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Unformatted text preview: © M. S. Shell 2009 1/16 last modified 11/12/2009 Statistical mechanics of classical systems ChE210A The classical canonical partition function When quantum effects are not significant, we can approximate the behavior of a system using a classical representation. Here, we derive the expressions for the partition functions in such cases. We will consider a singlecomponent system composed of sphericallysymmetric, single particle molecules. For the canonical partition function, the discrete version in general is gG¡,¢,£¤ = ¥ expG−¦§ ¨ ¤ all ¨ at ©,ª = ¥ ¥ … ¥ « ¬®Gall DOF¤ DOF for ª DOF for ¯ DOF for ° where ± is an index running over all microstates of the system and the sums indicate exhaustive enumeration of all possible values for each degree of freedom (DOF) of each particle. For a classical system of structureless particles, each microstate involves 6£ degrees of freedom— the 3 positions and 3 momenta for each particle. These variables are continuous, rather than discrete. We rewrite the summation over microstates as an integral over these variables: gG¡, ¢,£¤~²²…²« ¬³´µ ¶ ,· ¶ ¸ ¹µ ° ¹µ ¯ …¹µ ª ¹· ° ¹· ¯ …¹· ª Here, we have replaced the energy with the Hamiltonian function. The quantities · ª and µ ª are the vectors of all positions and momenta, respectively. The notation ¹µ indicates ¹º » ¹º ¼ ¹º ½ such that the integral over ¹µ for one particle is actually a threedimensional integral. A similar case exists for ¹· . Therefore, the total number of integrals is 6£ . We say that this is a 6£dimensional integral. We must first discuss the limits of integration: • For each momentum variable, the limits are −∞ < º < ∞ . In other words, the momen ta can take on any real value. These degrees of freedom and the associated integrals are unbounded . • For each position variable, the values it can take on depend on the volume of the con tainer into which it was placed. For a cubic box of side length ¾ , one has 0 < ¿ < ¾ . © M. S. Shell 2009 2/16 last modified 11/12/2009 Therefore these position variables are bounded and the integrals are definite. The lim its of the position integrals also introduce a dependence of the partition function on g . • Does the specific shape of the container matter? For large homogeneous systems, this integral is independent of the container shape, for the same average density. The rea son is that any unusual container shape can be broken down and approximated by a number of smaller subsystems of the same density that are each cubic in size, in a pix ilated fashion. Each pixel can still be large enough to be considered macroscopic in size, which means that boundary interactions between it and neighboring pixels are very small. Therefore, the net properties of the whole system are simply the sum of those of the small pixels, which is insensitive to how they are arranged (the overall shape)....
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 Fall '08
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 Energy, Kinetic Energy, Statistical Mechanics, M. S. Shell

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