Chemistry Reference
In-Depth Information
12
2.4
T
= 9
2.2
10
p
= 2
2.0
8
T
= 5
p
= 6
1.8
6
1.6
4
p
= 10
T
= 1
1.4
2
1.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
A
x
A
FIGURE 2.3
The volume of the mixture A and B as a function of the mole fraction of A, for
hard rods with σ
AA
= 1 and σ
BB
= 2.
The partial molar volumes of the two components are
A
=
(
−
B
=
(
−
1
1
V
σβ
p
,
V
σβ
p
(2.63)
A
B
The volume per particle of the mixture is
+
(
−
(
)
(
)
1
VxV
=
+
1
−
x
Vx
=
σ
+
1
−
x
σ
β
p
(2.64)
mAA
A
BAAA
B
B
Figure 2.3 shows the volume per particle of a mixture of HRs with diameters σ
AA
= 1
and σ
BB
= 2. Clearly, as the pressure increases, or as the temperature decreases, the
linear curves converge to the limiting linear curve,
(
)
Vx
=
σ
+
1
−
x
σ
(2.65)
mAAA
A
B
This is true either for
p
→ ∞ or for
T
→ 0. On the other hand, for
T
→ ∞ or for
p
→ 0,
we get the ideal gas behavior of the mixture.
The isothermal compressibility of the mixture, defined in Equation 1.5, is given by
1
κ
=
(2.66)
T
(
)
(
)
p
1
+
βσ
p x
+
1
−
x
σ
AAA
A
BB
Figure 2.4 shows the isothermal compressibility as a function of
x
A
, at a fixed tem-
perature (
k
B
T
= 1) and various pressures. As expected, the isothermal compress-
ibility is large at low pressures, but at high pressures it converges to zero, for any
composition of the system.
