24
1 Molecules and Intermolecular Interactions
sion” here is based on the fact that Eq. 1.74 holds in the limit of ρ → +0 and p → +0.
Terms from the second in the right hand sides can thus be regarded as successive
corrections arising from the non-ideality. The temperature where the lowest-order
non-ideality vanishes, i.e., B(T B ) = B p (T B ) = 0, is known as the Boyle temperature,
T B .
A systematic series expansion of the compressibility factor for non-ideal gas
consisting of spherical molecules has been derived assuming that the intermolecular
interaction is of two-body interaction [25].
6 First, the following function (the Mayer
f function) reflecting the two-body interaction u i j between ith and jth molecules is
defined:
f i j = exp
−
u i j
k B T
− 1.
(1.78)
The following function is further defined
β k =
1
k!
k+1≥ j>i≥1
f i j d r 2 · · · d r k;1 ,
(1.79)
where the sum runs over every irreducible graph
7 containing (k + 1) vertexes, connection between which corresponds to f i j . Note that this β k is obtained after integrating over (k + 1) molecules. Using this β k , the compressibility factor is given
as
p
ρk B T
= 1 −
∞
k=1
k
k + 1
β k ρ
k
.
(1.80)
Since the forms are exactly the same between Eqs. 1.76 and 1.80, the following
correspondence is evident:
B(T ) = −
1
2
β 1 = −
1
2
f 12 d r 2
(1.81)
C(T ) = −
2
3
β 2 = −
1
3
f 12 f 23 f 31 d r 2 d r 3
(1.82)
D(T ) = −
3
4
β 3
= −
1
8
d r 2 d r 3 d r 4
(1.83)
(3 f 12 f 23 f 34 f 41 + 6 f 12 f 23 f 34 f 41 f 13 + f 12 f 23 f 34 f 41 f 13 f 24 )
· · ·
6 Full description of the cluster expansion is left for the literature [26].
7 The term “irreducible graph” follows the convention in the formulation of statistical physics for
many-body systems, which is beyond the level of this book. It is essential here that the graph
corresponding to the two-body interaction is irreducible and survives accordingly.
1 Molecules and Intermolecular Interactions
sion” here is based on the fact that Eq. 1.74 holds in the limit of ρ → +0 and p → +0.
Terms from the second in the right hand sides can thus be regarded as successive
corrections arising from the non-ideality. The temperature where the lowest-order
non-ideality vanishes, i.e., B(T B ) = B p (T B ) = 0, is known as the Boyle temperature,
T B .
A systematic series expansion of the compressibility factor for non-ideal gas
consisting of spherical molecules has been derived assuming that the intermolecular
interaction is of two-body interaction [25].
6 First, the following function (the Mayer
f function) reflecting the two-body interaction u i j between ith and jth molecules is
defined:
f i j = exp
−
u i j
k B T
− 1.
(1.78)
The following function is further defined
β k =
1
k!
k+1≥ j>i≥1
f i j d r 2 · · · d r k;1 ,
(1.79)
where the sum runs over every irreducible graph
7 containing (k + 1) vertexes, connection between which corresponds to f i j . Note that this β k is obtained after integrating over (k + 1) molecules. Using this β k , the compressibility factor is given
as
p
ρk B T
= 1 −
∞
k=1
k
k + 1
β k ρ
k
.
(1.80)
Since the forms are exactly the same between Eqs. 1.76 and 1.80, the following
correspondence is evident:
B(T ) = −
1
2
β 1 = −
1
2
f 12 d r 2
(1.81)
C(T ) = −
2
3
β 2 = −
1
3
f 12 f 23 f 31 d r 2 d r 3
(1.82)
D(T ) = −
3
4
β 3
= −
1
8
d r 2 d r 3 d r 4
(1.83)
(3 f 12 f 23 f 34 f 41 + 6 f 12 f 23 f 34 f 41 f 13 + f 12 f 23 f 34 f 41 f 13 f 24 )
· · ·
6 Full description of the cluster expansion is left for the literature [26].
7 The term “irreducible graph” follows the convention in the formulation of statistical physics for
many-body systems, which is beyond the level of this book. It is essential here that the graph
corresponding to the two-body interaction is irreducible and survives accordingly.