Derive the Maxwell construction by considerations involving the minimization of the Gibbs potential instead of the...

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Physics

Derive the Maxwell construction by considerations involving theminimization of the Gibbs potential instead of the Helmholtz freeenergy.
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Helmholtz and Gibbs Free EnergyWhen we made the transformation from U to H bythe Legendre transformation7we remarked that V was not the most convenientindependent variable In the laboratory it is usually much easierto control pressure than it is to control V Since bothU and H are natural functions of entropy it isfair to ask how convenient it is to have S as a variableThe answer is that it is not at all convenient to control entropyor to have entropy as an independent variable We do not have ameter that reads entropy and we do not know how to hold entropyconstant as we change some other variable Recall that we cancontrol temperature pressure and volume So we make some moreLegendre transformationsWe will not give an extensive discussion of Legendretransformations here but we should point out that they are notarbitrary You cant just pick any two variables you wish and putthem together to make a Legendre transformation We could make theLegendre transformation from U to H by adding thepV term to U only because V is relatedto p and U through Equation 3b Using this as a guide itwould seem reasonable to use Equation 3a to change the variableS to T in the function U and useEquation 6a to change the variable S to T in thefunction H Lets try itDefine the Helmholtz free energy A as8Then9a b cEquation 9c tells us that the Helmholtz free energy is a See Answer

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