  Starting out with an entropy identity, the entropy flux, the entropy
production and the corresponding Gibbs and Gibbs-Duhem equations of
general-covariant conti\-nuum thermodynamics are established. Non-dissipative
materials and equilibria are investigated. It is proved that equilibrium
conditions only put on material properties cannot generate equilibria, rather
additionally, the Killing property of the 4-temperature is a necessary
condition for space-times in which equilibria are possible.
