Phys. 344 Ch 7 Lecture 8 Fri., April. 10 th,
|
|
- Clemence Hawkins
- 5 years ago
- Views:
Transcription
1 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, 009 Fri. 4/0 8. Ising Mdel f Ferrmagnets HW30 66, 74 Mn. 4/3 Review Sat. 4/8 3pm Exam 3 HW Mnday: Review fr est 3. See n-line practice test lecture-prep is t bring in questins 7.6 Bse-Einstein Cndensatin / Bsns with µ 0 his is kind f tricky stuff. Cnceptually: it appeals t the heart f the difference between distinguishable and indistinguishable particles and what we mean by temperature. Quantitatively: he math desn t let us cme up with a single, perfect mdel. S, we ll talk sme abut the cncepts and then we ll g abut building sme mathematical tls that d what we need. If yu feel like the mathematical mdel is kind f cbbled-tgether, yu re abslutely right. It isn t perfect; ur gal is just t see that the qualitative behavir we expect is in there Why Des it Happen Okay, what s special abut Bsns is that a) their indistinguishable and b) they can ccupy the same single-particle state as each ther. 0 S, bviusly, if yu take away all the energy yu can, every particle will happily chabitate in the single-particle grund state they have cmpletely degenerated, cndensed int a single state. Lw As yu add energy / raise the temperature, sme f the particles will rise t lwlying energy states, but there will still be a large ppulatin in the grund state, a large ppulatin in the cndensate, fr higher temperatures than ne wuld classically expect. In pint f fact, there will always be sme particles sharing the grund state; hwever, as increases, it becmes an insignificant fractin f the ppulatin. First, why des the grund-state ppulatin eventually becme insignificant? hink f the distributin f particles in terms f energy: he average ccupancy f a particular state depends n the energy/k f that state: n. hat bviusly tells us that a given high energy state is ( µ ) e less ppulated than a given lw energy state. Meanwhile, the average number f particles with a given energy als depends n hw many states have the same energy, i.e., the density f states: nw / g( ) d g d. his f curse grws with energy at higher energies, there are mre states with the same energy. S the prduct f these tw determines hw many particles have a given energy. S, while the grund state will
2 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, 009 always be the mst ppular single state, the mst ppular energy level will g be where ever g( ) n is peaked, s the grund state quickly ( µ ) e becmes nt s ppular an energy level, and the cndensate becmes insignificant. dg( ) n d he peak shuld ccur at g g e ( µ ) 0 ( µ ) µ e ( ) ( e ) ( µ ) ( ) e ) ( µ ) 0 ( e ) ( µ ) ( e ) hat said, why des the grund-state remain significant fr lw-ish temperatures? Let s think abut the ther tw kinds f particles fr cmparisn. Fermins Okay, these can t have multiply ccupied states, s the lwest energy / zer-temperature cnfiguratin fr a system f them is simply the first single-particle states being full. Distinguishable Particles vs. Bsns Like indistinguishable bsns, the lwest energy cnfiguratin f the system has all f the particles being in the lwest energy level the difference is that each particle is distinguishable, s while they all have the same energy, they are in distinct states. Anther difference is that this situatin mre rapidly fades int bscurity as temperature rises. Here s why: their distinguishability means that there are a lt mre unique states available when yu add just a little energy: will particle Bb be, r will Alice, r will Carl, r will Dug, the mst ppular energy level (where ( µ ) g( ) n g e peaks) shifts up higher faster with increasing temperature it des fr is Bsns. get deeper int this, we need t recall just what temperature means. If yu re in the habit f directly assciating temperature with average energy, this may S be a little hard t swallw, but remember: S U the temperature f a system depends nt just n hw much energy yu put in it, but als hw much disrder it induces (quantified in the entrpy).
3 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, Distinguishable system. Imagine yu have an -particle system f distinguishable particles and anther -particle system f bsns. If yu add, say units f energy t the distinguishable system, then there are + ( + )! ( + ) ΩDist(, q ) ( )!. S, the temperature assciated with adding ne unit is rughly U q dist S k ln (( + ) / ) Als nte that f all these pssibilities, f them have - particles remaining in the grund state;the remaining pssibilities have nly - particles remaining in the grund state. Put anther way, the prbability f having - particles in the grund state is Ω Pr( ) Ω ( + ) / ( + ) ttal Bsn System. In cntrast, if yu add tw units f energy t a Bsn system, there are nly tw ways they can be distributed: all t ne particle, r ne t ne, and ne t anther. ΩBse (, q ). S the temperature assciated with adding ne unit is rughly U q Bse which is a significantly S k ln higher temperature! Als, even at this higher temperature, the prbability f having - particles (rather than -) in the grund state is ½ - far larger than fr distinguishable particles. Cnclusin. It takes a higher temperature t frce the same amunt f energy int a Bse system, and even then, there s a greater ppulatin in the grund state. Density f states effect. Smething that isn t addressed in this simple illustratin is a higher energy level generally has a greater degeneracy, but this effects distinguishable particles and bsns equally. Pulling back a bit and summarizing: here s the cmpetitin between the prbability f being in a
4 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, Okay, let s get quantitative particular state (higher prbability fr lwer energy states) and the number f states with a particular energy. he higher the energy, the less prbable a particular state, but the mre individual states available. Fr Distinguishable particle, nt nly are there mre states available fr individual particles at higher energies, but there are even mre ways f chsing which particles will be in which f the ccupied states. Fr distinguishable particles then, it is preferable t have E energy accunted fr by having particles in different, medium energy states. Fr Bsns n the ther hand, yu re a little mre likely t have E energy accunted fr by having fewer particles in higher energy states leaving mre particle back in the grund state. 3 Regimes: 0, very lw/mderate, high (classical) We re interested in what fractin f the particles in ur system are in the grund state. In the Cndensatin regime, it s startlingly large. Grund State he average ccupancy f the grund state is n ( µ ). Since e there is nly ne grund state (degeneracy f ), the number f particles with this lwest energy is simply n ( µ ). e (Prep fr Pr. 66) Questin: Fr that matter, what wuld be the average ccupancy f ne f the st states? n ( µ ) e Fr a spin0 particle, hw many st states are there? lk in p(r n) space. 3. S the number in the first energy level is 3 3n ( µ ) e n µ What s m during cndensatin, i.e. Lw? We see it in n ( µ ). e At lw, we knw that where is generally n rder f 0 3. Lking at ( µ ), can get quite large nly if e ( µ ) the denminatr gets quite small, i.e., e appraches. S, ( µ ) expanding e ( µ ) arund gives e + ( µ ). te: at first blush, yu might think what the heck are yu ding with a aylr series, is huge, but fr physical
5 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, reasns we knw that the expnent must be tiny fr t be huge, s we can find what µ des that fr us. k + µ µ k (Prep fr Pr. 66) S, µ w, as drps and grws, µ clearly appraches frm belw. S it appraches the grund state energy frm belw. S, what s? Energy Fr an rder f magnitude calculatin, imagine a simple cube f h dimensins L. he smallest mmentum available is p x, L ditt fr the y and z cmpnents f mmentum. S the smallest h h h 3 energy available is h m + +. L L L 8mL 3 34 ( Js) his is n rder f 7 8 ( kg) J, m really small! A crrespnding temperature wuld be 0-8 K! te: fr a spin 0 particle, there is nly ne grund state. (Prep fr Pr. 66) Questin: Fr that matter, what wuld be the energy f the st state? 6 h h h h m L L L 8mL k Since is quite small, µ means that µ is negative fr elevated temperatures // when is small. Only when temperature drps and the ppulatin f the grund state grws appreciably will µ apprach the very small psitive value f. Cndensatin, dependence n. S, arund what temperature d the particles begin cndensing int the lwest energy state? ns + + sµ µ s µ e e e states states states Prep. Fr HW 74: Yu ll use Excel t d this sum fr the first ~ 00 terms. te that the degeneracy, n and energy structure are bth given in the previus prblem. w / Here, I ve explicitly separated ff the term fr the grund state ppulatin frm all the ther terms fr the states.
6 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, dnw/ µ n s. w/ s µ µ e e d e states his last step f curse is in preparatin fr replacing the sum with an integral, but we can nly get away with that if <<, r in ther wrds << k. he energy steps between states are n rder f which is pretty darned small, s we can get away with this fr mst temperatures. dns. w / g ( ) d µ µ d e e 3/ πm nspin d µ π h e Lk at the cmpetitin in the integrand. vs.. µ e While the lw energy states are by far the mst ppular (accrding t the secnd factr particles really want t be in thse states), they are als the fewest (accrding t the first factr there just aren t that many f these lw energy states), in the end, states with energy much less than k dn t really cntribute t the sum. hat s gd because, it allws us t play a little lse with the lwer end f the integratin withut significantly affecting the result. he tw apprximatins are that and µ are much smaller than the energy where the integrand becmes significant, i.e.,, µ 0. n n spin spin π πm h 3/ πmk π h 0 3/ 3/ 0 πmk πmk nspin.35 nspin.6 π h h 3 3 he integral was Γ ζ.6... Limit f Apprximatin: As we ve already seen at very lw temperatures, µ grws increasingly negative with increasing. S the apprximatin µ 0 is nly gd fr small and mderate temperatures. At high temperatures we can t neglect the chemical ptential. hreshld: Clearly, the apprximatin breaks dwn by the time it predicts that the number f particles in states exceeds the ttal number f particles in the system. S we can define a threshld by e ( ) d x dx x e 3/
7 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, / / m k h n v h mk n c spin Q c spin π π te: this is rughly when the vlume per particle is the quantum vlume when wavefunctins must just verlap. his can be rephrased in terms f the grund state energy: 3/. 0 k c, s it can be cnsiderably mre than the grund state energy (which itself is quite small and is abut between states). With this definitin in hand, we can re-write the number f particles as C 3/ Rughly speaking, the ppulatin f states ges like: Determine m frm cndensate ppulatin: Fr that matter, fr these mderate temperatures, / / / 3 / C k k C e e µ µ + 3/ ln C k µ c
8 Phys. 344 Ch 7 Lecture 8 Fri., April. 0 th, Assuming that is quite large, k µ 3/ C his has the same lw behavir as we d already predicted. Clearly, this gets ill-behaved as appraches c. m at > c Abve c, he ppulatin f the grund state is negligible, s 3/ πm nspin d µ π h is the e 0 defining equatin fr µ. One can use this t verify the basic frm in Figure Real Wrld Examples It s readily bserved in liquid 4 He. It takes sme ding, but it s als bserved in Rubidium gas.
Prof. Dr. I. Nasser Phys530, T142 3-Oct-17 Fermi_gases. 0 f e. and fall off exponentially like Maxwell-Boltzmaan distribution.
Pr. Dr. I. Nasser Phys, -Oct-7 FERMI_DIRAC GASSES Fermins: Are particles hal-integer spin that bey Fermi-Dirac statistics. Fermins bey the Pauli exclusin principle, which prhibits the ccupancy an available
More informationThermodynamics Partial Outline of Topics
Thermdynamics Partial Outline f Tpics I. The secnd law f thermdynamics addresses the issue f spntaneity and invlves a functin called entrpy (S): If a prcess is spntaneus, then Suniverse > 0 (2 nd Law!)
More informationAP Physics Kinematic Wrap Up
AP Physics Kinematic Wrap Up S what d yu need t knw abut this mtin in tw-dimensin stuff t get a gd scre n the ld AP Physics Test? First ff, here are the equatins that yu ll have t wrk with: v v at x x
More information, which yields. where z1. and z2
The Gaussian r Nrmal PDF, Page 1 The Gaussian r Nrmal Prbability Density Functin Authr: Jhn M Cimbala, Penn State University Latest revisin: 11 September 13 The Gaussian r Nrmal Prbability Density Functin
More informationThermodynamics and Equilibrium
Thermdynamics and Equilibrium Thermdynamics Thermdynamics is the study f the relatinship between heat and ther frms f energy in a chemical r physical prcess. We intrduced the thermdynamic prperty f enthalpy,
More informationLCAO APPROXIMATIONS OF ORGANIC Pi MO SYSTEMS The allyl system (cation, anion or radical).
Principles f Organic Chemistry lecture 5, page LCAO APPROIMATIONS OF ORGANIC Pi MO SYSTEMS The allyl system (catin, anin r radical).. Draw mlecule and set up determinant. 2 3 0 3 C C 2 = 0 C 2 3 0 = -
More information37 Maxwell s Equations
37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut
More information20 Faraday s Law and Maxwell s Extension to Ampere s Law
Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the case f a charged particle that is ming in the icinity f a ming bar magnet
More informationLecture 24: Flory-Huggins Theory
Lecture 24: 12.07.05 Flry-Huggins Thery Tday: LAST TIME...2 Lattice Mdels f Slutins...2 ENTROPY OF MIXING IN THE FLORY-HUGGINS MODEL...3 CONFIGURATIONS OF A SINGLE CHAIN...3 COUNTING CONFIGURATIONS FOR
More informationDispersion Ref Feynman Vol-I, Ch-31
Dispersin Ref Feynman Vl-I, Ch-31 n () = 1 + q N q /m 2 2 2 0 i ( b/m) We have learned that the index f refractin is nt just a simple number, but a quantity that varies with the frequency f the light.
More informationWe can see from the graph above that the intersection is, i.e., [ ).
MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Functin Arithmetic: With nt t much difficulty, we ntice that inputs f functins are numbers, and utputs f functins are numbers. S whatever we can d with
More informationMedium Scale Integrated (MSI) devices [Sections 2.9 and 2.10]
EECS 270, Winter 2017, Lecture 3 Page 1 f 6 Medium Scale Integrated (MSI) devices [Sectins 2.9 and 2.10] As we ve seen, it s smetimes nt reasnable t d all the design wrk at the gate-level smetimes we just
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More information**DO NOT ONLY RELY ON THIS STUDY GUIDE!!!**
Tpics lists: UV-Vis Absrbance Spectrscpy Lab & ChemActivity 3-6 (nly thrugh 4) I. UV-Vis Absrbance Spectrscpy Lab Beer s law Relates cncentratin f a chemical species in a slutin and the absrbance f that
More informationALE 21. Gibbs Free Energy. At what temperature does the spontaneity of a reaction change?
Name Chem 163 Sectin: Team Number: ALE 21. Gibbs Free Energy (Reference: 20.3 Silberberg 5 th editin) At what temperature des the spntaneity f a reactin change? The Mdel: The Definitin f Free Energy S
More informationPhysics 2B Chapter 23 Notes - Faraday s Law & Inductors Spring 2018
Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and
More informationLecture 17: Free Energy of Multi-phase Solutions at Equilibrium
Lecture 17: 11.07.05 Free Energy f Multi-phase Slutins at Equilibrium Tday: LAST TIME...2 FREE ENERGY DIAGRAMS OF MULTI-PHASE SOLUTIONS 1...3 The cmmn tangent cnstructin and the lever rule...3 Practical
More informationCHAPTER Read Chapter 17, sections 1,2,3. End of Chapter problems: 25
CHAPTER 17 1. Read Chapter 17, sectins 1,2,3. End f Chapter prblems: 25 2. Suppse yu are playing a game that uses tw dice. If yu cunt the dts n the dice, yu culd have anywhere frm 2 t 12. The ways f prducing
More information5 th grade Common Core Standards
5 th grade Cmmn Cre Standards In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin
More informationThis section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving.
Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus
More information1 The limitations of Hartree Fock approximation
Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants
More informationHow do scientists measure trees? What is DBH?
Hw d scientists measure trees? What is DBH? Purpse Students develp an understanding f tree size and hw scientists measure trees. Students bserve and measure tree ckies and explre the relatinship between
More information(2) Even if such a value of k was possible, the neutrons multiply
CHANGE OF REACTOR Nuclear Thery - Curse 227 POWER WTH REACTVTY CHANGE n this lessn, we will cnsider hw neutrn density, neutrn flux and reactr pwer change when the multiplicatin factr, k, r the reactivity,
More informationCompressibility Effects
Definitin f Cmpressibility All real substances are cmpressible t sme greater r lesser extent; that is, when yu squeeze r press n them, their density will change The amunt by which a substance can be cmpressed
More informationThe standards are taught in the following sequence.
B L U E V A L L E Y D I S T R I C T C U R R I C U L U M MATHEMATICS Third Grade In grade 3, instructinal time shuld fcus n fur critical areas: (1) develping understanding f multiplicatin and divisin and
More informationLecture 23: Lattice Models of Materials; Modeling Polymer Solutions
Lecture 23: 12.05.05 Lattice Mdels f Materials; Mdeling Plymer Slutins Tday: LAST TIME...2 The Bltzmann Factr and Partitin Functin: systems at cnstant temperature...2 A better mdel: The Debye slid...3
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationAP CHEMISTRY CHAPTER 6 NOTES THERMOCHEMISTRY
AP CHEMISTRY CHAPTER 6 NOTES THERMOCHEMISTRY Energy- the capacity t d wrk r t prduce heat 1 st Law f Thermdynamics: Law f Cnservatin f Energy- energy can be cnverted frm ne frm t anther but it can be neither
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical
More informationIntroduction to Spacetime Geometry
Intrductin t Spacetime Gemetry Let s start with a review f a basic feature f Euclidean gemetry, the Pythagrean therem. In a twdimensinal crdinate system we can relate the length f a line segment t the
More informationMath 105: Review for Exam I - Solutions
1. Let f(x) = 3 + x + 5. Math 105: Review fr Exam I - Slutins (a) What is the natural dmain f f? [ 5, ), which means all reals greater than r equal t 5 (b) What is the range f f? [3, ), which means all
More informationSolutions to the Extra Problems for Chapter 14
Slutins t the Extra Prblems r Chapter 1 1. The H -670. T use bnd energies, we have t igure ut what bnds are being brken and what bnds are being made, s we need t make Lewis structures r everything: + +
More informationWhat is Statistical Learning?
What is Statistical Learning? Sales 5 10 15 20 25 Sales 5 10 15 20 25 Sales 5 10 15 20 25 0 50 100 200 300 TV 0 10 20 30 40 50 Radi 0 20 40 60 80 100 Newspaper Shwn are Sales vs TV, Radi and Newspaper,
More informationLecture 02 CSE 40547/60547 Computing at the Nanoscale
PN Junctin Ntes: Lecture 02 CSE 40547/60547 Cmputing at the Nanscale Letʼs start with a (very) shrt review f semi-cnducting materials: - N-type material: Obtained by adding impurity with 5 valence elements
More informationPHYS 314 HOMEWORK #3
PHYS 34 HOMEWORK #3 Due : 8 Feb. 07. A unifrm chain f mass M, lenth L and density λ (measured in k/m) hans s that its bttm link is just tuchin a scale. The chain is drpped frm rest nt the scale. What des
More informationChapters 29 and 35 Thermochemistry and Chemical Thermodynamics
Chapters 9 and 35 Thermchemistry and Chemical Thermdynamics 1 Cpyright (c) 011 by Michael A. Janusa, PhD. All rights reserved. Thermchemistry Thermchemistry is the study f the energy effects that accmpany
More informationGeneral Chemistry II, Unit I: Study Guide (part I)
1 General Chemistry II, Unit I: Study Guide (part I) CDS Chapter 14: Physical Prperties f Gases Observatin 1: Pressure- Vlume Measurements n Gases The spring f air is measured as pressure, defined as the
More informationCHAPTER 24: INFERENCE IN REGRESSION. Chapter 24: Make inferences about the population from which the sample data came.
MATH 1342 Ch. 24 April 25 and 27, 2013 Page 1 f 5 CHAPTER 24: INFERENCE IN REGRESSION Chapters 4 and 5: Relatinships between tw quantitative variables. Be able t Make a graph (scatterplt) Summarize the
More informationLab 11 LRC Circuits, Damped Forced Harmonic Motion
Physics 6 ab ab 11 ircuits, Damped Frced Harmnic Mtin What Yu Need T Knw: The Physics OK this is basically a recap f what yu ve dne s far with circuits and circuits. Nw we get t put everything tgether
More informationChemistry 20 Lesson 11 Electronegativity, Polarity and Shapes
Chemistry 20 Lessn 11 Electrnegativity, Plarity and Shapes In ur previus wrk we learned why atms frm cvalent bnds and hw t draw the resulting rganizatin f atms. In this lessn we will learn (a) hw the cmbinatin
More informationCHM112 Lab Graphing with Excel Grading Rubric
Name CHM112 Lab Graphing with Excel Grading Rubric Criteria Pints pssible Pints earned Graphs crrectly pltted and adhere t all guidelines (including descriptive title, prperly frmatted axes, trendline
More informationExperiment #3. Graphing with Excel
Experiment #3. Graphing with Excel Study the "Graphing with Excel" instructins that have been prvided. Additinal help with learning t use Excel can be fund n several web sites, including http://www.ncsu.edu/labwrite/res/gt/gt-
More informationCAUSAL INFERENCE. Technical Track Session I. Phillippe Leite. The World Bank
CAUSAL INFERENCE Technical Track Sessin I Phillippe Leite The Wrld Bank These slides were develped by Christel Vermeersch and mdified by Phillippe Leite fr the purpse f this wrkshp Plicy questins are causal
More informationPart One: Heat Changes and Thermochemistry. This aspect of Thermodynamics was dealt with in Chapter 6. (Review)
CHAPTER 18: THERMODYNAMICS AND EQUILIBRIUM Part One: Heat Changes and Thermchemistry This aspect f Thermdynamics was dealt with in Chapter 6. (Review) A. Statement f First Law. (Sectin 18.1) 1. U ttal
More informationOur Lady Star of the Sea Religious Education CIRCLE OF GRACE LESSON PLAN - Grade 1
Our Lady Star f the Sea Religius Educatin CIRCLE OF GRACE LESSON PLAN - Grade 1 Opening Prayer: (ech prayer) Hly Spirit (ech) Shw us the way (ech) Be with us in all we think.. d and say (ech) Amen GETTING
More informationPhysics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1
Physics 1 Lecture 1 Tday's Cncept: Magnetic Frce n mving charges F qv Physics 1 Lecture 1, Slide 1 Music Wh is the Artist? A) The Meters ) The Neville rthers C) Trmbne Shrty D) Michael Franti E) Radiatrs
More informationThe Law of Total Probability, Bayes Rule, and Random Variables (Oh My!)
The Law f Ttal Prbability, Bayes Rule, and Randm Variables (Oh My!) Administrivia Hmewrk 2 is psted and is due tw Friday s frm nw If yu didn t start early last time, please d s this time. Gd Milestnes:
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More information[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )
(Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well
More informationTrigonometric Ratios Unit 5 Tentative TEST date
1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin
More informationNAME TEMPERATURE AND HUMIDITY. I. Introduction
NAME TEMPERATURE AND HUMIDITY I. Intrductin Temperature is the single mst imprtant factr in determining atmspheric cnditins because it greatly influences: 1. The amunt f water vapr in the air 2. The pssibility
More informationChem 75 February 16, 2017 Exam 2 Solutions
1. (6 + 6 pints) Tw quick questins: (a) The Handbk f Chemistry and Physics tells us, crrectly, that CCl 4 bils nrmally at 76.7 C, but its mlar enthalpy f vaprizatin is listed in ne place as 34.6 kj ml
More informationExample 1. A robot has a mass of 60 kg. How much does that robot weigh sitting on the earth at sea level? Given: m. Find: Relationships: W
Eample 1 rbt has a mass f 60 kg. Hw much des that rbt weigh sitting n the earth at sea level? Given: m Rbt = 60 kg ind: Rbt Relatinships: Slutin: Rbt =589 N = mg, g = 9.81 m/s Rbt = mrbt g = 60 9. 81 =
More informationFive Whys How To Do It Better
Five Whys Definitin. As explained in the previus article, we define rt cause as simply the uncvering f hw the current prblem came int being. Fr a simple causal chain, it is the entire chain. Fr a cmplex
More informationGeneral Chemistry II, Unit II: Study Guide (part 1)
General Chemistry II, Unit II: Study Guide (part 1) CDS Chapter 21: Reactin Equilibrium in the Gas Phase General Chemistry II Unit II Part 1 1 Intrductin Sme chemical reactins have a significant amunt
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationx x
Mdeling the Dynamics f Life: Calculus and Prbability fr Life Scientists Frederick R. Adler cfrederick R. Adler, Department f Mathematics and Department f Bilgy, University f Utah, Salt Lake City, Utah
More informationKinetic Model Completeness
5.68J/10.652J Spring 2003 Lecture Ntes Tuesday April 15, 2003 Kinetic Mdel Cmpleteness We say a chemical kinetic mdel is cmplete fr a particular reactin cnditin when it cntains all the species and reactins
More informationIn the OLG model, agents live for two periods. they work and divide their labour income between consumption and
1 The Overlapping Generatins Mdel (OLG) In the OLG mdel, agents live fr tw perids. When ung the wrk and divide their labur incme between cnsumptin and savings. When ld the cnsume their savings. As the
More informationStudy Group Report: Plate-fin Heat Exchangers: AEA Technology
Study Grup Reprt: Plate-fin Heat Exchangers: AEA Technlgy The prblem under study cncerned the apparent discrepancy between a series f experiments using a plate fin heat exchanger and the classical thery
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationLecture 5: Equilibrium and Oscillations
Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if
More informationHypothesis Tests for One Population Mean
Hypthesis Tests fr One Ppulatin Mean Chapter 9 Ala Abdelbaki Objective Objective: T estimate the value f ne ppulatin mean Inferential statistics using statistics in rder t estimate parameters We will be
More information11. DUAL NATURE OF RADIATION AND MATTER
11. DUAL NATURE OF RADIATION AND MATTER Very shrt answer and shrt answer questins 1. Define wrk functin f a metal? The minimum energy required fr an electrn t escape frm the metal surface is called the
More information/ / Chemistry. Chapter 1 Chemical Foundations
Name Chapter 1 Chemical Fundatins Advanced Chemistry / / Metric Cnversins All measurements in chemistry are made using the metric system. In using the metric system yu must be able t cnvert between ne
More informationLecture 12: Chemical reaction equilibria
3.012 Fundamentals f Materials Science Fall 2005 Lecture 12: 10.19.05 Chemical reactin equilibria Tday: LAST TIME...2 EQUATING CHEMICAL POTENTIALS DURING REACTIONS...3 The extent f reactin...3 The simplest
More informationPipetting 101 Developed by BSU CityLab
Discver the Micrbes Within: The Wlbachia Prject Pipetting 101 Develped by BSU CityLab Clr Cmparisns Pipetting Exercise #1 STUDENT OBJECTIVES Students will be able t: Chse the crrect size micrpipette fr
More informationPreparation work for A2 Mathematics [2017]
Preparatin wrk fr A2 Mathematics [2017] The wrk studied in Y12 after the return frm study leave is frm the Cre 3 mdule f the A2 Mathematics curse. This wrk will nly be reviewed during Year 13, it will
More informationCHEM 1001 Problem Set #3: Entropy and Free Energy
CHEM 1001 Prblem Set #3: Entry and Free Energy 19.7 (a) Negative; A liquid (mderate entry) cmbines with a slid t frm anther slid. (b)psitive; One mle f high entry gas frms where n gas was resent befre.
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 11: Mdeling with systems f ODEs In Petre Department f IT, Ab Akademi http://www.users.ab.fi/ipetre/cmpmd/ Mdeling with differential equatins Mdeling strategy Fcus
More informationWho is the Holy Spirit?
ill at w w this h t h in SS est abut erence u O q L G ka iff hink : As m t t es a d K S k A the n ma. wn help rmati ur Jesus. y f t u inf e life ab h iple in t alk a disc f T : RE ce as ece t i A p SH
More informationChE 471: LECTURE 4 Fall 2003
ChE 47: LECTURE 4 Fall 003 IDEL RECTORS One f the key gals f chemical reactin engineering is t quantify the relatinship between prductin rate, reactr size, reactin kinetics and selected perating cnditins.
More informationPrinciples of Organic Chemistry lecture 5, page 1
Principles f Organic Chemistry lecture 5, page 1 Bnding Mdels Fact: electrns hld mlecules tgether. Theries: mre than ne way t cnceptualize bnding. Let s fllw Carrll in the cnsideratin f tw theries f bnding.
More informationMatter Content from State Frameworks and Other State Documents
Atms and Mlecules Mlecules are made f smaller entities (atms) which are bnded tgether. Therefre mlecules are divisible. Miscnceptin: Element and atm are synnyms. Prper cnceptin: Elements are atms with
More informationLesson Plan. Recode: They will do a graphic organizer to sequence the steps of scientific method.
Lessn Plan Reach: Ask the students if they ever ppped a bag f micrwave ppcrn and nticed hw many kernels were unppped at the bttm f the bag which made yu wnder if ther brands pp better than the ne yu are
More informationDepartment of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets
Department f Ecnmics, University f alifrnia, Davis Ecn 200 Micr Thery Prfessr Giacm Bnann Insurance Markets nsider an individual wh has an initial wealth f. ith sme prbability p he faces a lss f x (0
More informationLecture 7: Damped and Driven Oscillations
Lecture 7: Damped and Driven Oscillatins Last time, we fund fr underdamped scillatrs: βt x t = e A1 + A csω1t + i A1 A sinω1t A 1 and A are cmplex numbers, but ur answer must be real Implies that A 1 and
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 3: Mdeling change (2) Mdeling using prprtinality Mdeling using gemetric similarity In Petre Department f IT, Ab Akademi http://www.users.ab.fi/ipetre/cmpmd/ http://users.ab.fi/ipetre/cmpmd/
More informationmaking triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y=
Intrductin t Vectrs I 21 Intrductin t Vectrs I 22 I. Determine the hrizntal and vertical cmpnents f the resultant vectr by cunting n the grid. X= y= J. Draw a mangle with hrizntal and vertical cmpnents
More informationUse a lens holder fabricated from SiC. SiC has a larger CTE than C-C, i.e. it is better matched to the SFL6.
Frm: Steve Sctt, Jinsek K, Syun ichi Shiraiwa T: MSE enthusiasts Re: MSE mem 101b: allwable thickness f Vitn sheet Nvember 25, 2008 Update frm MSE Mem 101b Let s assume: Vitn thickness = 1 mm Vitn mdulus
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationEquilibrium of Stress
Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small
More informationRigid Body Dynamics (continued)
Last time: Rigid dy Dynamics (cntinued) Discussin f pint mass, rigid bdy as useful abstractins f reality Many-particle apprach t rigid bdy mdeling: Newtn s Secnd Law, Euler s Law Cntinuus bdy apprach t
More informationLead/Lag Compensator Frequency Domain Properties and Design Methods
Lectures 6 and 7 Lead/Lag Cmpensatr Frequency Dmain Prperties and Design Methds Definitin Cnsider the cmpensatr (ie cntrller Fr, it is called a lag cmpensatr s K Fr s, it is called a lead cmpensatr Ntatin
More informationCS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007
CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is
More informationMANIPAL INSTITUTE OF TECHNOLOGY
MANIPAL INSTITUTE OF TECHNOLOGY MANIPAL UNIVERSITY, MANIPAL SECOND SEMESTER B.Tech. END-SEMESTER EXAMINATION - MAY 013 SUBJECT: ENGINEERING PHYSICS (PHY101/10) Time: 3 Hrs. Max. Marks: 50 Nte: Answer any
More informationk-nearest Neighbor How to choose k Average of k points more reliable when: Large k: noise in attributes +o o noise in class labels
Mtivating Example Memry-Based Learning Instance-Based Learning K-earest eighbr Inductive Assumptin Similar inputs map t similar utputs If nt true => learning is impssible If true => learning reduces t
More informationWhen a substance heats up (absorbs heat) it is an endothermic reaction with a (+)q
Chemistry Ntes Lecture 15 [st] 3/6/09 IMPORTANT NOTES: -( We finished using the lecture slides frm lecture 14) -In class the challenge prblem was passed ut, it is due Tuesday at :00 P.M. SHARP, :01 is
More informationGetting Involved O. Responsibilities of a Member. People Are Depending On You. Participation Is Important. Think It Through
f Getting Invlved O Literature Circles can be fun. It is exciting t be part f a grup that shares smething. S get invlved, read, think, and talk abut bks! Respnsibilities f a Member Remember a Literature
More informationRegents Chemistry Period Unit 3: Atomic Structure. Unit 3 Vocabulary..Due: Test Day
Name Skills: 1. Interpreting Mdels f the Atm 2. Determining the number f subatmic particles 3. Determine P, e-, n fr ins 4. Distinguish istpes frm ther atms/ins Regents Chemistry Perid Unit 3: Atmic Structure
More informationFind this material useful? You can help our team to keep this site up and bring you even more content consider donating via the link on our site.
Find this material useful? Yu can help ur team t keep this site up and bring yu even mre cntent cnsider dnating via the link n ur site. Still having truble understanding the material? Check ut ur Tutring
More informationChem 163 Section: Team Number: ALE 24. Voltaic Cells and Standard Cell Potentials. (Reference: 21.2 and 21.3 Silberberg 5 th edition)
Name Chem 163 Sectin: Team Number: ALE 24. Vltaic Cells and Standard Cell Ptentials (Reference: 21.2 and 21.3 Silberberg 5 th editin) What des a vltmeter reading tell us? The Mdel: Standard Reductin and
More informationAP Statistics Notes Unit Two: The Normal Distributions
AP Statistics Ntes Unit Tw: The Nrmal Distributins Syllabus Objectives: 1.5 The student will summarize distributins f data measuring the psitin using quartiles, percentiles, and standardized scres (z-scres).
More informationREADING STATECHART DIAGRAMS
READING STATECHART DIAGRAMS Figure 4.48 A Statechart diagram with events The diagram in Figure 4.48 shws all states that the bject plane can be in during the curse f its life. Furthermre, it shws the pssible
More informationPart a: Writing the nodal equations and solving for v o gives the magnitude and phase response: tan ( 0.25 )
+ - Hmewrk 0 Slutin ) In the circuit belw: a. Find the magnitude and phase respnse. b. What kind f filter is it? c. At what frequency is the respnse 0.707 if the generatr has a ltage f? d. What is the
More informationECEN 4872/5827 Lecture Notes
ECEN 4872/5827 Lecture Ntes Lecture #5 Objectives fr lecture #5: 1. Analysis f precisin current reference 2. Appraches fr evaluating tlerances 3. Temperature Cefficients evaluatin technique 4. Fundamentals
More informationUnit 14 Thermochemistry Notes
Name KEY Perid CRHS Academic Chemistry Unit 14 Thermchemistry Ntes Quiz Date Exam Date Lab Dates Ntes, Hmewrk, Exam Reviews and Their KEYS lcated n CRHS Academic Chemistry Website: https://cincchem.pbwrks.cm
More informationCHEM Thermodynamics. Change in Gibbs Free Energy, G. Review. Gibbs Free Energy, G. Review
Review Accrding t the nd law f Thermdynamics, a prcess is spntaneus if S universe = S system + S surrundings > 0 Even thugh S system
More informationBASD HIGH SCHOOL FORMAL LAB REPORT
BASD HIGH SCHOOL FORMAL LAB REPORT *WARNING: After an explanatin f what t include in each sectin, there is an example f hw the sectin might lk using a sample experiment Keep in mind, the sample lab used
More informationWork, Energy, and Power
rk, Energy, and Pwer Physics 1 There are many different TYPES f Energy. Energy is expressed in JOULES (J 419J 4.19 1 calrie Energy can be expressed mre specifically by using the term ORK( rk The Scalar
More information