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Unformatted text preview: Ben Sauerwine Practice for Qualifying Exams Solution based on advice from Diana. Problem Source: CMU February 2006 Qualifying Exam Consider an ideal gas of N atoms confined to a container of volume V and internal surface area A. Although you may neglect the interactions between atoms, there is an attraction between the atoms and the walls of the container that cannot be ignored. A simple model for the atoms adsorbed onto the surface is to treat them as a two-dimensional classical ideal gas, where the energy of an adsorbed atom is ( ) 2 2 ε ε − = m p p v v and p v is the two-dimensional momentum. Do not concern yourself with the details of the binding. Treat ε as a known parameter. (a) What is the classical partition function of the adsorbed atoms if N’ atoms are bound to the surface? The classical partition function is given for one atom by: ( ) ∫ ∫ ∞ ∞ − ∞ ∞ − − = q pd d e Z n n q p H v v , β In this case, then, I have: 2 2 2 2 1 2 2 2 2 2 2 2 βε β βε ε β ε β β π β β π π e m A Z dv e dr m r dv m r v rdr e Ae rdr e A p d e A Z v m r m r n m p = = = = = = = ∫ ∫ ∫ ∫ ∞...
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