7.5 The Liouville and Boltzmann Equations
387
potentials, while simultaneously restricting our consideration to central forces, i.e.,
potential energy functions that depend solely upon the distance between the centres
of interacting pairs of molecules. Such potential energy functions are said to be
isotropic, i.e., independent of angles.
These arguments allow us to write F
(N )
k · ∇ p k as
F
(N )
k · ∇ p k = X k · ∇ p k +
−∇ r k V (r 1 , · · · , r N ) · ∇ p k
or as
F
(N )
k · ∇ p k = X k · ∇ p k + F
int
k · ∇ p k ,
with the intermolecular force F int
k acting on molecule k defined as
F
int
k ≡ −∇ r k V (r 1 , · · · , r N ) ,
and representing the force exerted upon molecule k by all the other molecules in the
container.
We shall take a slight detour to consider in more detail the nature of the
intermolecular potential energy function and the intermolecular forces acting upon
the molecules during a binary molecular collision. We begin by examining the
potential energy function for the interaction between two molecules, i and j . If
this interaction energy depends only upon the distance between the two molecules,
then we may write it as
V (r i , r j ) = V (|r i − r j |) ,
(7.5.12)
which allows us to calculate the corresponding intermolecular force exerted upon
molecule i by molecule j as
F
int
ij = −∇ r i V (|r i − r j |)
= −
r i − r j
|r i − r j |
∂V
∂r ij
≡
r i − r j
|r i − r j |
F
int
ij ,
in which r ij ≡ |r i − r j | is the distance between molecules i and j and F ij ≡ −
∂V
∂r ij
is the magnitude of the force exerted by molecule j upon molecule i. Notice that
this force is directed from molecule j towards molecule i [i.e., it is the force exerted
upon molecule i by molecule j ]. A force that is equal in magnitude but opposite in
direction is exerted by molecule i upon molecule j .