We know that S (entropy) is additive and satisfies the relation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\lambda[/tex]S(U,V,N)=S([tex]\lambda[/tex]U,[tex]\lambda[/tex]V,[tex]\lambda[/tex]N)

(S is a homogeneous function of U, V and N)

I need to show that U is a homogeneous function of S, V and N

that is, to show that

[tex]\lambda[/tex]U(S,V,N)=U([tex]\lambda[/tex]S,[tex]\lambda[/tex]V,[tex]\lambda[/tex]N)

I tried to differentiate S by [tex]\lambda[/tex] and to represent U through derivatives of S but I only managed to get an implicit representation of U. I know that the solution should include differentiation by [tex]\lambda[/tex] and probably choosing a specific value for it (depending on some variable of the system). Moreover,the solution must include the fact that S is a monotonically increasing function of U (otherwise it is easy to show that the relation is not satisfied). Any ideas?

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# Homework Help: Thermodynamics: showing that U (energy) is a homogeneous function of S,V and N

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