Determine which of the following functions may represent the entropy of a system of positive temperature. In these expressions, E_0 and V_0 are constants representing an energy and a volume, respectively.
a) S (U, V,N) = NR \ln \biggl(1+\frac{U}{NE_0} \biggr)+\frac{RU}{E_0} \ln \biggl(1+\frac{NE_0}{U} \biggr).
b) S (U, V,N) = \frac{RU}{E_0} \exp \biggl(-\frac{UV}{NE_0}\biggr).
c) S (U, V,N) = \frac{NRU}{(V^3/V^3_0)E_0} .
d) S (U, V,N) = R \frac{U^{3/5}V^{2/5}}{E^{3/5}_0 V^{2/5}_0} .
Since the entropy S (U, V,N) is an extensive state function and the state variables U, V and N are also extensive, they have to satisfy the identity S (λ U, λ V, λ N) = λ S (U, V,N) where λ is a positive integer. Expressions b) and c) do not satisfy this identity. Moreover, since the partial derivative of the entropy with respect to the internal energy in expression b) is negative, this expression also needs to be rejected because it has a negative temperature. Thus, only expressions a) and d) may represent the entropy of a system of positive temperature.