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In this paper, a thermodynamic interpretation of the kinetic theory and Boltzmann's equation is explicitly obtained; the nonequilibrium thermodynamic space consists of the nonnegative distribution functions. Beginning from a molecular expression for entropy in the form o f Boltzmann's H functional, the systematic construction of thermodynamic potentials is demonstrated for gaseous systems beyond local equilibrium ("not intinitesimally near to equilibrium"). Since the nonequilibrium thermodynamic space provides natural comparison states for the principles of maximum entropy or minimum energy, a simple criterion for the choice of gas-state variables can be given which shows that the maximization of the Legendre transforms of entropy is logically equivalent to the minimization of the Legendre of energy. This criterion is sought after in such a way that the concepts of temperature and pressure need no reformulation out of equilibrium. After these preparations, the technique of functional differentiation is used to derive the generalized Gibbs equation (relation) for Boltzmann's entropy. Finally, the paper presents an analysis of how the functional representation of this equation relates to the method of moments.