Thermodynamic properties from pair distribution functions

4 Thermodynamic properties from pair distribution functions 

For any non-ideal system, the excess internal energy can be defined as 

The excess energy encompasses all the interactions between particles 

The pair distribution function gives the probability of finding a second particle in a given volume element at position r from a first particle. Specifically, in volume 

The contribution of this differential system to the potential energy is then 

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Thermodynamic properties from pair distribution functions 163 Assuming pairwise interactions, we write... 

For the case of a pure monatomic liquid, in the limit that there are only pair interactions, the pair distribution function provides a complete microscopic specification from which all thermodynamic properties can be calculated 2>. If there are (excess) three molecule interactions, then one must also know the triplet distribution function to complete the microscopic description the extension to still higher order (excess) interactions is obvious. 

How then can this information about the local structure of fluids be used to determine thermodynamic properties John Kirkwood was able to find an ingenious theoretical path from the pairwise potential of interaction between two particles, u(r), and the pair distribution function to the internal energy and the pressure of liquids. In Section 9.4, we briefly present this theoretical path. Note that it is beyond the scope of this text to discuss the details of Kirkwood s efforts to detennine g(r) with purely theoretical arguments. The interested reader should consult Kirkwood s papers for a detailed exposition of the theory (Further reading). We should also note that pair distribution functions of materials can be readily obtained with diffraction measurements and, as discussed later in the book, using computer simulations. 

The radial distribution function plays an important role in the study of liquid systems. In the first place, g(r) is a physical quantity that can be determined experimentally by a number of techniques, for instance X-ray and neutron scattering (for atomic and molecular fluids), light scattering and imaging techniques (in the case of colloidal liquids and other complex fluids). Second, g(r) can also be determined from theoretical approximations and from computer simulations if the pair interparticle potential is known. Third, from the knowledge of g(r) and of the interparticle interactions, the thermodynamic properties of the system can be obtained. These three aspects are discussed in more detail in the following sections. In addition, let us mention that the static structure is also important in determining physical quantities such as the dynamic an other transport properties. Some theoretical approaches for those quantities use as an input precisely this structural information of the system [15-17,30,31]. 

Two sets of methods for computer simulations of molecular fluids have been developed Monte Carlo (MC) and Molecular Dynamics (MD). In both cases the simulations are performed on a relatively small number of particles (atoms, ions, and/or molecules) of the order of 100simulation supercell. The interparticle interactions are represented by pair potentials, and it is generally assumed that the total potential energy of the system can be described as a sum of these pair interactions. Very large numbers of particle configurations are generated on a computer in both methods, and, with the help of statistical mechanics, many useful thermodynamic and structural properties of the fluid (pressure, temperature, internal energy, heat capacity, radial distribution functions, etc.) can then be directly calculated from this microscopic information about instantaneous atomic positions and velocities. 

CHECKPOINT The pair correlation function g(R) demonstrates a solution to the problem of describing the liquid. Instead of trying to represent the liquid exactly, we use an average distribution of the relative positions of the particles. This varies sufficiently from one liquid to another and from one set of conditions to another that we can learn a great deal about the nature of the liquid from g R). In Thermodynamics, Kinetics, and Statisticai mechanics, we show how the pair correlation function can be used to predict many bulk properties of liquids. 

The results of BOSS calculations can be analyzed In ChemEdit. Some plots from a BOSS calculation are shown in Figure 6. Information extracted from a BOSS output includes (a) geometries of the solutes (b) radial distribution functions (c) energy and energy pair distributions (d) average thermodynamic properties (e) components of the solvent-solute energy and (f) AH, AS, and AG for perturbations. 

We will demonstrate next that the thermodynamic properties of a fluid can be determined from knowledge of the radial distribution function, g(r), and the pair potential, T(r). To this purpose we will develop expressions for its equation of state and internal energy. 

We conclude that if the assumption of pairwise additivity for the potential energy, Eq. 17.3.8, is valid and the radial distribution function is available, the thermodynamic properties of a fluid can be determined from Eqs 17.4.3 through 17.4.5. provided, of course, that the pair potential be known.)... 

Uncertainties in the prediction of the radial distribution function at high densities from knowledge of the pair potential do not allow for the direct evaluation of thermodynamic properties in such densities through Eqs 17.4.3 and 17.4.5. Three general approaches have been used to circumvent this problem ... 

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