The following ten equations are purported to be fundamental equations of various thermodynamic systems. However, five are inconsistent with one or more of postulates II, III, and IV and consequently arc not physically acceptable. In each case qualitatively sketch the fundamental relationship between S and U (with N and Y constant). Find the five equations that are not physically permissible and indicate the postulates violated by each.

The quantities v _o , ?, and R are positive constants, and in all cases in which fractional exponents appear only the real positive root is to be taken.

R a) c) S = (R) (NVU)/3 (NVUY/x R 2/3 b) S = (^)^^ (NU) S= e) S-(A)^ (NU + RV/ R c) = ROV2 d) S = (R0) 1/NU - 1/5 R (1802) (0111/5 e) S f) S = NRIn(UV/NR0vo) 8) S (8)[NU/_exp(-V/2N23) A) S-(=) (NU)/"exp(- NRI) R UV h) 2 1) U = (1/) = 7 exp(S/NR) RV S 1) U = (RO)NV(1 + R) exp(-S/NR) NR