The instantaneous dipole moment of one molecule executes a correlated motion (it is like two dancers dancing at tandem) with the instantaneous dipole moment of the neighbouring molecule. The dispersion energy Udisp is distance (rij) dependent and is always attractive. A calculation on a simple model gives
Udisp = -1.5
(10.1)
Here
and
are the characteristic frequencies of molecules i and j and are almost equal to the frequency of the electronic transition from the lowest energy level in these molecules. The polarizabilities ( polarizability measures the extent of distortion of an electron cloud by an external electric field) of molecules i and j are
and
. This attractive dispersive force between molecules is also known as the van der Waals force. The above formula is valid at large separations rij. At short distances, there is always the Pauli repulsion between molecules. Although the form of this repulsion is exponential, power law repulsion forms such as r-12 or r-10 are commonly used. One of the most commonly used form for intermolecular forces is the Lennard-Jones potential ULJ(r) whose form is given by
ULJ(r) = 4
(10.2)
This is also called a 6-12 potential because of the r-6 and r-12 terms appearing in it. The attractive term (
/r)6 incorporates the asymptotic dispersion interaction. The contact distance between the molecules is given by
and is the depth of the potential. The values of parameters
and for some systems are given in Table 10.1
System
( / kB ) / K
/
Ar-Ar
145
3.8
Kr-Kr
201
3.6
Xe-Xe
288
3.9
Na-Ar
72
3.8
Li-Xe
175
3.8
CH4-CH4
145
4.2
Table 10.1 The parameters for the Lennard-Jones potentials for some systems.
These potentials are empirical, i.e., the values of the parameters are obtained by fitting the potential to the experimental data such as virial coefficients. The virial coefficients quantify the departure of the equation of state of a real system from the ideal gas behaviour. The first correction to the ideal gas behaviour is given by the second virial coefficient (B) defined through
PV / nRT = 1+ ( n / V ) B +...
(10.3)
The value of B is related to intermolecular potential [ u(r) ] through
B = - 2
NA [ 1 - e - U(r) / kBT ] r2 dr
(10.4)
where NA is the Avogadro number. The values of and
in Table 10.1 are obtained by fitting these parameters to the virial coefficient data. This is a typical example of how microscopic characteristics of molecules determine the macroscopic thermodynamic behaviour of gases and liquids.
and
are the characteristic frequencies of molecules i and j and are almost equal to the frequency of the electronic transition from the lowest energy level in these molecules. The polarizabilities ( polarizability measures the extent of distortion of an electron cloud by an external electric field) of molecules i and j are
and
. This attractive dispersive force between molecules is also known as the van der Waals force. The above formula is valid at large separations rij. At short distances, there is always the Pauli repulsion between molecules. Although the form of this repulsion is exponential, power law repulsion forms such as r-12 or r-10 are commonly used. One of the most commonly used form for intermolecular forces is the Lennard-Jones potential ULJ(r) whose form is given by 
/r)6 incorporates the asymptotic dispersion interaction. The contact distance between the molecules is given by
and 
and 
/ 
NA 
[ 1 - e - U(r) / kBT ] r2 dr 
in Table 10.1 are obtained by fitting these parameters to the virial coefficient data. This is a typical example of how microscopic characteristics of molecules determine the macroscopic thermodynamic behaviour of gases and liquids. 
