What is a Liquid State

Equation (271) shows that g(r) is the correction factor applied to the random probability (dV/V)2 for ideal molecules, if no intermolecular interactions takes place. In a similar manner, we can calculate the probability of finding a particular molecule in the dV volume element, at a distance r from the center of a fixed molecule 

The volume element, dV, can be deduced from the volume of molecules in a spherical shell of thickness, dr, given as dV = 4nr2dr for a spherical mathematical model, and the bulk molecular density = Nm/V, where Nm is the total number of molecules in a given volume V. To obtain the probability of finding Nm molecules in volume element dV, at a distance r from the center of a fixed molecule, we may write 

The radial distribution function, g(r), can be determined experimentally from X-ray diffraction patterns. Liquids scatter X-rays so that the scattered X-ray intensity is a function of angle, which shows broad maximum peaks, in contrast to the sharp maximum peaks obtained from solids. Then, g(r) can be extracted from these diffuse diffraction patterns. In Equation (273) there is an enhanced probability due to g(r) 1 for the first shell around the specified molecule at r = o, and a minimum probability, g(r) 1 between the first and the second shells at r = 1.5cr. Other maximum probabilities are seen at r = 2(7, r = 3 o, and so on. Since there is a lack of long-range order in liquids, g(r) approaches 1, as r approaches infinity. For a liquid that obeys the Lennard-Jones attraction-repulsion equation (Equation (97) in Section 2.7.3), a maximum value of g(r) = 3 is found for a distance of r = 7. If r cr, then g(r) rapidly goes to zero, as a result of intermolecular Pauli repulsion. 

On the other hand, more explicitly, the radial distribution function can also be shown as the ratio of local molecular density to the bulk molecular density, 

Although the above analysis has been limited to pure liquids containing spherical molecules, the same ideas can be applied to liquids having non-spherical molecules or liquid mixtures. For non-spherical molecules, the radial distribution function depends on the directional angles 0 and j from the central molecule as well as on r. However, when we consider non-spherical molecules, not only does the mathematical complexity increase but also much more detailed information on liquid structure and intermolecular forces is required. 

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