1 Exercise 4.1b pg 153

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1 Exercise 4.1b pg 153"

Transcription

1 In this solution set, an underline is used to show the last significant digit of numbers. For instance in x = the 2,5,1, and 6 are all significant. Digits to the right of the underlined digit, the 9 & 3 in the examle, are not significant and would be rounded off at the end of calculations. Carrying these extra digits for intermediate values in calculations reduces rounding errors and ensures we get the same answer regardless of the order of arithmetic stes. Numbers without underlines (including final answers) are shown with the roer number of sig figs. 1 Exercise 4.1b g 153 Question How many hases are resent at each of the oints marked in Fig. 4.23b? a. 1 - oint is entirely inside a single hase region (not on any boundaries). b. 3 - oint on a boundary where 3 hases meet. c. 3 - (same exlanation as b) d. 2 - oint occurs on boundary between two hases (on a single line). 1

2 2 Exercise 4.5b g 153 Given Iron is heated from T i = 25 C to T f = 100 C. Over this temerature range S m = 53 J K 1 1. In terms of given variables, this is written: T i = 25 C T f = 100 C S m = 53 J K 1 1 Find By how much does its chemical otential change? Strategy First, temeratures are converted to Kelvin. T i = C = 373 K T f = C = 1273 K We can use text book Equation 4.2 (g 143) to relate temerature changes to changes in chemical otential. which gives the differential ( ) µ = Sm dµ = S m Integrating dµ over the temerature range gives the change in chemical otential. Tf µ = Sm T i = Sm (T f T i ) J = 53 (1273 K 373 K) K = J = 47.7 kj where the integral has been easily solved, as we ve assumed S m is constant over this temerature range. 2

3 µ = 50 kj 3

4 3 Exercise 4.11b g 153 Given The vaour ressure of a liquid between 15 C and 35 C fits the exression log(/ torr) = /(T/ K) Find Calculate... (a) the enthaly of vaorization (b) the normal boiling oint of the liquid Strategy We ll start with Equation 4.11 (g 148). Through rearrangement we solve for va H. d ln = vah RT 2 va H = RT 2 d ln To find d ln, the given exression for log(/ torr) can be converted to an exression for ln(/ torr) using the change of base formula. and this gives the equation log b x = log k x log k b Differentiating this exression by T gives ln(/ torr) = ln(10) ( / (T/ K)) ln(d) = ln(10) 1625 K T 2 (The Torr units were discarded as the units of ressure would only lead to constant shift in the exression for ln() and this constant is lost on differentiation.) Lastly we can substitute this exression for ln(d) into our earlier exression for va H. 4

5 va H = R T 2 ln(10) 1625 K T 2 = R ln(10) 1625 K = J ln(10) 1625 K K = J = kj The normal boiling oint can be found by solving the given exression (T ) for the temerature at which (T ) = atmoshere where the atmosheric ressure atmoshere = 1.00 atm = 760 torr. log(760 torr/ torr) = /(T/ K) T = 1625 K log 760 = K (a) va H = kj (b) T = 280 K 5

6 4 Exercise 4.14b g 153 Given On a cold, dry morning after a frost, the temerature was T = 5 C and the artial ressure of water in the atmoshere fell to H2 O = 0.30 kpa. In terms of given variables, this is written: T = 5 C = 0.30 kpa Find (a) Will the frost sublime? (b) What artial ressure of water H2O would ensure that the frost remained? Strategy In this exercise, the second question answers the first question, in that once we know the artial ressure of water H2 O needed to ensure the frost remains, we know any H2 O below this will lead to frost sublimation. The artial ressure needed to revent sublimation is found by determing the solid-vaor ressure for water at this temerature; this is the ressure at this temerature on the coexistance curve for ice and water vaor. We know that if the atmoshere has a lower artial ressure of water than the solid-vaor ressure, then the water vaour will be favored over the solid and the ice will sublime. At higher artial ressures the solid hase is favored. We can find the solid-vaor ressure using Equation 4.12 (g 149) from our text book = e χ χ = subh R ( 1 T 1 ) T where we ve relaced va H in the original exression with sub H as we re concerned with the sublimation coexistence oint instead of the vaorization oint. The sublimation enthaly va H is found from the vaorization enthaly and the fusion enthaly, va H and fus H resectively. sub H = fus H + va H = kj kj 1 = kj 1 = J 1 Using Equation 4.12 we can calculate the ressure at temerature T when we know the reference ressure at the reference temerature T. As we re solving for a (T ) on the solid/gas coexistence curve, we ll need the reference to also fall on this curve. Therefore we ll use the the trile oint of water as our reference giving the following values: T = K 6

7 = kpa This allows us to find the solid-vaor ressure on the solid/gas coexistence curve for the temerature T = 5 C = 268 K. ( 1 T 1 ) T χ = subh R = J J K 1 1 = ( ) K K = e χ = kpa e = kpa Now that we know the solid-vaor ressure = kpa at the given temerature, we know the ice will sublime into any gas system with a artial ressure of water H2O < kpa. This gives us the answer to art b as H2 O = kpa. Additionally, can determine that the ice will sublime into the atmoshere as the artial ressure of water is H2O = 0.30 kpa. (a) Yes. (b) H2 O = kpa 7

8 5 Exercise 4.17b g 154 Question What fraction of the enthaly of vaorization va H of ethanol is sent on exanding its vaour? Strategy We can use the definition of enthaly, H = U + V (Equation 2.18 g 56) to decomose the enthaly of vaorization va H into an internal energy comonent va U and an exansion work comonent va (V ). va H = va U + va (V ) Additionally, we can assume the ressure is constant and the liquid volume V liq is negligible relative to the gas volume such that va (V ) = (V gas V liq ) V gas Next, we can use the ideal gas law V = nrt to relate V gas to RT for a ar quantity of gas by assuming the ethanol vaor is ideal. This gives va (V ) = RT And thereby the ratio of exansion work va (V ) to exansion enthaly va H is va (V ) va H = RT va H Using text book table 2.3 we find that ethanol vaorizes at T = 352 K and its exansion enthaly is 43.5 kj 1. Substituting these values gives the ratio value va (V ) va H RT = va H = J K K 43.5 kj J 1 kj = = 6.73% va (V ) va H = 6.73% 8

9 6 Problem 4.4 g 154 Question Calculate the difference in sloe of the chemical otential against temerature on either side of (a) the normal freezing oint of water and and (b) the normal boiling oint of water. (c) By how much does the chemical otential of water suer cooled to 5.0 C exceed that of ice at that temerature. Strategy To solve arts (a) and (b) we ll use Equation 4.13 (art 2 g 150) that relates how the temerature derivative (sloe) of chemical otential, µ, changes across a coexistence curve. ( ) µ (β) where α and β denote the two hases. ( ) µ (α) = S m (β) + S m (α) = trs S = trsh T trs For art (a) we re interested in the difference between the liquid l and solid s hases, the fusion transition fus H, which gives the following exression. ( ) µ (s) = S m (l) + S m (s) = fus S = fush T f Substituting in the enthaly and temerature of water s fusion transition, fus H = kj 1 and T f = K resectively as found in Table 2.3 of our text book, gives ( ) µ (s) = fush T f kj 1 = K = kj 1 K 1 = J 1 K 1 Likewise for art (b) we re interested in the difference between the gas g and the liquid l hases which is the vaorization transition va H. ( ) µ (g) = S m (g) + S m (l) = va S = vah T v Substituting in va H = kj 1 and T v = K from Table 2.3 gives ( ) µ (g) = vah T v kj 1 = K = kj 1 K 1 = J 1 K 1 9

10 For art (c) we re interested in calculating the difference in chemical otential of liquid water at 5 C, µ(l, 5 C) and the chemical otential of solid water at the same temerature µ(s, 5 C) and we ll call this quantity x. x = µ(l, 5 C) µ(s, 5 C) To calculate x we ll take advantage of the normal freezing oint of water, which imlies that at T = 0 C solid water and liquid water have the same chemical otential: µ(l, 0 C) = µ(s, 0 C) Therefore we can subtract µ(l, 0 C) µ(s, 0 C) (as this quantity is 0) from the exression we re working to solve. x = µ(l, 5 C) µ(s, 5 C) = µ(l, 5 C) µ(s, 5 C) [µ (l, 0 C) µ (s, 0 C)] = [µ (l, 5 C) µ (l, 0 C)] [µ (s, 5 C) µ (s, 0 C)] = µ(l) µ(s) In the third equation above we ve rearranged the righthand side so as to lace the difference in chemical otentials at the two temeratures in brackets for each hase. We call this quantity µ(α) where µ(α) = µ (α, 5 C)µ (α, 0 C) and α secifies either the liquid (l) or solid (s) hase. We can calculate µ(l) and µ(s) using Equation 4.2 (g 143) which gives the change in chemical otential with temerature. ( ) µ = S m Hence the change in chemical otential for the α hase can be calculated as where S m is assumed temerature indeendent. (α) = = Tf T i Tf T i ( ) µ S m = S m (T f T i ) Using T f T i = T = 5 K and the standard entroies of liquid water S m (l) and solid water S m (s) we get the following exressions for chemical otential changes (l) = S m (l) T (s) = S m (s) T Substituting these exression into our equation for x gives x = µ(l) µ(s) = [S m (l) S m (s)] T The above term in brackets, the difference in entroy of liquid and solid water, is just the oosite of the entroy of fusion fus S which we know from art (a). 10

11 x = [S m (l) S m (s)] T = fus S T ( = J 1 K 1 ) 5 K = 109 J 1 We are reminded that x is just the difference in chemical otentials that we were solving. µ(l, 5 C) µ(s, 5 C) = x = 109 J 1 This ositive difference in chemical otential between liquid and solid hases imlies higher free energy for the liquid hase relative to the solid hase (ice) and exlains why ice is favorable at this temerature. (a) (b) (c) ( ) µ (g) ( ) µ (s) = J 1 K 1 = J 1 K 1 µ(l, 5 C) µ(s, 5 C) = x = 100 J 1 11

- The value of a state function is independent of the history of the system. - Temperature is an example of a state function.

- The value of a state function is independent of the history of the system. - Temperature is an example of a state function. First Law of hermodynamics 1 State Functions - A State Function is a thermodynamic quantity whose value deends only on the state at the moment, i. e., the temerature, ressure, volume, etc - he value of

More information

be the mass flow rate of the system input stream, and m be the mass flow rates of the system output stream, then Vout V in in out out

be the mass flow rate of the system input stream, and m be the mass flow rates of the system output stream, then Vout V in in out out Chater 4 4. Energy Balances on Nonreactive Processes he general energy balance equation has the form Accumulation Inut Outut Heat added = + of Energy of Energy of Energy to System Work by done System Let

More information

Science Department Mark Erlenwein, Assistant Principal

Science Department Mark Erlenwein, Assistant Principal Staten Island Technical High School Vincent A. Maniscalco, Principal The Physical Setting: CHEMISTRY Science Department Mark Erlenwein, Assistant Principal - Unit 1 - Matter and Energy Lessons 9-14 Heat,

More information

The International Standard Atmosphere (ISA)

The International Standard Atmosphere (ISA) Nomenclature The International Standard Atmoshere (ISA) Mustafa Cavcar * Anadolu University, 2647 Eskisehir, Turkey a = seed of sound, m/sec g = acceleration of gravity, m/sec 2 h = altitude, m or ft =

More information

7. 1.00 atm = 760 torr = 760 mm Hg = 101.325 kpa = 14.70 psi. = 0.446 atm. = 0.993 atm. = 107 kpa 760 torr 1 atm 760 mm Hg = 790.

7. 1.00 atm = 760 torr = 760 mm Hg = 101.325 kpa = 14.70 psi. = 0.446 atm. = 0.993 atm. = 107 kpa 760 torr 1 atm 760 mm Hg = 790. CHATER 3. The atmosphere is a homogeneous mixture (a solution) of gases.. Solids and liquids have essentially fixed volumes and are not able to be compressed easily. have volumes that depend on their conditions,

More information

In order to solve this problem it is first necessary to use Equation 5.5: x 2 Dt. = 1 erf. = 1.30, and x = 2 mm = 2 10-3 m. Thus,

In order to solve this problem it is first necessary to use Equation 5.5: x 2 Dt. = 1 erf. = 1.30, and x = 2 mm = 2 10-3 m. Thus, 5.3 (a) Compare interstitial and vacancy atomic mechanisms for diffusion. (b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion. Solution (a) With vacancy diffusion,

More information

Free Software Development. 2. Chemical Database Management

Free Software Development. 2. Chemical Database Management Leonardo Electronic Journal of Practices and echnologies ISSN 1583-1078 Issue 1, July-December 2002. 69-76 Free Software Develoment. 2. Chemical Database Management Monica ŞEFU 1, Mihaela Ligia UNGUREŞAN

More information

= 1.038 atm. 760 mm Hg. = 0.989 atm. d. 767 torr = 767 mm Hg. = 1.01 atm

= 1.038 atm. 760 mm Hg. = 0.989 atm. d. 767 torr = 767 mm Hg. = 1.01 atm Chapter 13 Gases 1. Solids and liquids have essentially fixed volumes and are not able to be compressed easily. Gases have volumes that depend on their conditions, and can be compressed or expanded by

More information

SOLAR CALCULATIONS (2)

SOLAR CALCULATIONS (2) OLAR CALCULATON The orbit of the Earth is an ellise not a circle, hence the distance between the Earth and un varies over the year, leading to aarent solar irradiation values throughout the year aroximated

More information

GAS TURBINE PERFORMANCE WHAT MAKES THE MAP?

GAS TURBINE PERFORMANCE WHAT MAKES THE MAP? GAS TURBINE PERFORMANCE WHAT MAKES THE MAP? by Rainer Kurz Manager of Systems Analysis and Field Testing and Klaus Brun Senior Sales Engineer Solar Turbines Incororated San Diego, California Rainer Kurz

More information

Type: Single Date: Homework: READ 12.8, Do CONCEPT Q. # (14) Do PROBLEMS (40, 52, 81) Ch. 12

Type: Single Date: Homework: READ 12.8, Do CONCEPT Q. # (14) Do PROBLEMS (40, 52, 81) Ch. 12 Type: Single Date: Objective: Latent Heat Homework: READ 12.8, Do CONCEPT Q. # (14) Do PROBLEMS (40, 52, 81) Ch. 12 AP Physics B Date: Mr. Mirro Heat and Phase Change When bodies are heated or cooled their

More information

Energy Matters Heat. Changes of State

Energy Matters Heat. Changes of State Energy Matters Heat Changes of State Fusion If we supply heat to a lid, such as a piece of copper, the energy supplied is given to the molecules. These start to vibrate more rapidly and with larger vibrations

More information

ORGANIC LABORATORY TECHNIQUES 10 10.1. NEVER distill the distillation flask to dryness as there is a risk of explosion and fire.

ORGANIC LABORATORY TECHNIQUES 10 10.1. NEVER distill the distillation flask to dryness as there is a risk of explosion and fire. ORGANIC LABORATORY TECHNIQUES 10 10.1 DISTILLATION NEVER distill the distillation flask to dryness as there is a risk of explosion and fire. The most common methods of distillation are simple distillation

More information

EDEXCEL HIGHERS ENGINEERING THERMODYNAMICS H2 NQF LEVEL 4. TUTORIAL No. 1 PRE-REQUISITE STUDIES FLUID PROPERTIES

EDEXCEL HIGHERS ENGINEERING THERMODYNAMICS H2 NQF LEVEL 4. TUTORIAL No. 1 PRE-REQUISITE STUDIES FLUID PROPERTIES EDEXCEL HIGHERS ENGINEERING HERMODYNAMICS H NQF LEEL 4 UORIAL No. PRE-REQUISIE SUDIES FLUID PROPERIES INRODUCION Before you study the four outcomes that make u the module, you should be cometent in finding

More information

In this section, we will consider techniques for solving problems of this type.

In this section, we will consider techniques for solving problems of this type. Constrained optimisation roblems in economics typically involve maximising some quantity, such as utility or profit, subject to a constraint for example income. We shall therefore need techniques for solving

More information

10 g 5 g? 10 g 5 g. 10 g 5 g. scale

10 g 5 g? 10 g 5 g. 10 g 5 g. scale The International System of Units, or the SI Units Vs. Honors Chem 1 LENGTH In the SI, the base unit of length is the Meter. Prefixes identify additional units of length, based on the meter. Smaller than

More information

9.2 Summation Notation

9.2 Summation Notation 9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

More information

Sample Solutions for Assignment 2.

Sample Solutions for Assignment 2. AMath 383, Autumn 01 Sample Solutions for Assignment. Reading: Chs. -3. 1. Exercise 4 of Chapter. Please note that there is a typo in the formula in part c: An exponent of 1 is missing. It should say 4

More information

A Note on Integer Factorization Using Lattices

A Note on Integer Factorization Using Lattices A Note on Integer Factorization Using Lattices Antonio Vera To cite this version: Antonio Vera A Note on Integer Factorization Using Lattices [Research Reort] 2010, 12 HAL Id: inria-00467590

More information

TEMPERATURE 2008, 2004, 1990 by David A. Katz. All rights reserved.

TEMPERATURE 2008, 2004, 1990 by David A. Katz. All rights reserved. TEMPERATURE 2008, 2004, 10 by David A. Katz. All rights reserved. A BRIEF HISTORY OF TEMPERATURE MEASUREMENT Ancient people were physically aware of hot and cold and probably related temperature by the

More information

Electrochemical Kinetics ( Ref. :Bard and Faulkner, Oldham and Myland, Liebhafsky and Cairns) R f = k f * C A (2) R b = k b * C B (3)

Electrochemical Kinetics ( Ref. :Bard and Faulkner, Oldham and Myland, Liebhafsky and Cairns) R f = k f * C A (2) R b = k b * C B (3) Electrochemical Kinetics ( Ref. :Bard and Faulkner, Oldham and Myland, Liebhafsky and Cairns) 1. Background Consider the reaction given below: A B (1) If k f and k b are the rate constants of the forward

More information

PRIME NUMBERS AND THE RIEMANN HYPOTHESIS

PRIME NUMBERS AND THE RIEMANN HYPOTHESIS PRIME NUMBERS AND THE RIEMANN HYPOTHESIS CARL ERICKSON This minicourse has two main goals. The first is to carefully define the Riemann zeta function and exlain how it is connected with the rime numbers.

More information

Introduction to the Ideal Gas Law

Introduction to the Ideal Gas Law Course PHYSICS260 Assignment 5 Consider ten grams of nitrogen gas at an initial pressure of 6.0 atm and at room temperature. It undergoes an isobaric expansion resulting in a quadrupling of its volume.

More information

The Membrane Equation

The Membrane Equation The Membrane Equation Professor David Heeger September 5, 2000 RC Circuits Figure 1A shows an RC (resistor, capacitor) equivalent circuit model for a patch of passive neural membrane. The capacitor represents

More information

F inding the optimal, or value-maximizing, capital

F inding the optimal, or value-maximizing, capital Estimating Risk-Adjusted Costs of Financial Distress by Heitor Almeida, University of Illinois at Urbana-Chamaign, and Thomas Philion, New York University 1 F inding the otimal, or value-maximizing, caital

More information

"Essential Mathematics & Statistics for Science" by Dr G Currell & Dr A A Dowman (John Wiley & Sons) Answers to In-Text Questions

Essential Mathematics & Statistics for Science by Dr G Currell & Dr A A Dowman (John Wiley & Sons) Answers to In-Text Questions "Essential Mathematics & Statistics for Science" by Dr G Currell & Dr A A Dowman (John Wiley & Sons) Answers to In-Text Questions 5 Logarithmic & Exponential Functions To navigate, use the Bookmarks in

More information

MODELLING AND SIMULATION OF A DISH STIRLING SOLAR ENGINE. Sergio Bittanti Antonio De Marco Marcello Farina Silvano Spelta

MODELLING AND SIMULATION OF A DISH STIRLING SOLAR ENGINE. Sergio Bittanti Antonio De Marco Marcello Farina Silvano Spelta MODELLING AND SIMULATION OF A DISH STIRLING SOLAR ENGINE Sergio Bittanti Antonio De Marco Marcello Farina Silvano Selta Diartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio 34,

More information

Risk and Return. Sample chapter. e r t u i o p a s d f CHAPTER CONTENTS LEARNING OBJECTIVES. Chapter 7

Risk and Return. Sample chapter. e r t u i o p a s d f CHAPTER CONTENTS LEARNING OBJECTIVES. Chapter 7 Chater 7 Risk and Return LEARNING OBJECTIVES After studying this chater you should be able to: e r t u i o a s d f understand how return and risk are defined and measured understand the concet of risk

More information

United Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane

United Arab Emirates University College of Sciences Department of Mathematical Sciences HOMEWORK 1 SOLUTION. Section 10.1 Vectors in the Plane United Arab Emirates University College of Sciences Deartment of Mathematical Sciences HOMEWORK 1 SOLUTION Section 10.1 Vectors in the Plane Calculus II for Engineering MATH 110 SECTION 0 CRN 510 :00 :00

More information

Freeman, & Arya 1995). This could be interpreted as the ratio of specific heats for a gas of particles with n 4 degrees of

Freeman, & Arya 1995). This could be interpreted as the ratio of specific heats for a gas of particles with n 4 degrees of The Astrohysical Journal, 595:L57 L1, 003 Setember 0 003. The American Astronomical Society. All rights reserved. Printed in U.S.A. A THREE-DIMENSIONAL MODEL OF THE SOLAR WIND INCORPORATING SOLAR MAGNETOGRAM

More information

Large-Scale IP Traceback in High-Speed Internet: Practical Techniques and Theoretical Foundation

Large-Scale IP Traceback in High-Speed Internet: Practical Techniques and Theoretical Foundation Large-Scale IP Traceback in High-Seed Internet: Practical Techniques and Theoretical Foundation Jun Li Minho Sung Jun (Jim) Xu College of Comuting Georgia Institute of Technology {junli,mhsung,jx}@cc.gatech.edu

More information

Lecture 8: Binary Multiplication & Division

Lecture 8: Binary Multiplication & Division Lecture 8: Binary Multiplication & Division Today s topics: Addition/Subtraction Multiplication Division Reminder: get started early on assignment 3 1 2 s Complement Signed Numbers two = 0 ten 0001 two

More information

Prentice Hall. Chemistry (Wilbraham) 2008, National Student Edition - South Carolina Teacher s Edition. High School. High School

Prentice Hall. Chemistry (Wilbraham) 2008, National Student Edition - South Carolina Teacher s Edition. High School. High School Prentice Hall Chemistry (Wilbraham) 2008, National Student Edition - South Carolina Teacher s Edition High School C O R R E L A T E D T O High School C-1.1 Apply established rules for significant digits,

More information

Example 1: Suppose the demand function is p = 50 2q, and the supply function is p = 10 + 3q. a) Find the equilibrium point b) Sketch a graph

Example 1: Suppose the demand function is p = 50 2q, and the supply function is p = 10 + 3q. a) Find the equilibrium point b) Sketch a graph The Effect of Taxes on Equilibrium Example 1: Suppose the demand function is p = 50 2q, and the supply function is p = 10 + 3q. a) Find the equilibrium point b) Sketch a graph Solution to part a: Set the

More information

CQG Integrated Client Options User Guide. November 14, 2012 Version 13.5

CQG Integrated Client Options User Guide. November 14, 2012 Version 13.5 CQG Integrated Client Otions User Guide November 4, 0 Version 3.5 0 CQG Inc. CQG, DOMrader, Snarader, Flow, and FOBV are registered trademarks of CQG. able of Contents About this Document... Related Documents...

More information

What is Adverse Selection. Economics of Information and Contracts Adverse Selection. Lemons Problem. Lemons Problem

What is Adverse Selection. Economics of Information and Contracts Adverse Selection. Lemons Problem. Lemons Problem What is Adverse Selection Economics of Information and Contracts Adverse Selection Levent Koçkesen Koç University In markets with erfect information all rofitable trades (those in which the value to the

More information

The atomic packing factor is defined as the ratio of sphere volume to the total unit cell volume, or APF = V S V C. = 2(sphere volume) = 2 = V C = 4R

The atomic packing factor is defined as the ratio of sphere volume to the total unit cell volume, or APF = V S V C. = 2(sphere volume) = 2 = V C = 4R 3.5 Show that the atomic packing factor for BCC is 0.68. The atomic packing factor is defined as the ratio of sphere volume to the total unit cell volume, or APF = V S V C Since there are two spheres associated

More information

Lesson 1. Key Financial Concepts INTRODUCTION

Lesson 1. Key Financial Concepts INTRODUCTION Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have

More information

Heterogeneous Catalysis and Catalytic Processes Prof. K. K. Pant Department of Chemical Engineering Indian Institute of Technology, Delhi

Heterogeneous Catalysis and Catalytic Processes Prof. K. K. Pant Department of Chemical Engineering Indian Institute of Technology, Delhi Heterogeneous Catalysis and Catalytic Processes Prof. K. K. Pant Department of Chemical Engineering Indian Institute of Technology, Delhi Module - 03 Lecture 10 Good morning. In my last lecture, I was

More information

Introduction to NP-Completeness Written and copyright c by Jie Wang 1

Introduction to NP-Completeness Written and copyright c by Jie Wang 1 91.502 Foundations of Comuter Science 1 Introduction to Written and coyright c by Jie Wang 1 We use time-bounded (deterministic and nondeterministic) Turing machines to study comutational comlexity of

More information

NBER WORKING PAPER SERIES HOW MUCH OF CHINESE EXPORTS IS REALLY MADE IN CHINA? ASSESSING DOMESTIC VALUE-ADDED WHEN PROCESSING TRADE IS PERVASIVE

NBER WORKING PAPER SERIES HOW MUCH OF CHINESE EXPORTS IS REALLY MADE IN CHINA? ASSESSING DOMESTIC VALUE-ADDED WHEN PROCESSING TRADE IS PERVASIVE NBER WORKING PAPER SERIES HOW MUCH OF CHINESE EXPORTS IS REALLY MADE IN CHINA? ASSESSING DOMESTIC VALUE-ADDED WHEN PROCESSING TRADE IS PERVASIVE Robert Kooman Zhi Wang Shang-Jin Wei Working Paer 14109

More information

An optimal batch size for a JIT manufacturing system

An optimal batch size for a JIT manufacturing system Comuters & Industrial Engineering 4 (00) 17±136 www.elsevier.com/locate/dsw n otimal batch size for a JIT manufacturing system Lutfar R. Khan a, *, Ruhul. Sarker b a School of Communications and Informatics,

More information

PURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be

PURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our

More information

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

More information

arxiv:0711.4143v1 [hep-th] 26 Nov 2007

arxiv:0711.4143v1 [hep-th] 26 Nov 2007 Exonentially localized solutions of the Klein-Gordon equation arxiv:711.4143v1 [he-th] 26 Nov 27 M. V. Perel and I. V. Fialkovsky Deartment of Theoretical Physics, State University of Saint-Petersburg,

More information

( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those

( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those 1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make

More information

Web Application Scalability: A Model-Based Approach

Web Application Scalability: A Model-Based Approach Coyright 24, Software Engineering Research and Performance Engineering Services. All rights reserved. Web Alication Scalability: A Model-Based Aroach Lloyd G. Williams, Ph.D. Software Engineering Research

More information

Chapter 2 - Porosity PIA NMR BET

Chapter 2 - Porosity PIA NMR BET 2.5 Pore tructure Measurement Alication of the Carmen-Kozeny model requires recise measurements of ore level arameters; e.g., secific surface area and tortuosity. Numerous methods have been develoed to

More information

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks 6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

More information

Chemical Kinetics. Reaction Rate: The change in the concentration of a reactant or a product with time (M/s). Reactant Products A B

Chemical Kinetics. Reaction Rate: The change in the concentration of a reactant or a product with time (M/s). Reactant Products A B Reaction Rates: Chemical Kinetics Reaction Rate: The change in the concentration of a reactant or a product with time (M/s). Reactant Products A B change in number of moles of B Average rate = change in

More information

MAT 274 HW 2 Solutions c Bin Cheng. Due 11:59pm, W 9/07, 2011. 80 Points

MAT 274 HW 2 Solutions c Bin Cheng. Due 11:59pm, W 9/07, 2011. 80 Points MAT 274 HW 2 Solutions Due 11:59pm, W 9/07, 2011. 80 oints 1. (30 ) The last two problems of Webwork Set 03 Modeling. Show all the steps and, also, indicate the equilibrium solutions for each problem.

More information

The Basics of Interest Theory

The Basics of Interest Theory Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

More information

On Multicast Capacity and Delay in Cognitive Radio Mobile Ad-hoc Networks

On Multicast Capacity and Delay in Cognitive Radio Mobile Ad-hoc Networks On Multicast Caacity and Delay in Cognitive Radio Mobile Ad-hoc Networks Jinbei Zhang, Yixuan Li, Zhuotao Liu, Fan Wu, Feng Yang, Xinbing Wang Det of Electronic Engineering Det of Comuter Science and Engineering

More information

Computational study of wave propagation through a dielectric thin film medium using WKB approximation

Computational study of wave propagation through a dielectric thin film medium using WKB approximation AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH 0, Science Huβ, htt://www.scihub.org/ajsir ISSN: 53-649X doi:0.55/ajsir.0..4.547.55 Comutational study of wave roagation through a dielectric thin

More information

Chapter 7 : Simple Mixtures

Chapter 7 : Simple Mixtures Chapter 7 : Simple Mixtures Using the concept of chemical potential to describe the physical properties of a mixture. Outline 1)Partial Molar Quantities 2)Thermodynamics of Mixing 3)Chemical Potentials

More information

EFFECTS OF FEDERAL RISK MANAGEMENT PROGRAMS ON INVESTMENT, PRODUCTION, AND CONTRACT DESIGN UNDER UNCERTAINTY. A Dissertation SANGTAEK SEO

EFFECTS OF FEDERAL RISK MANAGEMENT PROGRAMS ON INVESTMENT, PRODUCTION, AND CONTRACT DESIGN UNDER UNCERTAINTY. A Dissertation SANGTAEK SEO EFFECTS OF FEDERAL RISK MANAGEMENT PROGRAMS ON INVESTMENT, PRODUCTION, AND CONTRACT DESIGN UNDER UNCERTAINTY A Dissertation by SANGTAEK SEO Submitted to the Office of Graduate Studies of Texas A&M University

More information

ES-7A Thermodynamics HW 1: 2-30, 32, 52, 75, 121, 125; 3-18, 24, 29, 88 Spring 2003 Page 1 of 6

ES-7A Thermodynamics HW 1: 2-30, 32, 52, 75, 121, 125; 3-18, 24, 29, 88 Spring 2003 Page 1 of 6 Spring 2003 Page 1 of 6 2-30 Steam Tables Given: Property table for H 2 O Find: Complete the table. T ( C) P (kpa) h (kj/kg) x phase description a) 120.23 200 2046.03 0.7 saturated mixture b) 140 361.3

More information

Large Sample Theory. Consider a sequence of random variables Z 1, Z 2,..., Z n. Convergence in probability: Z n

Large Sample Theory. Consider a sequence of random variables Z 1, Z 2,..., Z n. Convergence in probability: Z n Large Samle Theory In statistics, we are interested in the roerties of articular random variables (or estimators ), which are functions of our data. In ymtotic analysis, we focus on describing the roerties

More information

Steady Heat Conduction

Steady Heat Conduction Steady Heat Conduction In thermodynamics, we considered the amount of heat transfer as a system undergoes a process from one equilibrium state to another. hermodynamics gives no indication of how long

More information

2. Information Economics

2. Information Economics 2. Information Economics In General Equilibrium Theory all agents had full information regarding any variable of interest (prices, commodities, state of nature, cost function, preferences, etc.) In many

More information

The First Law of Thermodynamics

The First Law of Thermodynamics Thermodynamics The First Law of Thermodynamics Thermodynamic Processes (isobaric, isochoric, isothermal, adiabatic) Reversible and Irreversible Processes Heat Engines Refrigerators and Heat Pumps The Carnot

More information

Chapter 8 Maxwell relations and measurable properties

Chapter 8 Maxwell relations and measurable properties Chapter 8 Maxwell relations and measurable properties 8.1 Maxwell relations Other thermodynamic potentials emerging from Legendre transforms allow us to switch independent variables and give rise to alternate

More information

CHEMICAL EQUILIBRIUM Chapter 13

CHEMICAL EQUILIBRIUM Chapter 13 Page 1 1 hemical Equilibrium EMIAL EQUILIBRIUM hapter 1 The state where the concentrations of all reactants and products remain constant with time. On the molecular level, there is frantic activity. Equilibrium

More information

FREESTUDY HEAT TRANSFER TUTORIAL 3 ADVANCED STUDIES

FREESTUDY HEAT TRANSFER TUTORIAL 3 ADVANCED STUDIES FREESTUDY HEAT TRANSFER TUTORIAL ADVANCED STUDIES This is the third tutorial in the series on heat transfer and covers some of the advanced theory of convection. The tutorials are designed to bring the

More information

There is no such thing as heat energy

There is no such thing as heat energy There is no such thing as heat energy We have used heat only for the energy transferred between the objects at different temperatures, and thermal energy to describe the energy content of the objects.

More information

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max

More information

The Second Law of Thermodynamics

The Second Law of Thermodynamics Objectives MAE 320 - Chapter 6 The Second Law of Thermodynamics The content and the pictures are from the text book: Çengel, Y. A. and Boles, M. A., Thermodynamics: An Engineering Approach, McGraw-Hill,

More information

Implementation of Statistic Process Control in a Painting Sector of a Automotive Manufacturer

Implementation of Statistic Process Control in a Painting Sector of a Automotive Manufacturer 4 th International Conference on Industrial Engineering and Industrial Management IV Congreso de Ingeniería de Organización Donostia- an ebastián, etember 8 th - th Imlementation of tatistic Process Control

More information

New Tools for Project Managers: Evolution of S-Curve and Earned Value Formalism

New Tools for Project Managers: Evolution of S-Curve and Earned Value Formalism New Tools for Project Managers: Evolution of S-Curve and Earned Value Formalism A Presentation at the Third Caribbean & Latin American Conference On Project Management, 21 23 May 2003 Denis F. Cioffi,

More information

FURTHER VECTORS (MEI)

FURTHER VECTORS (MEI) Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

More information

Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis

Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................

More information

SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions

SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q. 1. Quadratic Extensions SOME PROPERTIES OF EXTENSIONS OF SMALL DEGREE OVER Q TREVOR ARNOLD Abstract This aer demonstrates a few characteristics of finite extensions of small degree over the rational numbers Q It comrises attemts

More information

II- Review of the Literature

II- Review of the Literature A Model for Estimating the Value Added of the Life Insurance Market in Egypt: An Empirical Study Dr. N. M. Habib Associate Professor University of Maryland Eastern Shore Abstract The paper is an attempt

More information

Unit 11 Practice. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Unit 11 Practice. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question. Name: Class: Date: Unit 11 Practice Multiple Choice Identify the choice that best completes the statement or answers the question. 1) Crystalline solids. A) have their particles arranged randomly B) have

More information

Re-Dispatch Approach for Congestion Relief in Deregulated Power Systems

Re-Dispatch Approach for Congestion Relief in Deregulated Power Systems Re-Disatch Aroach for Congestion Relief in Deregulated ower Systems Ch. Naga Raja Kumari #1, M. Anitha 2 #1, 2 Assistant rofessor, Det. of Electrical Engineering RVR & JC College of Engineering, Guntur-522019,

More information

RELEASED. Student Booklet. Chemistry. Fall 2014 NC Final Exam. Released Items

RELEASED. Student Booklet. Chemistry. Fall 2014 NC Final Exam. Released Items Released Items Public Schools of North arolina State oard of Education epartment of Public Instruction Raleigh, North arolina 27699-6314 Fall 2014 N Final Exam hemistry Student ooklet opyright 2014 by

More information

Vertically Integrated Systems in Stand-Alone Multistory Buildings

Vertically Integrated Systems in Stand-Alone Multistory Buildings 2005, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae. org). Rerinted by ermission from ASHRAE Journal, (Vol. 47, No. 6, June 2005). This article may not be

More information

Sample Test Questions

Sample Test Questions mathematics Numerical Skills/Pre-Algebra Algebra Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about

More information

Chapter 4: Transfer of Thermal Energy

Chapter 4: Transfer of Thermal Energy Chapter 4: Transfer of Thermal Energy Goals of Period 4 Section 4.1: To define temperature and thermal energy Section 4.2: To discuss three methods of thermal energy transfer. Section 4.3: To describe

More information

Integration of a fin experiment into the undergraduate heat transfer laboratory

Integration of a fin experiment into the undergraduate heat transfer laboratory Integration of a fin experiment into the undergraduate heat transfer laboratory H. I. Abu-Mulaweh Mechanical Engineering Department, Purdue University at Fort Wayne, Fort Wayne, IN 46805, USA E-mail: mulaweh@engr.ipfw.edu

More information

Teaching Notes. Contextualised task 27 Gas and Electricity

Teaching Notes. Contextualised task 27 Gas and Electricity Contextualised task 27 Gas and Electricity Teaching Notes This task is concerned with understanding gas and electricity bills, including an opportunity to read meters. It is made up of a series of 5 questions,

More information

All these models were characterized by constant returns to scale technologies and perfectly competitive markets.

All these models were characterized by constant returns to scale technologies and perfectly competitive markets. Economies of scale and international trade In the models discussed so far, differences in prices across countries (the source of gains from trade) were attributed to differences in resources/technology.

More information

PRESENT VALUE ANALYSIS. Time value of money equal dollar amounts have different values at different points in time.

PRESENT VALUE ANALYSIS. Time value of money equal dollar amounts have different values at different points in time. PRESENT VALUE ANALYSIS Time value of money equal dollar amounts have different values at different points in time. Present value analysis tool to convert CFs at different points in time to comparable values

More information

On the Accuracy of a Finite-Difference Method for Parabolic Partial Differential Equations with Discontinuous Boundary Conditions

On the Accuracy of a Finite-Difference Method for Parabolic Partial Differential Equations with Discontinuous Boundary Conditions This article was downloaded by: [University of Limerick], [S. L. Mitchell] On: 08 October 013, At: 07:59 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 107954

More information

The fast Fourier transform method for the valuation of European style options in-the-money (ITM), at-the-money (ATM) and out-of-the-money (OTM)

The fast Fourier transform method for the valuation of European style options in-the-money (ITM), at-the-money (ATM) and out-of-the-money (OTM) Comutational and Alied Mathematics Journal 15; 1(1: 1-6 Published online January, 15 (htt://www.aascit.org/ournal/cam he fast Fourier transform method for the valuation of Euroean style otions in-the-money

More information

Eurodollar Futures, and Forwards

Eurodollar Futures, and Forwards 5 Eurodollar Futures, and Forwards In this chapter we will learn about Eurodollar Deposits Eurodollar Futures Contracts, Hedging strategies using ED Futures, Forward Rate Agreements, Pricing FRAs. Hedging

More information

Computational Finance The Martingale Measure and Pricing of Derivatives

Computational Finance The Martingale Measure and Pricing of Derivatives 1 The Martingale Measure 1 Comutational Finance The Martingale Measure and Pricing of Derivatives 1 The Martingale Measure The Martingale measure or the Risk Neutral robabilities are a fundamental concet

More information

Consumer Price Index Dynamics in a Small Open Economy: A Structural Time Series Model for Luxembourg

Consumer Price Index Dynamics in a Small Open Economy: A Structural Time Series Model for Luxembourg Aka and Pieretii, International Journal of Alied Economics, 5(, March 2008, -3 Consumer Price Index Dynamics in a Small Oen Economy: A Structural Time Series Model for Luxembourg Bédia F. Aka * and P.

More information

Bachelor of Engineering (Honours) Degree in Electrical/Electronic Engineering

Bachelor of Engineering (Honours) Degree in Electrical/Electronic Engineering ERE60 Energy Resources & Engineering ermodynamics Control, D0 Bacelor of Engineering (Honours) Degree in Electrical/Electronic Engineering Scool of Electrical Engineering Systems Dublin Institute of ecnology

More information

The Solvency II Square-Root Formula for Systematic Biometric Risk

The Solvency II Square-Root Formula for Systematic Biometric Risk The Solvency II Square-Root Formula for Systematic Biometric Risk Marcus Christiansen, Michel Denuit und Dorina Lazar Prerint Series: 2010-14 Fakultät für Mathematik und Wirtschaftswissenschaften UNIVERSITÄT

More information

Ether boils at 34.6 C. This value only has meaning because it is a comparison to the temperature at which water freezes and boils.

Ether boils at 34.6 C. This value only has meaning because it is a comparison to the temperature at which water freezes and boils. Temperature Making a relative scale is simple. Create a device which reacts to changes in temperature. Pick one temperature and assign it a number. Next, pick a second temperature and assign it a number.

More information

SECTION 2 Transmission Line Theory

SECTION 2 Transmission Line Theory SEMICONDUCTOR DESIGN GUIDE Transmission Line Theory SECTION 2 Transmission Line Theory Introduction The ECLinPS family has pushed the world of ECL into the realm of picoseconds. When output transitions

More information

Math 115 HW #8 Solutions

Math 115 HW #8 Solutions Math 115 HW #8 Solutions 1 The function with the given graph is a solution of one of the following differential equations Decide which is the correct equation and justify your answer a) y = 1 + xy b) y

More information

CSI:FLORIDA. Section 4.4: Logistic Regression

CSI:FLORIDA. Section 4.4: Logistic Regression SI:FLORIDA Section 4.4: Logistic Regression SI:FLORIDA Reisit Masked lass Problem.5.5 2 -.5 - -.5 -.5 - -.5.5.5 We can generalize this roblem to two class roblem as well! SI:FLORIDA Reisit Masked lass

More information

DET: Mechanical Engineering Thermofluids (Higher)

DET: Mechanical Engineering Thermofluids (Higher) DET: Mechanical Engineering Thermofluids (Higher) 6485 Spring 000 HIGHER STILL DET: Mechanical Engineering Thermofluids Higher Support Materials *+,-./ CONTENTS Section : Thermofluids (Higher) Student

More information

OUTCOME 3 TUTORIAL 5 DIMENSIONAL ANALYSIS

OUTCOME 3 TUTORIAL 5 DIMENSIONAL ANALYSIS Unit 41: Fluid Mechanics Unit code: T/601/1445 QCF Level: 4 Credit value: 15 OUTCOME 3 TUTORIAL 5 DIMENSIONAL ANALYSIS 3 Be able to determine the behavioural characteristics and parameters of real fluid

More information

Mathematical Modeling and Engineering Problem Solving

Mathematical Modeling and Engineering Problem Solving Mathematical Modeling and Engineering Problem Solving Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University Reference: 1. Applied Numerical Methods with

More information

8 Speed control of Induction Machines

8 Speed control of Induction Machines 8 Speed control of Induction Machines We have seen the speed torque characteristic of the machine. In the stable region of operation in the motoring mode, the curve is rather steep and goes from zero torque

More information

A Certification Authority for Elliptic Curve X.509v3 Certificates

A Certification Authority for Elliptic Curve X.509v3 Certificates A Certification Authority for Ellitic Curve X509v3 Certificates Maria-Dolores Cano, Ruben Toledo-Valera, Fernando Cerdan Det of Information Technologies & Communications Technical University of Cartagena

More information

P h o t o g r a p h y. Vá c l a v J i r á s e k 瓦 茨 拉 夫 伊 拉 塞 克 I n f e c t i o n. I n d u s t r i a. U p s y c h 蔓. 工 业. 痴

P h o t o g r a p h y. Vá c l a v J i r á s e k 瓦 茨 拉 夫 伊 拉 塞 克 I n f e c t i o n. I n d u s t r i a. U p s y c h 蔓. 工 业. 痴 P h o t o g r a p h y Vá c l a v J i r á s e k 瓦 茨 拉 夫 伊 拉 塞 克 I n f e c t i o n. I n d u s t r i a. U p s y c h 蔓. 工 业. 痴 Vá c l a v J i r á s e k 瓦 茨 拉 夫 伊 拉 塞 克 I n f e c t i o n. I n d u s t r i a.

More information