Chemistry Reference
In-Depth Information
where the matrix elements are defined for each pair of species αβ by
*
12
/
12
/
*
Mp
()(
=
λ
/
Λ
)
(
λ
/
Λ
)
Ψ
()
p
(2.26)
αβ
αα
ββ
αβ
and
∞
∫
0
*
*
Ψ
αβ
()
p
=
exp
−
ββ
p rUrdr
−
( )
(2.27)
αβ
where Ψ
αβ
is the Laplace transform of exp[-β
U
αβ
(
r
)].
Assuming that the sum in Equation 2.25 converges, we rewrite it as
∞
∞
c
∞
∑
∑∑
()
−1
Ψ
()
=
N
N
*
Tr
M
=
γ
=
1
−
γ
p
(2.28)
j
j
N
=
0
Nj
=
0
=
1
j
=
1
where γ
j
is the
j
th eigenvalue of the matrix
M
. We note the dependence of γ
j
on the
variable
p
*
.
From the definition of Ψ(Ξ) in Equation 2.24, and from Equation 2.23, we have
∞
∫
*
Ψ()
=
exp[
−
β
Lp
(
−
pdL
)]
(2.29)
0
If we choose
p
*
as the thermodynamic pressure
p
, then Ψ(Ξ) is referred to as the general-
ized PF. It is clear from Equation 2.29 that the integral in this case diverges. The reason
is that
∑
=
∞
Ξ (,,{ }λ
is a function of the single extensive variable
L
. Transforming
L
into the thermodynamic intensive variable
p
gives a partition function that is a function
of intensive variables
T
,
L
, {λ} only. However, the Gibbs-Duhem relation (see Equation
1.11) indicates that the intensive variables
T
,
p,
{μ}
or
T
,
p,
{λ} are not independent. For
this reason, we have denoted by
p
*
the new variable in the Laplace transform taken in
Equation 2.24. Suppose that we choose
p
*
>
p
, then the integral in Equation 2.29 con-
verges, and we have
TL
N
0
n
c
1
∑
−
1
*
=
1
−
γ
()
p
(2.30)
j
*
β
(
pp
−
)
j
=
1
It is clear that since in the limit
p
*
→
p
the left-hand side of Equation 2.30 diverges,
there must be at least one of γ
j
(
p
*
) equal to 1. The secular equation of the matrix
M
is
M
−=
γ
0
(2.31)
where
I
is the unit matrix of the same dimensions as of
M
.
Since the elements of
M
are functions of
T
,
p
, and {λ}, we can use the implicit
Equation 2.31 to derive all thermodynamic quantities of interest.
We now discuss the special case of a two-component system of A and B for which
the secular equation is
