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Lecture 20 - Rewriting Van der Waals Eqn of State kT 2 av...

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Rewriting Van der Waals Eqn. of State 3 2 v v v 0 kT a ab b P P P (1) For c T T 3 solutions 1 2 3 (v , v , v ) for fixed P, T c T T 3 solutions coalesce at v v c i.e. for , C C P P T T LHS of Eqn. (1) is 3 C v v But 3 3 2 2 3 v- v v -3v v 3v v- v c c c c (2) Comparing (1) and (2) From (iii) and (ii) dividing Then 2 3 2 and 3 3 ( ) 3 3 3 ( ) 27 27 8 ( ) 27 c c c c c c c c c c c kT b i b v b P a ab a v ii P P b b ab a v iii T P bk Define reduced variables / , / , / c c c P P P T T T Universal from 1 2 8 3 1 3 (3) P T Law of Corresponding States
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Figure Caption: The liquid gas coexistence curves of many fluids can be virtually superimposed when the temperature and density are scaled by their corresponding critical values c T and c . Plots of this kind are manifestations of the “law of corresponding states”. This plot from Guggenheim (1945) played an important role in the development of the theory of critical phenomena by showing that the data cannot be fitted with a quadratic curves van der Waals theory implies, but require the cubic shown here, which corresponds to the fact that the order-parameter exponent 1/ 3 and not 1/2 as van der Waals theory implies.
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