Statistical Mechanics — DU B.Sc. (Hons) Physics Previous Year Question Papers

Name of the Department: Physics and Astrophysics Question paper Set No. : 11 Attempt five questions in all i) At a high temperature T, calculate the average energy per particle of a non-interacting gas of two-dimensional classical harmonic oscillators (No derivation required) ii) 5 distinguishable particles are to be distributed anongst three energy levels with energy 0, g and 2€. For a fixed energy of 5£, enumerate the number of macrostates. Identify which macrostate is the most probable? iii) Consider the blackbody radiation contalned in a cavity whose walls are at temperature T = 500 K. The radiation is in equilibrium with the walls of the cavity. If the temperature of the walls is increased to 1000 K and the radiation is allowed to come to equilibrium at the new temperature, by what factor will the entropy change for the blackbody radiation in the cavity. iv) Calculate the chemical potential of strongly degenerate system of N bosons (N >> 1) at temperature T (T < Tc), if the energy of the lowest state is €o. vi) Show graphically the variation of radius of white dwarf electrons) as a function of its mass. Chandrasekhar mass limit. star (consisting of relativistic Hence explain the physical significance, of vii) Calculate the value of +i;2 function. Explain the result. at 8 = pr, where//gJ represents Fermi-Dirac distribution 2(a)Consider an isolated system of IV >> 1 weakly interacting classical particles. These particles can be in any of the states with energies 0, €, 2€, 3€, „..The temperature of the system may be written as 1AS _T = EE where AS is the change in the entropy of the system corresponding to change in intemal energy of the system by AE. i) What is the entropy of the system when it is in its ground state. ii) If the total energy of the system is €, what will be its entropy? iii) Determine the change in entropy and the corresponding temperature when the energy of the system is increased fi.om € to 2€ ? (12) a) An isolated physical system consisted of N distinguishable particles is characterized by two energy states: a ground state of energy €i with a degeneracy gi and number of particles ni. An excited state is characterised by energy €2 with degeneracy g2 having number of particles n2. Show that the entropy of this system can be written as where Pi and P2 are the probabilities that the system will be found in the ground and the first excited states respectively (Here N = ni + n2). (6) 3 (a) A system of H-atoms is enclosed in a cavity of volume y at a very high temperature that results in the thermal ionization of the H atom into H+ ion and c -electron. Show that the degree of ionization x for this system in terms of the temperature T., pressurep and ionization energy £* can be expressed as: where g|, go are the ground state degeneracies of the ion and the atom respectively and 777e is the mass of electron. (12) (b) A system ofIV classical particles confined to a volulne 7'has the partition function given by