ChE 210B: Advanced Topics in Equilibrium
Statistical Mechanics
Glenn Fredrickson
Lecture 1
Reading: 3.13.5 Chandler, Chapters 1 and 2 McQuarrie
This course builds on the elementary concepts of statistical mechanics that
were introduced in ChE 210A and provides a more indepth exploration of topics
related to simple and complex ﬂuid systems at equilibrium
. ChE 210C intro
duces nonequilibrium statistical mechanics.
* Discuss course info sheet / oﬃce hrs / web site
* Discuss syllabus
Let’s now launch into our introductory lecture, reviewing the basic principles
of equilibrium statistical mechanics:
1. Introduction and Review of Equilibrium Sta
tistical Mechanics
A. Classical ﬂuid systems of interest
As chemical engineers or “complex ﬂuids” oriented materials scientists, you will
repeatedly encounter a variety of ﬂuid systems of varying complexity, e.g.,
•
Atomic liquids (1
˚
A): e.g. condensed noble gases A, Xe, ...
•
Diatomic liquids (23
˚
A): e.g. O
2
, CO, N
2
, ...
•
Molecular liquids (25
˚
A): e.g. H
2
O, CO
2
, CH
4
, ...
•
Polymers (50500
˚
A): e.g. polyethylene, polystyrene, ...
•
Colloids (0.110
µ
m): e.g. latex particles, clays, ...
All of these systems can be described quite accurately at thermodynamic
equilibrium by using the principles of
classical
(as opposed to quantum) statis
tical mechanics. We shall primarily focus on such systems and on the condensed
1
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liquid
state, rather than the gas state (which you have already treated in ChE
210A) and the solid state (which is treated in courses on solid state physics).
Note the “coarsegraining” of our “cartoons” as we move up in lengthscale.
B. Averages and Ensembles
Let us now review the way in which averages are computed in statistical me
chanics. Suppose that within a large volume of a gas or liquid, you had a smaller
“measurement volume” in which you could count the number of particles.
Imagine that you made a large number
M
of such measurements over a long
time period and the system was at equilibrium
. Then, you could compute the
“time average” of the number of particles
N
inside the permeable barrier by
¯
N
=
lim
M
→∞
1
M
M
i
=1
N
i
where
N
i
is the number of particles in
i
th observation.
We shall see later in the course that such time averages
, constructed by fol
lowing a particular dynamical trajectory of one
system, are the most convenient
way calculating equilibrium averages in computer simulations.
It is of course
also what we do in experimental measurements on ﬂuids!
There is another convenient way of averaging in statistical mechanics.
A
basic tenant of statistical mechanics is that after a long enough time, all micro
scopic states
ν
(quantum or classical) in a given system will be visited. Thus,
we should also be able to construct an “ensemble average”
N
=
ν
P
ν
N
ν
where
P
ν
is the probability of the
ν
th microstate occurring and
N
ν
is the
number of particles inside the barrier in this microstate.
This is called an
ensemble average
because the right hand side can be constructed by making
2
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 Spring '09
 Thermodynamics, Statistical Mechanics, Entropy, Canonical Ensemble, Equilibrium Statistical Mechanics

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