Lecture 11 : Non-covalent interactions 2 : Structures of Liquids
The bulk (number) density of a liquid is given by N/V where N is the number of atoms in volume V. The unit of this is number/ volume and not g/cc which is used in macroscopic or bulk measurements. We now define the radial distribution g(r) function as
g(r) = <(r)>/
(11.2)
where the angular brackets mean that we take the average value of (r). The average is taken over all the molecules and at different times as well. This average value does not depend on any specific atom chosen at the center or any specific instance of time and it is an equilibrium property. A typical sketch of g(r) is shown in Fig 11.2
Figure 11.2 The radial distribution function g (r).
The meaning of this graph can be understood as follows. Most atoms or molecules have a size of about 1 to 3 .If one is considering a distance of 0.1 or 0.5 from the center of an atom, there is no chance of finding an adjacent atom there and hence the local density (r) is zero at very short distances. When one reaches a value of 3 to 5 , adjacent molecules can be found and the local density when several neighbours are "touching" the central molecule is much higher than the bulk density and hence g(r) is much greater than one.
Between the first set of neighbours and the second set of neighbours there are fewer atoms /molecules and hence g(r) is less than one. The value of g(r) reaches 1 (the asymptotic value) for large distances. At larger r, the influence of the central atom is not significant and hence
(r) =
, the bulk density.
The importance of g(r) stems from the fact that the macroscopic properties of bulk fluids (gases and liquids) can be obtained in terms of g(r). It gives a link between the microscopic structure determined from intermolecular forces and the macroscopic properties such as (thermodynamic) energy (E), entropy (S) and pressure (p). The equation for energy and pressure are given below
E / N = 2
g(r) u (r) r2 dr
(11.3)
p / KBT = 1- ( 2 / 3kBT ) g(r) [ du(r)/dr ] r3 dr
(11.4)
Here E / N is the energy per particle resulting from intermolecular forces. On adding kinetic energy to it, we get the energy used in the first law of thermodynamics. The Boltzmann constant kB = 1.38 x 10-23 J / K, T is the absolute temperature and U(r) is the pair-wise interaction potential between two particles. It is assumed here that the interaction between N molecules U(1,2,...N) is pair-wise additive, i.e.
is given by N/V where N is the number of atoms in volume V. The unit of this 
is number/ volume and not g/cc which is used in macroscopic or bulk measurements. We now define the radial distribution g(r) function as 
(r)>/ 
(r). The average is taken over all the molecules and at different times as well. This average value does not depend on any specific atom chosen at the center or any specific instance of time and it is an equilibrium property. A typical sketch of g(r) is shown in Fig 11.2 
.If one is considering a distance of 0.1 or 0.5
(r) =
, the bulk density. 
g(r) u (r) r2 dr 
Ui j (r i j) 
