1. The problem statement, all variables and given/known data Consider the fundamental equation S = AUnVmNr where A is a positive constant. Evaluate the permissible values of the three constants n, m, and r if the fundamental equation is to satisfy the thermodynamic postulates and if, in addition, we wish to have P increase with U/V, at constant N. (This latter condition is an intuitive substitute for stability requirements to be studied in Chapter 8.) For definiteness, the zero of energy is to be taken as the energy of the zero-temperature state. 2. Relevant equations The thermodynamic postulates in question are - 1 there exist equilibrium states of simple systems macroscopically characterized completely by internal energy, volume, and the mole numbers of the chemical components 2 there exists some S, the entropy function, of the extensive parameters of any composite system, defined for all states of equilibrium, which has the property that the values assumed by the extensive parameters in the absence of an internal constraint are those that maximize, either positively or negatively, the entropy over the manifold of constrained equilibrium states 3 the entropy of a composite system is additive over the constituent subsystems, which implies that S =ΣS(α), where S(α) is the entropy of a subsystem, and n is the number of subsystems; and it is a continuous, differentiable, monotonically-increasing function of the energy, where the energy, U, is a single-valued, continuous, differentiable function of S, V, N1, …, Nr 4 the entropy of any system vanishes in the state for which temperature is zero, where the partial derivative of U with respect to S is zero. 3. The attempt at a solution I know that I need to develop mathematical constraints on n, m, and r.