Thermodynamics For Chemical Engineers

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A TEXTBOOK OF CHEMICAL ENGINEERING THERMODYNAMICS

Author : K. V. NARAYANAN
Publisher : PHI Learning Pvt. Ltd.
ISBN 13 : 8120347471
Total Pages : 528 pages
Book Rating : 4.1/5 (23 download)

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Book Synopsis A TEXTBOOK OF CHEMICAL ENGINEERING THERMODYNAMICS by : K. V. NARAYANAN

Download or read book A TEXTBOOK OF CHEMICAL ENGINEERING THERMODYNAMICS written by K. V. NARAYANAN and published by PHI Learning Pvt. Ltd.. This book was released on 2013-01-11 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: Designed as an undergraduate-level textbook in Chemical Engineering, this student-friendly, thoroughly class-room tested book, now in its second edition, continues to provide an in-depth analysis of chemical engineering thermodynamics. The book has been so organized that it gives comprehensive coverage of basic concepts and applications of the laws of thermodynamics in the initial chapters, while the later chapters focus at length on important areas of study falling under the realm of chemical thermodynamics. The reader is thus introduced to a thorough analysis of the fundamental laws of thermodynamics as well as their applications to practical situations. This is followed by a detailed discussion on relationships among thermodynamic properties and an exhaustive treatment on the thermodynamic properties of solutions. The role of phase equilibrium thermodynamics in design, analysis, and operation of chemical separation methods is also deftly dealt with. Finally, the chemical reaction equilibria are skillfully explained. Besides numerous illustrations, the book contains over 200 worked examples, over 400 exercise problems (all with answers) and several objective-type questions, which enable students to gain an in-depth understanding of the concepts and theory discussed. The book will also be a useful text for students pursuing courses in chemical engineering-related branches such as polymer engineering, petroleum engineering, and safety and environmental engineering. New to This Edition • More Example Problems and Exercise Questions in each chapter • Updated section on Vapour–Liquid Equilibrium in Chapter 8 to highlight the significance of equations of state approach • GATE Questions up to 2012 with answers

Engineering and Chemical Thermodynamics

Author : Milo D. Koretsky
Publisher : John Wiley & Sons
ISBN 13 : 0470259612
Total Pages : 724 pages
Book Rating : 4.4/5 (72 download)

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Book Synopsis Engineering and Chemical Thermodynamics by : Milo D. Koretsky

Download or read book Engineering and Chemical Thermodynamics written by Milo D. Koretsky and published by John Wiley & Sons. This book was released on 2012-12-17 with total page 724 pages. Available in PDF, EPUB and Kindle. Book excerpt: Chemical engineers face the challenge of learning the difficult concept and application of entropy and the 2nd Law of Thermodynamics. By following a visual approach and offering qualitative discussions of the role of molecular interactions, Koretsky helps them understand and visualize thermodynamics. Highlighted examples show how the material is applied in the real world. Expanded coverage includes biological content and examples, the Equation of State approach for both liquid and vapor phases in VLE, and the practical side of the 2nd Law. Engineers will then be able to use this resource as the basis for more advanced concepts.

Thermodynamics

Author : G. Astarita
Publisher : Springer Science & Business Media
ISBN 13 : 1489907718
Total Pages : 437 pages
Book Rating : 4.4/5 (899 download)

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Book Synopsis Thermodynamics by : G. Astarita

Download or read book Thermodynamics written by G. Astarita and published by Springer Science & Business Media. This book was released on 2013-11-11 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: If a Writer would know how to behave himself with relation to Posterity; let him consider in old Books, what he finds, that he is glad to know; and what Omissions he most laments. Jonathan Swift This book emerges from a long story of teaching. I taught chemical engineering thermodynamics for about ten years at the University of Naples in the 1960s, and I still remember the awkwardness that I felt about any textbook I chose to consider-all of them seemed to be vague at best, and the standard of logical rigor seemed immensely inferior to what I could find in books on such other of the students in my first class subjects as calculus and fluid mechanics. One (who is now Prof. F. Gioia of the University of Naples) once asked me a question which I have used here as Example 4. 2-more than 20 years have gone by, and I am still waiting for a more intelligent question from one of my students. At the time, that question compelled me to answer in a way I didn't like, namely "I'll think about it, and I hope I'll have the answer by the next time we meet. " I didn't have it that soon, though I did manage to have it before the end of the course.

Thermodynamics with Chemical Engineering Applications

Author : Elias I. Franses
Publisher : Cambridge University Press
ISBN 13 : 1107069750
Total Pages : 469 pages
Book Rating : 4.1/5 (7 download)

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Book Synopsis Thermodynamics with Chemical Engineering Applications by : Elias I. Franses

Download or read book Thermodynamics with Chemical Engineering Applications written by Elias I. Franses and published by Cambridge University Press. This book was released on 2014-08-25 with total page 469 pages. Available in PDF, EPUB and Kindle. Book excerpt: Master the principles of thermodynamics, and understand their practical real-world applications, with this deep and intuitive undergraduate textbook.

Thermodynamics for Chemical Engineers

Author : K. E. Bett
Publisher : MIT Press (MA)
ISBN 13 : 9780262523981
Total Pages : 518 pages
Book Rating : 4.5/5 (239 download)

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Book Synopsis Thermodynamics for Chemical Engineers by : K. E. Bett

Download or read book Thermodynamics for Chemical Engineers written by K. E. Bett and published by MIT Press (MA). This book was released on 2003-02-01 with total page 518 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook covers the thermodynamics needed by chemical engineers both in their engineering and in their chemistry; it is intended for use in all undergraduate and some graduate-level courses. The authors emphasize a rigorous yet concise presentation of the fundamental chemical concepts governing the behavior of single and multicomponent mixtures, including phase and chemical equilibria. In the application of these concepts, consideration is given to the presentation of experimentally measured thermodynamic properties, and to their prediction for real fluids and their mixtures using methods founded on statistical mechanics. Several applications involving the transfer of heat and work that are of special importance to chemical engineers are studied in detail to show the use of thermodynamics in improving performance. The book is written in SI units and contains worked examples, exercises, and problems.

Introductory Chemical Engineering Thermodynamics

Author : J. Richard Elliott
Publisher : Prentice Hall
ISBN 13 : 0132901099
Total Pages : 912 pages
Book Rating : 4.1/5 (329 download)

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Book Synopsis Introductory Chemical Engineering Thermodynamics by : J. Richard Elliott

Download or read book Introductory Chemical Engineering Thermodynamics written by J. Richard Elliott and published by Prentice Hall. This book was released on 2012-02-06 with total page 912 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Practical, Up-to-Date Introduction to Applied Thermodynamics, Including Coverage of Process Simulation Models and an Introduction to Biological Systems Introductory Chemical Engineering Thermodynamics, Second Edition, helps readers master the fundamentals of applied thermodynamics as practiced today: with extensive development of molecular perspectives that enables adaptation to fields including biological systems, environmental applications, and nanotechnology. This text is distinctive in making molecular perspectives accessible at the introductory level and connecting properties with practical implications. Features of the second edition include Hierarchical instruction with increasing levels of detail: Content requiring deeper levels of theory is clearly delineated in separate sections and chapters Early introduction to the overall perspective of composite systems like distillation columns, reactive processes, and biological systems Learning objectives, problem-solving strategies for energy balances and phase equilibria, chapter summaries, and “important equations” for every chapter Extensive practical examples, especially coverage of non-ideal mixtures, which include water contamination via hydrocarbons, polymer blending/recycling, oxygenated fuels, hydrogen bonding, osmotic pressure, electrolyte solutions, zwitterions and biological molecules, and other contemporary issues Supporting software in formats for both MATLAB® and spreadsheets Online supplemental sections and resources including instructor slides, ConcepTests, coursecast videos, and other useful resources

Thermodynamics: Fundamentals and Applications for Chemical Engineers

Author : Miguel T. Fleischer
Publisher :
ISBN 13 : 9781516526680
Total Pages : 592 pages
Book Rating : 4.5/5 (266 download)

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Book Synopsis Thermodynamics: Fundamentals and Applications for Chemical Engineers by : Miguel T. Fleischer

Download or read book Thermodynamics: Fundamentals and Applications for Chemical Engineers written by Miguel T. Fleischer and published by . This book was released on 2017-12-29 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt: Thermodynamics: Fundamentals and Applications for Chemical Engineers explores the concepts and properties of thermodynamics and illustrates how they can be applied to solve practical problems. The book introduces the fundamentals of thermodynamics for multi-phase, multi-component systems, providing a framework for dealing with problems in chemical engineering including mixing, compressing, and distilling fluids. The first eight chapters of Thermodynamics focus on single-component thermodynamics, introducing important concepts that will be referenced throughout subsequent chapters. Later chapters introduce modeling for multi-component systems. Topics covered include: properties as a function of state variables; first and second law of thermodynamics; power cycles, combustion, refrigeration cycles, and heat pumps; equilibrium phase relationships; correlations and calculations of vapor-liquid equilibrium data; elementary theories of solutions; and the efficiency of multicomponent separation and reaction processes. The Second Law of Thermodynamics, availability concepts, and process efficiency receive extensive coverage. The clear, well-organized sequence of the chapters helps students successfully learn and retain information. Each of the fifteen chapters includes updated sample problems that underline key principles and problem-solving steps. The book has numerous appendixes for quick reference on everything from conversion factors to Francis constants, and from properties of pure substances to thermodynamics tables and Diagrams. Thermodynamics can be used by chemical, petroleum, and mechanical engineering departments in introductory and intermediate courses on engineering thermodynamics and thermodynamics fundamentals. Born and raised in Chile, Miguel T. Fleischer earned his M.S. and Ph.D. in chemical engineering from the University of Houston where he is an adjunct professor and the undergraduate program director of the Chemical and Biomolecular Engineering Department. Dr. Fleischer worked at Royal Dutch Shell for more than 26 years in research and development, manufacturing, finance, and management. He began teaching when he was an undergraduate student in Chile where he developed a program sponsored by Universidad CatOlica de Chile to prepare high school students for college. He was the co-owner and CEO of Fleischer International Trading, a private enterprise that imported and distributed wines from all over the world for 13 years. He continued teaching while he was a graduate student at the University of Houston. He has received the Outstanding Lecturer award of the Cullen College of Engineering four times, the University's Teaching Excellence Award, the Cullen College of Engineering's Career Teaching Award, and the Cullen College of Engineering's Distinguished Engineering Alumni Award.

Chemical Energy and Exergy

Author : Norio Sato
Publisher : Elsevier
ISBN 13 : 0080501001
Total Pages : 161 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Chemical Energy and Exergy by : Norio Sato

Download or read book Chemical Energy and Exergy written by Norio Sato and published by Elsevier. This book was released on 2004-03-31 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a beginners introduction to chemical thermodynamics for engineers. In the textbook efforts have been made to visualize as clearly as possible the main concepts of thermodynamic quantities such as enthalpy and entropy, thus making them more perceivable. Furthermore, intricate formulae in thermodynamics have been discussed as functionally unified sets of formulae to understand their meaning rather than to mathematically derive them in detail. In this textbook, the affinity of irreversible processes, defined by the second law of thermodynamics, has been treated as the main subject, rather than the equilibrium of chemical reactions. The concept of affinity is applicable in general not only to the processes of chemical reactions but also to all kinds of irreversible processes. This textbook also includes electrochemical thermodynamics in which, instead of the classical phenomenological approach, molecular science provides an advanced understanding of the reactions of charged particles such as ions and electrons at the electrodes. Recently, engineering thermodynamics has introduced a new thermodynamic potential called exergy, which essentially is related to the concept of the affinity of irreversible processes. This textbook discusses the relation between exergy and affinity and explains the exergy balance diagram and exergy vector diagram applicable to exergy analyses in chemical manufacturing processes. This textbook is written in the hope that the readers understand in a broad way the fundamental concepts of energy and exergy from chemical thermodynamics in practical applications. Finishing this book, the readers may easily step forward further into an advanced text of their specified line. - Visualizes the main concepts of thermodynamics to show the meaning of the quantities and formulae. - Focuses mainly on the affinity of irreversible processes and the related concept of exergy. - Provides an advanced understanding of electrochemical thermodynamics.

Chemical and Engineering Thermodynamics

Author : Stanley I. Sandler
Publisher :
ISBN 13 :
Total Pages : 662 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Chemical and Engineering Thermodynamics by : Stanley I. Sandler

Download or read book Chemical and Engineering Thermodynamics written by Stanley I. Sandler and published by . This book was released on 1989 with total page 662 pages. Available in PDF, EPUB and Kindle. Book excerpt: A revised edition of the well-received thermodynamics text, this work retains the thorough coverage and excellent organization that made the first edition so popular. Now incorporates industrially relevant microcomputer programs, with which readers can perform sophisticated thermodynamic calculations, including calculations of the type they will encounter in the lab and in industry. Also provides a unified treatment of phase equilibria. Emphasis is on analysis and prediction of liquid-liquid and vapor-liquid equilibria, solubility of gases and solids in liquids, solubility of liquids and solids in gases and supercritical fluids, freezing point depressions and osmotic equilibria, as well as traditional vapor-liquid and chemical reaction equilibria. Contains many new illustrations and exercises.

Chemical Engineering Thermodynamics

Author : RAO
Publisher : Universities Press
ISBN 13 : 9788173710483
Total Pages : 638 pages
Book Rating : 4.7/5 (14 download)

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Book Synopsis Chemical Engineering Thermodynamics by : RAO

Download or read book Chemical Engineering Thermodynamics written by RAO and published by Universities Press. This book was released on 1997 with total page 638 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Chemical Thermodynamics for Process Simulation

Author : Jürgen Gmehling
Publisher : John Wiley & Sons
ISBN 13 : 3527343253
Total Pages : 819 pages
Book Rating : 4.5/5 (273 download)

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Book Synopsis Chemical Thermodynamics for Process Simulation by : Jürgen Gmehling

Download or read book Chemical Thermodynamics for Process Simulation written by Jürgen Gmehling and published by John Wiley & Sons. This book was released on 2019-06-10 with total page 819 pages. Available in PDF, EPUB and Kindle. Book excerpt: The only textbook that applies thermodynamics to real-world process engineering problems This must-read for advanced students and professionals alike is the first book to demonstrate how chemical thermodynamics work in the real world by applying them to actual engineering examples. It also discusses the advantages and disadvantages of the particular models and procedures, and explains the most important models that are applied in process industry. All the topics are illustrated with examples that are closely related to practical process simulation problems. At the end of each chapter, additional calculation examples are given to enable readers to extend their comprehension. Chemical Thermodynamics for Process Simulation instructs on the behavior of fluids for pure fluids, describing the main types of equations of state and their abilities. It discusses the various quantities of interest in process simulation, their correlation, and prediction in detail. Chapters look at the important terms for the description of the thermodynamics of mixtures; the most important models and routes for phase equilibrium calculation; models which are applicable to a wide variety of non-electrolyte systems; membrane processes; polymer thermodynamics; enthalpy of reaction; chemical equilibria, and more. -Explains thermodynamic fundamentals used in process simulation with solved examples -Includes new chapters about modern measurement techniques, retrograde condensation, and simultaneous description of chemical equilibrium -Comprises numerous solved examples, which simplify the understanding of the often complex calculation procedures, and discusses advantages and disadvantages of models and procedures -Includes estimation methods for thermophysical properties and phase equilibria thermodynamics of alternative separation processes -Supplemented with MathCAD-sheets and DDBST programs for readers to reproduce the examples Chemical Thermodynamics for Process Simulation is an ideal resource for those working in the fields of process development, process synthesis, or process optimization, and an excellent book for students in the engineering sciences.

Advanced Thermodynamics for Engineers

Author : D. Winterbone
Publisher : Butterworth-Heinemann
ISBN 13 : 0080523366
Total Pages : 399 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Advanced Thermodynamics for Engineers by : D. Winterbone

Download or read book Advanced Thermodynamics for Engineers written by D. Winterbone and published by Butterworth-Heinemann. This book was released on 1996-11-01 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: Although the basic theories of thermodynamics are adequately covered by a number of existing texts, there is little literature that addresses more advanced topics. In this comprehensive work the author redresses this balance, drawing on his twenty-five years of experience of teaching thermodynamics at undergraduate and postgraduate level, to produce a definitive text to cover thoroughly, advanced syllabuses. The book introduces the basic concepts which apply over the whole range of new technologies, considering: a new approach to cycles, enabling their irreversibility to be taken into account; a detailed study of combustion to show how the chemical energy in a fuel is converted into thermal energy and emissions; an analysis of fuel cells to give an understanding of the direct conversion of chemical energy to electrical power; a detailed study of property relationships to enable more sophisticated analyses to be made of both high and low temperature plant and irreversible thermodynamics, whose principles might hold a key to new ways of efficiently covering energy to power (e.g. solar energy, fuel cells). Worked examples are included in most of the chapters, followed by exercises with solutions. By developing thermodynamics from an explicitly equilibrium perspective, showing how all systems attempt to reach a state of equilibrium, and the effects of these systems when they cannot, the result is an unparalleled insight into the more advanced considerations when converting any form of energy into power, that will prove invaluable to students and professional engineers of all disciplines.

Statistical Thermodynamics and Stochastic Kinetics

Author : Yiannis N. Kaznessis
Publisher : Cambridge University Press
ISBN 13 : 0521765617
Total Pages : 329 pages
Book Rating : 4.5/5 (217 download)

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Book Synopsis Statistical Thermodynamics and Stochastic Kinetics by : Yiannis N. Kaznessis

Download or read book Statistical Thermodynamics and Stochastic Kinetics written by Yiannis N. Kaznessis and published by Cambridge University Press. This book was released on 2012 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides engineers with the knowledge they need to apply thermodynamics and solve engineering challenges at the molecular level.

Fundamentals of Chemical Engineering Thermodynamics

Author : Themis Matsoukas
Publisher : Pearson Education
ISBN 13 : 0132693062
Total Pages : 719 pages
Book Rating : 4.1/5 (326 download)

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Book Synopsis Fundamentals of Chemical Engineering Thermodynamics by : Themis Matsoukas

Download or read book Fundamentals of Chemical Engineering Thermodynamics written by Themis Matsoukas and published by Pearson Education. This book was released on 2013 with total page 719 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fundamentals of Chemical Engineering Thermodynamics is the clearest and most well-organized introduction to thermodynamics theory and calculations for all chemical engineering undergraduates. This brand-new text makes thermodynamics far easier to teach and learn. Drawing on his award-winning courses at Penn State, Dr. Themis Matsoukas organizes the text for more effective learning, focuses on "why" as well as "how," offers imagery that helps students conceptualize the equations, and illuminates thermodynamics with relevant examples from within and beyond the chemical engineering discipline. Matsoukas presents solved problems in every chapter, ranging from basic calculations to realistic safety and environmental applications.

Thermodynamics for the Practicing Engineer

Author : Louis Theodore
Publisher : John Wiley & Sons
ISBN 13 : 1118216318
Total Pages : 370 pages
Book Rating : 4.1/5 (182 download)

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Book Synopsis Thermodynamics for the Practicing Engineer by : Louis Theodore

Download or read book Thermodynamics for the Practicing Engineer written by Louis Theodore and published by John Wiley & Sons. This book was released on 2011-11-30 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Enables you to easily advance from thermodynamics principles to applications Thermodynamics for the Practicing Engineer, as the title suggests, is written for all practicing engineers and anyone studying to become one. Its focus therefore is on applications of thermodynamics, addressing both technical and pragmatic problems in the field. Readers are provided a solid base in thermodynamics theory; however, the text is mostly dedicated to demonstrating how theory is applied to solve real-world problems. This text's four parts enable readers to easily gain a foundation in basic principles and then learn how to apply them in practice: Part One: Introduction. Sets forth the basic principles of thermodynamics, reviewing such topics as units and dimensions, conservation laws, gas laws, and the second law of thermodynamics. Part Two: Enthalpy Effects. Examines sensible, latent, chemical reaction, and mixing enthalpy effects. Part Three: Equilibrium Thermodynamics. Addresses both principles and calculations for phase, vapor-liquid, and chemical reaction equilibrium. Part Four: Other Topics. Reviews such important issues as economics, numerical methods, open-ended problems, environmental concerns, health and safety management, ethics, and exergy. Throughout the text, detailed illustrative examples demonstrate how all the principles, procedures, and equations are put into practice. Additional practice problems enable readers to solve real-world problems similar to the ones that they will encounter on the job. Readers will gain a solid working knowledge of thermodynamics principles and applications upon successful completion of this text. Moreover, they will be better prepared when approaching/addressing advanced material and more complex problems.

Thermodynamics for Chemists, Physicists and Engineers

Author : Robert Hołyst
Publisher : Springer Science & Business Media
ISBN 13 : 9400729995
Total Pages : 345 pages
Book Rating : 4.4/5 (7 download)

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Book Synopsis Thermodynamics for Chemists, Physicists and Engineers by : Robert Hołyst

Download or read book Thermodynamics for Chemists, Physicists and Engineers written by Robert Hołyst and published by Springer Science & Business Media. This book was released on 2012-07-05 with total page 345 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook takes an interdisciplinary approach to the subject of thermodynamics and is therefore suitable for undergraduates in chemistry, physics and engineering courses. The book is an introduction to phenomenological thermodynamics and its applications to phase transitions and chemical reactions, with some references to statistical mechanics. It strikes the balance between the rigorousness of the Callen text and phenomenological approach of the Atkins text. The book is divided in three parts. The first introduces the postulates and laws of thermodynamics and complements these initial explanations with practical examples. The second part is devoted to applications of thermodynamics to phase transitions in pure substances and mixtures. The third part covers thermodynamic systems in which chemical reactions take place. There are some sections on more advanced topics such as thermodynamic potentials, natural variables, non-ideal mixtures and electrochemical reactions, which make this book of suitable also to post-graduate students.

Lectures in Classical Thermodynamics with an Introduction to Statistical Mechanics

Author : Daniel Blankschtein
Publisher : Springer Nature
ISBN 13 : 3030491986
Total Pages : 758 pages
Book Rating : 4.0/5 (34 download)

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Book Synopsis Lectures in Classical Thermodynamics with an Introduction to Statistical Mechanics by : Daniel Blankschtein

Download or read book Lectures in Classical Thermodynamics with an Introduction to Statistical Mechanics written by Daniel Blankschtein and published by Springer Nature. This book was released on 2021-03-15 with total page 758 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook facilitates students’ ability to apply fundamental principles and concepts in classical thermodynamics to solve challenging problems relevant to industry and everyday life. It also introduces the reader to the fundamentals of statistical mechanics, including understanding how the microscopic properties of atoms and molecules, and their associated intermolecular interactions, can be accounted for to calculate various average properties of macroscopic systems. The author emphasizes application of the fundamental principles outlined above to the calculation of a variety of thermodynamic properties, to the estimation of conversion efficiencies for work production by heat interactions, and to the solution of practical thermodynamic problems related to the behavior of non-ideal pure fluids and fluid mixtures, including phase equilibria and chemical reaction equilibria. The book contains detailed solutions to many challenging sample problems in classical thermodynamics and statistical mechanics that will help the reader crystallize the material taught. Class-tested and perfected over 30 years of use by nine-time Best Teaching Award recipient Professor Daniel Blankschtein of the Department of Chemical Engineering at MIT, the book is ideal for students of Chemical and Mechanical Engineering, Chemistry, and Materials Science, who will benefit greatly from in-depth discussions and pedagogical explanations of key concepts. Distills critical concepts, methods, and applications from leading full-length textbooks, along with the author’s own deep understanding of the material taught, into a concise yet rigorous graduate and advanced undergraduate text; Enriches the standard curriculum with succinct, problem-based learning strategies derived from the content of 50 lectures given over the years in the Department of Chemical Engineering at MIT; Reinforces concepts covered with detailed solutions to illuminating and challenging homework problems.