Set 11 Spring 2018 Solutions P V 2 3 < KE > where is the average kinetic energy of the gas olecules. P 1 3 ρ v2 KE 1 2 v2 ρ N V P V 1 3 N v2 2 3 N
More informationProblem Set 2. Chapter 1 Numerical:
Chapter 1 Nuerical: roble Set 16. The atoic radius of xenon is 18 p. Is that consistent with its b paraeter of 5.15 1 - L/ol? Hint: what is the volue of a ole of xenon atos and how does that copare to
More informationA simple phenomenologic model for particle transport in spaceperiodic potentials in underdamped systems
A siple phenoenologic odel for particle transport in spaceperiodic potentials in underdaped systes IG MARCHENKO 1,(a,b), II MARCHENKO 3, A ZHIGLO 1 1 NSC Kharov Institute of Physics and Technology, Aadeichesaya
More informationA nonstandard cubic equation
MATH-Jan-05-0 A nonstandard cubic euation J S Markoitch PO Box West Brattleboro, VT 050 Dated: January, 05 A nonstandard cubic euation is shown to hae an unusually econoical solution, this solution incorporates
More informationSpine Fin Efficiency A Three Sided Pyramidal Fin of Equilateral Triangular Cross-Sectional Area
Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 18-0, 006 (pp13-18) Spine Fin Efficiency A Three Sided Pyraidal Fin of Equilateral Triangular
More informationMomentum. February 15, Table of Contents. Momentum Defined. Momentum Defined. p =mv. SI Unit for Momentum. Momentum is a Vector Quantity.
Table of Contents Click on the topic to go to that section Moentu Ipulse-Moentu Equation The Moentu of a Syste of Objects Conservation of Moentu Types of Collisions Collisions in Two Diensions Moentu Return
More information3 Thermodynamics and Statistical mechanics
Therodynaics and Statistical echanics. Syste and environent The syste is soe ortion of atter that we searate using real walls or only in our ine, fro the other art of the universe. Everything outside the
More informationUSEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS. By: Ian Blokland, Augustana Campus, University of Alberta
1 USEFUL HINTS FOR SOLVING PHYSICS OLYMPIAD PROBLEMS By: Ian Bloland, Augustana Capus, University of Alberta For: Physics Olypiad Weeend, April 6, 008, UofA Introduction: Physicists often attept to solve
More informationChapter 1: Basics of Vibrations for Simple Mechanical Systems
Chapter 1: Basics of Vibrations for Siple Mechanical Systes Introduction: The fundaentals of Sound and Vibrations are part of the broader field of echanics, with strong connections to classical echanics,
More informationDeveloped Correlations for Prediction of The Enthalpies of Saturated Vapor Liquid Coexisting Phases
Nahrain University, College of Engineering Journal (NUCEJ) Vol.13 No.2, 2010 pp.116-128 Developed Correlations for Prediction of he Enthalpies of Saturated Vapor Liquid Coexisting Phases Mahoud Oar bdullah
More informationpoints Points <40. Results of. Final Exam. Grade C D,F C B
Results of inal Exa 5 6 7 8 9 points Grade C D, Points A 9- + 85-89 7-8 C + 6-69 -59 < # of students Proble (che. equilibriu) Consider the following reaction: CO(g) + H O(g) CO (g) + H (g) In equilibriu
More informationThe Characteristic Planet
The Characteristic Planet Brano Zivla, bzivla@gail.co Abstract: I have calculated a relation significant for planets fro a logical starting point that a whole and its parts are ianently depandant on each
More informationDaniel López Gaxiola 1 Student View Jason M. Keith
Suppleental Material for Transport Process and Separation Process Principles Chapter Principles of Moentu Transfer and Overall Balances In fuel cells, the fuel is usually in gas or liquid phase. Thus,
More informationPeriodic Motion is everywhere
Lecture 19 Goals: Chapter 14 Interrelate the physics and atheatics of oscillations. Draw and interpret oscillatory graphs. Learn the concepts of phase and phase constant. Understand and use energy conservation
More informationTitle. Author(s)Izumida, Yuki; Okuda, Koji. CitationPhysical review E, 80(2): Issue Date Doc URL. Rights. Type.
Title Onsager coefficients of a finite-tie Carnot cycle Author(s)Izuida, Yuki; Okuda, Koji CitationPhysical review E, 80(2): 021121 Issue Date 2009-08 Doc URL http://hdl.handle.net/2115/39348 Rights 2009
More informationPhysics 11 HW #7 Solutions
hysics HW #7 Solutions Chapter 7: Focus On Concepts: 2, 6, 0, 3 robles: 8, 7, 2, 22, 32, 53, 56, 57 Focus On Concepts 7-2 (d) Moentu is a ector quantity that has a agnitude and a direction. The agnitudes
More information8.012 Physics I: Classical Mechanics Fall 2008
MIT OpenCourseWare http://ocw.it.edu 8.012 Physics I: Classical Mechanics Fall 2008 For inforation about citing these aterials or our Ters of Use, isit: http://ocw.it.edu/ters. MASSACHUSETTS INSTITUTE
More informationPhysics 207 Lecture 18. Physics 207, Lecture 18, Nov. 3 Goals: Chapter 14
Physics 07, Lecture 18, Nov. 3 Goals: Chapter 14 Interrelate the physics and atheatics of oscillations. Draw and interpret oscillatory graphs. Learn the concepts of phase and phase constant. Understand
More informationGeneral Physical Chemistry I
General Physical Cheistry I Lecture 12 Aleksey Kocherzhenko Aril 2, 2015" Last tie " Gibbs free energy" In order to analyze the sontaneity of cheical reactions, we need to calculate the entroy changes
More information(a) Why cannot the Carnot cycle be applied in the real world? Because it would have to run infinitely slowly, which is not useful.
PHSX 446 FINAL EXAM Spring 25 First, soe basic knowledge questions You need not show work here; just give the answer More than one answer ight apply Don t waste tie transcribing answers; just write on
More informationThermodynamics. Temperature Scales Fahrenheit: t F. Thermal Expansion and Strss. Temperature and Thermal Equilibrium
herodynaics Fro the Greek theros eaning heat and dynais eaning power is a branch of physics that studies the effects of changes in teperature, pressure, and volue on physical systes at the acroscopic scale
More informationME 300 Thermodynamics II Exam 2 November 13, :00 p.m. 9:00 p.m.
ME 300 Therodynaics II Exa 2 Noveber 3, 202 8:00 p.. 9:00 p.. Nae: Solution Section (Circle One): Sojka Naik :30 a.. :30 p.. Instructions: This is a closed book/notes exa. You ay use a calculator. You
More informationFirst of all, because the base kets evolve according to the "wrong sign" Schrödinger equation (see pp ),
HW7.nb HW #7. Free particle path integral a) Propagator To siplify the notation, we write t t t, x x x and work in D. Since x i, p j i i j, we can just construct the 3D solution. First of all, because
More information1. (2.5.1) So, the number of moles, n, contained in a sample of any substance is equal N n, (2.5.2)
Lecture.5. Ideal gas law We have already discussed general rinciles of classical therodynaics. Classical therodynaics is a acroscoic science which describes hysical systes by eans of acroscoic variables,
More informationPHYS 1101 Practice problem set 5, Chapter 18: 4, 9, 15, 23, 27, 32, 40, 43, 55, 56, 59 1 = = = Nk T Nk T Nk T B 1 B 2 B 1
PHYS 0 Practice roble set, Chater 8: 4, 9,,, 7,, 40, 4,, 6, 9 8.4. Sole: (a he ean free ath of a olecule in a gas at teerature, olue V, and ressure is λ 00 n. We also know that λ λ V 4 π ( N V r Although,
More informationMeasuring Temperature with a Silicon Diode
Measuring Teperature with a Silicon Diode Due to the high sensitivity, nearly linear response, and easy availability, we will use a 1N4148 diode for the teperature transducer in our easureents 10 Analysis
More informationCurrent, Resistance Electric current and current density
General Physics Current, Resistance We will now look at the situation where charges are in otion - electrodynaics. The ajor difference between the static and dynaic cases is that E = 0 inside conductors
More information12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015
18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring 2015 12 Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tie-independent) elastic systes.
More informationProblem Set 14: Oscillations AP Physics C Supplementary Problems
Proble Set 14: Oscillations AP Physics C Suppleentary Probles 1 An oscillator consists of a bloc of ass 050 g connected to a spring When set into oscillation with aplitude 35 c, it is observed to repeat
More informationOne Dimensional Collisions
One Diensional Collisions These notes will discuss a few different cases of collisions in one diension, arying the relatie ass of the objects and considering particular cases of who s oing. Along the way,
More information2.003 Engineering Dynamics Problem Set 2 Solutions
.003 Engineering Dynaics Proble Set Solutions This proble set is priarily eant to give the student practice in describing otion. This is the subject of kineatics. It is strongly recoended that you study
More information9 HOOKE S LAW AND SIMPLE HARMONIC MOTION
Experient 9 HOOKE S LAW AND SIMPLE HARMONIC MOTION Objectives 1. Verify Hoo s law,. Measure the force constant of a spring, and 3. Measure the period of oscillation of a spring-ass syste and copare it
More informationSupporting information for Self-assembly of multicomponent structures in and out of equilibrium
Supporting inforation for Self-assebly of ulticoponent structures in and out of equilibriu Stephen Whitela 1, Rebecca Schulan 2, Lester Hedges 1 1 Molecular Foundry, Lawrence Berkeley National Laboratory,
More information5.60 Thermodynamics & Kinetics Spring 2008
MIT OpenCourseWare http://ocw.it.edu 5.60 Therodynaics & Kinetics Spring 2008 For inforation about citing these aterials or our Ters of Use, visit: http://ocw.it.edu/ters. 1 Enzye Catalysis Readings: SAB,
More informationIncluded in this hand-out are five examples of problems requiring the solution of a system of linear algebraic equations.
he Lecture Notes, Dept. of heical Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updated /) Eaple pplications of systes of linear equations Included in this hand-out are five eaples of probles
More informationPh 20.3 Numerical Solution of Ordinary Differential Equations
Ph 20.3 Nuerical Solution of Ordinary Differential Equations Due: Week 5 -v20170314- This Assignent So far, your assignents have tried to failiarize you with the hardware and software in the Physics Coputing
More informationCharacteristics of Low-Temperature Plasmas Under Nonthermal Conditions A Short Summary
1 1 Characteristics of Low-Teperature Plasas Under Nontheral Conditions A Short Suary Alfred Rutscher 1.1 Introduction The concept of a plasa dates back to Languir (1928) and originates fro the fundaental
More informationincreases. In part (b) the impulse and initial momentum are in opposite directions and the velocity decreases.
8IDENTIFY and SET U: p = K = EXECUTE: (a) 5 p = (, kg)( /s) = kg /s 5 p kg /s (b) (i) = = = 6 /s (ii) kg =, so T T SUV SUV, kg ( /s) 68 /s T SUV = T = = SUV kg EVALUATE:The SUV ust hae less speed to hae
More informationSUPERPOSITION AND STANDING WAVES 16
SUPERPOSITION AND STANDING WAVES 6 Q6.. Reason: Where there is a change in ediu in particular a change in the wae speed then reflection can occur. Assess: Light traels at different speeds in water and
More informationCrystallization of Supercooled Liquid Elements Induced by Superclusters Containing Magic Atom Numbers Abstract: Keywords: 1.
Crystallization of Supercooled Liquid Eleents Induced by Superclusters Containing Magic Ato Nubers Robert F. Tournier, CRETA /CNRS, Université Joseph Fourier, B.P. 166, 804 Grenoble cedex 09, France. E-ail:
More informationThermodynamics. Temperature Scales Fahrenheit: t F. Thermal Expansion and Stress. Temperature and Thermal Equilibrium
herodynaics Fro the Greek theros eaning heat and dynais eaning power is a branch of physics that studies the effects of changes in teperature, pressure, and volue on physical systes at the acroscopic scale
More informationPhys463.nb. Many electrons in 1D at T = 0. For a large system (L ), ΕF =? (6.7) The solutions of this equation are plane waves (6.
â â x Ψn Hx Ε Ψn Hx 35 (6.7) he solutions of this equation are plane waves Ψn Hx A exphä n x (6.8) he eigen-energy Εn is n (6.9) Εn For a D syste with length and periodic boundary conditions, Ψn Hx Ψn
More informationI. Concepts and Definitions. I. Concepts and Definitions
F. Properties of a syste (we use the to calculate changes in energy) 1. A property is a characteristic of a syste that can be given a nuerical value without considering the history of the syste. Exaples
More informationarxiv:cond-mat/ v1 [cond-mat.stat-mech] 11 Jan 2007
Transport and Helfand oents in the Lennard-Jones fluid. II. Theral conductivity arxiv:cond-at/7125v1 [cond-at.stat-ech] 11 Jan 27 S. Viscardy, J. Servantie, and P. Gaspard Center for Nonlinear Phenoena
More informationElastic Force: A Force Balance: Elastic & Gravitational Force: Force Example: Determining Spring Constant. Some Other Forces
Energy Balance, Units & Proble Solving: Mechanical Energy Balance ABET Course Outcoes: 1. solve and docuent the solution of probles involving eleents or configurations not previously encountered (e) (e.g.
More informationToday s s topics are: Collisions and Momentum Conservation. Momentum Conservation
Today s s topics are: Collisions and P (&E) Conservation Ipulsive Force Energy Conservation How can we treat such an ipulsive force? Energy Conservation Ipulsive Force and Ipulse [Exaple] an ipulsive force
More informationSPH4U. Conservation of Energy. Review: Springs. More Spring Review. 1-D Variable Force Example: Spring. Page 1. For a spring we recall that F x = -kx.
-D Variable Force Exaple: Spring SPH4U Conseration of Energ For a spring we recall that F x = -kx. F(x) x x x relaxe position -kx F = - k x the ass F = - k x Reiew: Springs Hooke s Law: The force exerte
More information