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Thermodynamics

Hello, I'm getting too confused on a homework question.. (it's not as long as it looks! :smile:)

Q: Consider a system whose energetic fundamental relation is given by
U(S,V,N)=α(SaVbNc)1dU(S,V,N)=\alpha(S^a V^b N^c )^{\frac{1}{d}}
for some positive constant α\alpha.

(i) What is the condition on the parameters {a,b,c,d} for which this gives an acceptable energetic fundamental relation?
(ii) Find the equations of state
T=T(S,V,N),P=P(S,V,N),μ=μ(S,V,N) T=T(S,V,N), P=P(S,V,N), \mu=\mu(S,V,N)
(iii) Find the expression for μ(T,P)\mu(T,P). Does this require any additional constraints on the parameters {a,b,c,d}?
(iv) Consider the system at fixed N. Find the expressions for the adiabats (fixed entropy) and isotherms (fixed temperature) in the (P,V) plane.

(i) I have that a+b+c=d

(ii) I have used
∂U∂S=T \frac{\partial U}{\partial S} = T
−∂U∂V=P-\frac{\partial U}{\partial V} = P
∂U∂N=μ\frac{\partial U}{\partial N} = \mu
to find that
T=aαdSa−1(SaVbNc)1d−1T=\frac{a\alpha}{d} S^{a-1}(S^aV^bN^c)^{\frac{1}{d}-1}
T=−bαdVb−1(SaVbNc)1d−1T=\frac{-b\alpha}{d}V^{b-1}(S^aV^bN^c)^{\frac{1}{d}-1}
T=cαdSc−1(SaVbNc)1d−1T=\frac{c\alpha}{d}S^{c-1}(S^aV^bN^c)^{\frac{1}{d}-1}

(iii)&(iv) I'm really not sure on these two.
(iii) I've tried using the above equations but no luck.
(iv) What equations would I use to then set dS=0 and dT=0??

Thank you!
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