DOCSLIB.ORG

Bsc Chemistry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bsc Chemistry ____________________________________________________________________________________________________ Subject Physics Paper No and Title P10 Statistical Physics Module No and Title Module 14Ensemble Theory(classical)-IV(Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States) Module Tag Phy_P10_M14.doc Content Writer Prof. P.K. Ahluwalia Physics Department Himachal Pradesh University Shimla 171 005 TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Canonical Ensemble(More Applications): 3.1 A System of Harmonic Oscillators (Classical and Quantum Mechanical Treatment) 3.2 Para-magnetism (Classical and Quantum Mechanical Treatment) 4. Equipartition Theorem 5. Virial Theorem 6. Density of Statesover Energy 7. Summary Appendices A1 Expansion of a logarithmic function and 퐜퐨퐭퐡 풙 A2 Spreadsheet for plotting Langevin and Brillouin functions Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States) ____________________________________________________________________________________________________ 1. Learning Outcomes After studying this module, you shall be able to Apply the approach of canonical ensemble via partition function to two very important prototype systems in physics via partition function o A collection of harmonic oscillators (classical and quantum mechanical treatment) o Para-magnetism (classical and quantum mechanical treatment). Compare the classical and quantum mechanical results obtained in the above two examples and see under what conditions quantum mechanical results approach classical results. To appreciate equipartition theorem and virial theorems as classical canonical ensemble 흏푯 averages of the quantity 푿풂 , where 푿풂 and 푿풃 are any two of the phase space co- 흏푿풃 ordinates of the system. To apply equipartition theorem to derive some well-known results for systems such as monoatomic gas, a diatomic gas, a harmonic oscillator and a crystalline solid. Derive virial theorem. Understand the importance of virial in computation of equation of state of a system and paving way for calculating pressure of the system. Calculate density of states over energy for a three dimensional system of free particles in non-relativistic, relativistic and ultra-relativistic regime. 2. Introduction In this module we carry forward the applications of canonical ensemble two most important problems of the physics: a collection of harmonic oscillators and classical model of para-magnetism treated as a system of N magnetic dipoles. Also we shall encounter two beautiful theorems called virial theorem of Clausius and equipartition theorem of Boltzmann. These theorems have many applications to arrive at some useful results in thermodynamics and statistical mechanics. 3. Canonical Ensemble (More Applications) Now we will look at some more interesting applicationsof canonical ensemble and simplicity with results can be obtained using partition function approach. 3.1 A System of Harmonic Oscillators Let us now examine a collection of N independent identical Linear harmonic oscillators using partition function approach as an application of canonical ensemble which forms a precursor to the study the statistical mechanics of a collection of photons in black body radiation and a collection of phonons in theory of lattice vibrations in solids. We treat the oscillators as first as classical entities and later as quantum mechanical entities to get the thermodynamics of the system and compare the results for any change in going from classical system of harmonic oscillators to quantum mechanical system of harmonic oscillators. Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States) ____________________________________________________________________________________________________ The Hamiltonian for a this set of identical linear harmonic oscillators, each having mass m and frequency 흎 is given by 푵 풑ퟐ ퟏ (1) 푯 = 풊 + 풎흎ퟐ풒ퟐ ퟐ풎 ퟐ 풊 풊=ퟏ (a) Classical Approach Now we are ready to write a one particle partition function for the recalling the definition in the Module 13. ퟏ (2) 풁 = 풆−휷푯풊 풒풊,풑풊 풅ퟑ풒 풅ퟑ퐩 ퟏ 풉 풊 퐢 ퟐ 휷풑 ퟏ ퟏ − 풊 − 휷풎흎ퟐ풒ퟐ = 풆 ퟐ풎 풅ퟑ풑 풆 ퟐ 풊 풅ퟑ퐪 풉 풊 퐢 ퟏ ퟏ (3) ퟏ ퟐ풎흅 ퟐ ퟐ흅 ퟐ ퟏ 풁 = = ퟏ 풉 휷 휷풎흎ퟐ 휷ℏ흎 ퟏ Here 휷 = . The partition function for the collection of independent N linear 풌푩푻 harmonic oscillators can be written as 푵 −푵 풁 = 풁ퟏ = 휷ℏ흎 (4) Now all thermodynamic properties can be derived Helmholtz free energy (F) ℏ흎 (5) 푭 = −풌푩푻 퐥퐧 풁 = 푵풌푩푻 퐥퐧 휷ℏ흎 = 푵풌푩푻 퐥퐧 풌푩푻 Entropy (S) 흏푭 ℏ흎 (6) 푺 = − = 푵풌푩 ퟏ − 퐥퐧 흏푻 푵,푽 풌푩푻 Internal Energy (E=F+TS) 푬 = 푵풌푩푻 (7) Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States) ____________________________________________________________________________________________________ This result is in accordance with law of equipartition of energy, according to which each quadratic term in the Hamiltonian for each harmonic oscillator contributes a ퟏ term 풌 푻. ퟐ 푩 Pressure (P) 흏푬 (8) 푷 = − = ퟎ 흏푽 푵,푻 Chemical Potential (흁) 흏푭 ℏ흎 (9) 흁 = = 푵풌푩푻 퐥퐧 흏푵 푽,푻 풌푩푻 Specific heat (푪풑 = 푪푽) 흏푬 (10) 푪푷 = 푪푽 = = 풌푩푻 흏푻 푵,푽 (b) Quantum mechanical approach Quantum mechanically, each linear harmonic oscillator is characterized by energy eigen value given by ퟏ (11) 흐 = 풏 + ℏ흎 , 풏 = ퟎ, ퟏ, ퟐ, ퟑ … … … … 풏 ퟐ Once again single oscillator partition function can be written as ퟏ ∞ ∞ − 휷ℏ흎 (12) ퟏ ퟐ − 휷ℏ흎 풆 풛 = 풆−휷흐풏 = 풆 ퟐ 풆−휷풏ℏ흎 = ퟏ ퟏ − 풆−휷풏ℏ흎 풏=ퟎ 풏=ퟎ ퟏ −ퟏ = ퟐ 퐬퐢퐧퐡 휷ℏ흎 ퟐ The partition function for the N Oscillator system is given by Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States) ____________________________________________________________________________________________________ ퟏ −푵 (13) 풁 = 풁 푵 = ퟐ 퐬퐢퐧퐡 휷ℏ흎 ퟏ ퟐ Now we can derive all thermodynamic properties Helmholtz free energy (F) ퟏ (14) 푭 = −풌 푻 퐥퐧 풁 = 푵풌 푻 퐥퐧 ퟐ 퐬퐢퐧퐡 휷ℏ흎 푩 푩 ퟐ ퟏ = 푵 ℏ흎 + 풌 푻 퐥퐧 ퟏ − 풆−휷ℏ흎 ퟐ 푩 Entropy (S) 흏푭 (15) 푺 = − 흏푻 푵,푽 ퟏ ퟏ = 푵풌 휷ℏ흎 퐜퐨퐭퐡 휷ℏ흎 푩 ퟐ ퟐ ퟏ − 퐥퐧 ퟐ 퐬퐢퐧퐡 휷ℏ흎 ퟐ Internal Energy (푬 = 푭 + 푻푺) ퟏ ퟏ ퟏ ℏ흎 (16) 푬 = 푭 + 푻푺 = 푵ℏ흎 퐜퐨퐭퐡 휷ℏ흎 = 푵 ℏ흎 + ퟐ ퟐ ퟐ 풆휷ℏ흎 − ퟏ This result is in not in accordance with law of equipartition of energy, according to which each quadratic term in the Hamiltonian for each harmonic oscillator ퟏ contributes a term 풌 푻. ퟐ 푩 Pressure (P) 흏푬 (17) 푷 = − = ퟎ 흏푽 푵,푻 Chemical Potential (흁) 흏푭 ퟏ −휷ℏ흎 (18) 흁 = = ℏ흎 + 풌푩푻 퐥퐧 ퟏ − 풆 흏푵 푽,푻 ퟐ Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States) ____________________________________________________________________________________________________ Specific heat (푪풑 = 푪푽) ퟐ (19) 흏푬 ퟏ ퟐ ퟏ 푪푷 = 푪푽 = = 푵풌푩 ℏ흎휷 퐜퐨퐬퐞퐜퐡 휷ℏ흎 흏푻 푵,푽 ퟐ ퟐ 풆휷ℏ흎 = 푵풌 ℏ흎휷 ퟐ 푩 풆휷ℏ흎 − ퟏ ퟐ All these results from (5) to (9) approach classical results for 풌푩푻 ≫ ℏ흎 3.1 Paramagnetism Let us now examine a collection of N independent identical spins sitting at different points using partition function approach. Each of these spins can be treated as magnetic dipoles with a magnetic moment 흁. These are distinguishable as they are sitting at different locations. These form a non-interacting system in the sense that each magnetic dipole does not experience the field of the other dipoles. These magnetic dipoles can be orientedin any direction with the application of external magnetic field. Since the spins are localized they have no kinetic energy. The Hamiltonian of such a system consists, therefore, of magnetic potential energy in the presence of magnetic field 횮 only which can be written as 푵 (20) 푯 = − 흁 풊 . 횮 풊=ퟏ Here 흁풊 is the magnetic moment of the ith magnetic dipole. We look at the system both classically and quantum mechanically and try to get some well-knownresult such as Curie Law for paramagnetic systems. (i) A classical Treatment In this case equation (20) can be written explicitly in terms of orientation 휽풊 of each magnetic dipole as 푵 (21) 푯 = − 흁풊 횮 퐜퐨퐬 휽풊 = −푵흁횮 퐜퐨퐬 휽 풊=ퟏ 휽 Now the single particle partition function can be written, remembering that three dimensional orientation is decided by 휽 and 흓 Physics PAPER No. 10 : Statistical Mechanics MODULE No.14 : Ensemble Theory(classical)-IV (Canonical Ensemble( More Applications), Equipartition Theorem Virial Theorem and Density of States) ____________________________________________________________________________________________________ So that single dipole partition function can be written as ퟐ흅 흅 (22) (휷흁횮 퐜퐨퐬 휽) (휷흁횮 퐜퐨퐬 휽) 풁ퟏ = 풆 = 풆 퐬퐢퐧 휽 풅휽풅흓 휽 ퟎ ퟎ Putting 퐱 = 퐜퐨퐬 휽 we solve equation (22) to get ퟐ흅 +ퟏ (23) (휷흁횮 풙) 퐬퐢퐧퐡(휷흁 횮) 풁ퟏ = 풆 풅풙풅흓 = ퟒ흅 ퟎ −ퟏ (훃훍횮) So the partition function of the system can be written as 퐬퐢퐧퐡(휷흁 횮) 푵 (24) 풁 = 풁 푵 = ퟒ흅 ퟏ 훃훍횮 Now we can calculate the thermodynamic properties using the results derived in module XIII. Helmholtz free energy (F) 퐬퐢퐧퐡(휷흁 횮) (25) 푭 = −푵 퐤 퐓 퐥퐧 ퟒ흅 퐁 훃훍횮 Magnetization 푴 of the system 흏푭 훛(퐥퐧 퐬퐢퐧퐡(휷흁 횮) − 퐥퐧 훃훍횮 ) (26) 푴 = − = 푵퐤퐁퐓 = 흏횮 푻,푵 흏횮 ퟏ (27) 푴 = 푵 흁 퐜퐨퐭퐡 휷흁횮 − = 푵흁푳 휷흁횮 훃훍횮 The mean magnetization in the direction of the magnetic field is 푴 ퟏ (28) 흁 = = 흁 퐜퐨퐭퐡 휷흁횮 − 푵 훃훍횮 ퟏ Where 푳 풙 = 퐜퐨퐭퐡 풙 − is the Langevin function,
Load more

Recommended publications
  • Dark Matter 1 Introduction 2 Evidence of Dark Matter
    Proceedings Astronomy from 4 perspectives 1. Cosmology Dark matter Marlene G¨otz (Jena) Dark matter is a hypothetical form of matter. It has to be postulated to describe phenomenons, which could not be explained by known forms of matter. It has to be assumed that the largest part of dark matter is made out of heavy, slow moving, electric and color uncharged, weakly interacting particles. Such a particle does not exist within the standard model of particle physics. Dark matter makes up 25 % of the energy density of the universe. The true nature of dark matter is still unknown. 1 Introduction Due to the cosmic background radiation the matter budget of the universe can be divided into a pie chart. Only 5% of the energy density of the universe is made of baryonic matter, which means stars and planets. Visible matter makes up only 0,5 %. The influence of dark matter of 26,8 % is much larger. The largest part of about 70% is made of dark energy. Figure 1: The universe in a pie chart [1] But this knowledge was obtained not too long ago. At the beginning of the 20th century the distribution of luminous matter in the universe was assumed to correspond precisely to the universal mass distribution. In the 1920s the Caltech professor Fritz Zwicky observed something different by looking at the neighbouring Coma Cluster of galaxies. When he measured what the motions were within the cluster, he got an estimate for how much mass there was. Then he compared it to how much mass he could actually see by looking at the galaxies.
    [Show full text]
  • S Chandrasekhar: His Life and Science
    REFLECTIONS S Chandrasekhar: His Life and Science Virendra Singh 1. Introduction Subramanyan Chandrasekhar (or `Chandra' as he was generally known) was born at Lahore, the capital of the Punjab Province, in undivided India (and now in Pakistan) on 19th October, 1910. He was a nephew of Sir C V Raman, who was the ¯rst Asian to get a science Nobel Prize in Physics in 1930. Chandra also went on to win the Nobel Prize in Physics in 1983 for his early work \theoretical studies of physical processes of importance to the structure and evolution of the stars". Chandra discovered that there is a maximum mass limit, now called `Chandrasekhar limit', for the white dwarf stars around 1.4 times the solar mass. This work was started during his sea voyage from Madras on his way to Cambridge (1930) and carried out to completion during his Cambridge period (1930{1937). This early work of Chandra had revolutionary consequences for the understanding of the evolution of stars which were not palatable to the leading astronomers, such as Eddington or Milne. As a result of controversy with Eddington, Chandra decided to shift base to Yerkes in 1937 and quit the ¯eld of stellar structure. Chandra's work in the US was in a di®erent mode than his initial work on white dwarf stars and other stellar-structure work, which was on the frontier of the ¯eld with Chandra as a discoverer. In the US, Chandra's work was in the mode of a `scholar' who systematically explores a given ¯eld. As Chandra has said: \There is a complementarity between a systematic way of working and being on the frontier.
    [Show full text]
  • Virial Theorem and Gravitational Equilibrium with a Cosmological Constant
    21 VIRIAL THEOREM AND GRAVITATIONAL EQUILIBRIUM WITH A COSMOLOGICAL CONSTANT Marek N owakowski, Juan Carlos Sanabria and Alejandro García Departamento de Física, Universidad de los Andes A. A. 4976, Bogotá, Colombia Starting from the Newtonian limit of Einstein's equations in the presence of a positive cosmological constant, we obtain a new version of the virial theorem and a condition for gravitational equi­ librium. Such a condition takes the form P > APvac, where P is the mean density of an astrophysical system (e.g. galaxy, galaxy cluster or supercluster), A is a quantity which depends only on the shape of the system, and Pvac is the vacuum density. We conclude that gravitational stability might be infiuenced by the presence of A depending strongly on the shape of the system. 1. Introd uction Around 1998 two teams (the Supernova Cosmology Project [1] and the High- Z Supernova Search Team [2]), by measuring distant type la Supernovae (SNla), obtained evidence of an accelerated ex­ panding universe. Such evidence brought back into physics Ein­ stein's "biggest blunder", namely, a positive cosmological constant A, which would be responsible for speeding-up the expansion of the universe. However, as will be explained in the text below, the A term is not only of cosmological relevance but can enter also the do­ main of astrophysics. Regarding the application of A in astrophysics we note that, due to the small values that A can assume, i s "repul­ sive" effect can only be appreciable at distances larger than about 22 1 Mpc. This is of importance if one considers the gravitational force between two bodies.
    [Show full text]
  • Scientometric Portrait of Nobel Laureate S. Chandrasekhar
    :, Scientometric Portrait of Nobel Laureate S. Chandrasekhar B.S. Kademani, V. L. Kalyane and A. B. Kademani S. ChandntS.ekhar, the well Imown Astrophysicist is wide!)' recognised as a ver:' successful Scientist. His publications \\'ere analysed by year"domain, collaboration pattern, d:lannels of commWtications used, keywords etc. The results indicate that the temporaJ ,'ari;.&-tjon of his productivit." and of the t."pes of papers p,ublished by him is of sudt a nature that he is eminent!)' qualified to be a role model for die y6J;mf;ergene-ration to emulate. By the end of 1990, he had to his credit 91 papers in StelJJlrSlructllre and Stellar atmosphere,f. 80 papers in Radiative transfer and negative ion of hydrogen, 71 papers in Stochastic, ,ftatisticql hydromagnetic problems in ph}'sics and a,ftrono"9., 11 papers in Pla.fma Physics, 43 papers in Hydromagnetic and ~}.droa:rnamic ,S"tabiJjty,42 papers in Tensor-virial theorem, 83 papers in Relativi,ftic a.ftrophy,fic.f, 61 papers in Malhematical theory. of Black hole,f and coUoiding waves, and 19 papers of genual interest. The higb~t Collaboration Coefficient \\'as 0.5 during 1983-87. Producti"it." coefficient ,,.as 0.46. The mean Synchronous self citation rate in his publications \\.as 24.44. Publication densi~. \\.as 7.37 and Publication concentration \\.as 4.34. Ke.}.word.f/De,fcriptors: Biobibliometrics; Scientometrics; Bibliome'trics; Collaboration; lndn.idual Scientist; Scientometric portrait; Sociolo~' of Science, Histor:. of St.-iencc. 1. Introduction his three undergraduate years at the Institute for Subrahmanvan Chandrasekhar ,,.as born in Theoretisk Fysik in Copenhagen.
    [Show full text]
  • Equipartition Theorem - Wikipedia 1 of 32
    Equipartition theorem - Wikipedia 1 of 32 Equipartition theorem In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion. The equipartition theorem makes quantitative Thermal motion of an α-helical predictions. Like the virial theorem, it gives the total peptide. The jittery motion is random average kinetic and potential energies for a system at a and complex, and the energy of any given temperature, from which the system's heat particular atom can fluctuate wildly. capacity can be computed. However, equipartition also Nevertheless, the equipartition theorem allows the average kinetic gives the average values of individual components of the energy of each atom to be energy, such as the kinetic energy of a particular particle computed, as well as the average or the potential energy of a single spring. For example, potential energies of many it predicts that every atom in a monatomic ideal gas has vibrational modes. The grey, red an average kinetic energy of (3/2)kBT in thermal and blue spheres represent atoms of carbon, oxygen and nitrogen, equilibrium, where kB is the Boltzmann constant and T respectively; the smaller white is the (thermodynamic) temperature. More generally, spheres represent atoms of equipartition can be applied to any classical system in hydrogen.
    [Show full text]
  • Astronomy Astrophysics
    A&A 470, 449–466 (2007) Astronomy DOI: 10.1051/0004-6361:20077443 & c ESO 2007 Astrophysics A CFH12k lensing survey of X-ray luminous galaxy clusters II. Weak lensing analysis and global correlations S. Bardeau1,2,G.Soucail1,J.-P.Kneib3,4, O. Czoske5,6,H.Ebeling7,P.Hudelot1,5,I.Smail8, and G. P. Smith4,9 1 Laboratoire d’Astrophysique de Toulouse-Tarbes, CNRS-UMR 5572 and Université Paul Sabatier Toulouse III, 14 avenue Belin, 31400 Toulouse, France e-mail: [email protected] 2 Laboratoire d’Astrodynamique, d’Astrophysique et d’Aéronomie de Bordeaux, CNRS-UMR 5804 and Université de Bordeaux I, 2 rue de l’Observatoire, BP 89, 33270 Floirac, France 3 Laboratoire d’Astrophysique de Marseille, OAMP, CNRS-UMR 6110, Traverse du Siphon, BP 8, 13376 Marseille Cedex 12, France 4 Department of Astronomy, California Institute of Technology, Mail Code 105-24, Pasadena, CA 91125, USA 5 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany 6 Kapteyn Astronomical Institute, PO Box 800, 9700 AV Groningen, The Netherlands 7 Institute for Astronomy, University of Hawaii, 2680 Woodlawn Dr, Honolulu, HI 96822, USA 8 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK 9 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Received 9 March 2007 / Accepted 9 May 2007 ABSTRACT Aims. We present a wide-field multi-color survey of a homogeneous sample of eleven clusters of galaxies for which we measure total masses and mass distributions from weak lensing.
    [Show full text]
  • Virial Theorem for a Molecule Approved
    VIRIAL THEOREM FOR A MOLECULE APPROVED: (A- /*/• Major Professor Minor Professor Chairman' lof the Physics Department Dean 6f the Graduate School Ranade, Manjula A., Virial Theorem For A Molecule. Master of Science (Physics), May, 1972, 120 pp., bibliog- raphy, 36 titles. The usual virial theorem, relating kinetic and potential energy, is extended to a molecule by the use of the true wave function. The virial theorem is also obtained for a molecule from a trial wave function which is scaled separately for electronic and nuclear coordinates. A transformation to the body fixed system is made to separate the center of mass motion exactly. The virial theorems are then obtained for the electronic and nuclear motions, using exact as well as trial electronic and nuclear wave functions. If only a single scaling parameter is used for the electronic and the nuclear coordinates, an extraneous term is obtained in the virial theorem for the electronic motion. This extra term is not present if the electronic and nuclear coordinates are scaled differently. Further, the relation- ship between the virial theorems for the electronic and nuclear motion to the virial theorem for the whole molecule is shown. In the nonstationary state the virial theorem relates the time average of the quantum mechanical average of the kinetic energy to the radius vector dotted into the force. VIRIAL THEOREM FOR A MOLECULE THESIS Presented to the Graduate Council of the North Texas State University in Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE By Manjula A. Ranade Denton, Texas May, 1972 TABLE OF CONTENTS Page I.
    [Show full text]
  • 8.901 Lecture Notes Astrophysics I, Spring 2019
    8.901 Lecture Notes Astrophysics I, Spring 2019 I.J.M. Crossfield (with S. Hughes and E. Mills)* MIT 6th February, 2019 – 15th May, 2019 Contents 1 Introduction to Astronomy and Astrophysics 6 2 The Two-Body Problem and Kepler’s Laws 10 3 The Two-Body Problem, Continued 14 4 Binary Systems 21 4.1 Empirical Facts about binaries................... 21 4.2 Parameterization of Binary Orbits................. 21 4.3 Binary Observations......................... 22 5 Gravitational Waves 25 5.1 Gravitational Radiation........................ 27 5.2 Practical Effects............................ 28 6 Radiation 30 6.1 Radiation from Space......................... 30 6.2 Conservation of Specific Intensity................. 33 6.3 Blackbody Radiation......................... 36 6.4 Radiation, Luminosity, and Temperature............. 37 7 Radiative Transfer 38 7.1 The Equation of Radiative Transfer................. 38 7.2 Solutions to the Radiative Transfer Equation........... 40 7.3 Kirchhoff’s Laws........................... 41 8 Stellar Classification, Spectra, and Some Thermodynamics 44 8.1 Classification.............................. 44 8.2 Thermodynamic Equilibrium.................... 46 8.3 Local Thermodynamic Equilibrium................ 47 8.4 Stellar Lines and Atomic Populations............... 48 *[email protected] 1 Contents 8.5 The Saha Equation.......................... 48 9 Stellar Atmospheres 54 9.1 The Plane-parallel Approximation................. 54 9.2 Gray Atmosphere........................... 56 9.3 The Eddington Approximation................... 59 9.4 Frequency-Dependent Quantities.................. 61 9.5 Opacities................................ 62 10 Timescales in Stellar Interiors 67 10.1 Photon collisions with matter.................... 67 10.2 Gravity and the free-fall timescale................. 68 10.3 The sound-crossing time....................... 71 10.4 Radiation transport.......................... 72 10.5 Thermal (Kelvin-Helmholtz) timescale............... 72 10.6 Nuclear timescale..........................
    [Show full text]
  • 1. Hydrostatic Equilibrium and Virial Theorem Textbook: 10.1, 2.4 § Assumed Known: 2.1–2.3 §
    1. Hydrostatic equilibrium and virial theorem Textbook: 10.1, 2.4 § Assumed known: 2.1–2.3 § Equation of motion in spherical symmetry d2r GM ρ dP ρ = r (1.1) dt2 − r2 − dr Hydrostatic equilibrium dP GM ρ = r (1.2) dr − r2 Mass conservation dM r =4πr2ρ (1.3) dr Virial Theorem 1 E = 2E or E = E (1.4) pot − kin tot 2 pot Derivation for gaseous spheres: multiply equation of hydrostatic equilibrium by r on both sides, integrate over sphere, and relate pressure to kinetic energy (easiest to verify for ideal gas). For next time – Read derivation of virial theorem for set of particles ( 2.4). – Remind yourself of interstellar dust and gas, and extinction§ ( 12.1). – Remind yourself about thermodynamics, in particular adiabatic§ processes (bottom of p. 317 to p. 321). Fig. 1.1. The HRD of nearby stars, with colours and distances measured by the Hipparcos satellite. Taken from Verbunt (2000, first-year lecture notes, Utrecht University). Fig. 1.2. Observed HRD of the stars in NGC 6397. Taken from D’Antona (1999, in “The Galactic Halo: from Globular Clusters to Field Stars”, 35th Liege Int. Astroph. Colloquium). Fig. 1.3. HRD of the brightest stars in the LMC, with observed spectral types and magnitudes transformed to temperatures and luminosities. Overdrawn is the empirical upper limit to the luminosity, as well as a theoretical main sequence. Taken from Humphreys & Davidson (1979, ApJ 232, 409). 2. Star formation Textbook: 12.2 § Jeans Mass and Radius 1/2 3/2 3/2 3/2 1/2 3 5k T 2 T n − MJ = 1/2 = 29 M µ− 4 3 , (2.1) 4π GµmH ρ ⊙ 10K 10 cm− 1/2 1/2 1/2 1/2 3 5k T 1 T n − RJ = 1/2 =0.30pc µ− 4 3 .
    [Show full text]
  • Principle of Equipartition of Energy -Dr S P Singh (Dept of Chemistry, a N College, Patna)
    Principle of Equipartition of Energy -Dr S P Singh (Dept of Chemistry, A N College, Patna) Historical Background 1843: The equipartition of kinetic energy was proposed by John James Waterston. 1845: more correctly proposed by John James Waterston. 1859: James Clerk Maxwell argued that the kinetic heat energy of a gas is equally divided between linear and rotational energy. ∑ Experimental observations of the specific heat capacities of gases also raised concerns about the validity of the equipartition theorem. ∑ Several explanations of equipartition's failure to account for molar heat capacities were proposed. 1876: Ludwig Boltzmann expanded this principle by showing that the average energy was divided equally among all the independent components of motion in a system. ∑ Boltzmann applied the equipartition theorem to provide a theoretical explanation of the Dulong-Petit Law for the specific heat capacities of solids. 1900: Lord Rayleigh instead put forward a more radical view that the equipartition theorem and the experimental assumption of thermal equilibrium were both correct; to reconcile them, he noted the need for a new principle that would provide an "escape from the destructive simplicity" of the equipartition theorem. 1906: Albert Einstein provided that escape, by showing that these anomalies in the specific heat were due to quantum effects, specifically the quantization of energy in the elastic modes of the solid. 1910: W H Nernst’s measurements of specific heats at low temperatures supported Einstein's theory, and led to the widespread acceptance of quantum theory among physicists. Under the head we deal with the contributions of translational and vibrational motions to the energy and heat capacity of a molecule.
    [Show full text]
  • Arxiv:2006.01326V1 [Astro-Ph.GA] 2 Jun 2020 58090 Morelia, Michoacan, Mexico
    Noname manuscript No. (will be inserted by the editor) From diffuse gas to dense molecular cloud cores Javier Ballesteros-Paredes · Philippe Andre,´ · Patrick Hennebelle, · Ralf S. Klessen, · Shu-ichiro Inutsuka, · J. M. Diederik Kruijssen, · Melanie´ Chevance, · Fumitaka Nakamura, · Angela Adamo · Enrique Vazquez-Semadeni Received: 2020-01-31 / Accepted: 2020-05-17 Abstract Molecular clouds are a fundamental ingredient of galaxies: they are the channels that transform the diffuse gas into stars. The detailed process of how they Javier Ballesteros-Paredes Instituto de Radioastronom´ıa y Astrof´ısica, UNAM, Campus Morelia, Antigua Carretera a Patzcuaro 8701. 58090 Morelia, Michoacan, Mexico. E-mail: [email protected] Philippe Andre´ Laboratoire d’Astrophysique (AIM), CEA/DRF, CNRS, Universite´ Paris-Saclay, Universite´ Paris Diderot, Sorbonne Paris Cite,91191´ Gif-sur-Yvette, France Patrick Hennebelle AIM, CEA, CNRS, Universite´ Paris-Saclay, Universite´ Paris Diderot, Sorbonne Paris Cite,´ 91191, Gif- sur-Yvette, France Ralf S. Klessen Universitat¨ Heidelberg, Zentrum fur¨ Astronomie, Institut fur¨ Theoretische Astrophysik, Albert-Ueberle- Str. 2, 69120 Heidelberg, Germany, Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan J. M. Diederik Kruijssen Astronomisches Rechen-Institut, Zentrum fur¨ Astronomie der Universitat¨ Heidelberg, Monchhofstraße¨ 12- 14, 69120 Heidelberg, Germany, Melanie´ Chevance Astronomisches Rechen-Institut, Zentrum fur¨ Astronomie der Universitat¨ Heidelberg,
    [Show full text]
  • 10. Timescales in Stellar Interiors This Separation Is Pleasant Because It
    10.Timescales inStellarInteriors This separation is pleasant because it means whenever we consider one timescale, we can assume that the faster processes are in equilibrium while the slower processes are static. Much excitement ensues when this hierarchy breaks down. For example, we see convection occur onτ dyn which then fundamentally changes the ther- mal transport. Or in the cores of stars near the end of their life,τ nuc becomes much shorter. If it gets shorter thanτ dyn, then the star has no time to settle into equilibrium – it may collapse. 10.8 The Virial Theorem In considering complex systems as a whole, it becomes easier to describe im- portant properties of a system in equilibrium in terms of its energy balance rather than its force balance. For systems in equilibrium– not just a star now, or even particles in a gas, but systems as complicated as planets in orbit, or clusters of stars and galaxies– there is a fundamental relationship between the internal, kinetic energy of the system and its gravitational binding energy. This relationship can be derived in a fairly complicated way by taking several time derivatives of the moment of inertia of a system, and applying the equations of motion and Newton’s laws. We will skip this derivation, the result of which can be expressed as: d2 I (208) = 2 K + U , dt2 � � � � where K is the time-averaged kinetic energy, and U is the time-averaged � � � � 2 gravitational potential energy. For a system in equilibrium, d I is zero, yielding dt2 the form more traditionally used in astronomy: 1 (209) K = U � � − 2 � � The relationship Eq.
    [Show full text]