9 Kinetic Theory of Gases

Size: px
Start display at page:

Download "9 Kinetic Theory of Gases"

Transcription

1 Contnt 9 Kintic hory of Gass By Liw Sau oh 9. Ial gas quation 9. rssur of a gas 9. Molcular kintic nrgy 9.4 h r..s. sp of olculs 9.5 Dgrs of fro an law of quipartition of nrgy 9.6 Intrnal nrgy of an ial gas Objctis (a) us th quation of ial gas, p = nr (b) stat th assuptions of th kintic thory of an ial gas (c) ri an us th quation for th prssur xrt by an ial gas, p = / <c > () stat an us th rlationship btwn th Boltzann constant an olar gas constant k = R / A () ri an us th xprssion for th an translational kintic nrgy of a olcul, ½ <c > = / k (f) calculat th r..s. sp of gas olculs Objctis g) sktch th olcular sp istribution graph an xplain th shap of th graph (scription of th xprint is not rquir) h) prict th ariation of olcular sp istribution with tpratur i) fin th grs of fro of a gas olcul j) intify th nubr of grs of fro of a onatoic, iatoic or polyatoic olcul at roo tpratur k) xplain th ariation in th nubr of grs of fro of a iatoic olcul ranging fro ry low to ry high tpraturs 4 Objctis l) stat an apply th law of quipartition of nrgy ) istinguish btwn an ial gas an a ral gas n) xplain th concpt of intrnal nrgy of an ial gas o) ri an us th rlationship btwn th intrnal nrgy an th nubr of grs of fro. 9. Ial gas quation 9. Ial gas quation Gass at low prssurs ar foun to oby th ial gas law: nr at constant tpratur is inrsly if = constant 5 9. Ial gas quation Equation abo also can b writ as = Whr = initial prssur = final prssur = initial olu = final olu 6 7 8

2 / 9 Stats : constant prssur is irctly proportional to its if = constant, thus constant Whr = initial absolut olu, = initial absolut tpratur, = final olu, = final tpratur -7.5 / o C / K 4 Gay- Stats : constant olu is irctly proportional to its if = constant Equation abo also can b writ as / = constant or / = / whr Graphs of Gay- : initial absolut tpratur : final absolut tpratur : initial prssur : final prssur -7.5 / o C / K 5 6

3 Equation of Stat for an Ial Gas An ial gas is fin as a prfct gas which Gay- Equation of Stat for an Ial Gas Consir an ial gas in a containr changs its prssur, olu an tpratur as shown in figur blow. st stag ' n stag 7 8 Equation of Stat for an Ial Gas Equation of Stat for an Ial Gas st stag ' n stag st stag ' n stag st stag, tpratur is kpt at n stag, prssur is kpt constant at, ' ' 9 ' ' Equation of Stat for an Ial Gas Equation of Stat for an Ial Gas hus Or constant nc, R For n ol of an ial gas, th quation of stat is writtn as nr Equation of Stat for an Ial Gas If th Boltzann constant, k is fin as k A thn th quation of stat bcos R.8x J K k Whr k = R/ A = nr = (/ A )R = (R/ A ) = k nr Whr n : th nubr of ol gas n n M A whr whr : ass of thgas M : olcular ass :nubr of olculs A : Aogaro' s constant 6. x ol Ral Gass Assuptions of ral gass by an r Waals: - h olu of th olculs ay not b ngligibl in rlation to th olu occupi by th gas. h attracti forcs btwn th olculs ay not ngligibl. hrfor th quation of stat for an ial gas has to b oifi i.. nr 4

4 Ral Gass Graphs rprsnting th ral gass an ( nb) nr > 4 > an r Waals quation of stat h constants a an b ar pirical constants, iffrnt for iffrnt gass. Whr a olu of ol of th gas olculs & pns on th attracti introlculs forcs nb total olu of th olculs Graphs coparing th ral-ial gass J K ol 9. rssur of a gas 8. Ial gas n / ol rssur of a gas h acroscopic bhaiour of an ial gas can b scrib by using th quation of stat but th icroscopic bhaiour only can b scrib by kintic thory of gass. Kintic hory of Gass Assuptions 9 Kintic hory of Gass Assuptions h ain assuptions of th kintic thory of gass ar: a) All gass ar a up of intical atos or olculs. b) All atos or olculs o ranoly an haphazarly. c) h olu of th atos or olculs is ngligibl whn copar with th olu occupi by th gas. 9. rssur of a gas ) h introlcular forcs ar ngligibl xcpt uring collisions. ) Intr-atoic or olcular collisions ar lastic. f) h uration of a collision is ngligibl copar with th ti spnt tralling btwn collisions. g) Atos an olculs o with constant locity btwn collisions. Graity has no rssur of a gas, is fin as: ow to gt this? ffct on olcular otion. whr c c :prssur by gas :nsity of thgas : an squar locity of thgas olculs

5 9. rssur of a gas Consir th olculs ar contain in a cubic box with th si lngth as shown in figur (a). Assu th locity of a olcul h locity, can b rsol into coponnts of x, y an z. 9. rssur of a gas If a olcul of ass, collis with wall A hnc it will bounc off in opposit irction with locity, -x bcaus of lastic collision as shown in figur (b) rssur of a gas 9. rssur of a gas hrfor th chang in linar ontu for x- coponnt : x x x x x 5 By assuing th olcul o fro wall A to B an bounc back to wall A without collis with othr olculs, th ti takn for that ont is t x 6 9. rssur of a gas 9. rssur of a gas Fro th finition of th ipuls, J F t x x x x Fx t x Fx F x x x 9. rssur of a gas whr F x is th arag forc of on olcul. so th x-coponnt for total forc xrt on th wall of th cubic box: Fx x h locity is rsol into x, y an z, hnc x y z thn x y z 7 9 For olculs of ial gas in th cubic box, Fx x x... x Fx x x... x h an squar of x is x x... x 9. rssur of a gas x Sinc th locitis of th olculs in th ial gas ar assu to b rano, thr is no prfrnc to on irction or anothr. nc x y x x z 8 4

6 9. rssur of a gas By substituting th rlationship abo in th quation for total forc, Fx hnc th total forc xrt on th wall in all irction is gin by F 4 9. rssur of a gas Fro th finition of prssur, F A bcaus whr A an hn F whr : ass of thgas in thbox rssur of a gas Sinc th nsity of th gas, whr hnc quation (5.) can b writtn as OR :prssur : an squar locity of is gin by c by gas :nsity of thgas thgas olculs 9. Molcular Kintic Enrgy Molcular Kintic Enrgy Rarrang quation into his quation shows that incrass ( ) Whn incrass an 9. Molcular Kintic Enrgy incrass Molcular Kintic Enrgy Rarrang q. into Fro th quation of stat in trs of Boltzann constant, k : By quating q. with q. k 9. Molcular Kintic Enrgy k 46 thus k an K tr k K tr whr :arag translational kintic nrgy of a olcul : absolut tpratur K tr For olculs of ial gas in th cubic box, th total arag (an) translational kintic nrgy, E is gin by E E K tr k nr E k 47 48

7 Distribution of olcular sps 9.4 h root an squar (R..s.) sp of olculs ot all th olculs ha th sa sp Apparatus for stuying olcular sp istribution (8-879) 49 5 Distribution of olcular sps Maxwll Distribution Gas olculs constantly colli lastically with ach othr an with th wall of th containr Unr th collision, kintic nrgy transfr fro on olcul to anothr, hnc th kintic nrgy of on olcul incrass whil th othr on crass Molculs o in iffrnc sp as th sp changs aftr ach collision h istribution of olcular sp is known as Maxwll istribution () fraction of olculs with sps in th rang fro to + istribution law: / M M ( ) 4 R R () fraction of olculs with sps in th rang fro to + Jas Clrk Maxwll (8-879) 5 5 Maxwll Distribution ubr of olculs, n () < < rs = Most probabl sp = an sp rs = root an squar sp () fraction of olculs with sps in th rang fro to + 5 rs Sp, h istribution of sps for nitrogn gas olculs at thr iffrnt tpraturs 54 u rs = R 55 56

8 Maxwll istribution h istribution of sps of thr iffrnt gass at th sa tpratur ubr of olculs, n () > Sp, h r..s. sp of olculs Bcaus of thus hn rs k k or k rs whr rs :root an squar locity (sp) : ass of a olcul gas M : rlati olcular ass of gas 9.4 h r..s. sp of olculs Sinc thrfor th quation of root an squar locity also can b writtn as :absolut tpratur 59 6 rs thus 9.4 Distribution of olcular sps 9.4 Distribution of olcular sps Distribution function is noraliz to : Arag sp: Root an squar sp: ( ) 8R ag ( ) M R rs ( ) M 6 Most probabl sp: R M Gas iffusion is th graual ixing of olculs of on gas with olculs of anothr by irtu of thir kintic proprtis Dgrs of fro 9.5 Dgrs of fro an law of quipartition of nrgy Dfinition is fin as th nubr of inpnnt ways in which an ato or olcul can absorb or rlas or stor nrgy 6 64

9 Exapl Exapl Monatoic gas (.g., on, Argon) h nubr of grs of fro is i.. thr irction of translational otion whr contribut translational kintic nrgy. 65 Diatoic gas (.g., O, ) h nubr of grs of fro is: ranslational kintic nrgy = Rotational kintic nrgy = /5 66 Exapl Exapl: grs of fro olyatoic gas (.g. O, CO, ) h nubr of grs of fro is ranslational kintic nrgy Rotational kintic nrgy = /6 Molcul Eg. 5 6 k k Dgrs of Fro (f) ranslat ional Rotati onal ot al Monatoic Diatoic 5 olyatoic O 6 Arag kintic nrgy pr olcul,<k> k 5 k 6 k k Dgrs of fro 9.5 Dgrs of fro Dgrs of fro pn on th absolut tpratur of th gass. For xapl : Diatoic gas () ibration yrogn gas ha th ibrational kintic nrgy as shown in figur abo whr contribut grs of fro which corrspon to th kintic nrgy an th potntial nrgy associat with ibrations along th bon btwn th atos At 5 K f = At tpratur (5 75 K) f = At tpratur >75 K f = nrgy of ry grs of fro of a olcul is k or R f f K k R whr f K :arag kintic nrgy of a olcul gas of fro : absolut tpratu r : grs k :Boltzann constant R:olar gas constant Intrnal Enrgy of An Ial Gas 7

10 9.6 Intrnal Enrgy of An Ial Gas Dfinition is fin as th su of total kintic nrgy an total potntial nrgy of th gas olculs. But in ial gas, introlcular forcs ar assu to b ngligibl hnc th potntial of th olculs can b nglct. hus for olculs, U K.E. 9.6 Intrnal Enrgy of An Ial Gas U K.E. f U k an k R A f U nr whr U :intrnal nrgy of thgas 7 74 Exapl A quantity of plasa is copos of hyrogn ions (protons) an lctrons in thral quilibriu. Both th protons an lctrons ar assu to bha lik olculs of an ial gas. h r..s. sp of an lctron in th plasa is stiat to b x 6 s -. a. Dtrin th r..s. sp of th hyrogn ions. b. Estiat th tpratur of th plasa. (Gin ass of lctron = 9. x - kg, ass of hyrogn ion =.67 x -7 kg, Boltzann constant, k =.8 x - J K - ) Solution thn q. () ii by q. (), thus ( ( rs rs rs ) ) 4 7.x s 75 Solution ( rs ) = x 6 s - a. By using th quation of rs, lctron : k ( rs ) hyrogn ion : Solution ( rs ) k b. h tpratur of th plasa is gin by k ( rs) ( rs ) k.98x 5 ( ) K Suary Kintic hory of Gass Ial Gas Equation Molcular Kintic Enrgy R.M.S. Sp = nr = (/ A )R ½ c = / k Molcular sp istribution & graph rssur of a Gas = / <c > En of opic Dgr of Fro & Law of Equipartition of Enrgy Intrnal Enrgy K.E. Molcul = f/ k Intral nrgy of n ols, U = f/ nr 79 8

Maxwellian Collisions

Maxwellian Collisions Maxwllian Collisions Maxwll ralizd arly on that th particular typ of collision in which th cross-sction varis at Q rs 1/g offrs drastic siplifications. Intrstingly, this bhavior is physically corrct for

More information

Elements of Statistical Thermodynamics

Elements of Statistical Thermodynamics 24 Elmnts of Statistical Thrmodynamics Statistical thrmodynamics is a branch of knowldg that has its own postulats and tchniqus. W do not attmpt to giv hr vn an introduction to th fild. In this chaptr,

More information

fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics

fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics Atoic and olcular Physics JEST Q. Th binding nrgy of th hydrogn ato (lctron bound to proton) is.6 V. Th binding nrgy of positroniu (lctron bound to positron) is (a).6 / V (b).6 / 8 V (c).6 8 V (d).6 V.6

More information

Problem Set 4 Solutions Distributed: February 26, 2016 Due: March 4, 2016

Problem Set 4 Solutions Distributed: February 26, 2016 Due: March 4, 2016 Probl St 4 Solutions Distributd: Fbruary 6, 06 Du: March 4, 06 McQuarri Probls 5-9 Ovrlay th two plots using Excl or Mathatica. S additional conts blow. Th final rsult of Exapl 5-3 dfins th forc constant

More information

PHYS ,Fall 05, Term Exam #1, Oct., 12, 2005

PHYS ,Fall 05, Term Exam #1, Oct., 12, 2005 PHYS1444-,Fall 5, Trm Exam #1, Oct., 1, 5 Nam: Kys 1. circular ring of charg of raius an a total charg Q lis in th x-y plan with its cntr at th origin. small positiv tst charg q is plac at th origin. What

More information

Introduction to the quantum theory of matter and Schrödinger s equation

Introduction to the quantum theory of matter and Schrödinger s equation Introduction to th quantum thory of mattr and Schrödingr s quation Th quantum thory of mattr assums that mattr has two naturs: a particl natur and a wa natur. Th particl natur is dscribd by classical physics

More information

SPH4U Electric Charges and Electric Fields Mr. LoRusso

SPH4U Electric Charges and Electric Fields Mr. LoRusso SPH4U lctric Chargs an lctric Fils Mr. LoRusso lctricity is th flow of lctric charg. Th Grks first obsrv lctrical forcs whn arly scintists rubb ambr with fur. Th notic thy coul attract small bits of straw

More information

Chemistry 342 Spring, The Hydrogen Atom.

Chemistry 342 Spring, The Hydrogen Atom. Th Hyrogn Ato. Th quation. Th first quation w want to sov is φ This quation is of faiiar for; rca that for th fr partic, w ha ψ x for which th soution is Sinc k ψ ψ(x) a cos kx a / k sin kx ± ix cos x

More information

CHAPTER 5 FREE ELECTRON THEORY

CHAPTER 5 FREE ELECTRON THEORY CHAPTER 5 REE ELECTRON THEORY r Elctron Thory Many solids conduct lctricity. Thr ar lctrons that ar not bound to atos but ar abl to ov through th whol crystal. Conducting solids fall into two ain classs;

More information

ES 330 Electronics II Homework # 9 (Fall 2017 Due Monday, December 4, 2017)

ES 330 Electronics II Homework # 9 (Fall 2017 Due Monday, December 4, 2017) Pag1 Na OLUTON E 330 Elctronics Howork # 9 (Fall 017 Du Monday, Dcbr 4, 017) Probl 1 (14 points) Dsign a MO diffrntial aplifir illsuratd in th schatic blow to oprat at O = 0.5 olt with a transconductanc

More information

Unit 7 Charge-to-mass ratio of the electron

Unit 7 Charge-to-mass ratio of the electron Unit 7 Charg-to-ass ratio of th lctron Kywords: J. J. Thoson, Lorntz Forc, Magntic Filds Objctiv: Obsrv th rsults of lctron ba influncd by th agntic fild and calculat th charg-to-ass ratio of th lctron.

More information

School of Electrical Engineering and Telecommunications

School of Electrical Engineering and Telecommunications School of Elctrical Enginring an Tlcounications owr Syst Stability Dr Jayashri Ravishankar School of Elctrical Enginring & tlcounications Stability finition owr syst stability is th ability of an lctric

More information

Collisions. In had on lastic collision of two bodis of qual ass ) Th fastr body spds up furthr and th slowr body slows down. ) Th fastr body slows down and th slowr body spds up. 3) Both of th abov. 4)

More information

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory

Ch. 24 Molecular Reaction Dynamics 1. Collision Theory Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic

More information

Gradebook & Midterm & Office Hours

Gradebook & Midterm & Office Hours Your commnts So what do w do whn on of th r's is 0 in th quation GmM(1/r-1/r)? Do w nd to driv all of ths potntial nrgy formulas? I don't undrstand springs This was th first lctur I actually larnd somthing

More information

Physics. X m (cm)

Physics. X m (cm) Entranc xa 006-007 Physics Duration: hours I- [ pts] An oscillator A chanical oscillator (C) is ford of a solid (S), of ass, attachd to th xtrity A of a horizontal spring of stiffnss (constant) = 80 N/

More information

Collisions between electrons and ions

Collisions between electrons and ions DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision

More information

Gases and Vapour Mixtures

Gases and Vapour Mixtures 9 Gass and aour Mixturs 9.. Introduction. 9.. Dalton s law and Gibbs-Dalton law. 9.. olutric analysis of a gas ixtur. 9.4. h aarnt olcular wight and gas constant. 9.. Scific hats of a gas ixtur. 9.6. Adiabatic

More information

Background: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.

Background: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals. Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby

More information

Classical Magnetic Dipole

Classical Magnetic Dipole Lctur 18 1 Classical Magntic Dipol In gnral, a particl of mass m and charg q (not ncssarily a point charg), w hav q g L m whr g is calld th gyromagntic ratio, which accounts for th ffcts of non-point charg

More information

A RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES

A RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES A RELATIVISTIC LAGRANGIAN FOR MULTIPLE CHARGED POINT-MASSES ADRIAAN DANIËL FOKKER (1887-197) A translation of: Ein invariantr Variationssatz für i Bwgung mhrrr lctrischr Massntilshn Z. Phys. 58, 386-393

More information

Multiple Short Term Infusion Homework # 5 PHA 5127

Multiple Short Term Infusion Homework # 5 PHA 5127 Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300

More information

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches.

22/ Breakdown of the Born-Oppenheimer approximation. Selection rules for rotational-vibrational transitions. P, R branches. Subjct Chmistry Papr No and Titl Modul No and Titl Modul Tag 8/ Physical Spctroscopy / Brakdown of th Born-Oppnhimr approximation. Slction ruls for rotational-vibrational transitions. P, R branchs. CHE_P8_M

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f

More information

The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function

The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function. The Transfer Function A gnraliation of th frquncy rsons function Th convolution sum scrition of an LTI iscrt-tim systm with an imuls rsons h[n] is givn by h y [ n] [ ] x[ n ] Taing th -transforms of both sis w gt n n h n n

More information

Brief Introduction to Statistical Mechanics

Brief Introduction to Statistical Mechanics Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.

More information

PHYS 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 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 information

Model neurons!!the membrane equation!

Model neurons!!the membrane equation! Modl nurons!!th bran quation! Suggstd rading:! Chaptr 5.1-5.3 in Dayan, P. & Abbott, L., Thortical Nuroscinc, MIT Prss, 2001.! Modl nurons: Th bran quation! Contnts:!!!!!! Ion channls Nnst quation Goldan-Hodgkin-Katz

More information

Chapter 7. A Quantum Mechanical Model for the Vibration and Rotation of Molecules

Chapter 7. A Quantum Mechanical Model for the Vibration and Rotation of Molecules Chaptr 7. A Quantu Mchanica Mo for th Vibration an Rotation of Mocus Haronic osciator: Hook s aw: F k is ispacnt Haronic potntia: V F k k is forc constant: V k curvatur of V at quiibriu Nwton s quation:

More information

(A) (C) relation for the Legendre polynomial is α given by Pm. (A) σ = m. (B) σ 2 = m (C) σ + m = 0 (D) σ = m

(A) (C) relation for the Legendre polynomial is α given by Pm. (A) σ = m. (B) σ 2 = m (C) σ + m = 0 (D) σ = m . h atrix i Only Hritian i is Only Unitary Hritian and Unitary Nithr Hritian nor Unitary. What is th product of ign valus of 6. h first proprty of th orthogonality rlation for th Lgndr polynoial is α 0

More information

Title: Vibrational structure of electronic transition

Title: Vibrational structure of electronic transition Titl: Vibrational structur of lctronic transition Pag- Th band spctrum sn in th Ultra-Violt (UV) and visibl (VIS) rgions of th lctromagntic spctrum can not intrprtd as vibrational and rotational spctrum

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat

More information

SOLAR SYSTEM STABILITY EXPLAINED UNDER THE N-BODY PROBLEM SOLUTION

SOLAR SYSTEM STABILITY EXPLAINED UNDER THE N-BODY PROBLEM SOLUTION SOLAR SYSTEM STABILITY EXPLAINED UNDER THE N-BODY PROBLEM SOLUTION Jorg A Franco R E-ail: gorgafr@gailco Abstract: Th priction of th ovnt of a group of N gravitationally attracting bois aroun its cntr

More information

Addition of angular momentum

Addition of angular momentum Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th

More information

Chemical Engineering 412

Chemical Engineering 412 Chical Enginring 4 Introductory Nuclar Enginring Lctur 6 Nuclar Radiation Typs Ky oints Typs of cay Na roprtis athatical scriptions Cavats cay Charts (KNOW HOW TO USE!) Nuclar Equation for cay -Valus for

More information

ME 300 Exam 1 October 9, :30 p.m. to 7:30 p.m.

ME 300 Exam 1 October 9, :30 p.m. to 7:30 p.m. CIRCLE YOUR LECTURE BELOW: First Na Last Na 10:0 a.. 1:0 p.. Naik Gor ME 00 Exa 1 Octobr 9, 014 6:0 p.. to 7:0 p.. INSTRUCTIONS 1. This is a closd book and closd nots xaination. You ar providd with an

More information

Thomas Whitham Sixth Form

Thomas Whitham Sixth Form Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv

More information

Molecular Speeds. Real Gasses. Ideal Gas Law. Reasonable. Why the breakdown? P-V Diagram. Using moles. Using molecules

Molecular 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 information

0WAVE PROPAGATION IN MATERIAL SPACE

0WAVE PROPAGATION IN MATERIAL SPACE 0WAVE PROPAGATION IN MATERIAL SPACE All forms of EM nrgy shar thr fundamntal charactristics: 1) thy all tral at high locity 2) In traling, thy assum th proprtis of was 3) Thy radiat outward from a sourc

More information

nd the particular orthogonal trajectory from the family of orthogonal trajectories passing through point (0; 1).

nd the particular orthogonal trajectory from the family of orthogonal trajectories passing through point (0; 1). Eamn EDO. Givn th family of curvs y + C nd th particular orthogonal trajctory from th family of orthogonal trajctoris passing through point (0; ). Solution: In th rst plac, lt us calculat th di rntial

More information

First order differential equation Linear equation; Method of integrating factors

First order differential equation Linear equation; Method of integrating factors First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial

More information

1 Input-Output Stability

1 Input-Output Stability Inut-Outut Stability Inut-outut stability analysis allows us to analyz th stability of a givn syst without knowing th intrnal stat x of th syst. Bfor going forward, w hav to introduc so inut-outut athatical

More information

5.80 Small-Molecule Spectroscopy and Dynamics

5.80 Small-Molecule Spectroscopy and Dynamics MIT OpnCoursWar http://ocw.mit.du 5.80 Small-Molcul Spctroscopy and Dynamics Fall 008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. Lctur # 3 Supplmnt Contnts

More information

Linear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let

Linear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr

More information

Constants and Conversions:

Constants and Conversions: EXAM INFORMATION Radial Distribution Function: P 2 ( r) RDF( r) Br R( r ) 2, B is th normalization constant. Ordr of Orbital Enrgis: Homonuclar Diatomic Molculs * * * * g1s u1s g 2s u 2s u 2 p g 2 p g

More information

Module 8 Non equilibrium Thermodynamics

Module 8 Non equilibrium Thermodynamics Modul 8 Non quilibrium hrmodynamics ctur 8.1 Basic Postulats NON-EQUIIRIBIUM HERMODYNAMICS Stady Stat procsss. (Stationary) Concpt of ocal thrmodynamic qlbm Extnsiv proprty Hat conducting bar dfin proprtis

More information

Hydrogen Atom and One Electron Ions

Hydrogen Atom and One Electron Ions Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial

More information

Chapter 8: Electron Configurations and Periodicity

Chapter 8: Electron Configurations and Periodicity Elctron Spin & th Pauli Exclusion Principl Chaptr 8: Elctron Configurations and Priodicity 3 quantum numbrs (n, l, ml) dfin th nrgy, siz, shap, and spatial orintation of ach atomic orbital. To xplain how

More information

PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101. Plan for Lecture 4:

PHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101. Plan for Lecture 4: PHY 114 A Gnral Physics II 11 AM-1:15 PM TR Olin 101 Plan for Lctur 4: 1. Introuction to th lctric potntial.rlationship btwn th lctric potntial an th lctric fil 1/31/01 PHY 114 A Spring 01 -- Lctur 4 1

More information

Artificial Noise Based Secure Transmission Scheme in Multiple Antenna Systems

Artificial Noise Based Secure Transmission Scheme in Multiple Antenna Systems Intrnational Journal of Appli Enginring Rsarch ISSN 973-46 Volu, Nubr (6) pp 76-7 Rsarch Inia Publications http://wwwripublicationco Artificial Nois Bas Scur Transission Sch in ultipl Antnna Systs Bangwon

More information

a 1and x is any real number.

a 1and x is any real number. Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars

More information

Statistical Thermodynamics: Sublimation of Solid Iodine

Statistical Thermodynamics: Sublimation of Solid Iodine c:374-7-ivap-statmch.docx mar7 Statistical Thrmodynamics: Sublimation of Solid Iodin Chm 374 For March 3, 7 Prof. Patrik Callis Purpos:. To rviw basic fundamntals idas of Statistical Mchanics as applid

More information

Exam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.

Exam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator. Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r

More information

CS 491 G Combinatorial Optimization

CS 491 G Combinatorial Optimization CS 49 G Cobinatorial Optiization Lctur Nots Junhui Jia. Maiu Flow Probls Now lt us iscuss or tails on aiu low probls. Thor. A asibl low is aiu i an only i thr is no -augnting path. Proo: Lt P = A asibl

More information

Lecture 28 Title: Diatomic Molecule : Vibrational and Rotational spectra

Lecture 28 Title: Diatomic Molecule : Vibrational and Rotational spectra Lctur 8 Titl: Diatomic Molcul : Vibrational and otational spctra Pag- In this lctur w will undrstand th molcular vibrational and rotational spctra of diatomic molcul W will start with th Hamiltonian for

More information

Chemical Physics II. More Stat. Thermo Kinetics Protein Folding...

Chemical Physics II. More Stat. Thermo Kinetics Protein Folding... Chmical Physics II Mor Stat. Thrmo Kintics Protin Folding... http://www.nmc.ctc.com/imags/projct/proj15thumb.jpg http://nuclarwaponarchiv.org/usa/tsts/ukgrabl2.jpg http://www.photolib.noaa.gov/corps/imags/big/corp1417.jpg

More information

Differential Equations

Differential Equations UNIT I Diffrntial Equations.0 INTRODUCTION W li in a world of intrrlatd changing ntitis. Th locit of a falling bod changs with distanc, th position of th arth changs with tim, th ara of a circl changs

More information

MA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c.

MA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c. MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ

More information

High Energy Physics. Lecture 5 The Passage of Particles through Matter

High Energy Physics. Lecture 5 The Passage of Particles through Matter High Enrgy Physics Lctur 5 Th Passag of Particls through Mattr 1 Introduction In prvious lcturs w hav sn xampls of tracks lft by chargd particls in passing through mattr. Such tracks provid som of th most

More information

The failure of the classical mechanics

The failure of the classical mechanics h failur of th classical mchanics W rviw som xprimntal vidncs showing that svral concpts of classical mchanics cannot b applid. - h blac-body radiation. - Atomic and molcular spctra. - h particl-li charactr

More information

A 1 A 2. a) Find the wavelength of the radio waves. Since c = f, then = c/f = (3x10 8 m/s) / (30x10 6 Hz) = 10m.

A 1 A 2. a) Find the wavelength of the radio waves. Since c = f, then = c/f = (3x10 8 m/s) / (30x10 6 Hz) = 10m. 1. Young s doubl-slit xprint undrlis th instrunt landing syst at ost airports and is usd to guid aircraft to saf landings whn th visibility is poor. Suppos that a pilot is trying to align hr plan with

More information

Introduction to Condensed Matter Physics

Introduction to Condensed Matter Physics Introduction to Condnsd Mattr Physics pcific hat M.P. Vaughan Ovrviw Ovrviw of spcific hat Hat capacity Dulong-Ptit Law Einstin modl Dby modl h Hat Capacity Hat capacity h hat capacity of a systm hld at

More information

2. Finite Impulse Response Filters (FIR)

2. Finite Impulse Response Filters (FIR) .. Mthos for FIR filtrs implmntation. Finit Impuls Rspons Filtrs (FIR. Th winow mtho.. Frquncy charactristic uniform sampling. 3. Maximum rror minimizing. 4. Last-squars rror minimizing.. Mthos for FIR

More information

de/dx Effectively all charged particles except electrons

de/dx Effectively all charged particles except electrons de/dx Lt s nxt turn our attntion to how chargd particls los nrgy in mattr To start with w ll considr only havy chargd particls lik muons, pions, protons, alphas, havy ions, Effctivly all chargd particls

More information

Mathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination

Mathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts

More information

Math 34A. Final Review

Math 34A. Final Review Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right

More information

Pair (and Triplet) Production Effect:

Pair (and Triplet) Production Effect: Pair (and riplt Production Effct: In both Pair and riplt production, a positron (anti-lctron and an lctron (or ngatron ar producd spontanously as a photon intracts with a strong lctric fild from ithr a

More information

Coupled Pendulums. Two normal modes.

Coupled Pendulums. Two normal modes. Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring Start Swinging Som tim latr - swings with full amplitud. stationary M +n L M +m Elctron

More information

UGC POINT LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM. are the polar coordinates of P, then. 2 sec sec tan. m 2a m m r. f r.

UGC POINT LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM. are the polar coordinates of P, then. 2 sec sec tan. m 2a m m r. f r. UGC POINT LEADING INSTITUE FOR CSIR-JRF/NET, GATE & JAM Solution (TEST SERIES ST PAPER) Dat: No 5. Lt a b th adius of cicl, dscibd by th aticl P in fig. if, a th ola coodinats of P, thn acos Diffntial

More information

AS 5850 Finite Element Analysis

AS 5850 Finite Element Analysis AS 5850 Finit Elmnt Analysis Two-Dimnsional Linar Elasticity Instructor Prof. IIT Madras Equations of Plan Elasticity - 1 displacmnt fild strain- displacmnt rlations (infinitsimal strain) in matrix form

More information

ECE507 - Plasma Physics and Applications

ECE507 - Plasma Physics and Applications ECE507 - Plasma Physics and Applications Lctur 7 Prof. Jorg Rocca and Dr. Frnando Tomasl Dpartmnt of Elctrical and Computr Enginring Collisional and radiativ procsss All particls in a plasma intract with

More information

Principles of Humidity Dalton s law

Principles of Humidity Dalton s law Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

More information

Problem 22: Journey to the Center of the Earth

Problem 22: Journey to the Center of the Earth Problm : Journy to th Cntr of th Earth Imagin that on drilld a hol with smooth sids straight through th ntr of th arth If th air is rmod from this tub (and it dosn t fill up with watr, liquid rok, or iron

More information

Y 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall

Y 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)

More information

5.62 Physical Chemistry II Spring 2008

5.62 Physical Chemistry II Spring 2008 MIT OpnCoursWar http://ocw.mit.du 5.62 Physical Chmistry II Spring 2008 For information about citing ths matrials or our Trms of Us, visit: http://ocw.mit.du/trms. 5.62 Lctur #7: Translational Part of

More information

Electrochemical Energy Systems Spring 2014 MIT, M. Z. Bazant. Midterm Exam

Electrochemical Energy Systems Spring 2014 MIT, M. Z. Bazant. Midterm Exam 10.66 Elctrochmical Enrgy Systms Spring 014 MIT, M. Z. Bazant Midtrm Exam Instructions. This is a tak-hom, opn-book xam du in Lctur. Lat xams will not b accptd. You may consult any books, handouts, or

More information

v d = (VII) (II) (IV)

v d = (VII) (II) (IV) P7..1.4 Pag 1/5 Objcts of th xprints 1. Masuring of th Hall voltag as function of th currnt at a constant agntic fild: dtrination of th dnsity and obility of charg carrirs.. Masuring of th Hall voltag

More information

Case Study 4 PHA 5127 Aminoglycosides Answers provided by Jeffrey Stark Graduate Student

Case Study 4 PHA 5127 Aminoglycosides Answers provided by Jeffrey Stark Graduate Student Cas Stuy 4 PHA 527 Aminoglycosis Answrs provi by Jffry Stark Grauat Stunt Backgroun Gntamicin is us to trat a wi varity of infctions. Howvr, u to its toxicity, its us must b rstrict to th thrapy of lif-thratning

More information

PHY 171. Lecture 14. (February 16, 2012)

PHY 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 information

Einstein Equations for Tetrad Fields

Einstein Equations for Tetrad Fields Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for

More information

Lecture # 12: Shock Waves and De Laval Nozzle

Lecture # 12: Shock Waves and De Laval Nozzle ArE 311L & ArE343L Lctur Nots Lctur # 1: Shock Wavs and D Laval Nozzl Dr. Hui H Hu Dpartmnt of Arospac Enginring Iowa Stat Univrsity Ams, Iowa 50011, U.S.A ArE311L Lab#3: rssur Masurmnts in a d Laval Nozzl

More information

Intro to QM due: February 8, 2019 Problem Set 12

Intro to QM due: February 8, 2019 Problem Set 12 Intro to QM du: Fbruary 8, 9 Prob St Prob : Us [ x i, p j ] i δ ij to vrify that th anguar ontu oprators L i jk ɛ ijk x j p k satisfy th coutation rations [ L i, L j ] i k ɛ ijk Lk, [ L i, x j ] i k ɛ

More information

Electrical Energy and Capacitance

Electrical Energy and Capacitance haptr 6 Elctrical Enrgy and apacitanc Quick Quizzs. (b). Th fild xrts a forc on th lctron, causing it to acclrat in th dirction opposit to that of th fild. In this procss, lctrical potntial nrgy is convrtd

More information

The van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012

The van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012 Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor

More information

Case Study Vancomycin Answers Provided by Jeffrey Stark, Graduate Student

Case Study Vancomycin Answers Provided by Jeffrey Stark, Graduate Student Cas Stuy Vancomycin Answrs Provi by Jffry Stark, Grauat Stunt h antibiotic Vancomycin is liminat almost ntirly by glomrular filtration. For a patint with normal rnal function, th half-lif is about 6 hours.

More information

Schematic of a mixed flow reactor (both advection and dispersion must be accounted for)

Schematic of a mixed flow reactor (both advection and dispersion must be accounted for) Cas stuy 6.1, R: Chapra an Canal, p. 769. Th quation scribin th concntration o any tracr in an lonat ractor is known as th avction-isprsion quation an may b writtn as: Schmatic o a mi low ractor (both

More information

electron -ee mrw o center of atom CLASSICAL ELECTRON THEORY Lorentz' classical model for the dielectric function of insulators

electron -ee mrw o center of atom CLASSICAL ELECTRON THEORY Lorentz' classical model for the dielectric function of insulators CLASSICAL ELECTRON THEORY Lorntz' claical odl for th dilctric function of inulator In thi odl th lctron ar aud to b bound to th nuclu ith forc obying Hook la. Th forc ar aud to b iotropic and daping can

More information

The pn junction: 2 Current vs Voltage (IV) characteristics

The pn junction: 2 Current vs Voltage (IV) characteristics Th pn junction: Currnt vs Voltag (V) charactristics Considr a pn junction in quilibrium with no applid xtrnal voltag: o th V E F E F V p-typ Dpltion rgion n-typ Elctron movmnt across th junction: 1. n

More information

Deepak Rajput

Deepak Rajput Q Prov: (a than an infinit point lattic is only capabl of showing,, 4, or 6-fold typ rotational symmtry; (b th Wiss zon law, i.. if [uvw] is a zon axis and (hkl is a fac in th zon, thn hu + kv + lw ; (c

More information

11/13/17. directed graphs. CS 220: Discrete Structures and their Applications. relations and directed graphs; transitive closure zybooks

11/13/17. directed graphs. CS 220: Discrete Structures and their Applications. relations and directed graphs; transitive closure zybooks dirctd graphs CS 220: Discrt Strctrs and thir Applications rlations and dirctd graphs; transiti closr zybooks 9.3-9.6 G=(V, E) rtics dgs dgs rtics/ nods Edg (, ) gos from rtx to rtx. in-dgr of a rtx: th

More information

Problem Set 6 Solutions

Problem Set 6 Solutions 6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr

More information

Analysis of Algorithms - Elementary graphs algorithms -

Analysis of Algorithms - Elementary graphs algorithms - Analysis of Algorithms - Elmntary graphs algorithms - Anras Ermahl MRTC (Mälaralns Ral-Tim Rsach Cntr) anras.rmahl@mh.s Autumn 00 Graphs Graphs ar important mathmatical ntitis in computr scinc an nginring

More information

Notes on Differential Geometry

Notes on Differential Geometry Nots from phz 6607, Spcial an Gnral Rlativity Univrsity of Floria, Fall 2004, Dtwilr Nots on Diffrntial Gomtry Ths nots ar not a substitut in any mannr for class lcturs. Plas lt m know if you fin rrors.

More information

Magnetic vector potential. Antonio Jose Saraiva ; -- Electric current; -- Magnetic momentum; R Radius.

Magnetic vector potential. Antonio Jose Saraiva ; -- Electric current; -- Magnetic momentum; R Radius. Magnti vtor potntial Antonio Jos araiva ajps@hotail.o ; ajps137@gail.o A I.R A Magnti vtor potntial; -- auu prability; I -- ltri urrnt; -- Magnti ontu; R Radius. un agnti ronntion un tru surfa tpratur

More information

Study of QCD critical point at high temperature and density by lattice simulations

Study of QCD critical point at high temperature and density by lattice simulations Stuy of QCD critical point at high tmpratur an nsity by lattic simulations Shinji Ejiri (Brookhavn ational Laboratory) Canonical partition function an finit nsity phas transition in lattic QCD arxiv:84.7

More information

Electrochemistry L E O

Electrochemistry L E O Rmmbr from CHM151 A rdox raction in on in which lctrons ar transfrrd lctrochmistry L O Rduction os lctrons xidation G R ain lctrons duction W can dtrmin which lmnt is oxidizd or rducd by assigning oxidation

More information

Exam 2 Thursday (7:30-9pm) It will cover material through HW 7, but no material that was on the 1 st exam.

Exam 2 Thursday (7:30-9pm) It will cover material through HW 7, but no material that was on the 1 st exam. Exam 2 Thursday (7:30-9pm) It will covr matrial through HW 7, but no matrial that was on th 1 st xam. What happns if w bash atoms with lctrons? In atomic discharg lamps, lots of lctrons ar givn kintic

More information

Byeong-Joo Lee

Byeong-Joo Lee OSECH - MSE calphad@postch.ac.kr Equipartition horm h avrag nrgy o a particl pr indpndnt componnt o motion is ε ε ' ε '' ε ''' U ln Z Z ε < ε > U ln Z β ( ε ' ε '' ε ''' / Z' Z translational kintic nrgy

More information

6. The Interaction of Light and Matter

6. The Interaction of Light and Matter 6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd

More information