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139
0
1 2
ln
!
number of microstates
! !...
!
n
S k
N
N N N
=
Ω
Ω
=
=
Example:
A system with 4 molecules, 2 of each type:
N
0
= N
1
+ N
2
+ … = 4 (molecules)
N
1
= 2
N
2
= 2
4 3 2 1
6
(2 1) (2 1)
⋅ ⋅ ⋅
Ω
=
=
⋅ ⋅ ⋅
Δ
S
mix
=
S
(solution)
−
S
(purecomponents)
=
S
(solution)
=
k
ln
Ω
Using Stirling's formula (eq. 10.25)
⇒
mix
S
ln
ln
i
i
i
i
k N X R n X
Δ
=
−
=
−
∑
∑
mix
: moles
: mole fraction
Note: S 0
i
i
n
X
Δ
>
Combining the expressions, we obtain, for an ideal solution: