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 . A beautiful historical perspective on the importance of contribution of Otto Sackur and Hugo Tetrode independently in using the idea of discretization of phase space for massive particles as , where h is Plancks constant. It put to rest the adhoc resolution of Gibbs paradox. Sackur Tetrode equation and measure of entropy, a pre print using sackur tetrode equation with the idea of discretized phase space http://diposit.ub.edu/dspace/bitstream/2445/59903/1/TFG- Pa%C3%B1os-Exp%C3%B3sito-FranciscoJos%C3%A9.pdf A site devoted to teaching Statistical and thermal physics . It has many links to simulations, research articles, demos and talks http://stp.darku.edu/ Glossary B Boltzmann Counting- Boltzmann Counting Process amounts to diminishing the number of microstates of the monoatomic ideal gas having atoms by a factor of . If we recall problems which we have done in permutations and combinations which allows us to count without counting, the number of permutations of N different things taken N at a time is given by N!, but if all the N things are identical it should lead to 1 possibility only this requires us to divide N! by N!. C Chemical Potential- 5 Statistical Mechanics Physics Gibbs Paradox and Indistinguishability of Particles Chemical potential is a generalised potential in physics which tells which way particles (atoms, molecules, electrons etc.) in a system tend to change in a physical process. For example in diffusion particles tend to move from higher concentration to lower concentration. Particle number and chemical potential form a pair of conjugate thermodynamic variables. For a particular species in a system it is defined in terms of Gibbs free energy as , when temperature and volume are kept constant it is defined in terms of Helmholtz free energy as . E Entropy of Mixing- It is variation of entropy involved when a number of species are mixed together to form a mixture. It is given , where is the number of species mixed together to form mixture. Extensity- It is defined as a quantity which is proportional to the dimensions of a system, for example entropy has extensity or extensiveness. G Gibbs Paradox- Unexpected non- zero entropy of mixing of two ideal monoatomic gases of the same type of atoms, resulting because of wrong counting of number of microstates because of indistinguishability of atoms is known as Gibbs Paradox. Gibbs Free Energy- Gibbs free energy is the thermodynamic potential of a system for constant and . Gibbs defined it as , where is enthalpy. H Helmholtz Free Energy- Helmholtz free energy is the thermodynamic potential of a system for constant and . It is defined it as , where is internal energy. I Indistinguishability- Physical inability of labelling identical particles by any means is called indistiguisability and has interesting consequences such as drastically reducing the number of possible microstates for a system of identical particles. T Thermodynamic Wavelength – It is the quantity also known as thermal de Broglie wavelength which has dimensions of momentum divided by Planck’s constant. At low temperatures size of thermodynamic wave length is of the same order as the distance between the atoms/molecules and quantum mechanics and quantum statistics need to be invoked to understand the properties of thermo-dynamical systems. 6 Statistical Mechanics Physics Gibbs Paradox and Indistinguishability of Particles Did you know? Description Image Sackur Tetrode formula has a very interesting Indian connection when it was used by Indian Physicist Megh Nad Saha to arrive at has famous Saha equation to calculate the fraction x of the ionised atoms of calcium which paved way for establishing a theory to assign temperatures to stars of various spectral classes, by obtaining theoretical expressions fo the temperatures at which lines of various ionized species of atoms appear. The importance of this formula was recognized by famous astrophysicist E.A. Milne in 1921 who predicted the significance of this theory which found application in many problems in astrophysics such as differences in spectra of giant stars and dwarfs. In Megh Nad Saha’s own words: It was while pondering over the problems of astrophysics, and teaching thermodynamics and spectroscopy to the MSc classes that the theory of thermal ionization took a definite shape in my mind in 1919. I was a regular reader of German Journals, which had just started coming after four years of first world war, and in course of these studies, I came across a paper by J.Eggert in the Physikalische zeitschrifts (p.573) Dec. 1919, “ Uber den Dissociationzustand der Fixterngase” in which he applied Nernst’s Heat Theorem to explain the high ionization in stars due to high temperatures, postulated by Eddington in course of his studies on stellar structure. Eggert, who was a pupil of Nernst and was at the time his assistant, had Megh Nad Saha given a formula for thermal ionization, but it is rather strange that he missed the significance of ionization potential of atoms. Importance of which was apparent from the theoretical work of Bohr, and practical work of Franck and Hertz which was attracting a good deal of attention in those days…Eggert used Sackur’s formula of the chemical constant for calculating that of the electron, but in trying to account for multiple ionization of iron atoms in the interior of stars on this basis, he used very artificial values of ionization potential. While reading Eggert’s paper I saw at once the importance of introducing the value of ionization potential in the formula of Eggert, for calculating accurately the ionization, single or multiple, of any particular element under any combination of temperature and pressure. I thus arrived at the formula which now goes by my name. Owing to my previous acquaintance with 7 Statistical Mechanics Physics Gibbs Paradox and Indistinguishability of Particles chromospheric and stellar problems, I could at once see its application”. Paying tribute to him Jayant Narlikar, famous astrophysicists wrote in his book Scientific Edge: "Meghnad Saha’s ionization equation (c. 1920), which opened the door to stellar astrophysics” was one of the top ten achievements of 20th century Indian science and could be considered in the Nobel Prize class." Sources: 1. http://www.physics.utah.edu/~kieda/SahaEquation.pdf 2. Biography of Megh Nad Saha http://www.vigyanprasar.gov.in/scientists/saha/sahanew.htm 3. 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 4. https://upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Dr-Meghnad-Saha.jpg/220px-Dr- Meghnad-Saha.jpg 5. Narlikar, J. “The Scientific Edge,” Penguin Books, 2003 . 8 Statistical Mechanics Physics Gibbs Paradox and Indistinguishability of Particles .