Lecture 14
5.60/
20
.110/2.772
1
Why
Ω
works for large N
Derivation of the Boltzmann Distribution Law
Partition Function
•
Why
Ω
works for large N
We have seen that a system will vary its degrees of freedom in order to maximize
Ω
and thus S. A
system has a higher probability of being in a state due to it being more probable. This allows us to
simply count states and see which one is more likely.
The lattice model of mixing gases had only N=8 particles. Is this approach still justified when we look
at a larger number of particles, like N
A
? It turns out the most probable state at low N becomes even
more likely at very high N.
Consider: coin flips
n
H
Ω
S =kln
Ω
4
(
)
1
!
0
!
4
!
4
!
!
!
=
=
−
=
Ω
n
N
n
N
0
3
4
!
1
!
3
!
4
=
==
Ω
1.386k
2
6
!
2
!
2
!
4
=
==
Ω
1.792k
1
4
!
3
!
1
!
4
=
==
Ω
1.386k
0
1
!
4
!
0
!
4
=
==
Ω
0
Then do for N = 10, 100, 1000
n
N=4
Ω
max
0
1
2
3
4
Ω
max
n
N=100
0
50
100
Ω
max
n
N=1000
0
500
1000
Ω
max
n
N=10
0
5
10
Ω
becomes increasingly narrower as N
↑
. Compare numerically:
20.110J / 2.772J / 5.601J
Thermodynamics of Biomolecular Systems
Instructors: Linda G. Griffith, Kimberly Hamad-Schifferli, Moungi G. Bawendi, Robert W. Field

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