Multivariable Calculus
|
|
- Ruby Mitchell
- 6 years ago
- Views:
Transcription
1 Multivariable Calculus In thermodynamics, we will frequently deal with functions of more than one variable e.g., P PT, V, n, U UT, V, n, U UT, P, n U = energy n = # moles etensive variable: depends on the size of the system intensive variable: independent of the size of the system V, n are etensive, P, T, molar volume V/n are intensive change volume at fied T, n pressure changes P dp V nt, dv these variables are held constant cylinder with piston
2 Suppose we have an ideal gas PV nrt P V P T P n nt, Vn, VT, nrt V nr V RT V subscripts refer to variables that are held constant now suppose we want to see how P changes when both V and T change P P dp dv dt V T Tn, Vn, Note that this is fully consistent with Taylor series epansions If n changes as well need to add a P n TV, dn term
3 In general, y y,, dy n i1 Ideal gas 1 y n ' n d, i ' hold fied all variables ecept i P P P dp dt dv dn T V n nv, T, n TV, nr nrt RT dp dt dv dn V V V for small finite changes nr nrt RT P T V n V V V
4 These equations are most useful when we don't have an analytical function for the quantity of interest. Consider U U( T, P, n) U U U du dt dp dn T P n Pn, Tn, T, P The derivatives are often available eperimentally. We can also write U U T, V, n U U U du dt dv dn T V n Vn, Tn, TV, Note, in general, U U T T P, n V, n
5 Eample 3 z a bycy z a by y Suppose u y then cu z a bu z cu a 3 u Comment: The z, y in the tet should be z z, y Comment: if you work out eample 8.4, note that when it refers to eample 8., it should be 8.3. Also, in the last term of the answer, the u should be u. Above, we saw U U T T P V We may want to know how these two quantities are related (variable change identity)
6 Consider U U T, P, n U U U du dt dp dn T P n Pn, Tn, T, P du U U dp U dn dt T P dt n dt Pn, Tn, T, P U U U P U n T T P T n T Vn, Pn, Tn, Vn, TP, Vn, Note: This essentially follows from a Taylor U series UT, P, n U0 T T0 T Pn, U U U P T T, P, T, Vn, Pn Tn Vn 0 for small changes T T0 dt We have now obtained the relation between the two derivatives of interest.
7 Reciprocal identity y 1 zu, y zu, P 1 e.g., V V nt, P nt, Eample z sin y arcsin z y Show dz 1 in this case d y z Second derivatives E. z z is y y y z z y y U V VT V T y n Vn, Tn, y
8 Euler Reciprocity z y z y It is normal to suppress the info on the variable(s) held constant Mawell relations Consider du TdS PdV U T S V U P V S You will derive this in P Chem. S = entropy, n fied, system assumed to be reversible T U V V S S P U S S V V T P V S Sn, Vn, T P is much easier to measure than V S Sn, Vn,
9 da SdT PdV Helmholtz free energy A U TS da SdT PdV A A S P T V V S A V V T S P P A V T T T V T V T Tn, Vn, Tet describes a rule for finding these derivative relationships but it is confusing. It is easier to start from the definitions of U, H, A, G energies.
10 Cycle rule Show that y z z y z y 1 Start with y y dy d dz z z choose d and dz so that dy = 0 y y z 0 z z y or y z 1 z y z y Chain rule Suppose z = z(u,,y) and = ( u,v,y) z z y y uv, uv, uv,
11 Important thermodynamic quantities C C K K P V T S H S T T T Pn, Pn, U S T T T Vn, Vn, 1 V V P 1 V V P 1 V V T dq ds T rev Pn, Tn, Sn, Note: There are constant volume and constant pressure heat capacities H U PV enthalpy isothermal compressibility adiabatic compressibility coefficient of thermal epansion definition of entropy Eample 8.10 C C P V K K T S S S P S V V T T P T P S P S S V V T V T T V T T P P P T S T T T T V T S S S S S
12 Eact and Ineact Differentials Suppose,, du M y d N y dy This is called an eact differential if M u u and N y y If M and N are as above M u N u y y y y Then we have an eact differential If there is no function u for which this is true, du is an ineact differential
13 Eample dz y d dy y y Note: Typo in tet 9 9 y y y y 3 9 y y 3 Since these are equal, we are dealing with an eact differential More general case For ;,,,,, dum yzdn yzdyp yzdz Note: Typo in tet Eact differential if M N M P, y z z, N P z y y, z, yz, y, yz,
14 Heat + work under reversible conditions heat transferred = dq rev work done on system = dw rev if work is associated only with a volume change dwrev PdV Check to see if dw is an eact differential dw MdT NdV but we have already said that for a reversible process, this is 0, so dw rev is not an eact differential V, T, P. n are state functions w and q are not state functions If only work that is due to volume change du dq dw state function
15 Integrating Factors Sometimes, one can find a factor that, when multiplying an ineact differential, makes it eact Eample: du Md Ndy a by d b cy dy 1 is an integrating factor for du du a byd b cydy y a by b b cy b So du is an eact differential dq rev As it turns out, S is an eact differential, where S is the entropy T
16 Maima and Minima of functions of more than one variable y y Maimum in the direction but minimum in the y direction (saddle point) Maimum in both the and y directions f f Given f, y, we need to search for 0 and 0 y y This tells us that we found a point where the slope = 0, but does not tell us what sort of special point this is.
17 nd derivative matri f f y f y f y f f f D y y f 1 D 0, and 0 local minimum f f 0 0 y f D 0, and 0 3 D 0, local maimum neither maimum nor minimum f f 0 0 y 4 D 0, cannot tell what sort of point it is
18 Constrained optimization Eample:, y f y e subject to constraint g y, y1 y 1 f ( 1) 4 e ( 1) f e e e e e which then gives y 1 1 = is also a solution 1 1 f, The unconstrained maimum of f is 1.
19 An alternative approach Lagrange multipliers find minimum or maimum of f y, subject to the constraint g, y 0 Define a function u such that u, y f, y g, y u f g 0 y y y u u g 0 y y y solve these together with g = 0 Eample: y f e, g y1 y u e y1 u y u y y e 0 y ye 0 y ye y y 0 ye 0 0 y y g y , y Note: this is a simpler derivation of the approach taken in the tet
20 A more general case f f, y, z subject to two constraints g g 1, y, z 0, y, z 0 u f g g u f g g yz, yz, yz, yz, u f g g y y y y z, z, z, z, u f g g 0 z, 1 1 y, z y, z y, z y together with g1 0, g 0
21 Vector operators Gradient ˆ f ˆ f i j f kˆ f y z Note typo in book Gradient of a scalar direction in which function increases most rapidly magnitude is the rate of change in that direction Eample: 3 bz g a ye g 3a iˆe ˆjbye k bz bz ˆ F V, V potential, F = force Gravitational potential, V Gm1m r r y z Gm1m F V iˆ yj ˆkzˆ 3 r 1 1 y z y z 3/
22 Divergence operating on a vector F F F F y z y scalar z ˆ ˆ i j kˆ if ˆ ˆjF kf ˆ y z y z E.g., Continuity Eq. v t = density, v velocity Eample ˆ ˆ ˆ kz F i jyz y z F z y
23 Curl ˆ F F z y F Fz ˆ Fy F F i j k y z z y Laplacian recall C AB iˆab ˆ ˆ y z AB z y j AB z AB z k AB y AB y f f f y z f Spherical coordinates dr e dr e rd e r d ˆ ˆ ˆ sin r z φ θ r y dsr dr ds rd ds r d sin f 1 f 1 f f eˆ ˆ ˆ r e e r r rsin Divergence Curve Laplacian can all be represented in spherical coordinates f r f sin f f r r r r sin r sin
Chemistry. 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 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 informationChapter 3. Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc.
Chapter 3 Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc. Concepts Energy functions F and G Chemical potential, µ Partial Molar properties
More informationThermodynamics of phase transitions
Thermodynamics of phase transitions Katarzyna Sznajd-Weron Department of Theoretical of Physics Wroc law University of Science and Technology, Poland March 12, 2017 Katarzyna Sznajd-Weron (WUST) Thermodynamics
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 informationThe Euler Equation. Using the additive property of the internal energy U, we can derive a useful thermodynamic relation the Euler equation.
The Euler Equation Using the additive property of the internal energy U, we can derive a useful thermodynamic relation the Euler equation. Let us differentiate this extensivity condition with respect to
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 informationI.G Approach to Equilibrium and Thermodynamic Potentials
I.G Approach to Equilibrium and Thermodynamic otentials Evolution of non-equilibrium systems towards equilibrium is governed by the second law of thermodynamics. For eample, in the previous section we
More informationChapter 4: Partial differentiation
Chapter 4: Partial differentiation It is generally the case that derivatives are introduced in terms of functions of a single variable. For example, y = f (x), then dy dx = df dx = f. However, most of
More informationWHY SHOULD WE CARE ABOUT THERMAL PHENOMENA? they can profoundly influence dynamic behavior. MECHANICS.
WORK-TO-HEAT TRANSDUCTION IN THERMO-FLUID SYSTEMS ENERGY-BASED MODELING IS BUILT ON THERMODYNAMICS the fundamental science of physical processes. THERMODYNAMICS IS TO PHYSICAL SYSTEM DYNAMICS WHAT GEOMETRY
More informationIntroduction. Statistical physics: microscopic foundation of thermodynamics degrees of freedom 2 3 state variables!
Introduction Thermodynamics: phenomenological description of equilibrium bulk properties of matter in terms of only a few state variables and thermodynamical laws. Statistical physics: microscopic foundation
More informationChapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics.
Chapter 3. Entropy, temperature, and the microcanonical partition function: how to calculate results with statistical mechanics. The goal of equilibrium statistical mechanics is to calculate the density
More informationI.G Approach to Equilibrium and Thermodynamic Potentials
I.G Approach to Equilibrium and Thermodynamic Potentials Evolution of non-equilibrium systems towards equilibrium is governed by the second law of thermodynamics. For eample, in the previous section we
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 informationMATTER TRANSPORT (CONTINUED)
MATTER TRANSPORT (CONTINUED) There seem to be two ways to identify the effort variable for mass flow gradient of the energy function with respect to mass is matter potential, µ (molar) specific Gibbs free
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 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 informationChemistry 223: State Functions, Exact Differentials, and Maxwell Relations David Ronis McGill University
Chemistry 223: State Functions, Exact Differentials, and Maxwell Relations David Ronis McGill University Consider the differential form: df M(x, y)dx + N(x, y)dy. (1) If can we define a single-valued,
More informationNENG 301 Week 8 Unary Heterogeneous Systems (DeHoff, Chap. 7, Chap )
NENG 301 Week 8 Unary Heterogeneous Systems (DeHoff, Chap. 7, Chap. 5.3-5.4) Learning objectives for Chapter 7 At the end of this chapter you will be able to: Understand the general features of a unary
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 informationEffect of adding an ideal inert gas, M
Effect of adding an ideal inert gas, M Add gas M If there is no change in volume, then the partial pressures of each of the ideal gas components remains unchanged by the addition of M. If the reaction
More informationThe Standard Gibbs Energy Change, G
The Standard Gibbs Energy Change, G S univ = S surr + S sys S univ = H sys + S sys T S univ = H sys TS sys G sys = H sys TS sys Spontaneous reaction: S univ >0 G sys < 0 More observations on G and Gº I.
More informationApplied Thermodynamics for Marine Systems Prof. P. K. Das Department of Mechanical Engineering Indian Institute of Technology, Kharagpur
Applied Thermodynamics for Marine Systems Prof. P. K. Das Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 8 Introduction to Vapour Power Cycle Today, we will continue
More informationChapter 19 The First Law of Thermodynamics
Chapter 19 The First Law of Thermodynamics The first law of thermodynamics is an extension of the principle of conservation of energy. It includes the transfer of both mechanical and thermal energy. First
More informationProperties of Entropy
Properties of Entropy Due to its additivity, entropy is a homogeneous function of the extensive coordinates of the system: S(λU, λv, λn 1,, λn m ) = λ S (U, V, N 1,, N m ) This means we can write the entropy
More informationBrief introduction to groups and group theory
Brief introduction to groups and group theory In physics, we often can learn a lot about a system based on its symmetries, even when we do not know how to make a quantitative calculation Relevant in particle
More informationChapter 3 - First Law of Thermodynamics
Chapter 3 - dynamics The ideal gas law is a combination of three intuitive relationships between pressure, volume, temp and moles. David J. Starling Penn State Hazleton Fall 2013 When a gas expands, it
More informationMore Thermodynamics. Specific Specific Heats of a Gas Equipartition of Energy Reversible and Irreversible Processes
More Thermodynamics Specific Specific Heats of a Gas Equipartition of Energy Reversible and Irreversible Processes Carnot Cycle Efficiency of Engines Entropy More Thermodynamics 1 Specific Heat of Gases
More informationMath 212-Lecture 8. The chain rule with one independent variable
Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle
More informationMME 2010 METALLURGICAL THERMODYNAMICS II. Partial Properties of Solutions
MME 2010 METALLURGICAL THERMODYNAMICS II Partial Properties of Solutions A total property of a system consisting of multiple substances is represented as nm = n i M i If the system consists of a liquid
More informationU = 4.18 J if we heat 1.0 g of water through 1 C. U = 4.18 J if we cool 1.0 g of water through 1 C.
CHAPER LECURE NOES he First Law of hermodynamics: he simplest statement of the First Law is as follows: U = q + w. Here U is the internal energy of the system, q is the heat and w is the work. CONVENIONS
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 informationUNIVERSITY OF SOUTHAMPTON
UNIVERSITY OF SOUTHAMPTON PHYS1013W1 SEMESTER 2 EXAMINATION 2014-2015 ENERGY AND MATTER Duration: 120 MINS (2 hours) This paper contains 8 questions. Answers to Section A and Section B must be in separate
More informationThermodynamics (Classical) for Biological Systems Prof. G. K. Suraishkumar Department of Biotechnology Indian Institute of Technology Madras
Thermodynamics (Classical) for Biological Systems Prof. G. K. Suraishkumar Department of Biotechnology Indian Institute of Technology Madras Module No. # 02 Additional Thermodynamic Functions Lecture No.
More information140a Final Exam, Fall 2006., κ T 1 V P. (? = P or V ), γ C P C V H = U + PV, F = U TS G = U + PV TS. T v. v 2 v 1. exp( 2πkT.
40a Final Exam, Fall 2006 Data: P 0 0 5 Pa, R = 8.34 0 3 J/kmol K = N A k, N A = 6.02 0 26 particles/kilomole, T C = T K 273.5. du = TdS PdV + i µ i dn i, U = TS PV + i µ i N i Defs: 2 β ( ) V V T ( )
More informationProblem: Calculate the entropy change that results from mixing 54.0 g of water at 280 K with 27.0 g of water at 360 K in a vessel whose walls are
Problem: Calculate the entropy change that results from mixing 54.0 g of water at 280 K with 27.0 g of water at 360 K in a vessel whose walls are perfectly insulated from the surroundings. Is this a spontaneous
More informationThe Chemical Potential
CHEM 331 Physical Chemistry Fall 2017 The Chemical Potential Here we complete our pivot towards chemical thermodynamics with the introduction of the Chemical Potential ( ). This concept was first introduced
More informationConservation of Energy
Conservation of Energy Energy can neither by created nor destroyed, but only transferred from one system to another and transformed from one form to another. Conservation of Energy Consider at a gas in
More information4.1 Constant (T, V, n) Experiments: The Helmholtz Free Energy
Chapter 4 Free Energies The second law allows us to determine the spontaneous direction of of a process with constant (E, V, n). Of course, there are many processes for which we cannot control (E, V, n)
More informationMATHEMATICS 200 April 2010 Final Exam Solutions
MATHEMATICS April Final Eam Solutions. (a) A surface z(, y) is defined by zy y + ln(yz). (i) Compute z, z y (ii) Evaluate z and z y in terms of, y, z. at (, y, z) (,, /). (b) A surface z f(, y) has derivatives
More informationCHAPTER 3 LECTURE NOTES 3.1. The Carnot Cycle Consider the following reversible cyclic process involving one mole of an ideal gas:
CHATER 3 LECTURE NOTES 3.1. The Carnot Cycle Consider the following reversible cyclic process involving one mole of an ideal gas: Fig. 3. (a) Isothermal expansion from ( 1, 1,T h ) to (,,T h ), (b) Adiabatic
More informationPreliminary Examination - Day 2 August 16, 2013
UNL - Department of Physics and Astronomy Preliminary Examination - Day August 16, 13 This test covers the topics of Quantum Mechanics (Topic 1) and Thermodynamics and Statistical Mechanics (Topic ). Each
More informationEnthalpy and Adiabatic Changes
Enthalpy and Adiabatic Changes Chapter 2 of Atkins: The First Law: Concepts Sections 2.5-2.6 of Atkins (7th & 8th editions) Enthalpy Definition of Enthalpy Measurement of Enthalpy Variation of Enthalpy
More informationLecture 7: Kinetic Theory of Gases, Part 2. ! = mn v x
Lecture 7: Kinetic Theory of Gases, Part 2 Last lecture, we began to explore the behavior of an ideal gas in terms of the molecules in it We found that the pressure of the gas was: P = N 2 mv x,i! = mn
More informationENGI Gradient, Divergence, Curl Page 5.01
ENGI 94 5. - Gradient, Divergence, Curl Page 5. 5. The Gradient Operator A brief review is provided here for the gradient operator in both Cartesian and orthogonal non-cartesian coordinate systems. Sections
More informationQuantities and Variables in Thermodynamics. Alexander Miles
Quantities and Variables in Thermodynamics Alexander Miles AlexanderAshtonMiles@gmail.com Written: December 8, 2008 Last edit: December 28, 2008 Thermodynamics has a very large number of variables, spanning
More informationMAC2313 Final A. (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative.
MAC2313 Final A (5 pts) 1. How many of the following are necessarily true? i. The vector field F = 2x + 3y, 3x 5y is conservative. ii. The vector field F = 5(x 2 + y 2 ) 3/2 x, y is radial. iii. All constant
More informationCHAPTER 2: Partial Derivatives. 2.2 Increments and Differential
CHAPTER : Partial Derivatives.1 Definition of a Partial Derivative. Increments and Differential.3 Chain Rules.4 Local Etrema.5 Absolute Etrema 1 Chapter : Partial Derivatives.1 Definition of a Partial
More informationFirst Law of Thermodynamics
First Law of Thermodynamics E int = Q + W other state variables E int is a state variable, so only depends on condition (P, V, T, ) of system. Therefore, E int only depends on initial and final states
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More informationLecture 4 Clausius Inequality
Lecture 4 Clausius Inequality Entropy distinguishes between irreversible and reversible processes. irrev S > 0 rev In a spontaneous process, there should be a net increase in the entropy of the system
More informationSolutions to Sample Questions for Final Exam
olutions to ample Questions for Final Exam Find the points on the surface xy z 3 that are closest to the origin. We use the method of Lagrange Multipliers, with f(x, y, z) x + y + z for the square of the
More informationLecture 3 Clausius Inequality
Lecture 3 Clausius Inequality Rudolf Julius Emanuel Clausius 2 January 1822 24 August 1888 Defined Entropy Greek, en+tropein content transformative or transformation content The energy of the universe
More information1 mol ideal gas, PV=RT, show the entropy can be written as! S = C v. lnt + RlnV + cons tant
1 mol ideal gas, PV=RT, show the entropy can be written as! S = C v lnt + RlnV + cons tant (1) p, V, T change Reversible isothermal process (const. T) TdS=du-!W"!S = # "Q r = Q r T T Q r = $W = # pdv =
More informationSOME PROBLEMS YOU SHOULD BE ABLE TO DO
OME PROBLEM YOU HOULD BE ABLE TO DO I ve attempted to make a list of the main calculations you should be ready for on the exam, and included a handful of the more important formulas. There are no examples
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
More informationChapter 3 - Vector Calculus
Chapter 3 - Vector Calculus Gradient in Cartesian coordinate system f ( x, y, z,...) dr ( dx, dy, dz,...) Then, f f f f,,,... x y z f f f df dx dy dz... f dr x y z df 0 (constant f contour) f dr 0 or f
More informationWork and heat. Expansion Work. Heat Transactions. Chapter 2 of Atkins: The First Law: Concepts. Sections of Atkins
Work and heat Chapter 2 of Atkins: The First Law: Concepts Sections 2.3-2.4 of Atkins Expansion Work General Expression for Work Free Expansion Expansion Against Constant Pressure Reversible Expansion
More informationYou MUST sign the honor pledge:
CHEM 3411 MWF 9:00AM Fall 2010 Physical Chemistry I Exam #2, Version B (Dated: October 15, 2010) Name: GT-ID: NOTE: Partial Credit will be awarded! However, full credit will be awarded only if the correct
More informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More information4) It is a state function because enthalpy(h), entropy(s) and temperature (T) are state functions.
Chemical Thermodynamics S.Y.BSc. Concept of Gibb s free energy and Helmholtz free energy a) Gibb s free energy: 1) It was introduced by J.Willard Gibb s to account for the work of expansion due to volume
More informationMATH H53 : Final exam
MATH H53 : Final exam 11 May, 18 Name: You have 18 minutes to answer the questions. Use of calculators or any electronic items is not permitted. Answer the questions in the space provided. If you run out
More informationLecture 4 Clausius Inequality
Lecture 4 Clausius Inequality We know: Heat flows from higher temperature to lower temperature. T A V A U A + U B = constant V A, V B constant S = S A + S B T B V B Diathermic The wall insulating, impermeable
More informationMinimum Bias Events at ATLAS
Camille Bélanger-Champagne McGill University Lehman College City University of New York Thermodynamics Charged Particle and Statistical Correlations Mechanics in Minimum Bias Events at ATLAS Thermodynamics
More informationThe first law of thermodynamics continued
Lecture 7 The first law of thermodynamics continued Pre-reading: 19.5 Where we are The pressure p, volume V, and temperature T are related by an equation of state. For an ideal gas, pv = nrt = NkT For
More information2 Equations of Motion
2 Equations of Motion system. In this section, we will derive the six full equations of motion in a non-rotating, Cartesian coordinate 2.1 Six equations of motion (non-rotating, Cartesian coordinates)
More informationLesson 50 Integration by Parts
5/3/07 Lesson 50 Integration by Parts Lesson Objectives Use the method of integration by parts to integrate simple power, eponential, and trigonometric functions both in a mathematical contet and in a
More informationGeneral Physics I (aka PHYS 2013)
General Physics I (aka PHYS 2013) PROF. VANCHURIN (AKA VITALY) University of Minnesota, Duluth (aka UMD) OUTLINE CHAPTER 12 CHAPTER 19 REVIEW CHAPTER 12: FLUID MECHANICS Section 12.1: Density Section 12.2:
More informationLast Name or Student ID
10/06/08, Chem433 Exam # 1 Last Name or Student ID 1. (3 pts) 2. (3 pts) 3. (3 pts) 4. (2 pts) 5. (2 pts) 6. (2 pts) 7. (2 pts) 8. (2 pts) 9. (6 pts) 10. (5 pts) 11. (6 pts) 12. (12 pts) 13. (22 pts) 14.
More informationMajor Ideas in Calc 3 / Exam Review Topics
Major Ideas in Calc 3 / Exam Review Topics Here are some highlights of the things you should know to succeed in this class. I can not guarantee that this list is exhaustive!!!! Please be sure you are able
More information( ) ( ) = (2) u u u v v v w w w x y z x y z x y z. Exercise [17.09]
Eercise [17.09] Suppose is a region of (Newtonian) space, defined at a point in time t 0. Let T(t,r) be the position at time t of a test particle (of insignificantly tiny mass) that begins at rest at point
More information[S R (U 0 ɛ 1 ) S R (U 0 ɛ 2 ]. (0.1) k B
Canonical ensemble (Two derivations) Determine the probability that a system S in contact with a reservoir 1 R to be in one particular microstate s with energy ɛ s. (If there is degeneracy we are picking
More informationThe Gibbs Phase Rule F = 2 + C - P
The Gibbs Phase Rule The phase rule allows one to determine the number of degrees of freedom (F) or variance of a chemical system. This is useful for interpreting phase diagrams. F = 2 + C - P Where F
More informationln( P vap(s) / torr) = T / K ln( P vap(l) / torr) = T / K
Chem 4501 Introduction to Thermodynamics, 3 Credits Kinetics, and Statistical Mechanics Fall Semester 2017 Homework Problem Set Number 9 Solutions 1. McQuarrie and Simon, 9-4. Paraphrase: Given expressions
More informationUseful Mathematics. 1. Multivariable Calculus. 1.1 Taylor s Theorem. Monday, 13 May 2013
Useful Mathematics Monday, 13 May 013 Physics 111 In recent years I have observed a reticence among a subpopulation of students to dive into mathematics when the occasion arises in theoretical mechanics
More informationThe mathematical description of the motion of Atoms, Molecules & Other Particles. University of Rome La Sapienza - SAER - Mauro Valorani (2007)
The mathematical description of the motion of Atoms, Molecules Other Particles Particle Dynamics Mixture of gases are made of different entities: atoms, molecules, ions, electrons. In principle, the knowledge
More informationIntroduction to Vector Calculus (29) SOLVED EXAMPLES. (d) B. C A. (f) a unit vector perpendicular to both B. = ˆ 2k = = 8 = = 8
Introduction to Vector Calculus (9) SOLVED EXAMPLES Q. If vector A i ˆ ˆj k, ˆ B i ˆ ˆj, C i ˆ 3j ˆ kˆ (a) A B (e) A B C (g) Solution: (b) A B (c) A. B C (d) B. C A then find (f) a unit vector perpendicular
More informationAnnouncement. Physical Chemistry I for Biochemists. Chem340. Lecture 9 (1/31/11) Yoshitaka Ishii. Homework 4 is uploaded at the web site
hsical Chemistr I or Biochemists artial Derivatives Ch3.-3.3 HW3 Continued Chem34 Lecture 9 /3/ Yoshitaka Ishii Announcement Homework 4 is uploaded at the web site Monda ep = e in case that ou do not know
More informationENGI Partial Differentiation Page y f x
ENGI 3424 4 Partial Differentiation Page 4-01 4. Partial Differentiation For functions of one variable, be found unambiguously by differentiation: y f x, the rate of change of the dependent variable can
More informationSolutions to the Exercises of Chapter 8
8A Domains of Functions Solutions to the Eercises of Chapter 8 1 For 7 to make sense, we need 7 0or7 So the domain of f() is{ 7} For + 5 to make sense, +5 0 So the domain of g() is{ 5} For h() to make
More informationHOMOGENEOUS CLOSED SYSTEM
CHAE II A closed system is one that does not exchange matter with its surroundings, although it may exchange energy. W n in = 0 HOMOGENEOUS CLOSED SYSEM System n out = 0 Q dn i = 0 (2.1) i = 1, 2, 3,...
More informationOptimization Methods: Optimization using Calculus - Equality constraints 1. Module 2 Lecture Notes 4
Optimization Methods: Optimization using Calculus - Equality constraints Module Lecture Notes 4 Optimization of Functions of Multiple Variables subect to Equality Constraints Introduction In the previous
More informationFinal Review Solutions
Final Review Solutions Jared Pagett November 30, 206 Gassed. Rapid Fire. We assume several things when maing the ideal gas approximation. With inetic molecular theory, we model gas molecules as point particles
More informationSummary for Vector Calculus and Complex Calculus (Math 321) By Lei Li
Summary for Vector alculus and omplex alculus (Math 321) By Lei Li 1 Vector alculus 1.1 Parametrization urves, surfaces, or volumes can be parametrized. Below, I ll talk about 3D case. Suppose we use e
More informationCHEMICAL ENGINEERING THERMODYNAMICS. Andrew S. Rosen
CHEMICAL ENGINEERING THERMODYNAMICS Andrew S. Rosen SYMBOL DICTIONARY 1 TABLE OF CONTENTS Symbol Dictionary... 3 1. Measured Thermodynamic Properties and Other Basic Concepts... 5 1.1 Preliminary Concepts
More informationWhat is thermodynamics? and what can it do for us?
What is thermodynamics? and what can it do for us? The overall goal of thermodynamics is to describe what happens to a system (anything of interest) when we change the variables that characterized the
More informationPhysical Biochemistry. Kwan Hee Lee, Ph.D. Handong Global University
Physical Biochemistry Kwan Hee Lee, Ph.D. Handong Global University Week 3 CHAPTER 2 The Second Law: Entropy of the Universe increases What is entropy Definition: measure of disorder The greater the disorder,
More informationdg = V dp - S dt (1.1) 2) There are two T ds equations that are useful in the analysis of thermodynamic systems. The first of these
CHM 3410 Problem Set 5 Due date: Wednesday, October 7 th Do all of the following problems. Show your work. "Entropy never sleeps." - Anonymous 1) Starting with the relationship dg = V dp - S dt (1.1) derive
More informationIdentify the intensive quantities from the following: (a) enthalpy (b) volume (c) refractive index (d) none of these
Q 1. Q 2. Q 3. Q 4. Q 5. Q 6. Q 7. The incorrect option in the following table is: H S Nature of reaction (a) negative positive spontaneous at all temperatures (b) positive negative non-spontaneous regardless
More informationPhysics 408 Final Exam
Physics 408 Final Exam Name You are graded on your work (with partial credit where it is deserved) so please do not just write down answers with no explanation (or skip important steps)! Please give clear,
More informationName: Discussion Section:
CBE 141: Chemical Engineering Thermodynamics, Spring 2018, UC Berkeley Midterm 1 February 13, 2018 Time: 80 minutes, closed-book and closed-notes, one-sided 8 ½ x 11 equation sheet allowed Please show
More informationand ( x, y) in a domain D R a unique real number denoted x y and b) = x y = {(, ) + 36} that is all points inside and on
Mat 7 Calculus III Updated on 10/4/07 Dr. Firoz Chapter 14 Partial Derivatives Section 14.1 Functions o Several Variables Deinition: A unction o two variables is a rule that assigns to each ordered pair
More informationwhere R = universal gas constant R = PV/nT R = atm L mol R = atm dm 3 mol 1 K 1 R = J mol 1 K 1 (SI unit)
Ideal Gas Law PV = nrt where R = universal gas constant R = PV/nT R = 0.0821 atm L mol 1 K 1 R = 0.0821 atm dm 3 mol 1 K 1 R = 8.314 J mol 1 K 1 (SI unit) Standard molar volume = 22.4 L mol 1 at 0 C and
More informationLecture 6 Free Energy
Lecture 6 Free Energy James Chou BCMP21 Spring 28 A quick review of the last lecture I. Principle of Maximum Entropy Equilibrium = A system reaching a state of maximum entropy. Equilibrium = All microstates
More informationMAT 211 Final Exam. Spring Jennings. Show your work!
MAT 211 Final Exam. pring 215. Jennings. how your work! Hessian D = f xx f yy (f xy ) 2 (for optimization). Polar coordinates x = r cos(θ), y = r sin(θ), da = r dr dθ. ylindrical coordinates x = r cos(θ),
More informationOutline Review Example Problem 1 Example Problem 2. Thermodynamics. Review and Example Problems. X Bai. SDSMT, Physics. Fall 2013
Review and Example Problems SDSMT, Physics Fall 013 1 Review Example Problem 1 Exponents of phase transformation 3 Example Problem Application of Thermodynamic Identity : contents 1 Basic Concepts: Temperature,
More informationName: SOLUTIONS Date: 11/9/2017. M20550 Calculus III Tutorial Worksheet 8
Name: SOLUTIONS Date: /9/7 M55 alculus III Tutorial Worksheet 8. ompute R da where R is the region bounded by x + xy + y 8 using the change of variables given by x u + v and y v. Solution: We know R is
More informationHeat and Thermodynamics. February. 2, Solution of Recitation 2. Consider the first case when air is allowed to expand isothermally.
Heat and Thermodynamics. February., 0 Solution of Recitation Answer : We have given that, Initial volume of air = = 0.4 m 3 Initial pressure of air = P = 04 kpa = 04 0 3 Pa Final pressure of air = P =
More informationCHAPTER 6 CHEMICAL EQUILIBRIUM
CHAPTER 6 CHEMICAL EQUILIBRIUM Spontaneous process involving a reactive mixture of gases Two new state functions A: criterion for determining if a reaction mixture will evolve towards the reactants or
More information