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

1. Mar 22, 2010

### maria clara

We know that S (entropy) is additive and satisfies the relation:
$$\lambda$$S(U,V,N)=S($$\lambda$$U,$$\lambda$$V,$$\lambda$$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
$$\lambda$$U(S,V,N)=U($$\lambda$$S,$$\lambda$$V,$$\lambda$$N)

I tried to differentiate S by $$\lambda$$ 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 $$\lambda$$ 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?