26
1 Molecules and Intermolecular Interactions
Fig. 1.6 Pressure as a
function of volume described
by van der Waals equation of
states in terms of reduced
variables. The upper position
of the curve corresponds to
the higher temperature
0
1
2
3
4
0
1
2
V / V c
p
/ p
c
vdW equation
5
gas and liquid cannot be realized. The temperature is known as a (liquid-gas) critical
temperature. In reality, not only the temperature but also pressure and volume are
fixed. Thus, the state is called a critical point. Figure 1.6 indicates that the following
should hold at the critical points: (∂ p/∂V ) T = 0 and (∂
2 p/∂V
2
) T = 0. Quantities
at the critical point are thus obtained as
V c = 3N b
(1.87)
p c =
1
27
a
b 2
(1.88)
RT c =
8
27
a
b
.
(1.89)
Then, by introducing the reduced variables, φ R = V /V c , π R = p/ p c , and θ R =
T /T c , the van der Waals equation of state is simply written as
π R +
3
φ
2
R
φ R −
1
3
=
8
3
θ R .
(1.90)
This simple form implies that the states of matter are “common” if expressed in terms
of a distance from the critical point. Although the numerical values of coefficients
(such as
8
3
) are different from Eq. 1.90 reflecting the approximate nature of van der
Waals’s treatment, it has been shown that a unified description holds well for some
real systems. The correspondence between states in different materials implied by
Eq. 1.90 is known as the principle of corresponding states [28], which is one of the
best examples of a unified understanding of matter, though its validity is limited to
simple molecular systems at most. The van der Waals equation of state does not say
anything about crystallization but clearly indicates that the intermolecular attraction
induces liquefaction.
1 Molecules and Intermolecular Interactions
Fig. 1.6 Pressure as a
function of volume described
by van der Waals equation of
states in terms of reduced
variables. The upper position
of the curve corresponds to
the higher temperature
0
1
2
3
4
0
1
2
V / V c
p
/ p
c
vdW equation
5
gas and liquid cannot be realized. The temperature is known as a (liquid-gas) critical
temperature. In reality, not only the temperature but also pressure and volume are
fixed. Thus, the state is called a critical point. Figure 1.6 indicates that the following
should hold at the critical points: (∂ p/∂V ) T = 0 and (∂
2 p/∂V
2
) T = 0. Quantities
at the critical point are thus obtained as
V c = 3N b
(1.87)
p c =
1
27
a
b 2
(1.88)
RT c =
8
27
a
b
.
(1.89)
Then, by introducing the reduced variables, φ R = V /V c , π R = p/ p c , and θ R =
T /T c , the van der Waals equation of state is simply written as
π R +
3
φ
2
R
φ R −
1
3
=
8
3
θ R .
(1.90)
This simple form implies that the states of matter are “common” if expressed in terms
of a distance from the critical point. Although the numerical values of coefficients
(such as
8
3
) are different from Eq. 1.90 reflecting the approximate nature of van der
Waals’s treatment, it has been shown that a unified description holds well for some
real systems. The correspondence between states in different materials implied by
Eq. 1.90 is known as the principle of corresponding states [28], which is one of the
best examples of a unified understanding of matter, though its validity is limited to
simple molecular systems at most. The van der Waals equation of state does not say
anything about crystallization but clearly indicates that the intermolecular attraction
induces liquefaction.