Chemistry Reference
In-Depth Information
or, equivalently,
2
=
MMM
AB
(2.40)
AA
BB
In this case, the equation of state reduces to
fTp
(,,
λ
AB
,
)
=
2
(
M
+
M
)
−=
2
0
(2.41)
AA
BB
The density of A is now
βλ
2
f
Ψ
A
AA
ρ
=−
(2.42)
A
Λ
∂∂
/
p
A
and the mole fraction of A is
ρ
ρρ
λΨ
Λ
A
A
A
x
A
=
=
(2.43)
+
AB
A
This may be written in a more familiar form as
o
µ
=
kT
ln(
ΛΨ
/
)
+
kT
ln
x
=+
µ
k Tx
ln
(2.44)
A
B
A
AA
B
AA
B
A
Equation 2.44 is the familiar form of the chemical potential of a symmetric ideal
solution. We shall see in Section 2.6 that a mixture of hard rods always forms a sym-
metric ideal solution.
2.4 CALCULATION OF THE GLOBAL QUANTITIES
IN A ONE-DIMENSIONAL SYSTEM
First, we calculate the chemical potential from the equation of state (Equation 2.35).
This is calculated using a series of steps.
The number densities are calculated from
the equation of state (Equation 2.35),
=−
∂∂
∂∂
f
/
µ
α
ρ
(2.45)
α
fp
/
Since
f
in Equation 2.35 is given as a function of
T
,
p
, λ
A
, λ
B
, the densities in Equation
2.45 are also functions of these variables. However, these are not independent vari-
ables. We can eliminate λ
B
from the equation of state to obtain
ΛΛ Ψ
ΨΛ ΨΨ
−
+−
(
λ
)
BA
A
A
λ
=
(2.46)
B
2
λ
(
λ
)
A
B
A
AAA
B
This can be substituted on the right-hand side of Equation 2.45 to obtain the density
ρ
A
as a function of
T
,
p
, λ
A
. The latter can be inverted to obtain λ
A
= λ
A
(
T
,
p
, ρ
A
). The
