Appendix II Summary of Important Equations
|
|
- Nancy Shepherd
- 5 years ago
- Views:
Transcription
1 W. M. Whte Geochemstry Equatons of State: Ideal GasLaw: Coeffcent of Thermal Expanson: Compressblty: Van der Waals Equaton: The Laws of Thermdynamcs: Frst Law: Appendx II Summary of Important Equatons PV = NRT α 1 V β 1 V V T V P P= RT V b - a V 2 U = Q + W 1 wrtten n dfferental form: du = dq + dw 2 work done on the system and heat added to the system are postve. The frst law states the equvalence of heat and work and the conservaton of energy. Second Law: dq rev = TdS 3 Two ways of statng the second law are Every system left to tself wll, on average, change to a condton of maxmum probablty and Heat cannot be extracted from a body and turned entrely nto work. Thrd Law: lm S = 0 T 0 3 Ths follows from the facts that S = R ln Ω and Ω =1 at T = 0 for a perfectly crystallne pure substance. Prmary Varables of Thermodynamcs The leadng thermodynamc propertes of a flud are determned by the relatons whch exst between the volume, pressure, termperature, energy and entropy of a gven mass of flud n a state of thermodynamc equlbrum - J. W. Gbbs The prmary varables of thermodynamcs are P, V, T, U, and S. Other thermodynamc functons can be stated n terms of these varables. For varous combnaton of these varables there are 1
2 W. M. Whte Geochemstry Appendx II Equaton Summary characterstcs functons. The characterstc functon for S and V s one of the prmary varables: U. Thus du = TdS + PdV 5 Other Important Thermodynamc Functons What then s the use of thermodynamc equatons? They are useful precsely because some quanttes are easer to measure than others. M. L. McGlashan Enthalpy: H U + PV 6 In dfferental form n terms of ts characterstc varables: dh = TdS + VdP 7 Helmholtz Free Energy: A U - TS 8 and: da = -PdV -SdT 9 Gbbs Free Energy: G H - TS 10 The Gbbs Free Energy change of a reacton at constant temperature and pressure s: G r = H r T S r and: dg = VdP - SdT 11 Your choce of whch of these functons to use should depend on what the ndependent varables n your system are. In geochemstry, P and T are the most common ndependent varables, so the Gbbs Free Energy s often the functon of choce. Exact Dfferentals and the Maxwell Relatons Any expresson that may be wrtten: M(x,y)dx+N(x,y)dy 12 s an exact dfferental f there exsts a functon z = f(x,y) such that f(x,y) = M(x,y)dx+N(x,y)dy 13 The total dfferental of the functon z(x,y) s wrtten: If dz s an exact dfferental, then whch s equvalent to: dz = z x y dx + z y x dy = Mdx + Ndy 14 M y = N x M y y = N x x 16 10a 15 2
3 W. M. Whte Geochemstry Appendx II Equaton Summary All thermodynamc varables of state are exact dfferentals. Thus the practcal applcaton of the propertes of exact dfferentals can be llustrated as follows. Equaton 11 (dg = VdP - SdT) has the form dz = M(x,y)dx+N(x,y)dy snce V and S are functons of temperature and pressure. Equaton 11 may also be wrtten as dg = G dp + G dt 17 P T T P and comparng equatons 11 and 16, we conclude that G P T =V and G T P = S 18, 19 Applyng the rule emboded n Equaton 15, we can conclude that: V = 20 T P P T Playng smlar games wth Equatons 5 through 9, we can develop a seres of relatonshps: from de from dh from da T = P 21 V S V T P V = P T V = Equatons are known as the Maxwell Relatons. V P 22 V T 23 Dervatves of Entropy pressure: P T = αv 24 temperature: T V = C V T and T P = C P T 25, 26 volume V T = α β 27 Dervatves of Enthalpy pressure temperature H P T = V(1 αt) 28 H T P =C P 29 Dervatves of Energy temperature: U T V =C V and U T P =C P PαV 30, 31 3
4 W. M. Whte Geochemstry Appendx II Equaton Summary volume: U V T = Tα β P 32 Dfference between C P and C V C P C V = TVα2 β 33 The Gbbs Phase Rule: The Gbbs Phase Rule s a rule for determnng the degrees of freedom of a system. f = c - p f s the number of degrees of freedom, c s the number of components, and p s the number of phases. The mnmum number of components needed to descrbe a system s: c = N - R where N s the number of speces, and R s the number of reactons possble between these speces. The Clapeyron Equaton The slope of a phase boundary n P-T space s: dt dp Solutons Raoult s Law: apples to deal solutons: = V r S r 35 P = X P total 36 Henry s Law: apples to very dlute solutons, and state that the partal pressure of a component n soluton s proportonal to t mole fracton: Chemcal Potental Chemcal potental s defned as: P =hx for X << 1 37 µ = G n P,T,n 38 where n s the number of moles of the th component. In multcomponent systems, the full expresson for the Gbbs Free Energy s: dg = VdP - SdT + µ dn 39 4
5 W. M. Whte Geochemstry Appendx II The Gbbs-Duhem Relaton At equlbrum and at constant pressure and temperature: Equaton Summary n dµ =0 40 Thermodynamc Varables n Ideal Solutons µ, deal = µ 0 +RTlnX 41 V deal mxng = 0 and therefore: V deal = X v H deal mxng = 0 and therefore: H deal = X h S deal mxng = -R S deal soluton = G deal mxng =RT G deal soluton = X ln X X S -R X ln X o X µ +RT X ln X X ln X = X V = X H Thermodynamc Varables n Non-Ideal Solutons Fugacty: Fugacty can be thought of as the escapng tendency of a gas n non-deal solutons. Because systems tend toward deal at low pressure, t has the property: lm ƒ P 0 =1 45 P and Actvty: Actvty s defned as: µ = µ o +RTln ƒ ƒ o 46 a f f o 47 hence: µ = µ o +RTlna 48 5
6 W. M. Whte Geochemstry Appendx II The actvty n an deal soluton s: The actvty coeffcent, λ, s defned as: Equaton Summary a,deal = X 49 a = X λ 50 When Henry's Law law holds: λ = h 51 The Debye-Hückel equaton s used to calculate actvty coeffcents n aqueous solutons. It s: 2 log 10 γ = -Az I 1+Bå I where z s charge, I s onc strength, å s the hydrated onc radus (sgnfcantly larger than onc radus), and A and B are solvent parameters. I s calculated as: I= 1 2 Excess Free Energy and actvty coeffcents: G excess =RT m z 2 X lnλ Excess Free Energy and Margules Parameters of a Regular Soluton: G ex =X 1 X 2 W G 55 Excess Free Energy and Margules Parameters of an Asymmetrc Soluton: G excess =W G1 X 2 +W G2 X 1 X 1 X 2 56 Equlbrum Constant The equlbrum constant s defned as: ν K= a It s related to the Gbbs Free Energy change of the reacton by: 57 K=e G /RT 58 It s related to enthalphy and entropy changes of the reacton by: o o ln K = H r RT + S r R Pressure and temperature dependences of the equlbrum constant are: o ln K P = V r T RT Oxdaton and Reducton: The redox potental s related to the Gbbs Free Energy change of reacton as: G = -nfe 61 6
7 W. M. Whte Geochemstry Appendx II Equaton Summary where E s the redox potental, n s the number of electrons exchanged and F s the Faraday constant. The Nernst Equaton s: E = E RT nf ln Πa ν 62 The pε s defned as: and s related to hydrogen scale redox potental, E H, as: pε = -log a e 63 pε = FE H 2.303RT 64 Knetcs Reacton Rates: For a reacton such as: aa + bb cc + dd A general form for the rate of a reacton s: 1 a da dt = 1 b db dt = 1 c dc dt = 1 d dd dt = k An A B n B C n c D n D 65 where n A, etc. can be any number. For an elementary reacton, ths reduces to: 1 a da dt = 1 b db dt = 1 c dc dt = 1 d dd dt = k Aa B b 66 The temperature dependence of the rate constant s gven by the Arrhenus Relaton: k = A exp - E B 67 RT Rate constants of elementary reactons are related to the equlbrum constant as: k + = [B] eq =K app 68 k - [A] eq Dffuson: Fck s Frst Law s: J= D c x 69 where J s the dffuson flux and D s the dffuson coeffcent. Fck s Second Law s: c =D t x 2 c x 2 t 70 7
8 W. M. Whte Geochemstry Appendx II Equaton Summary The temperature dependence of the dffuson coeffcent s: D=D o exp - E A RT 71 Dagentc Equaton: C t x = F + ΣR 72 x t Trace Elements Equlbrum or Batch Partal Meltng: C o = 1 C D s/ (1 F) + F 73 Fractonal Partal Meltng: C C o = 1 D ( 1 F) 1/D 1 74 Zone Refnng: C C o = 1 D (1 D 1)e DR 75 Equlbrum Crystallzaton: C l C o = 1 DX + (1 X) 76 Fractonal Crystallzaton: C l C o = 1 X D 1 77 Isotope Geochemstry Bndng Energy per Nucleon: E b = W M c 2 78 A Basc Equaton of Radoactve Decay: dn = λn 79 dt Isotope Growth (or Isochron) Equaton: R = R 0 + R P/D (e λt 1) 80 8
NAME and Section No. it is found that 0.6 mol of O
NAME and Secton No. Chemstry 391 Fall 7 Exam III KEY 1. (3 Ponts) ***Do 5 out of 6***(If 6 are done only the frst 5 wll be graded)*** a). In the reacton 3O O3 t s found that.6 mol of O are consumed. Fnd
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More information...Thermodynamics. If Clausius Clapeyron fails. l T (v 2 v 1 ) = 0/0 Second order phase transition ( S, v = 0)
If Clausus Clapeyron fals ( ) dp dt pb =...Thermodynamcs l T (v 2 v 1 ) = 0/0 Second order phase transton ( S, v = 0) ( ) dp = c P,1 c P,2 dt Tv(β 1 β 2 ) Two phases ntermngled Ferromagnet (Excess spn-up
More informationThermodynamics II. Department of Chemical Engineering. Prof. Kim, Jong Hak
Thermodynamcs II Department of Chemcal Engneerng Prof. Km, Jong Hak Soluton Thermodynamcs : theory Obectve : lay the theoretcal foundaton for applcatons of thermodynamcs to gas mxture and lqud soluton
More informationLecture. Polymer Thermodynamics 0331 L Chemical Potential
Prof. Dr. rer. nat. habl. S. Enders Faculty III for Process Scence Insttute of Chemcal Engneerng Department of Thermodynamcs Lecture Polymer Thermodynamcs 033 L 337 3. Chemcal Potental Polymer Thermodynamcs
More information4.2 Chemical Driving Force
4.2. CHEMICL DRIVING FORCE 103 4.2 Chemcal Drvng Force second effect of a chemcal concentraton gradent on dffuson s to change the nature of the drvng force. Ths s because dffuson changes the bondng n a
More informationIntroduction to Statistical Methods
Introducton to Statstcal Methods Physcs 4362, Lecture #3 hermodynamcs Classcal Statstcal Knetc heory Classcal hermodynamcs Macroscopc approach General propertes of the system Macroscopc varables 1 hermodynamc
More informationEnergy, Entropy, and Availability Balances Phase Equilibria. Nonideal Thermodynamic Property Models. Selecting an Appropriate Model
Lecture 4. Thermodynamcs [Ch. 2] Energy, Entropy, and Avalablty Balances Phase Equlbra - Fugactes and actvty coeffcents -K-values Nondeal Thermodynamc Property Models - P-v-T equaton-of-state models -
More informationReview of Classical Thermodynamics
Revew of Classcal hermodynamcs Physcs 4362, Lecture #1, 2 Syllabus What s hermodynamcs? 1 [A law] s more mpressve the greater the smplcty of ts premses, the more dfferent are the knds of thngs t relates,
More informationCHEMICAL REACTIONS AND DIFFUSION
CHEMICAL REACTIONS AND DIFFUSION A.K.A. NETWORK THERMODYNAMICS BACKGROUND Classcal thermodynamcs descrbes equlbrum states. Non-equlbrum thermodynamcs descrbes steady states. Network thermodynamcs descrbes
More informationIf two volatile and miscible liquids are combined to form a solution, Raoult s law is not obeyed. Use the experimental data in Table 9.
9.9 Real Solutons Exhbt Devatons from Raoult s Law If two volatle and mscble lquds are combned to form a soluton, Raoult s law s not obeyed. Use the expermental data n Table 9.3: Physcal Chemstry 00 Pearson
More informationa for save as PDF Chemistry 163B Introduction to Multicomponent Systems and Partial Molar Quantities
a for save as PDF Chemstry 163B Introducton to Multcomponent Systems and Partal Molar Quanttes 1 the problem of partal mmolar quanttes mx: 10 moles ethanol C 2 H 5 OH (580 ml) wth 1 mole water H 2 O (18
More informationNAME and Section No.
Chemstry 391 Fall 2007 Exam I KEY (Monday September 17) 1. (25 Ponts) ***Do 5 out of 6***(If 6 are done only the frst 5 wll be graded)*** a). Defne the terms: open system, closed system and solated system
More information3. Be able to derive the chemical equilibrium constants from statistical mechanics.
Lecture #17 1 Lecture 17 Objectves: 1. Notaton of chemcal reactons 2. General equlbrum 3. Be able to derve the chemcal equlbrum constants from statstcal mechancs. 4. Identfy how nondeal behavor can be
More informationChemical Equilibrium. Chapter 6 Spontaneity of Reactive Mixtures (gases) Taking into account there are many types of work that a sysem can perform
Ths chapter deals wth chemcal reactons (system) wth lttle or no consderaton on the surroundngs. Chemcal Equlbrum Chapter 6 Spontanety of eactve Mxtures (gases) eactants generatng products would proceed
More informationV T for n & P = constant
Pchem 365: hermodynamcs -SUMMARY- Uwe Burghaus, Fargo, 5 9 Mnmum requrements for underneath of your pllow. However, wrte your own summary! You need to know the story behnd the equatons : Pressure : olume
More informationand Statistical Mechanics Material Properties
Statstcal Mechancs and Materal Propertes By Kuno TAKAHASHI Tokyo Insttute of Technology, Tokyo 15-855, JAPA Phone/Fax +81-3-5734-3915 takahak@de.ttech.ac.jp http://www.de.ttech.ac.jp/~kt-lab/ Only for
More informationChapter 18, Part 1. Fundamentals of Atmospheric Modeling
Overhead Sldes for Chapter 18, Part 1 of Fundamentals of Atmospherc Modelng by Mark Z. Jacobson Department of Cvl & Envronmental Engneerng Stanford Unversty Stanford, CA 94305-4020 January 30, 2002 Types
More informationSolution Thermodynamics
Soluton hermodynamcs usng Wagner Notaton by Stanley. Howard Department of aterals and etallurgcal Engneerng South Dakota School of nes and echnology Rapd Cty, SD 57701 January 7, 001 Soluton hermodynamcs
More informationSupplementary Notes for Chapter 9 Mixture Thermodynamics
Supplementary Notes for Chapter 9 Mxture Thermodynamcs Key ponts Nne major topcs of Chapter 9 are revewed below: 1. Notaton and operatonal equatons for mxtures 2. PVTN EOSs for mxtures 3. General effects
More informationOsmotic pressure and protein binding
Osmotc pressure and proten bndng Igor R. Kuznetsov, KochLab Symposum talk 5/15/09 Today we take a closer look at one of the soluton thermodynamcs key ponts from Steve s presentaton. Here t s: d[ln(k off
More informationElectrochemical Equilibrium Electromotive Force
CHM465/865, 24-3, Lecture 5-7, 2 th Sep., 24 lectrochemcal qulbrum lectromotve Force Relaton between chemcal and electrc drvng forces lectrochemcal system at constant T and p: consder Gbbs free energy
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationSolution Thermodynamics
CH2351 Chemcal Engneerng Thermodynamcs II Unt I, II www.msubbu.n Soluton Thermodynamcs www.msubbu.n Dr. M. Subramanan Assocate Professor Department of Chemcal Engneerng Sr Svasubramanya Nadar College of
More informationIntroduction to Interfacial Segregation. Xiaozhe Zhang 10/02/2015
Introducton to Interfacal Segregaton Xaozhe Zhang 10/02/2015 Interfacal egregaton Segregaton n materal refer to the enrchment of a materal conttuent at a free urface or an nternal nterface of a materal.
More informationName: SID: Discussion Session:
Name: SID: Dscusson Sesson: Chemcal Engneerng Thermodynamcs 141 -- Fall 007 Thursday, November 15, 007 Mdterm II SOLUTIONS - 70 mnutes 110 Ponts Total Closed Book and Notes (0 ponts) 1. Evaluate whether
More informationUniversity of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014
Lecture 16 8/4/14 Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 214. Real Vapors and Fugacty Henry s Law accounts or the propertes o extremely dlute soluton. s shown n Fgure
More informationMass Transfer Processes
Mass Transfer Processes S. Majd Hassanzadeh Department of Earth Scences Faculty of Geoscences Utrecht Unversty Outlne: 1. Measures of Concentraton 2. Volatlzaton and Dssoluton 3. Adsorpton Processes 4.
More informationChapter 5. Simple Mixtures Fall Semester Physical Chemistry 1 (CHM2201)
Chapter 5. Simple Mixtures 2011 Fall Semester Physical Chemistry 1 (CHM2201) Contents The thermodynamic description of mixtures 5.1 Partial molar quantities 5.2 The thermodynamic of Mixing 5.3 The chemical
More informationThe ChemSep Book. Harry A. Kooijman Consultant. Ross Taylor Clarkson University, Potsdam, New York University of Twente, Enschede, The Netherlands
The ChemSep Book Harry A. Koojman Consultant Ross Taylor Clarkson Unversty, Potsdam, New York Unversty of Twente, Enschede, The Netherlands Lbr Books on Demand www.bod.de Copyrght c 2000 by H.A. Koojman
More informationChemistry. Lecture 10 Maxwell Relations. NC State University
Chemistry Lecture 10 Maxwell Relations NC State University Thermodynamic state functions expressed in differential form We have seen that the internal energy is conserved and depends on mechanical (dw)
More informationThermodynamics General
Thermodynamcs General Lecture 1 Lecture 1 s devoted to establshng buldng blocks for dscussng thermodynamcs. In addton, the equaton of state wll be establshed. I. Buldng blocks for thermodynamcs A. Dmensons,
More informationI wish to publish my paper on The International Journal of Thermophysics. A Practical Method to Calculate Partial Properties from Equation of State
I wsh to publsh my paper on The Internatonal Journal of Thermophyscs. Ttle: A Practcal Method to Calculate Partal Propertes from Equaton of State Authors: Ryo Akasaka (correspondng author) 1 and Takehro
More informationTP A SOLUTION. For an ideal monatomic gas U=3/2nRT, Since the process is at constant pressure Q = C. giving ) =1000/(5/2*8.31*10)
T A SOLUTION For an deal monatomc gas U/nRT, Snce the process s at constant pressure Q C pn T gvng a: n Q /( 5 / R T ) /(5/*8.*) C V / R and C / R + R 5 / R. U U / nr T (/ ) R T ( Q / 5 / R T ) Q / 5 Q
More informationy i x P vap 10 A T SOLUTION TO HOMEWORK #7 #Problem
SOLUTION TO HOMEWORK #7 #roblem 1 10.1-1 a. In order to solve ths problem, we need to know what happens at the bubble pont; at ths pont, the frst bubble s formed, so we can assume that all of the number
More informationModelli Clamfim Equazioni differenziali 22 settembre 2016
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 22 settembre 2016 professor Danele Rtell danele.rtell@unbo.t 1/22? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More information( ) 1/ 2. ( P SO2 )( P O2 ) 1/ 2.
Chemstry 360 Dr. Jean M. Standard Problem Set 9 Solutons. The followng chemcal reacton converts sulfur doxde to sulfur troxde. SO ( g) + O ( g) SO 3 ( l). (a.) Wrte the expresson for K eq for ths reacton.
More information3) Thermodynamic equation to characterize interfaces
3) Thermodynamc equaton to characterze nterfaces 3.1) Gbbs Model Realty: rapd contnuous change of chemcal and thermodynamc propertes Replaced by model (constant propertes up to the surface) uv bulk uv
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationNon-Ideality Through Fugacity and Activity
Non-Idealty Through Fugacty and Actvty S. Patel Deartment of Chemstry and Bochemstry, Unversty of Delaware, Newark, Delaware 19716, USA Corresondng author. E-mal: saatel@udel.edu 1 I. FUGACITY In ths dscusson,
More information5.60 Thermodynamics & Kinetics Spring 2008
MIT OpenCourseWare http://ocw.mt.edu 5.60 Thermodynamcs & Knetcs Sprng 2008 For nformaton about ctng these materals or our Terms of Use, vst: http://ocw.mt.edu/terms. 5.60 Sprng 2008 Lecture #29 page 1
More informationGeneral Thermodynamics for Process Simulation. Dr. Jungho Cho, Professor Department of Chemical Engineering Dong Yang University
General Thermodynamcs for Process Smulaton Dr. Jungho Cho, Professor Department of Chemcal Engneerng Dong Yang Unversty Four Crtera for Equlbra μ = μ v Stuaton α T = T β α β P = P l μ = μ l1 l 2 Thermal
More informationThermodynamic Variables and Relations
MME 231: Lecture 10 Thermodynamic Variables and Relations A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Today s Topics Thermodynamic relations derived from the Laws of Thermodynamics Definitions
More informationA quote of the week (or camel of the week): There is no expedience to which a man will not go to avoid the labor of thinking. Thomas A.
A quote of the week (or camel of the week): here s no expedence to whch a man wll not go to avod the labor of thnkng. homas A. Edson Hess law. Algorthm S Select a reacton, possbly contanng specfc compounds
More informationbetween standard Gibbs free energies of formation for products and reactants, ΔG! R = ν i ΔG f,i, we
hermodynamcs, Statstcal hermodynamcs, and Knetcs 4 th Edton,. Engel & P. ed Ch. 6 Part Answers to Selected Problems Q6.. Q6.4. If ξ =0. mole at equlbrum, the reacton s not ery far along. hus, there would
More informationThermodynamics and statistical mechanics in materials modelling II
Course MP3 Lecture 8/11/006 (JAE) Course MP3 Lecture 8/11/006 Thermodynamcs and statstcal mechancs n materals modellng II A bref résumé of the physcal concepts used n materals modellng Dr James Ellott.1
More information,..., k N. , k 2. ,..., k i. The derivative with respect to temperature T is calculated by using the chain rule: & ( (5) dj j dt = "J j. k i.
Suppleentary Materal Dervaton of Eq. 1a. Assue j s a functon of the rate constants for the N coponent reactons: j j (k 1,,..., k,..., k N ( The dervatve wth respect to teperature T s calculated by usng
More informationModelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationEnvr 210, Chapter 3, Intermolecular forces and partitioning Free energies and equilibrium partitioning chemical potential fugacity activity coef.
Envr 20, Chapter 3, Intermolecular forces and parttonng Free energes and equlbrum parttonng chemcal potental fugacty actvty coef. phase transfer- actvty coef and fugactes more on free energes and equlbrum
More informationIntroduction to geochemical modeling
Introducton to geochemcal modelng Prof. Dr. Broder J. Merkel Char of Hydrogeology Technsche Unverstät Bergakademe Freberg Germany What determnes the dstrbuton of aquatc speces? Interactons of aquatc speces
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be well-organzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationTopic 3 : Thermodynamics
GEOL360 LECTURE NOTES: T3 : THERMODYNAMICS 1/20 GEOL360 Topc 3 : Thermodynamcs 3.1 Introducton and vocabulary Thermodynamcs deals wth the physcal and chemcal changes of matter due to work and hear flow
More informationOCN 623: Thermodynamic Laws & Gibbs Free Energy. or how to predict chemical reactions without doing experiments
OCN 623: Thermodynamic Laws & Gibbs Free Energy or how to predict chemical reactions without doing experiments Definitions Extensive properties Depend on the amount of material e.g. # of moles, mass or
More information10.40 Appendix Connection to Thermodynamics and Derivation of Boltzmann Distribution
10.40 Appendx Connecton to Thermodynamcs Dervaton of Boltzmann Dstrbuton Bernhardt L. Trout Outlne Cannoncal ensemble Maxmumtermmethod Most probable dstrbuton Ensembles contnued: Canoncal, Mcrocanoncal,
More information#64. ΔS for Isothermal Mixing of Ideal Gases
#64 Carnot Heat Engne ΔS for Isothermal Mxng of Ideal Gases ds = S dt + S T V V S = P V T T V PV = nrt, P T ds = v T = nr V dv V nr V V = nrln V V = - nrln V V ΔS ΔS ΔS for Isothermal Mxng for Ideal Gases
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationChemistry 163B Free Energy and Equilibrium E&R ( ch 6)
Chemstry 163B Free Energy and Equlbrum E&R ( ch 6) 1 ΔG reacton and equlbrum (frst pass) 1. ΔG < spontaneous ( natural, rreversble) ΔG = equlbrum (reversble) ΔG > spontaneous n reverse drecton. ΔG = ΔHΔS
More informationBasic concept of reactive flows. Basic concept of reactive flows Combustion Mixing and reaction in high viscous fluid Application of Chaos
Introducton to Toshhsa Ueda School of Scence for Open and Envronmental Systems Keo Unversty, Japan Combuston Mxng and reacton n hgh vscous flud Applcaton of Chaos Keo Unversty 1 Keo Unversty 2 What s reactve
More informationUniversity of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2014
Lecture 12 7/25/14 ERD: 7.1-7.5 Devoe: 8.1.1-8.1.2, 8.2.1-8.2.3, 8.4.1-8.4.3 Unversty o Washngton Department o Chemstry Chemstry 452/456 Summer Quarter 2014 A. Free Energy and Changes n Composton: The
More informationThe Second Law of Thermodynamics (Chapter 4)
The Second Law of Thermodynamics (Chapter 4) First Law: Energy of universe is constant: ΔE system = - ΔE surroundings Second Law: New variable, S, entropy. Changes in S, ΔS, tell us which processes made
More informationChapters 18 & 19: Themodynamics review. All macroscopic (i.e., human scale) quantities must ultimately be explained on the microscopic scale.
Chapters 18 & 19: Themodynamcs revew ll macroscopc (.e., human scale) quanttes must ultmately be explaned on the mcroscopc scale. Chapter 18: Thermodynamcs Thermodynamcs s the study o the thermal energy
More informationUniversity of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2013
Lecture 8/8/3 Unversty o Washngton Departent o Chestry Chestry 45/456 Suer Quarter 3 A. The Gbbs-Duhe Equaton Fro Lecture 7 and ro the dscusson n sectons A and B o ths lecture, t s clear that the actvty
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationII. Equilibrium Thermodynamics. Lecture 8: The Nernst Equation
II. Equlbrum Thermodynamcs Lecture 8: The Nernst Equaton Notes by MIT Student, re-wrtten by MZB 2014 February 27, 2014 1 Chemcal Actvty The fundamental thermodynamc quantty controllng transport and reactons
More informationV. Electrostatics. Lecture 25: Diffuse double layer structure
V. Electrostatcs Lecture 5: Dffuse double layer structure MIT Student Last tme we showed that whenever λ D L the electrolyte has a quas-neutral bulk (or outer ) regon at the geometrcal scale L, where there
More informationSome properties of the Helmholtz free energy
Some properties of the Helmholtz free energy Energy slope is T U(S, ) From the properties of U vs S, it is clear that the Helmholtz free energy is always algebraically less than the internal energy U.
More informationMaximum Likelihood Estimation of Binary Dependent Variables Models: Probit and Logit. 1. General Formulation of Binary Dependent Variables Models
ECO 452 -- OE 4: Probt and Logt Models ECO 452 -- OE 4 Maxmum Lkelhood Estmaton of Bnary Dependent Varables Models: Probt and Logt hs note demonstrates how to formulate bnary dependent varables models
More informationModule 5 : Electrochemistry Lecture 21 : Review Of Thermodynamics
Module 5 : Electrochemistry Lecture 21 : Review Of Thermodynamics Objectives In this Lecture you will learn the following The need for studying thermodynamics to understand chemical and biological processes.
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationLecture 16 Statistical Analysis in Biomaterials Research (Part II)
3.051J/0.340J 1 Lecture 16 Statstcal Analyss n Bomaterals Research (Part II) C. F Dstrbuton Allows comparson of varablty of behavor between populatons usng test of hypothess: σ x = σ x amed for Brtsh statstcan
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationThermodynamics of Materials
Thermdynamcs f Materals 14th Lecture 007. 4. 8 (Mnday) FUGACITY dg = Vd SdT dg = Vd at cnstant T Fr an deal gas dg = (RT/)d = RT dln Ths s true fr deal gases nly, but t wuld be nce t have a smlar frm fr
More informationGouy-Chapman model (1910) The double layer is not as compact as in Helmholtz rigid layer.
CHE465/865, 6-3, Lecture 1, 7 nd Sep., 6 Gouy-Chapman model (191) The double layer s not as compact as n Helmholtz rgd layer. Consder thermal motons of ons: Tendency to ncrease the entropy and make the
More informationEntropy generation in a chemical reaction
Entropy generaton n a chemcal reacton E Mranda Área de Cencas Exactas COICET CCT Mendoza 5500 Mendoza, rgentna and Departamento de Físca Unversdad aconal de San Lus 5700 San Lus, rgentna bstract: Entropy
More informationLecture 2 Grand Canonical Ensemble GCE
Lecture 2 Grand Canoncal Ensemble GCE 2.1 hermodynamc Functons Contnung on from last day we also note that thus, dω = df dµ µd = Sd P dv dµ (2.1) P = V = S = From the expresson for the entropy, we therefore
More informationComputation of Phase Equilibrium and Phase Envelopes
Downloaded from orbt.dtu.dk on: Sep 24, 2018 Computaton of Phase Equlbrum and Phase Envelopes Rtschel, Tobas Kasper Skovborg; Jørgensen, John Bagterp Publcaton date: 2017 Document Verson Publsher's PDF,
More informationAPPROXIMATE PRICES OF BASKET AND ASIAN OPTIONS DUPONT OLIVIER. Premia 14
APPROXIMAE PRICES OF BASKE AND ASIAN OPIONS DUPON OLIVIER Prema 14 Contents Introducton 1 1. Framewor 1 1.1. Baset optons 1.. Asan optons. Computng the prce 3. Lower bound 3.1. Closed formula for the prce
More informationElectrochemistry Thermodynamics
CHEM 51 Analytcal Electrochemstry Chapter Oct 5, 016 Electrochemstry Thermodynamcs Bo Zhang Department of Chemstry Unversty of Washngton Seattle, WA 98195 Former SEAC presdent Andy Ewng sellng T-shrts
More information10, Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics)
Subect Chemistry Paper No and Title Module No and Title Module Tag 0, Physical Chemistry- III (Classical Thermodynamics, Non-Equilibrium Thermodynamics, Surface chemistry, Fast kinetics) 0, Free energy
More informationProcess Modeling. Improving or understanding chemical process operation is a major objective for developing a dynamic process model
Process Modelng Improvng or understandng chemcal process operaton s a major objectve for developng a dynamc process model Balance equatons Steady-state balance equatons mass or energy mass or energy enterng
More informationG4023 Mid-Term Exam #1 Solutions
Exam1Solutons.nb 1 G03 Md-Term Exam #1 Solutons 1-Oct-0, 1:10 p.m to :5 p.m n 1 Pupn Ths exam s open-book, open-notes. You may also use prnt-outs of the homework solutons and a calculator. 1 (30 ponts,
More informationPHY688, Statistical Mechanics
Department of Physcs & Astronomy 449 ESS Bldg. Stony Brook Unversty January 31, 2017 Nuclear Astrophyscs James.Lattmer@Stonybrook.edu Thermodynamcs Internal Energy Densty and Frst Law: ε = E V = Ts P +
More informationUniversity of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015
Lecture 2. 1/07/15-1/09/15 Unversty of Washngton Department of Chemstry Chemstry 453 Wnter Quarter 2015 We are not talkng about truth. We are talkng about somethng that seems lke truth. The truth we want
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationA Self-Consistent Gibbs Excess Mixing Rule for Cubic Equations of State: derivation and fugacity coefficients
A Self-Consstent Gbbs Excess Mxng Rule for Cubc Equatons of State: dervaton and fugacty coeffcents Paula B. Staudt, Rafael de P. Soares Departamento de Engenhara Químca, Escola de Engenhara, Unversdade
More informationChapter One Mixture of Ideal Gases
herodynacs II AA Chapter One Mxture of Ideal Gases. Coposton of a Gas Mxture: Mass and Mole Fractons o deterne the propertes of a xture, we need to now the coposton of the xture as well as the propertes
More informationThe underlying prerequisite to the application of thermodynamic principles to natural systems is that the system under consideration should be at equilibrium. http://eps.mcgill.ca/~courses/c220/ Reversible
More informationChE 512: Topic 1 Reactions at a fluid non-porous solid interface. P.A. Ramachandran
he 512: Topc 1 Reactons at a flud non-porous sold nterface P.. Ramachandran rama@wustl.edu OUTLIE External Transport: Flm oncept Mass transfer coeffcents Effect of transport on reacton multaneous heat
More informationLecture 8. Chapter 7. - Thermodynamic Web - Departure Functions - Review Equations of state (chapter 4, briefly)
Lecture 8 Chapter 5 - Thermodynamc Web - Departure Functons - Revew Equatons of state (chapter 4, brefly) Chapter 6 - Equlbrum (chemcal potental) * Pure Component * Mxtures Chapter 7 - Fugacty (chemcal
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More informationLecture 7: Boltzmann distribution & Thermodynamics of mixing
Prof. Tbbtt Lecture 7 etworks & Gels Lecture 7: Boltzmann dstrbuton & Thermodynamcs of mxng 1 Suggested readng Prof. Mark W. Tbbtt ETH Zürch 13 März 018 Molecular Drvng Forces Dll and Bromberg: Chapters
More informationThermodynamics. Section 4
Secton 4 Thermodynamcs Hendrck C. Van Ness, D.Eng., Howard. Isermann Department of Chemcal Engneerng, Rensselaer olytechnc Insttute; Fellow, Amercan Insttute of Chemcal Engneers; Member, Amercan Chemcal
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationStatistical mechanics handout 4
Statstcal mechancs handout 4 Explan dfference between phase space and an. Ensembles As dscussed n handout three atoms n any physcal system can adopt any one of a large number of mcorstates. For a quantum
More informationFormal solvers of the RT equation
Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed mult-partcle systems! Internal and external forces! Laws of acton and
More informationESCI 341 Atmospheric Thermodynamics Lesson 12 The Energy Minimum Principle
ESCI 341 Atmospheric Thermodynamics Lesson 12 The Energy Minimum Principle References: Thermodynamics and an Introduction to Thermostatistics, Callen Physical Chemistry, Levine THE ENTROPY MAXIMUM PRINCIPLE
More information