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# ch1 - ChE 210B Advanced Topics in Equilibrium Statistical...

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ChE 210B: Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 1 Reading: 3.1-3.5 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 in-depth exploration of topics related to simple and complex ﬂuid systems at equilibrium . ChE 210C intro- duces non-equilibrium 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 (2-3 ˚ A): e.g. O 2 , CO, N 2 , ... Molecular liquids (2-5 ˚ A): e.g. H 2 O, CO 2 , CH 4 , ... Polymers (50-500 ˚ A): e.g. polyethylene, polystyrene, ... Colloids (0.1-10 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 “coarse-graining” 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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ch1 - ChE 210B Advanced Topics in Equilibrium Statistical...

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