CH131_13B-14A

CH131_13B-14A - Summary Entropy S measure of disorder of a...

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Summary Entropy S measure of disorder of a system S = dq rev T i f = q rev T q rev => max amount of heat that can be absorbed by system & fundamental factor that governs the disorder of system Second Law of Thermo: The entropy of the universe (total) increases ( S > 0) for any spontaneous process & the entropy of the universe is constant ( S =0) for any reversible process. Isothermal process S = nR ln V f V i = nR ln P i P f rev. process at constant T S surr = - q sys T at constant P=> S surr = - D H sys T S universe > 0 spontaneous process S universe = 0 reversible process (equil.) S universe < 0 impossible (non-spont.) Total entropy must increase for process to proceed spontaneously
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Statistical Interpretation of S So far all comments about S based on macroscopic properties, e.g. P, V, T, n; no mention of microscopic => yet we have equated S with disorder; inherently a microscopic property ….so how do we connect the two points of view? For example: We’ve said as V of gas increases, disorder, S, increases. S = nR ln V f V i for reversible isothermal process How do we relate this expression to “disorder” of system?
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Consider an 8 x 8 “checker board” One checker can occupy just a single box First => all 8 confined to the first column only ….only one way to draw picture (1 possible configuration) Second => let 8 identical checkers occupy first 2 columns; here’s one possible configuration 16 15 14 13 12 11 10 9 8! =12,870 Let’s equate disorder => # of configurations (called microstates) Let = # configurations/arrangements/microstates There are a total of 12,870 unique configurations
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Let disorder ln so for 1 column system (# arrangements) = 1 disorder ln 1 = 0 no disorder (perfectly ordered system!) For the 2 column system: disorder ln 12,870 = 9.46 If entropy (S) disorder, and disorder ln , then S ln and we turn this relationship into an equality by: S = k ln where k = Boltzmann constant (R/N o )
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For the 3 column system (1 configuration is pictured below) = 735,471 ln = 13.51 Just like gas particles in a container of fixed V => As V increases more ways for particles to arrange themselves. It is possible that as V increases all gas particles would move to just one side of container (this is a possible configuration) but HIGHLY improbable. Much more likely, on the basis of statistics, that gas will occupy entire V, corresponding to many more possible arrangements.
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S for this process from this statistical interpretation of point of view: S = nR ln V f V i Previously, we saw S for an isothermal expansion/compression for an ideal gas given by: S = S final - S initial = k lnW f - k lnW i = k ln W f W i So, we need the ( f / I ) ratio… Consider one particle => V (double V => twice as many sites) where = # configurations/microstates/arrangements So for one particle
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CH131_13B-14A - Summary Entropy S measure of disorder of a...

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