3
So far, we have dealt with pure substances
, where
G = G(T, P, n)
Intuitively with a mixture G would also depend on composition
Chemical Potential
Intuitively, with a mixture, G would also depend on composition
of the mixture:
G = G(T, P, n
1
, n
2
, …, n
k
)
where
n
i
is the mole number of component
i
.
,,
ii
i
i
Pn
Tn
i
TPn
GG
G
dG
dT
dP
dn
TP
n
⎛⎞
∂∂
∂
=++
⎜⎟
∂
⎝⎠
∑
j i
≠
Chemical potential:
ji
i
i
G
n
μ
≠
∂
≡
∂
For a pure substance:
,
i
G
G
n
∂
≡
=
∂
The chemical potential
is equivalent to molar
Gibbs free energy
If we know the chemical potential (
i
) for each constituent species in a mixture, what’s the
total Gibbs free energy?
To address this question, let’s consider a hypothetic process where the mixture is generated
from nothing.
n
1
, n
2
, n
3
, …
Almost
nothing
mixture
Then the question becomes: what’s
Δ
G for this “creation” process? Since G is a state
variable, it doesn’t matter which path we choose to accomplish the creation process. In
particular, we can choose a path we add each species incrementally, where each
increment is proportional to the species’ final mole number Thus the concentration of each
increment is proportional to the species final mole number. Thus the concentration of each
remains the same throughout the process. This way, the chemical potential of each
component will also remain a constant! Therefore:
(
)
11
00
if
;
where
varies from 0 to 1
Thus:
i
i
i
dn
n dx
x
dG
dn
n dx
Gd
G
n
d
x
nd
x
n
μμ
=
==
⎡⎤
=
=
⎢⎥
⎣⎦
∑∑
∑
∫∫
∫
i
Gn
=
∑
For a mixture