[QUE/TH-09001] TH-PROBLEM

Consider the following expressions for entropy. Which ones can possibly be a fundamental equation and which ones violate one or more of postulates II,III and IV?

• $S\,=\,K_1\left(NVU\right)^{1/3}$
• $ S\,=\,K_2\left(\frac{NU}{V}\right)^{2/3} $
• $ S\,=\,K_3\left(\frac{V^3}{NU}\right)$
• $ S\,=\,N{\rm{\ln}}\left(\frac{UV}{N^2K_4}\right)$

$K_i$'s are positive constants so that dimensions match. $S,U,N,V $ are the entropy, internal energy, number of particles and volume respectively.