Van der Waals Model

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  • Topic: Statistical mechanics, Thermodynamics, Crystal structure
  • Pages : 7 (1672 words )
  • Download(s) : 285
  • Published : May 29, 2012
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Adam
 De
 Jesus
  ID:
 387876
  Thermodynamics
 262
 
Crystalline Solids Structure and Perturbation Approximation Methods Voices from the past rarely resonant as loudly in the present, however, for the Van der Waals (VDW) theoretical perspective, its voice has found substantially significant vibrancy in today’s modern scientific community. The advent of computer simulations has lead to the creation of complex theoretical models that can be calculated and visualized like never before. Although the prevailing perspective of the VDW model (to a lowly undergraduate anyway) may seem out-of-date, David chandler and his group breathe life into the VDW theory with the reintroduction the this theory for solids and liquids, emphasizing the qualitative and new quantitative approaches that are possible. The emphasis on the short ranged, highly repulsive intermolecular interactions as the dominating factor in the close packing of liquid and solid phase phenomena is highlighted in this review. Taking the VDW model to the theoretical limit when the attractive force has no effect the attractive force is infinitely weak and infinitely long ranged. This unrealistic limit can be relaxed to simulate more realistic liquid conditions (high density) and the attractive force is still seen to be miniscule, owning to the dominating—highly overshadowing—short ranged repulsive forces, allowing for the mean field approximation for the repulsive forces to be valid for many cases. The definition of a VDW material is one which is composed of hard core molecules, with size and shape parameterized to mimic the repulsive branches of the intermolecular potentials held at a certain potential by the mean field attractive potential. Computer simulations were performed using the radial distribution function, g(r), which gives relative information about the pair interactions of atoms. It is shown to be highly convincing that the attractive forces are small after comparing the g0(r) (function including repulsive and attractive factors) with the g(r) (the repulsive factors alone) in figure 1. The two functions are seen to correspond beautifully at the first main peak and provide convincing evidence that the attractive force plays a minimal role and can be approximated by a background potential. There is also close approximation of the hard sphere model to the radial distribution function with separation d (diameter of the molecular spheres). It is possible to find a value of d which fits the liquid structure factor—S(k) fitted by Sd(k)—and in essence find the size of the particles under observation, which points to the validity of the VDW qualitative perspective. The authors of the paper expand on the VDW model and introduce the WCA theory of liquids; the main results of which are exemplified by figure 1. The model splits the intermolecular forces into short and long (repulsive and attractive) and ultimately formulates equation 1. In equation 1 the uo is the LennardJones potential of the molecules, yd is a function of the difference between g0(r) and gd(r), and the exponential term is the Boltzmann factor, and the resulting graph is indistinguishable from that of the solid line in figure 1. This model is self contained and easier to calculate than most other models that involve correction terms.   g r ≈ g! r ≈ e ! ! ! ! ! ! !! ! !

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Adam
 De
 Jesus
  ID:
 387876
  Thermodynamics
 262
 
In quantum mechanics, perturbation theory is an essential tool in calculating fundamental properties of a system. Thermodynamic perturbation theory is a directly applicable result of the structural theory proposed by the WCA model. In this way the WCA model becomes a gateway for the calculation of the thermodynamic properties of a liquid system. When the approximation of g(r) ≅ g0(r) is made the thermodynamic properties can be obtained from first order perturbation theory, and an estimation of the attractive forces...
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