SOLUTIONS MANUAL FOR. Thermodynamics in Material Science, Second Edition. Robert DeHoff

Transcription

1 SOLUTIONS MANUAL FOR Thermodynamics in Material Science, Second Edition by Robert DeHoff

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3 SOLUTIONS MANUAL FOR Thermodynamics in Material Science, Second Edition by Robert DeHoff

4 CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper International Standard Book Number-13: (Softcover) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The Authors and Publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access ( or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at and the CRC Press Web site at

5 text: This Manual is a compilation of solutions to the homework problems presented in the Thermodynamics in Materials Science, Second Edition, CRC Press, Taylor and Francis Group, Publishers, ISBN , (2006). In preparing this manual the data used in the solutions are consistent with that presented in the Appendices of the text. The reader should be aware that thermochemical data continues to evolve, so that the numbers may change with time, sometimes significantly. In any real world application in industry or research it is incumbent upon the investigator to seek to obtain the latest information from appropriate data bases. Working equations developed in this process use notation that is consistent with that used in the text. All numerical problems have been worked in MathCad TM, a mathematics software program, so that every attempt has been made to produce correct answers from the input data and the strategy devised to solve the problem. All graphical presentations that involve numerical calculations were also produced in MathCad and copied into this manual. This includes a number of quantitative surface plots, which are particularly easily generated in this software. Robert T. DeHoff, July, 2006

6 Chapter 2. The Structure of Thermodynamics 1 Chapter 2. Structure of Thermodynamics 2.1. Classify the following thermodynamic systems in the five categories defined in Section 2.1: rpm. a. A solid bar of copper. b. A glass of ice water. c. A yttria stabilized zirconia furnace tube. d. A styrofoam coffee cup. e. A eutectic alloy turbine blade rotating at 20,000 If you find it necessary to qualify your answer by defining the system more precisely, state your assumptions. Answer to 2.1. In each case it is necessary to make some assumptions about the kind of processes that may be of interest in the problem. The simplest situation is assumed in each of the following. a. Pure solid copper is a unary, homogeneous, closed, nonreacting otherwise simple system. b. Ice water consists of two phases, solid and liquid; both phases are fixed composition. This system may be treated as a unary, heterogeneous, closed, non-reacting otherwise simple system. c. In this material system the yttria (Y 2 O 3 ) is present as second phase particles distributed throughout a matrix of ZrO 2. It may be treated as a multicomponent (binary if the components are taken to be Y 2 O 3 and ZrO 2 ) heterogeneous (two phases), closed, non-reacting otherwise simple system. d. Styrofoam is a polymer of fixed composition. It may be appropriate to treat it as a unary, homogeneous, closed, nonreacting otherwise simple system. If, for example, the system is defined to include the porosity then additional (gaseous) components will be present in these two phases. e. A eutectic system consists of alternating layers of two phases, each of which is a solid solution. The problem also imposes a centrifugal field on the rotating part. Thus, this is a multicomponent, heterogeneous, closed, non-reacting system in a centrifugal field It is not an overstatement to say that without state functions thermodynamics would be useless. Discuss this assertion. Answer to 2.2. If there were no state functions (like T, P, V, composition), i.e., properties that depend only upon the current condition of the system, and not on how it arrived at that condition) then the behavior of all aspects of matter would depend explicitly upon the history of the system. There would be no variables that, by themselves, explicitly describe the current condition of any system. Thus, even the history experienced by the system

7 could not be described in terms of some sequence of change of its properties Determine which of the following properties of a thermodynamic system are extensive properties and which are intensive. a. The mass density. b. The molar density. c. The number of gram atoms of aluminum in a chunk of alumina. d. The potential energy of the system in a gravitational field. e. The molar concentration of NaCl in a salt solution. f. The heat absorbed by a the gas in a cylinder when it is compressed. Answer to 2.3. a. The mass density is the ratio of the mass of a system to its volume; it is intensive. b. The molar density is the ratio of the number of moles to the volume; it is intensive. c. The number of gram atoms is a property of the system as a whole; it is an extensive property. d. The potential energy of the system is a property of the system as a whole; it is thus an extensive property. e. The molar concentration is the ratio of number of moles to the volume if the system; it is an intensive property. f. The heat absorbed is a property of the system as a whole; it is an extensive property. All of the properties identified as intensive also share the characteristic that they may be defined at a point within the system, and indeed may vary from point to point Why is heat a process variable? Answer to 2.4. Heat is fundamentally a flow of energy. Heat is transferred between two systems, or between parts of the same system; this rearrangement of the distribution of energy is necessarily accompanied by changes in at least some of the properties of the systems involved. Such a change is by definition a process Write the total differential of the function a. Identify the coefficients of the three differentials in 2

8 this expression as appropriate partial derivatives. b. Show that three Maxwell relations hold among these coefficients Answer to 2.5. a. Write the total differential of the function z: b. Evaluate the cross derivatives:

9 2.6. Describe what the notion of equilibrium means to you. List as many attributes as you can think of that would be exhibited by a system that has come to equilibrium. Why do you think these characteristics of a system in equilibrium are important in thermodynamics? Answer to 2.6. Attributes of equilibrium: 1. A state of rest: state of the system does not change with time. 2. A stable state: if the state is displaced from the equilibrium state, it will return to it. 3. A state of internal uniformity; (in the absence of external fields) gradients of intensive properties vanish. The equilibrium state is the final state of every process. The primary goal of thermodynamics is the prediction of the properties of the final equilibrium state for any given initial condition of any system. "How far the system is" from the equilibrium state is a measure of the driving force for processes changing the system toward equilibrium, and controls the rate of approach to the final state of rest

10 Chapter 3. The Laws of Thermodynamics 5 Answer to 3.2. Chapter 3. The Laws of Thermodynamics Electrical energy (the spark) combines with chemical energy (in the fuel) to produce heat and mechanical work that drives the pistons in the engine block; the mechanical work is then transmitted through the crankshaft and transmission to the axle and the differential that ultimately turns the wheels The laws of thermodynamics are "pervasive". Explain in detail the meaning of this important statement List the kinds of energy conversions involved in operating a hand calculator. Answer to 3.1. "Pervasive" means that the laws apply a. In every system b. At every instant in time c. In every volume element in which change is occurring. This characteristic emphasizes the generality, and thus the power, of the laws of thermodynamics. Answer to 3.3. Chemical energy stored in a battery is converted to electrical energy that flows through the connectors and integrated circuits along paths determined by mechanical input through the keypad. Depending upon the nature of the display, the pattern of output electrical energy may be used to alter the structure of molecules in a liquid crystal or convert to light energy in an LED array List the kinds of energy conversions involved in using your arm and hand to turn the page in this text List the kinds of energy conversions involved in propelling an automobile. Answer to 3.4.

11 Chapter 3. The Laws of Thermodynamics 6 Stored chemical energy derived from food products and oxygen is converted to electrical energy which generates patterns in the brain that are transmitted through the neural network to muscles in the arm and hand. These signals induce chemical and electrical changes in the required muscles causing them to contract appropriately; this contraction is converted to the mechanical work involved in the motion of the arm and hand as they move their weight and that of the paper in a gravitational field Suppose the convention were adopted that defines W and W' in the first law of thermodynamics to be the "work done by the system on the surroundings" Give five examples of the operation of the second law of thermodynamics in your daily experience; they must be different from those given in the text. Answer to The morning coffee cools with time. 2. Sugar dissolves in hot coffee. 3. Left to itself, a pendulum will slow to a stop. 4. Organisms die. 5. An expanding gas cools a. Write the first law with this alternate convention. b. Why do the signs change? 3.7. Biological systems, - organelles, cells, organs, plants and animals,- are highly ordered, yet form spontaneously. Does the formation and growth of biological systems violate the second law of thermodynamics? Explain your answer. Answer to 3.5. a. Answer to 3.7. b. With this convention W is defined to be positive when it is transmitted from the system to the surroundings; thus if W is positive, the internal energy of the system decreases. No, it does not violate the second law of thermodynamics. Their formation and growth are accompanied by changes in their surroundings such that the total change in entropy of the biological system plus its surroundings is positive "Irreversible" is an awkward adjective. Why is this term so appropriate in its application to the description of processes

12 Chapter 3. The Laws of Thermodynamics 7 in thermodynamics? Suggest two or three alternate words or phrases that might be used to replace "irreversible" in these contexts. Answer to 3.8. a. In the thermally insulated case there is no entropy transfer; the total entropy change in this case is entropy production. b. Slow expansion minimizes dissipation effects, and thus is accompanied by a small production of entropy; most of the entropy change in this case is entropy transfer. Irreversible means: "Incapable of being reversed" "The reverse process cannot happen" "Permanent change process" "Irretracable" The word is appropriate because all real processes are accompanied by permanent change in the universe. By definition, these permanent changes cannot be undone by reversing the influences driving the process Contrast the relative magnitudes of the entropy transfer versus entropy production in the following processes: a. A thermally insulated container has two compartments of equal size. Initially one side is filled with a gas and the other is evacuated. A valve is opened and the gas expands to fill both compartments. b. A gas contained in a steel cylinder is slowly expanded to twice its volume. Answer to Consider an isolated system (no heat, matter or work may be exchanged with the surroundings) consisting of three internal compartments A, B and C, of equal volumes. The compartments are separated by partitions; each partition has a valve which may be opened remotely. Initially the central volume B is filled with a gas at 298K (25 o C) and the outer two are evacuated. Consider the following two processes: a. The valve to the A side is opened, the gas expands freely into the compartment A, and the system comes to equilibrium. Then the valve to the C side is opened, and the system again comes to equilibrium. b. Both valves are opened simultaneously, the gas expands freely into both compartments, and the system comes to its equilibrium. Which of these processes produces more entropy? Answer to The initial states are the same for these processes; the final states are also the same. Since entropy is a state function, the entropy change for both processes must be the same.

13 Chapter 3. The Laws of Thermodynamics Answer It will be shown in chapter 4 that the change in entropy )S associated with process a in problem 3.6 is 4.60 (J/mol-K), and that the initial and final states will be at the same temperature. Application of equation (3.10) suggests that the heat absorbed by the system during this process is a. Heat flow is crucial in many industrial refining operations in which chemical reactions produce large quantities of heat. The dispersion of heat through the system and surroundings is a dissipation that cannot be recovered. b. Most microstructural changes in materials involve a redistribution of the chemical components called diffusion. Like heat flow, the rearrangement of the spatial distribution of the atoms through the system is a dissipation that cannot be recovered. Yet the description of the system says it is isolated from its surroundings so that Q = 0. Explain this apparent contradiction. c. Ions implanted in the fabrication of dopant layers in thin films in integrated circuits are not removed by simply reversing the electrical field. Removal of the ions could only be accomplished by evaporating the whole film and redepositing it Answer to Equation (3.10) applies only to reversible processes. It therefore does not apply to the processes described in Problem The notion of a "reversible" process is a fiction in the real world. What makes this concept, which at first glance would appear to be only of academic interest, so useful in applying thermodynamics to real world "irreversible" processes? Give three examples of processes that are important in materials science that are thermodynamically "irreversible". Speculate briefly about the nature of the dissipations in these processes that contribute to the production of entropy. Answer to If the property changes that are of interest in a real irreversible process are state functions, then their changes will be the same for every process that connects the initial and final states of the process. In particular, these changes will be the same as for some reversible process that takes the system between these

14 Chapter 3. The Laws of Thermodynamics 9 two states. Calculation of changes for a reversible process is very much simpler than for irreversible processes because internal intensive properties like temperature and pressure are uniform in the system during the change. Thus calculations of changes in state functions for irreversible processes are made accessible through the useful fiction of the reversible processes. Answer to The combined statement is used to evaluate du, which is a state function. Thus, the combined statement may be used to calculate the change in internal energy for any irreversible process because )U is the same as for a reversible process joining the same initial and final states. In addition, for reversible processes general relationships are available for computing W and Q from state function information by integrating Describe the kinds of experimental observations that have been invoked to support the hypothesis that the entropy of all substances is the same at absolute zero. along the simplest path that connects the initial and final states. These results may be used to estimate properties of real evolving systems that are carried out slowly, near equilibrium. They also provide estimates of the maxim heat absorbed or work that could be done on the system Answer to Entropy changes associated with heating or cooling the pure elements and compounds may be computed from heat capacity measurements as a function of temperature for these substances. Entropy changes for reactions between these substances may be evaluated from heats of reaction or measurements equilibrium compositions The combined statement of the first and second laws of thermodynamics, equation (3.15), evaluates the heat absorbed and mechanical work done on a system with relationships that are only valid for reversible processes. Since reversible processes do not occur in the real world, how is it possible for the combined statement to play an important role in the analysis of practical "irreversible" processes encountered in nature and in technology? Use the following values of absolute entropies of elements and compounds at 298K to compute the standard entropy changes associated with as many chemical reactions as you can generate from this list.

15 Chapter 3. The Laws of Thermodynamics 10 Al 28.3 CO The following list of reactions can be generated among these components is not exhaustive. REACTION )S o (J/mol K) Formation Reactions: C(gr) 5.69 CO Si Al 2 O O SiC SiO Answer to For any reaction of the form Linear Combinations: where l, m, r, s,... are stoichiometric coefficients for the components L, M, R, S,..., the corresponding change in entropy for the transformation from pure reactants to pure products at 298 K (the "standard entropy change") is given by A large number of linear combinations of the formation reactions may be constructed, including, e.g.,