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Maxwell equal area rule

Below the critical temperature (T¡¯ < 1) an isotherm of the Van der Waals equation oscillates as shown.
Maxwell's rule eliminates the oscillating behavior of the isotherm in the phase transition zone by defining it as a certain isobar in that zone.

Along the red portion of the isotherm \scriptstyle \left({{\partial P'}/{\partial V'}}\right)_{T'}>0 which is unstable; the Van der Waals equation fails to describe real substances in this region. To fix this problem James Clerk Maxwell (1875) replaced the isotherm between a and c with a horizontal line positioned so that the areas of the two hatched regions are equal. The flat line portion of the isotherm now corresponds to liquid-vapor equilibrium. The portions a¨Cd and c¨Ce are interpreted as metastable states of super-heated liquid and super-cooled vapor respectively.[3]

Maxwell justified the rule by saying that work done on the system in going from c to b should equal work released on going from a to b. (Area on a PV diagram corresponds to mechanical work). That¡¯s because the change in the free energy function A(T,V) equals the work done during a reversible process the free energy function being a state variable should take on a unique value regardless of path. In particular, the value of A at point b should calculate the same regardless of whether the path came from left or right, or went straight across the horizontal isotherm or around the original Van der Waals isotherm. Maxwell¡¯s argument is not totally convincing since it requires a reversible path through a region of thermodynamic instability. Nevertheless, more subtle arguments based on modern theories of phase equilibrium seem to confirm the Maxwell Equal Area construction and it remains a valid modification of the Van der Waals equation of state.[4]

The Maxwell equal area rule can be derived from an assumption of equal chemical potential ¦Ì of coexisting liquid and vapour phases[5].
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http://en.wikipedia.org/wiki/Van_der_Waals_equation

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The Maxwell equal area rule can be derived from an assumption of equal chemical potential ¦Ì of coexisting liquid and vapour phases[5].

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A E Elhassan, R J B Craven, K M de Reuck, The Area Method for pure fluids and an analysis of the two-phase region, Fluid Phase Equilibria 130 (1997) 167-187.

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Originally posted by Î÷ºþ¿áÓã at 2010-10-18 14:17:11:
http://en.wikipedia.org/wiki/Van_der_Waals_equation

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