Physical Chemistry - Fermi-Dirac Statistics - University of Mysore

University of Mysore Yuvaraja’s college Physical Chemistry Topic : Fermi-Dirac statistics Presented Guided by, by, Madhushree K S H.P.Jayadevappa 1ST M.Sc Chemistry Department of Chemistry...

University of Mysore Yuvaraja’s college Physical Chemistry Topic : Fermi-Dirac statistics Presented Guided by, by, Madhushree K S H.P.Jayadevappa 1ST M.Sc Chemistry Department of Chemistry 1 Content/Index  Introduction Thermodynamic probability Types of Statistics, 1. Maxwell- Boltzmann statistics 2. Bose-Einstein statistics 3. Fermi-Dirac statistics 2 Introduction Statistical thermodynamics or statistical mechanics provides a link between quantum mechanics (or wave mechanics) and classical thermodynamics.  Classical thermodynamic: It deals with macroscopic properties of matter and describes the behavior of large number of molecules in terms of properties such as pressure, volume, temperature, composition etc  Quantum mechanics : It deal almost exclusively with matter at the microscopic level. It tells us each microscopic system can be described by a wave function. Neither classical thermodynamics nor quantum mechanics is able to calculate the macroscopic properties of matter from the microscopic structure of individual molecules. 3 The discipline which deals with the computation of the macroscopic properties of matter from the data on the microscopic properties of individual atom (or molecules) is called Statistical mechanics or Statistical thermodynamics 4 Terms of Statistical thermodynamics Phase space : It is an imaginary 6-dimensional space that assists the particle is called Phase space. Ensemble : A large no virtual copy’s or large no of replica’s of phase space which are in thermodynamic equilibrium are called as ensembles. There are 3 types of ensembles: 1. Micro canonical ensemble : It is the collection of large no of essentially independent system having the same energy(E), volume(V) and same no of particles(N). 2. Canonical ensemble : The collection of large no of essentially independent system having the same temperature(T),volume(V) and same no of particles(N). 3. Grand-canonical ensemble : The collection of large no of essentially independent systems having the same temperature(T), volume(V) and chemical potential(µ) 5  Macroscopic property In macroscopic approach, a certain quantity of matter is considered without events occurring at molecular level. The structure of matter is not considered Only few variables are needed to describe the state of system. It is also called as classical thermodynamics. It provides a direct and easy way to the solution of problem Microscopic property In microscopic approach, the properties of the system are based on the average behavior of large group of molecule under consideration. The knowledge of the structure of matter is necessary. A large number of variables 6 are needed to describe  Thermodynamic probability The number of microstate corresponding to a given macrostate is called the “Thermodynamic probability”. It is denoted by “W”. The equation for thermodynamic probability is, When entropy is more probability also more.Therefore Entropy is the function of probability The relation between W and S is S=K lnW Where, S=entropy , K=boltzmann entropy, W=probability. 7 Types of statistics Different types or physical situation encountered in nature are described by three types of statistics through Maxwell- Boltzmann (or M-B) statistics, the Bose-Einstein (or B- E)statistics and the Fermi-Dirac(or F-D)statistics The M-B statistics, developed long before the advent of quantum mechanics, is also called classical statistics. The Bose-Einstein statistics and the Fermi-Dirac statistics are collectively called quantum statistics. 8 1. Maxwell-Boltzmann statistic The particles are assumed to be distinguishable and any number of particles may occupy the same energy level.particles obeying Maxwell-Boltzmann statistics are called maxwellons or boltzons. The Maxwell-Boltzmann distribution law equation is given by, 9 Limitation of M-B statistics Its assumes that particles are point like and do not interact with each other, which is not for real gases. It assumes that the gas is homogeneous and isotropic meaning that it has the same properties in all direction. This is not for real gases in the presence of external field. It does not take into account quantum effects, which become important at low temperature or high density. This limitation makes the Maxwell-Boltzmann statistics inadequate for describing certain types of gases and systems , such as those involving strong interactions, high density or low temperature. 10 2.Bose-Einstein statistics. The particles are indistinguishable and any number of particles may occupy given energy level. This statistics is obeyed by particles having integral spin , such as hydrogen (H2), deuterium(D2), nitrogen(N2), helium-4(4He) and photons. Particles obeying Bose-Einstein statistics are called bosons. The Bose-Einstein distribution law equation is given by, 11 Limitation of Bose-Einstein statistics Assumes non-interacting particles: It assumes that the particles are non interacting ,which is mot always the case in real system. Only applicable to bosons : It is specially designed for bosons and cannot be used to describe fermions, which are particles with half integral spin. Limitation to system with a fixed number of particles : It is typically used to describe system with a fixed number where the no of particles is not conserved. 12 Fermi-Dirac statistics The particles are indistinguishable but only one particle may occupy a given energy level. This statistics is obeyed by particles having half-integral spin. E.g. : The electron , proton , helium-3 , NO(nitric oxide) Particles obeying F-D statistics are called as Fermions. 13 Derivatio n Consider that ni particles are distributed among the gi states where gi is the degeneracy of the i th energy level. Imagine that particle are indistinguishable. ni is the no particles It follows the Pauli's exclusive principle Therefore the no of arrangement of ni particles in the i th energy level is given by, Thus , the thermodynamic probability W for the system of N particles( i.e the number of ways distributing N 14 particles among the various energy level) is given by, 15 16 17 Eq © is the Fermi-Dirac Statistics equation that gives the arrangement of particles in energy levels. 18 Conclusion Comparison between Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. Maxwell-Boltzmann statistics : It is applicable to identical, distinguisable particles of any type spin. The molecule of gas are particles of this type. Bose-Einstein Statistics : It is applicable to identical, indistinguishable particles of integral spin. These particles are called Bosons. Example: Photons, helium atom. Fermi-Dirac statistics : It is applicable to the identical, indistinguishable particles of half integral spin. These particles obey Pauli's exclusive principle. Example: electron, proton etc. 19 20