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Unformatted text preview: 4.3. LEGENDRE TRANSFORMATIONS 111 4.3 Legendre transformations The Gibbs equation (4.1), du = Pdv + Tds , is the fundamental equation of classical thermodynamics. It is a canonical form which suggests the most natural set of variables in which to express internal energy u are s and v : u = u ( v,s ) . (4.77) However, v and s may not be convenient for a particular problem. There may be other combinations of variables whose canonical form gives a more convenient set of independent variables for a particular problem. An example is the enthalpy: h u + Pv. (4.78) Differentiating the enthalpy gives dh = du + Pdv + vdP. (4.79) Use now Eq. (4.79) to eliminate du in the Gibbs equation to give dh Pdv vdP bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright = du = Pdv + Tds. (4.80) So dh = Tds + vdP. (4.81) So the canonical variables for h are s and P . One then expects h = h ( s,P ) . (4.82) This exercise can be systematized with the Legendre transformation, 1...
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- Fall '11