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stp_chap8

# stp_chap8 - Chapter 8 Classical Gases and Liquids c 2010 by...

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Chapter 8 Classical Gases and Liquids c c 2010 by Harvey Gould and Jan Tobochnik 8 June 2010 We discuss approximation techniques for interacting classical particle systems such as dense gases and liquids. 8.1 Introduction Because there are only a few problems in statistical mechanics that can be solved exactly, we need to Fnd approximate solutions. We introduce several perturbation methods that are applicable when there is a small expansion parameter. Our discussion of interacting classical particle systems will involve some of the same considerations and di±culties that are encountered in quantum Feld theory (no knowledge of the latter is assumed). ²or example, we will introduce diagrams that are analogous to ²eynman diagrams and Fnd divergences analogous to those found in quantum electrodynamics. We also discuss the spatial correlations between particles due to their interactions and the use of hard spheres as a reference system for understanding the properties of dense ³uids. 8.2 Density Expansion Consider a gas of N identical particles each of mass m at density ρ = N/V and temperature T . We will assume that the total potential energy U is a sum of two-body interactions u ij = u ( | r i r j | ), and write U as U = N s i<j u ij . (8.1) The exact form of u ( r ) for electrically neutral molecules and atoms must be constructed by a Frst principles quantum mechanical calculation. Such a calculation is very di±cult, and for many purposes it is su±cient to choose a simple phenomenological form for u ( r ). The most important 387

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CHAPTER 8. CLASSICAL GASES AND LIQUIDS 388 u LJ (r) r 2.0 σ 3.0 σ 1.0 σ ε Figure 8.1: Plot of the Lennard-Jones potential u LJ ( r ), where r is the distance between the parti- cles. The potential is characterized by a length σ and an energy ǫ . features of u ( r ) are a strong repulsion for small r and a weak attraction at large r . A common phenomenological form of u ( r ) is the Lennard-Jones or 6-12 potential shown in Figure 8.1: u LJ ( r ) = 4 ǫ bp σ r P 12 p σ r P 6 B . (8.2) The values of σ and ǫ for argon are σ = 3 . 4 × 10 10 m and ǫ = 1 . 65 × 10 21 J. The attractive 1 /r 6 contribution to the Lennard-Jones potential is due to the induced dipole- dipole interaction of two neutral atoms. 1 The resultant attractive interaction is called the van der Waals potential. The rapidly increasing repulsive interaction as the separation between atoms is decreased for small r is a consequence of the Pauli exclusion principle. The 1 /r 12 form of the repulsive potential in (8.2) is chosen only for convenience. The existence of many calculations and simulation results for the Lennard-Jones potential encourages us to use it even though there are more accurate forms of the interparticle potential for modeling the interactions in real ±uids, proteins, and other complex molecules.
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stp_chap8 - Chapter 8 Classical Gases and Liquids c 2010 by...

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