The Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases.
|
|
- Stanley Harper
- 5 years ago
- Views:
Transcription
1 Section 1 Section 2 The module clcultes nd plots isotherml 1-, 2- nd 3-metl predominnce re digrms. ccesses only compound dtbses. Tble of Contents Tble of Contents Opening the module Section 3 Stoichiometric rections nd innce Are digrms Section 4 Specifying clssicl predominnce re digrm: Fe-S 2 (g)-o 2 (g) for T=1000 K Section 5 Grphicl output using the Figure module Section 6 Numericl results from predomince re clcultion Section 7 Using overlys (Cu-SO 2 -O 2 t different tempertures) Section 8 One-metl predominnce digrm with four elements: Fe-S-O-Cl Section 9 Vrious predominnce digrm with four elements Section 10 Bsic computtionl procedure in the construction of predominnce digrm 1
2 The module Click on in the min FctSge window. 2
3 Using the Rection progrm to identify the most probble rection Question: A gs mixture, 80% SO % O 2 t 1 tm, is equilibrted with Fe t 1000 K. The possible products re FeO, Fe 2 O 3, Fe 3 O 4, FeS, FeS 2, FeSO 4 nd Fe 2 (SO 4 ) 3. Which product is the most the stble? Answer: Although some SO 3 (g) forms, for simplicity we will ssume tht SO 2 nd O 2 re t equilibrium: i.e., P SO2 = 0.8 tm nd P O2 = 0.2 tm. In such cse, it cn be shown tht the equilibrium prtil pressure of sulfur is P S2 = tm. Although the vlue is smll, this chemicl potentil is useful for thermodynmic clcultions. From the Rection progrm, the following vlues of DG re clculted: (note: 1 mol Fe(s) rectnt in ll cses) 1 : Fe(s) O 2 (0.2 tm) = FeO DG= kj 2 : Fe(s) O 2 (0.2 tm) = 0.5 Fe 2 O 3 DG= kj 3 : Fe(s) + (2/3) O 2 (0.2 tm) = (1/3) Fe 3 O 4 DG= kj 4 : Fe(s) S 2 (10-29 tm) = FeS DG= kj 5 : Fe(s) + S 2 (10-29 tm) = FeS 2 DG= kj 6 : Fe(s) + 2 O 2 (0.2 tm) + S 2 (10-29 tm) = FeSO 4 DG= kj 7 : Fe(s) + 3 O 2 (0.2 tm) S 2 (10-29 tm) = 0.5 Fe 2 (SO 4 ) 3 DG= kj 3.1
4 Rection progrm: 7 possible isotherml isobric rections nd 7 vlues of DG Rection 1 Rection 2 Rection 3 3.2
5 methodology Rection 4 Rection 5 Rection 6 Rection 7 Fe 2 (SO 4 ) 3 is the stble product since DG is the most negtive. For exmple, FeSO 4 could not be the most stble since combining rections 6 nd 7, we hve: FeSO 4 + O 2 (0.2 tm) S 2 (10-29 tm) = 0.5 Fe 2 (SO 4 ) 3 DG = ( J) - ( J) kj This methodology is used by the progrm to locte the domins of stbility of ech phse s function of gs potentil. 3.3
6 Specifying clssicl predominnce re digrm: Fe-S 2 (g)-o 2 (g) t 1000 K 1. Specify the metllic nd the non-metllic elements. 2. Press Next >> to ctivte the clcultion. 3. Select the vribles: Prmeters: Pressure Constnts Axes Lbels nd Disply Species: gs, liquids nd solids Clculte: digrm All exmples shown here re stored in FctSge - click on: File > Directories > Slide Show Exmples 4. Press Clculte >> 4.1
7 Figure disply of the predominnce re digrm of Fe-S 2 (g)-o 2 (g) t 1000 K The rrow is pointing to the Fe 2 (SO 4 ) 3 domin where: x = log 10 P O2 = P O2 0.2 tm y = log 10 P S2 = P S tm Figure cretes predominnce digrm tht cn be edited, mnipulted nd stored in vriety of wys. 5.1
8 Revised digrm displyed in Figure for Fe-SO 2 (g)-o 2 (g) t 1000 K Chnging the X nd Y xes. And Clculte >> revised digrm. Note: The Fe-S 2 (g)-o 2 (g) digrm (previous pge) nd the Fe-SO 2 (g)-o 2 (g) digrm (here) re topologiclly equivlent, i.e. the sme combintion of species coexists t the invrint points. 5.2 Intersection t: x = log 10 P O2 = P O2 0.2 tm y = log 10 P SO2 = P SO2 0.8 tm
9 Invrint points of Fe-SO 2 (g)-o 2 (g) t 1000 K. Phse Rule: F = C - P + 2 F : degrees of freedom C : number of components P : number of phses Univrint line (/): FeS/Fe 3 O 4 /gs. From the phse rule, F = 2. Hence, t 1000 K only one of SO 2 or O 2 cn be fixed. At the rrow, C = 3 (Fe,S,O) nd P = 4 (3 solids nd 1 gs); hence F = 1. At 1000 K, the system is invrint. (Note tht the totl pressure is not specified.) 6.1
10 Detiled point clcultion t P SO2 =0.8 tm nd P O2 =0.2 tm for Fe-SO 2 (g)-o 2 (g) t 1000K 18 gs species, 6 liquid species nd 21 solids species (totl: 46 species) in the FACT compound dtbse contining Fe, S nd/or O. Click on Dt Serch to include or exclude dtbse in the serch. Here, only the FACT compound dtbse is included. Point clcultion dt entry: P SO2 = 0.8 tm, P O2 = 0.2 tm 6.2
11 Detiled point clcultion t P SO2 =0.8 tm nd P O2 =0.2 tm for Fe-SO 2 (g)-o 2 (g) t 1000K Fe 2 (SO 4 ) 3 (s) is the stble species: unit ctivity, C p vlues not extrpolted A/P/M: Activity/Prtil pressure/mollity «T» indictes extrpolted dt Minly SO 2 (0.8 tm), O 2 (0.2 tm) nd SO 3 (0.65 tm, see code 39). 6.3
12 digrm for Cu-SO 2 (g)-o 2 (g) t 1000 K 1. Specify the metllic elements (Cu) nd the nonmetllic elements (S nd O) in the Elements frme. 2. Press Next >> to serch through the selected Compound dtbse(s) (here, FACT ) nd ctivte the clcultion. 3. Specifying n isobr, P = 0.01 tm, t T = 1000 K. 4. Select digrm 5. Press Clculte>> 6. In Figure, edit the digrm nd sve it. 7.1
13 Figure: Superimposed digrms for Cu-SO 2 (g)-o 2 (g) t 1000, 1100 nd 1200 K 7. Repet steps 3 (but uncheck the isobr checkbox), 5 nd 6 for T=1100 K nd 1200 K 8. Use the superimpose figure function in the Figure progrm to edit this predominnce re digrm t 1000 K, 1100 K nd 1200 K. 7.2
14 One-metl predominnce digrm with four elements: Fe-S-O-Cl Specifying the isotherml predominnce re digrm for: Fe-S 2 (g)-o 2 (g)- S 2 Cl 2 (g) t 1000 K nd P S2 Cl 2 = 0.1 tm 1. Enter the elements. This is one-metl Fe system with S, O nd Cl non-metllic elements. 2. Press Next >>. 3. Note: you must enter 2 constnts. T = 1000 K Z = P S2 Cl 2 = 10-1 tm 4. Select wht you wish to clculte nd press Clculte>>. 8.1
15 Invrint points tble nd predominnce digrm for Fe-S 2 (g)-o 2 (g)-s 2 Cl 2 (g) Appliction: Chlorintion of FeS At 1000 K nd P S2 Cl 2 = 0.1 tm 8.2
16 Two-metl predominnce digrm with four elements: Fe-Cr-C-O Fe-rich side of the iron-chronium 2-metl predominnce re digrm Appliction: Pssivity of Fe-Cr lloys 9.1
17 Two-metl predominnce digrm with four elements: Fe-Cr-C-O Cr-rich side of the iron-chronium 2-metl predominnce re digrm Appliction: Pssivity of Fe-Cr lloys 9.2
18 One-metl predominnce digrm with four elements: Fe-Cr-C-O Fe-Cr-C-O with Fe s the 1-metl element, Cr(s) = 1 Appliction: Pssivity of Fe-Cr lloys 9.3
19 One-metl predominnce digrm with four elements: Cr-Fe-C-O Cr-Fe-C-O with Cr s the 1-metl element, Fe(s) = 1 Appliction: Pssivity of Fe-Cr lloys 9.4
20 Chromium-Crbon-Oxygen innce Digrm Appliction: Decrburiztion of Chromium Use of the surimpose function of Figure (see Figure help, section 14) mkes it esy to compre the Cr-Fe-C-O digrm (previous slide) with this Cr-C-O digrm to show the role of Fe in the formtion of FeCr 2 O 4 9.5
21 Bsic computtionl procedure in the construction of predominnce digrm The following five slides give detiled explntion on the stoichiometric reltionships of the rections tht govern the phse boundries in predominnce re digrm. In principle there is no limittion in this pproch s to the number of system components. However, there is one mjor restriction in the entire pproch of using stoichiometric rections: it is not suited for the tretment of systems with solution phses. 10.0
22 Bsic computtionl procedure in the construction of predominnce digrm Isotherml predominnce digrm estblishes t ech log P S2 nd log P O2 prticulr compound MS O b with the lowest DG of formtion (bsed on 1 mole of M). M S O M S O b b log P S2 M 2 S t MSO 4 D G D G R T M S O b b o M S O ln M S O b b 2 2 P P M S O 2 2 M 2 O p log P O2 At point p, DG (for formtion) of M 2 O is most negtive (per mole of M). At point t, DG (for formtion) of M 2 O, M 2 S nd MSO 4 re eqully negtive. Notes: It is not necessry to identify possible equilibri mong phses by this method. The user my set MS O b or P MS O b to other thn 1 for some species. Useful especilly when MS O b is gseous species. 10.1
23 Compounds s species represented on xes Use of species other thn elements for prtil pressure ssocited with the xes. Select S-O species with different rtios of S/O for ech xis. For exmple SO 2 nd SO 3 : S O O S O S O 2S O S At ny coordinte, p, on the digrm, the prtil pressure or chemicl potentil of elementl species O 2 or S 2 my be determined. The bsic lgorithm on the previous pge cn now be pplied: log P SO3 M 2 S t M 2 O p MSO 4 log P SO2 Notes: Species on xes must not contin the bse element M Combintion of species t triple point is independent of choice of S/O species for xes 10.2
24 Point clcultion At ny prticulr point such s p in known domin ctivity of M cn be determined. M S O M S O b b log P S2 M 2 S Insert M from bove into eqution below t M 2 O MSO 4 p DG R T ln M S O b b 2 2 P P M S O 2 2 log P O2 To find the ctivity of ech MS O b : M S O M S O b b Activities nd prtil pressures of bse-element-contining species t ny point in the digrm. The ctivity of the prticulr species MS O b identifying tht domin is the set vlue (usully 1). 10.3
25 More thn 3 Elements Involved in Digrm 2 wys to proceed: ) Fix dditionl chemicl potentils (eg. P Cl2 ) Formulte formtion rections to construct the predominnce re digrm shown in 15 s follows: M S b O c C l M S O C l b c One cn specify P Cl2 for entire digrm, or s in 15, specify prtil pressure or ctivity of ny Cl-S-O species (eg. S 2 Cl 2 ) S ( g ) C l ( g ) S C l ( g ) By this equilibrium, one cn find Cl 2 for ny S 2 prtil pressure 10.4
26 More thn 3 Elements Involved in Digrm b) Introduce n dditionl bse element (eg. N) 1 r M rn S O g M N S O h M N S O b w x c d y z e f Bse elements (specify rnge in which r my be found) Combintion of two M nd N contining compounds which stisfy the mss blnces with non negtive vlues for g nd h for specified vlue of r Note: In the cse of two bse element digrm ech domin is doubly lbelled. In the specil cse of there being no compounds contining both M nd N, the two bse element digrm my be regrded simply s the superimposition of the two one bse element digrms for M nd N. 10.5
UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction
Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from
More informationAcid-Base Equilibria
Tdeusz Górecki Ionic Equiliri Acid-Bse Equiliri Brønsted-Lory: n cid is proton, se is. Acid Bse ( 3 PO 4, O), ( N 4 ) nd ( PO - 4 ) cn ll ehve s cids. Exmple: 4 N N3 Sustnces hich cn ehve oth s cids nd
More informationModule 2: Rate Law & Stoichiomtery (Chapter 3, Fogler)
CHE 309: Chemicl Rection Engineering Lecture-8 Module 2: Rte Lw & Stoichiomtery (Chpter 3, Fogler) Topics to be covered in tody s lecture Thermodynmics nd Kinetics Rection rtes for reversible rections
More informationTests for the Ratio of Two Poisson Rates
Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationFundamentals of Analytical Chemistry
Homework Fundmentls of nlyticl hemistry hpter 9 0, 1, 5, 7, 9 cids, Bses, nd hpter 9(b) Definitions cid Releses H ions in wter (rrhenius) Proton donor (Bronsted( Lowry) Electron-pir cceptor (Lewis) hrcteristic
More informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationHydronium or hydroxide ions can also be produced by a reaction of certain substances with water:
Chpter 14 1 ACIDS/BASES Acids hve tste, rect with most metls to produce, rect with most crbontes to produce, turn litmus nd phenolphthlein. Bses hve tste rect very well well with most metls or crbontes,
More informationCHM Physical Chemistry I Chapter 1 - Supplementary Material
CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion
More informationStrategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?
University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationTerminal Velocity and Raindrop Growth
Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,
More informationUNIVERSITY OF MALTA DEPARTMENT OF CHEMISTRY. CH237 - Chemical Thermodynamics and Kinetics. Tutorial Sheet VIII
UNIVERSITY OF MALTA DEPARTMENT OF CHEMISTRY CH237 - Chemicl Thermodynmics nd Kinetics Tutoril Sheet VIII 1 () (i) The rte of the rection A + 2B 3C + D ws reported s 1.0 mol L -1 s -1. Stte the rtes of
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationChem 130 Second Exam
Nme Chem 130 Second Exm On the following pges you will find questions tht cover the structure of molecules, ions, nd solids, nd the different models we use to explin the nture of chemicl bonding. Red ech
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationThe International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O
IAPWS R-7 The Interntionl Assocition for the Properties of Wter nd Stem Lucerne, Sitzerlnd August 7 Relese on the Ioniztion Constnt of H O 7 The Interntionl Assocition for the Properties of Wter nd Stem
More informationCHAPTER 08: MONOPROTIC ACID-BASE EQUILIBRIA
Hrris: Quntittive Chemicl Anlysis, Eight Edition CHAPTER 08: MONOPROTIC ACIDBASE EQUILIBRIA CHAPTER 08: Opener A CHAPTER 08: Opener B CHAPTER 08: Opener C CHAPTER 08: Opener D CHAPTER 08: Opener E Chpter
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationMatrix Solution to Linear Equations and Markov Chains
Trding Systems nd Methods, Fifth Edition By Perry J. Kufmn Copyright 2005, 2013 by Perry J. Kufmn APPENDIX 2 Mtrix Solution to Liner Equtions nd Mrkov Chins DIRECT SOLUTION AND CONVERGENCE METHOD Before
More informationCBE 291b - Computation And Optimization For Engineers
The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b - Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn
More informationStudent Activity 3: Single Factor ANOVA
MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether
More informationFe = Fe + e MnO + 8H + 5e = Mn
Redox Titrtions Net trnsfer of electrons during the rection. xidtion numbers of two/more species chnge. Stisfies requirements for rections in quntittion;. lrge K b. fst rection Blncing redox rections:
More informationOrganic Acids - Carboxylic Acids
Orgnic Acids - rboxylic Acids Orgnic cids - crboxylic cid functionl group rboxylic cids re redily deprotonted by bses such s NO eg 3 O O - + O - + O 3 O O Acid Bse onjugte Bse onjugte Acid This rection
More informationMathcad Lecture #1 In-class Worksheet Mathcad Basics
Mthcd Lecture #1 In-clss Worksheet Mthcd Bsics At the end of this lecture, you will be ble to: Evlute mthemticl epression numericlly Assign vrible nd use them in subsequent clcultions Distinguish between
More informationFreely propagating jet
Freely propgting jet Introduction Gseous rectnts re frequently introduced into combustion chmbers s jets. Chemicl, therml nd flow processes tht re tking plce in the jets re so complex tht nlyticl description
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationCu 3 (PO 4 ) 2 (s) 3 Cu 2+ (aq) + 2 PO 4 3- (aq) circle answer: pure water or Na 3 PO 4 solution This is the common-ion effect.
CHEM 1122011 NAME: ANSWER KEY Vining, Exm # SHORT ANSWER QUESTIONS ============= 4 points ech ============ All work must be shown. 1. Wht re [H O ] nd [OH ] for solution tht hs ph of 9.0? Choose one. ()
More informationFirst Law of Thermodynamics. Control Mass (Closed System) Conservation of Mass. Conservation of Energy
First w of hermodynmics Reding Problems 3-3-7 3-0, 3-5, 3-05 5-5- 5-8, 5-5, 5-9, 5-37, 5-0, 5-, 5-63, 5-7, 5-8, 5-09 6-6-5 6-, 6-5, 6-60, 6-80, 6-9, 6-, 6-68, 6-73 Control Mss (Closed System) In this section
More informationProblems for HW X. C. Gwinn. November 30, 2009
Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object
More informationLecture 6: Diffusion and Reaction kinetics
Lecture 6: Diffusion nd Rection kinetics 1-1-1 Lecture pln: diffusion thermodynmic forces evolution of concentrtion distribution rection rtes nd methods to determine them rection mechnism in terms of the
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More information12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS
1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility
More informationPsychrometric Applications
Psychrometric Applictions The reminder of this presenttion centers on systems involving moist ir. A condensed wter phse my lso be present in such systems. The term moist irrefers to mixture of dry ir nd
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationThe solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr
Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the e-vectors nd e-vlues
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationLecture 13 - Linking E, ϕ, and ρ
Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More informationWhich of the following describes the net ionic reaction for the hydrolysis. Which of the following salts will produce a solution with the highest ph?
95. Which of the following descries the net ionic rection for the hydrolysis of NH4Cl( s)? A. NH4 ( q) Cl & ( q) NH4Cl( s) B. NH Cl & 4 ( s) NH4 ( q) Cl ( q) C. Cl ( q) H O & 2 ( l) HCl( q) OH ( q) D.
More informationCHEMICAL KINETICS
CHEMICAL KINETICS Long Answer Questions: 1. Explin the following terms with suitble exmples ) Averge rte of Rection b) Slow nd Fst Rections c) Order of Rection d) Moleculrity of Rection e) Activtion Energy
More informationWe will see what is meant by standard form very shortly
THEOREM: For fesible liner progrm in its stndrd form, the optimum vlue of the objective over its nonempty fesible region is () either unbounded or (b) is chievble t lest t one extreme point of the fesible
More informationTImath.com Algebra 2. Constructing an Ellipse
TImth.com Algebr Constructing n Ellipse ID: 9980 Time required 60 minutes Activity Overview This ctivity introduces ellipses from geometric perspective. Two different methods for constructing n ellipse
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationScientific notation is a way of expressing really big numbers or really small numbers.
Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific
More information- 5 - TEST 2. This test is on the final sections of this session's syllabus and. should be attempted by all students.
- 5 - TEST 2 This test is on the finl sections of this session's syllbus nd should be ttempted by ll students. Anything written here will not be mrked. - 6 - QUESTION 1 [Mrks 22] A thin non-conducting
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationHeteroclinic cycles in coupled cell systems
Heteroclinic cycles in coupled cell systems Michel Field University of Houston, USA, & Imperil College, UK Reserch supported in prt by Leverhulme Foundtion nd NSF Grnt DMS-0071735 Some of the reserch reported
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More information9-1 (a) A weak electrolyte only partially ionizes when dissolved in water. NaHCO 3 is an
Chpter 9 9- ( A ek electrolyte only prtilly ionizes hen dissolved in ter. NC is n exmple of ek electrolyte. (b A Brønsted-ory cid is cule tht dontes proton hen it encounters bse (proton cceptor. By this
More informationChemistry Department. The Islamic University of Gaza. General Chemistry B.(CHEMB 1301) Time:2 hours الرقم الجامعي... اسم المدرس...
The Islmic University of Gz Chemistry Deprtment Generl Chemistry B.(CHEMB 1301) Time:2 hours 60 اسم الطالب... الرقم الجامعي... اسم المدرس... R = 8.314 J/mol.K, or = 0.0821 L.tm/mol.K Q1- True ( ) or flse(
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationCONVERSION AND REACTOR SIZING (2) Marcel Lacroix Université de Sherbrooke
CONVERSION ND RECTOR SIZING (2) Marcel Lacroix Université de Sherbrooke CONVERSION ND RECTOR SIZING: OBJECTIVES 1. TO DEINE CONVERSION j. 2. TO REWRITE THE DESIGN EQUTIONS IN TERMS O CONVERSION j. 3. TO
More informationCalculus - Activity 1 Rate of change of a function at a point.
Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationMath 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationDIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
More informationProblem Set 3 Solutions
Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,
More informationThe use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.
ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More informationEcuaciones Algebraicas lineales
Ecuciones Algebrics lineles An eqution of the form x+by+c=0 or equivlently x+by=c is clled liner eqution in x nd y vribles. x+by+cz=d is liner eqution in three vribles, x, y, nd z. Thus, liner eqution
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationThe Thermodynamics of Aqueous Electrolyte Solutions
18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong
More informationMATH 13 FINAL STUDY GUIDE, WINTER 2012
MATH 13 FINAL TUY GUI, WINTR 2012 This is ment to be quick reference guide for the topics you might wnt to know for the finl. It probbly isn t comprehensive, but should cover most of wht we studied in
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationUsing QM for Windows. Using QM for Windows. Using QM for Windows LEARNING OBJECTIVES. Solving Flair Furniture s LP Problem
LEARNING OBJECTIVES Vlu%on nd pricing (November 5, 2013) Lecture 11 Liner Progrmming (prt 2) 10/8/16, 2:46 AM Olivier J. de Jong, LL.M., MM., MBA, CFD, CFFA, AA www.olivierdejong.com Solving Flir Furniture
More informationLecture 7 notes Nodal Analysis
Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationSections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation
Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q
More informationOrdinary Differential Equations- Boundary Value Problem
Ordinry Differentil Equtions- Boundry Vlue Problem Shooting method Runge Kutt method Computer-bsed solutions o BVPFD subroutine (Fortrn IMSL subroutine tht Solves (prmeterized) system of differentil equtions
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationCHAPTER 4a. ROOTS OF EQUATIONS
CHAPTER 4. ROOTS OF EQUATIONS A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskk Spring 00 ENCE 03 - Computtion Methods in Civil Engineering II Deprtment
More informationSynoptic Meteorology I: Finite Differences September Partial Derivatives (or, Why Do We Care About Finite Differences?
Synoptic Meteorology I: Finite Differences 16-18 September 2014 Prtil Derivtives (or, Why Do We Cre About Finite Differences?) With the exception of the idel gs lw, the equtions tht govern the evolution
More informationENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions
ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner non-repeted ctors. The cover-up rule my be used: 4 4 4
More information1 Online Learning and Regret Minimization
2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
More informationHeat flux and total heat
Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce
More informationEstimation of Binomial Distribution in the Light of Future Data
British Journl of Mthemtics & Computer Science 102: 1-7, 2015, Article no.bjmcs.19191 ISSN: 2231-0851 SCIENCEDOMAIN interntionl www.sciencedomin.org Estimtion of Binomil Distribution in the Light of Future
More informationWhen a force f(t) is applied to a mass in a system, we recall that Newton s law says that. f(t) = ma = m d dt v,
Impulse Functions In mny ppliction problems, n externl force f(t) is pplied over very short period of time. For exmple, if mss in spring nd dshpot system is struck by hmmer, the ppliction of the force
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory 2. Diffusion-Controlled Reaction
Ch. 4 Moleculr Rection Dynmics 1. Collision Theory. Diffusion-Controlle Rection Lecture 17 3. The Mteril Blnce Eqution 4. Trnsition Stte Theory: The Eyring Eqution 5. Trnsition Stte Theory: Thermoynmic
More informationELE B7 Power System Engineering. Unbalanced Fault Analysis
Power System Engineering Unblnced Fult Anlysis Anlysis of Unblnced Systems Except for the blnced three-phse fult, fults result in n unblnced system. The most common types of fults re single lineground
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2009
University of Wshington Deprtment of Chemistry Chemistry Winter Qurter 9 Homework Assignment ; Due t pm on //9 6., 6., 6., 8., 8. 6. The wve function in question is: ψ u cu ( ψs ψsb * cu ( ψs ψsb cu (
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationMATH 3795 Lecture 18. Numerical Solution of Ordinary Differential Equations.
MATH 3795 Lecture 18. Numericl Solution of Ordinry Differentil Equtions. Dmitriy Leykekhmn Fll 2008 Gols Introduce ordinry differentil equtions (ODEs) nd initil vlue problems (IVPs). Exmples of IVPs. Existence
More informationRel Gses 1. Gses (N, CO ) which don t obey gs lws or gs eqution P=RT t ll pressure nd tempertures re clled rel gses.. Rel gses obey gs lws t extremely low pressure nd high temperture. Rel gses devited
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More information13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More information4 The dynamical FRW universe
4 The dynmicl FRW universe 4.1 The Einstein equtions Einstein s equtions G µν = T µν (7) relte the expnsion rte (t) to energy distribution in the universe. On the left hnd side is the Einstein tensor which
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More information