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First Order Phase Transitions

Physics 127b: Statistical MechanicsLecture 3: First Order Phase TransitionsThe van der Waals equation for a gas ishPCaV2i[V b]DNkBT:(1)(The variableais proportional toN2andbtoN, ;Nbconstants).It can be motivated by rewriting it in the formPDNkBTV b aV2(2)TheV bterm comes from estimating the free volume available for the molecules by excludinga hard core contribution, and thea=V2is a reduction in the pressure proportional to the densitysquared, representing the attractive interaction of the molecules. InLecture 2we derived expressionsforNaandNbin terms of the pair corresponding free energy isADAideal NkTln 1 bV aV(3)withAidealthe ideal gas expression. (Actually, if we integratePD .@A=@V/N;Tto getAthereis an integration constant ;T/, and we fix this by comparing with the ideal gas expressionforV!1.) The second term is Ttimes theentropy correctionfrom the excluded volume, andthe third term is theenergy correctionfrom the attractive > Visotherms do not look much different from those for theideal gas.

provides a single constraint on the two variables T;P, i.e. two phase coexistence occurs along a line in the P;Tplane—an example of the Gibbs phase rule. Using this expression at nearby points on the coexistence line and the thermodynamic identity d DdgD−sdTCvdP (9)

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