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Redalyc.On the Sackur-Tetrode Equation in an Expanding Universe
Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Pereira, S.H. On the Sackur-Tetrode equation in an expanding universe Revista Mexicana de Física, vol. 57, núm. 1, junio, 2011, pp. 11-15 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57024209002 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative ENSENANZA˜ REVISTA MEXICANA DE FISICA´ E 57 (1) 11–15 JUNIO 2011 On the Sackur-Tetrode equation in an expanding universe S.H. Pereira Universidade Federal de Itajuba,´ Campus Itabira, Rua Sao˜ Paulo, 377 – 35900-373, Itabira, MG, Brazil, e-mail: [email protected] Recibido el 5 de abril de 2010; aceptado el 2 de febrero de 2011 In this work we investigate the thermodynamic properties satisfied by an expanding universe filled with a monoatomic ideal gas. We show that the equations for the energy density, entropy density and chemical potential remain the same of an ideal gas confined to a constant volume V . In particular the Sackur-Tetrode equation for the entropy of the ideal gas is also valid in the case of an expanding universe, provided that the constant value that represents the current entropy of the universe is appropriately chosen. Keywords: Expanding universe; ideal gas; Sackur-Tetrode equation. En el presente trabajo investigamos las propriedades termodinamicas´ que son satisfechas por un universo en expansion,´ el cual es lleno por un gas ideal monoatomico.´ Se prueba que las ecuaciones para la densidad de la energ´ıa, la densidad de la entrop´ıa y el potencial qu´ımico son las mismas que las de un gas ideal, el cual se encuentra confinado en un volumen V. -
MOTION MOUNTAIN the Adventure of Physics – Vol.Iv Quantum Theory: the Smallest Change
Christoph Schiller MOTION MOUNTAIN the adventure of physics – vol.iv quantum theory: the smallest change www.motionmountain.net Christoph Schiller Motion Mountain The Adventure of Physics Volume IV Quantum Theory: The Smallest Change Edition 24.1, available as free pdf at www.motionmountain.net Editio vicesima tertia. Proprietas scriptoris © Christophori Schiller secundo anno Olympiadis vicesimae nonae. Omnia proprietatis iura reservantur et vindicantur. Imitatio prohibita sine auctoris permissione. Non licet pecuniam expetere pro aliquo, quod partem horum verborum continet; liber pro omnibus semper gratuitus erat et manet. Twenty-third edition, ISBN 978-300-021946-7. Copyright © 2009 by Christoph Schiller, the second year of the 29th Olympiad. This pdf file is licensed under the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Germany Licence,whosefulltextcanbefoundonthewebsite creativecommons.org/licenses/by-nc-nd/3.0/de, with the additional restriction that reproduction, distribution and use, in whole or in part, in any product or service, be it commercial or not, is not allowed without the written consent of the copyright owner. The pdf file was and remains free for everybody to read, store and print for personal use, and to distribute electronically, but only in unmodified form and at no charge. To Britta, Esther and Justus Aaron τῷ ἐμοὶ δαὶμονι Die Menschen stärken, die Sachen klären. PREFACE Primum movere, deinde docere.* Antiquity “ ” Motion Mountain – The Adventure of Physics pdf file available free of charg This book is written for anybody who is curious about nature and motion. Have you ever asked: Why do people, animals, things, images and space move? The answer leads to many adventures; this volume presents those due the discovery that there is a smallest change in nature. -
Planck's Quantum Theory of Ideal
Pursuing an Idea: Planck’s Quantum Theory of Ideal Gas ∗ Massimiliano Badino† 1 Prelude: One Hundred Years of ‘Quantitude’ Many years later, as he faced the firing squad, Erwin Planck was to remember that distant afternoon when his father took him to discover the quantum. It was indeed in a day at the beginning of the last century, during a walk in the Grunewald Forest, that Max Planck in- formed his youngest son that his recent discovery was the greatest since the time of Newton. After that day, physics was never the same again. From the mysterious realm of the interac- tion between matter and radiation, the quantum spread out both in the theory of radiation and in the theory of matter and eventually changed our picture of the physical world. Planck played a foremost role in this revolution till the mid-1920 and especially his work on the ap- plication of quantum hypothesis to the ideal gas was instrumental in setting the stage for quantum mechanics. In this paper I will give a cursory outline of this work and, centering on the argumen- tative structure, I will argue that Planck’s theory of gas was framed according to a theoretical strategy he had already adopted in his previous work on radiation theory. The thrust of this strategy consists in focussing on the general features of the system and leaving aside spe- cific assumptions on the micro-processes. Throughout the years Planck reconfigured and reorganized his arguments and the great flexibility of his theoretical strategy allowed him to maintain a consistent outlook on the problem. -
TERMODYNAMIK En Kort Historik Christoffer Norberg
ISRN LUTMDN/TMHP-08/3032-SE ISSN 0282-1990 Institutionen f¨or Energivetenskaper TERMODYNAMIK en kort historik Christoffer Norberg Joules skovelanordning fr˚an 1845/7 f¨or att best¨amma den mekaniska v¨armeekvivalenten. Phil. Trans. Roy. Soc. 140 (1850). januari 2008 F¨orord Denna skrift g¨or inga anspr˚ak p˚aatt vara komplett eller utt¨ommande. D¨aremot har jag i m¨ojligaste m˚an f¨ors¨okt vara korrekt n¨ar det g¨aller ˚artal, biografiska data och prioritet av originalarbeten. F¨or en mer utt¨ommande beskrivning (fram till 1800-talets slut) re- kommenderas From Watt to Clausius av Donald Cardwell ([5]). Kommentarer och f¨orslag till korrigeringar emottages tacksamt. Portr¨att ¨ar huvudsakligen h¨amtade fr˚an Internet samt [2, 6, 25, 28, 20, 27], biografiska data v¨asentligen ur [1, 28, 7, 12, 21, 26, 30] och originalreferenser mestadels ur bibliotekss¨okningar, tillg¨angliga tidskrifter inom LU-n¨atet samt [23, 30]. 8 januari 20081 Christoffer Norberg Tel. 046-2228606 Christoff[email protected] Levnads˚ar f¨or 35 pionj¨arer inom termodynamikens historiska utveckling. Tjocka linjer motsvarar ˚aldern 20–65 ˚ar. “But although, as a matter of history, statistical mechanics owes its origin to investigations in thermodynamics, it seems eminently worthy of an independent development, both on account of the elegance and simplicity of its principles, and because it yields new results and places old truths in a new light in departments quite outside of thermodynamics.” Willard Gibbs 1Sedan tryckningen fr˚an januari 2008 har det gjorts ett par uppdateringar av biografiska data, liksom sm¨arre justeringar och till¨agg i texten samt i den bibliografiska delen; 15 december 2013. -
On the 100Th Anniversary of the Sackur–Tetrode Equation
On the 100th Anniversary of the Sackur–Tetrode Equation Walter Grimus (University of Vienna) Seminar Particle Physics March 8, 2012, University of Vienna Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Sackur–Tetrode equation Entropy of a monoatomic ideal gas: 3 E V S(E, V , N)= kN ln + ln + s 2 N N 0 1912: Otto Sackur and Hugo Tetrode independently determined 3 4πm 5 s = ln + 0 2 3h2 2 Sackur–Tetrode equation = absolute entropy of a monoatomic ideal gas Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Contents 1 Motivation 2 Tetrode’s derivation 3 Sackur’s derivation 4 Test of the Sackur–Tetrode equation 5 Concluding remarks Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Motivation Absolute entropy: Boltzmann (1875), Planck (1900): S = k ln W + const. Argument: Nernst’s heat theorem (1906) (third law of thermodynamics) S should be calculable without any additive constant ⇒ Massive particles: phase space volume of “elementary cells” unknown Sackur (1911): Entropy of a monoatomic ideal gas as a function of the volume of elementary cell Walter Grimus (University of Vienna) On the 100th Anniversary of the Sackur–Tetrode Equation Tetrode’s derivation Ansatz: n degrees of freedom ⇒ n Elementary domains of volume σ = δq δp δqn δpn = (zh) 1 1 · · · N identical particles ⇒ W ′ S = k ln N! W ′ = number of configurations in phase space Total admissible volume in phase space space given E, V , N: 3 3 3 3 (E, V , N)= -
On the 100Th Anniversary of the Sackur-Tetrode Equation
UWThPh-2011-34 On the 100th anniversary of the Sackur–Tetrode equation W. Grimus∗ University of Vienna, Faculty of Physics Boltzmanngasse 5, A–1090 Vienna, Austria 23 January 2013 Abstract In 1912, Otto Sackur and Hugo Tetrode independently put forward an equation for the absolute entropy of a monoatomic ideal gas and published it in “Annalen der Physik.” The grand achievement in the derivation of this equation was the discretization of phase space for massive particles, expressed as δqδp = h, where q and p are conjugate variables and h is Planck’s constant. Due to the dependence of the absolute entropy on Planck’s constant, Sackur and Tetrode were able to devise a test of their equation by applying it to the monoatomic vapor of mercury; from the satisfactory numerical comparison of h obtained from thermodynamic data on mercury with Planck’s value from black-body radiation, they inferred the correctness of their equation. In this review we highlight this almost forgotten episode of physics, discuss the arguments leading to the derivation of the Sackur–Tetrode equation and outline the method how this equation was tested with thermodynamic data. arXiv:1112.3748v2 [physics.hist-ph] 23 Jan 2013 PACS: 05.70.-a, 51.30.+i ∗E-mail: [email protected] 1 1 Introduction The formula for the absolute entropy of a monoatomic ideal gas is named after Otto Sackur and Hugo Tetrode who independently derived it in 1912 [1, 2, 3]. In classical thermodynamics the entropy of a monoatomic ideal gas is 3 E V S(E,V,N)= kN ln + ln + s0 , (1) 2 N N where E, V and N are the kinetic energy, the volume and the number of atoms, respec- tively. -
A Equação De Sackur Tetrode Ou Como a Entropia Se Encontrou Com
A equação de Sackur-Tetrode ou como a entropia se encontrou com a mecânica quântica 3=2 5=2 2 3=2 5=2 S = Nk ln M T =P + ln 2=NAvh k + 5=2 n h i o No início do século XX, cientistas ilustres lutavam para entender mais profundamente o conceito de entropia. Esse era o conceito central da toda poderosa “segunda lei da termodinâmica”, que já tinha sido utilizada para estabelecer uma escala absoluta de temperaturas, mas que ainda demandava boa dose de esclarecimento. Uma questão inquietante permanecia sem resposta. Será que o valor ab- soluto da entropia poderia ser determinado? Ou será que a entropia somente poderia ser determinada a menos de uma constante aditiva? Os cientistas daquela época passaram a prestar maior atenção às ideias de Ludwig Boltzmann. O extenso trabalho de Boltzmann nesse problema pode ser resumido pela inscrição no seu túmulo, no cemitério central de Viena, S = k ln W: Essa equação expressa a entropia S em termos do logaritmo do número W de estados disponíveis para o movimento, multiplicado pelo constante k, batizada mais tarde em homenagem a Boltzmann. No entanto, de acordo com a física clássica, não há limite para a proximidade dos estados mecânicos no espaço de fase. Portanto, não há limite para o número de estados, e nem seria possível enumerá-los. Como é então que seria possível obter um valor de W bem de…nido? Como é que obteríamos a constante aditiva da entropia? A resposta a essas questões foi dada em dois artigos separados, que apare- ceram nos Annalen der Physik, principal revista alemã de física. -
The Gibbs Paradox: Early History and Solutions
entropy Article The Gibbs Paradox: Early History and Solutions Olivier Darrigol UMR SPHere, CNRS, Université Denis Diderot, 75013 Paris, France; [email protected] Received: 9 April 2018; Accepted: 28 May 2018; Published: 6 June 2018 Abstract: This article is a detailed history of the Gibbs paradox, with philosophical morals. It purports to explain the origins of the paradox, to describe and criticize solutions of the paradox from the early times to the present, to use the history of statistical mechanics as a reservoir of ideas for clarifying foundations and removing prejudices, and to relate the paradox to broad misunderstandings of the nature of physical theory. Keywords: Gibbs paradox; mixing; entropy; irreversibility; thermochemistry 1. Introduction The history of thermodynamics has three famous “paradoxes”: Josiah Willard Gibbs’s mixing paradox of 1876, Josef Loschmidt reversibility paradox of the same year, and Ernst Zermelo’s recurrence paradox of 1896. The second and third revealed contradictions between the law of entropy increase and the properties of the underlying molecular dynamics. They prompted Ludwig Boltzmann to deepen the statistical understanding of thermodynamic irreversibility. The Gibbs paradox—first called a paradox by Pierre Duhem in 1892—denounced a violation of the continuity principle: the mixing entropy of two gases (to be defined in a moment) has the same finite value no matter how small the difference between the two gases, even though common sense requires the mixing entropy to vanish for identical gases (you do not really mix two identical substances). Although this paradox originally belonged to purely macroscopic thermodynamics, Gibbs perceived kinetic-molecular implications and James Clerk Maxwell promptly followed him in this direction. -
James, Steinhauser, Hoffmann, Friedrich One Hundred Years at The
James, Steinhauser, Hoffmann, Friedrich One Hundred Years at the Intersection of Chemistry and Physics Published under the auspices of the Board of Directors of the Fritz Haber Institute of the Max Planck Society: Hans-Joachim Freund Gerard Meijer Matthias Scheffler Robert Schlögl Martin Wolf Jeremiah James · Thomas Steinhauser · Dieter Hoffmann · Bretislav Friedrich One Hundred Years at the Intersection of Chemistry and Physics The Fritz Haber Institute of the Max Planck Society 1911–2011 De Gruyter An electronic version of this book is freely available, thanks to the support of libra- ries working with Knowledge Unlatched. KU is a collaborative initiative designed to make high quality books Open Access. More information about the initiative can be found at www.knowledgeunlatched.org Aut ho rs: Dr. Jeremiah James Prof. Dr. Dieter Hoffmann Fritz Haber Institute of the Max Planck Institute for the Max Planck Society History of Science Faradayweg 4–6 Boltzmannstr. 22 14195 Berlin 14195 Berlin [email protected] [email protected] Dr. Thomas Steinhauser Prof. Dr. Bretislav Friedrich Fritz Haber Institute of the Fritz Haber Institute of the Max Planck Society Max Planck Society Faradayweg 4–6 Faradayweg 4–6 14195 Berlin 14195 Berlin [email protected] [email protected] Cover images: Front cover: Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry, 1913. From left to right, “factory” building, main building, director’s villa, known today as Haber Villa. Back cover: Campus of the Fritz Haber Institute of the Max Planck Society, 2011. The Institute’s his- toric buildings, contiguous with the “Röntgenbau” on their right, house the Departments of Physical Chemistry and Molecular Physics. -
Otto Sackur's Groping for a Quantum Theory of Gases
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by MPG.PuRe Much Polyphony but Little Harmony: Otto Sackur's Groping for a Quantum Theory of Gases Massimiliano Badino1 and Bretislav Friedrich2 The endeavor of Otto Sackur (1880-1914) was driven, on the one hand, by his interest in Nernst's theorem, statistical mechanics, and the problem of chemical equilibrium, and, on the other, his goal to shed light on classical mechanics from the quantum vantage point. Inspired by the interplay between classical physics and quantum theory, Sackur chanced to expound his personal take on the role of the quantum in the changing landscape of physics in the turbulent 1910s. In this paper, we tell the story of this enthusiastic practitioner of the old quantum theory and early contributor to quantum statistical mechanics, whose scientific ontogenesis provides a telling clue about the phylogenesis of his contemporaries. Key words: Otto Sackur, Richard Abegg, Fritz Haber, Max Planck, Ludwig Boltzmann, Max Born, Peter Debye, Walther Nernst, Otto Stern, Josiah Willard Gibbs, Arnold Eucken, Jacobus Henricus van't Hoff, Cato Maximillian Guldberg, Peter Waage, Hugo Tetrode, affin- ity, heat theorem/third law of thermodynamics, quantization, phase space, entropy, quan- tum theory of gases, quantum statistical mechanics, black body radiation, heat capacity, Sackur-Tetrode equation, law of mass action, chemical constant, Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry, University of Breslau, First Solvay Conference. 1Max-Planck-Institut f¨urWissenschaftsgeschichte, Boltzmannstraße 24, D-14195 Berlin, Germany. Cur- rent address: Centre d’Hist`oriade la Ci`encia,Facultat de Ci`encies,Universitat Autonoma de Barcelona, Cerdanyola del Valles, 08193 Bellaterra (Barcelona), Spain 2Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany 1 Introduction Upon its inception within the context of radiation theory, the quantum hypothesis rapidly diffused into other fields. -
Development Team
Physics Paper No. : Statistical Mechanics Module : Gibbs Paradox and Indistinguishability of Particles Development Team Prof. Vinay Gupta, Department of Physics and Astrophysics, Principal Investigator University of Delhi, Delhi Prof. P.K. Ahluwalia, Physics Department, Himachal Pradesh Paper Coordinator University, Shimla 171005 Conten t Writer Prof. P.K. Ahluwalia, Physics Department, Himachal Pradesh University, Shimla 171005 Content Reviewer Prof. P. N. Kotru , Department of Physics, university of Jammu 1 Statistical Mechanics Physics Gibbs Paradox and Indistinguishability of Particles Description of Module Subject Name Physics Paper Name Statistical Mechanics Module Name/Title Equilibrium, Thermodynamic Parameters and Response Functions Module Id M9 2 Statistical Mechanics Physics Gibbs Paradox and Indistinguishability of Particles Know-More Suggested Readings Title of Book Image Pathria R.K. and Beale P. D., “Statistical Mechanics”, 3rd Edition, Oxford: Elsiever Ltd., 2011 Landau L.D., Lifshitz E.M., “ Statistical Physics Part 1,” 3rd Edition, Oxford: Pergamon Press., 1982 3 Statistical Mechanics Physics Gibbs Paradox and Indistinguishability of Particles Panat P.V., “Thermodynamics and Statistical Mechanics”, New Delhi: Narosa Publishing House Pvt. Ltd., 2008 Pal B.P., “ An Introductory Course of Statistical Mechanics, New Delhi: Narosa Publishing House Pvt. Ltd., 2008 Yoshioka D., “Statistical Physics An Introduction,” Berlin Heidelberg: Springer-Verlag, 2007 4 Statistical Mechanics Physics Gibbs Paradox and Indistinguishability of Particles Brush S.G., “Statistical Physics and the Atomic Theory of Matter, from Boyle and Newton to Landau and Onsager”, New Jersey: Princeton University Press, 1983. Weblinks On the 100th anniversary of the Sackur–Tetrode equation W. Grimus∗ University of Vienna, Faculty of Physics Boltzmanngasse 5, A–1090 Vienna, Austria http://arxiv.org/pdf/1112.3748.pdf . -
" the Universal Meaning of the Quantum of Action", by Jun Ishiwara
The universal meaning of the quantum of action ∗ by Jun Ishiwara† [Submitted on April 4th, 1915.] Since the theory of radiation demanded the existence of a universal action quantum h, different developmental phases have taken place. It would currently be most appropriate to select the following point of view of a universal meaning of h: to explain the occurrence of h by the existence of certain finite elementary regions of the state space 1, whose size for all different elementary regions of equal probability is h [Planck 1913, Sommerfeld 1911]2. However, the constant h relates not only to stationary radiation processes, but also must be regarded as manifestation of a 107 h universal molecular property. Of great interest in this context is the Bohr atomic model, where to the quantity 2π is attributed a new meaning as the quantum of angular momentum of the rotating electrons [Bohr 1913a, Bohr 1913b, Bohr 1913c, Bohr 1914] 3. However, there has been as yet no theoretical attempt to identify a possible link between the two mentioned meanings of the same constant. Also, arisen quite independently, is the Sommerfeld hypothesis h that the action integral occurring in the Hamiltonian principle is equal to 2π for every pure molecular process [Debye 1913, Sommerfeld 1911]. I would like to propose the following question: Are those different interpretations of h identical to each other? If not, it should be asked: Which assessment of the universal meaning of h is the right one? Should all phenomena be explained by one and the same basic assumption? This paper will hopefully provide some answers.