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02_562ln08 - MIT OpenCourseWare http/ocw.mit.edu 5.62...

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MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.62 Lecture #2: E, A, and S: Macroscopic Properties from Microscopic { P i } Probabilities Problem: How do we calculate a macroscopic property, which is constant in time, from a microscopic property that fluctuates in time? Example: Pressure, which is a macroscopic property that arises from the microscopic impulses of each molecule impacting the vessel's walls. The positions and velocities of each molecule change on a 10 –13 s time scale (the duration of a collision)! Possible Solution: TIME AVERAGE the microscopic variable f obs is the observed macroscopic property f(q 3N ,p 3N ) is the instantaneous value of the sum over all ~ ~ microscopic contributions to the macroscopic property f obs = lim 1 τ τ f q ( 3N ,p 3N ) d τ′ this is how a classical mechanical time average is defined τ→∞ 0 ~ ~ But this calculation is impossible because it requires knowledge of the time dependence of a very large number, N, of q ~ i ,p ~ i . Instead, we make use of ENSEMBLE THEORY, developed by J. Willard Gibbs (1839- 1903) (founder of Stat. Mech.) ENSEMBLE A COLLECTION OF ALL “POSSIBLE” STATES OF AN ASSEMBLY system (e.g. a molecule) assembly of systems ensemble In thermodynamics, the word “system” is used to specify the macroscopic object under construction.
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5.62 Spring 2008 Lecture 2, Page 2 Example: (1) Quantum – assembly consisting of 2 particles only state n 1x n 1y n 1z n 2x n 2y n 2z Constant E ensemble α 2 1 1 1 1 1 with E = 9 ε 0 β 1 2 1 1 1 1 E = ε i i = 1 N γ δ 1 1 1 1 2 1 1 2 1 1 1 1 ε i = h 2 8m i a 2 n ix 2 + n iy 2 + n iz 2 ε η 1 1 1 1 1 1 1 1 2 1 1 2 h 2 ε 0 = 8m i a 2
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