A ΓΏrst principle analysis of an open system thermodynamical Carnot cycle is provided and the results are compared to those proposed by Gibbs for open systems. The Kelvin-Clausius statement concerning heat transfer for reversible cycles is taken as an axiom, from which several rigorous theorems are proven. An equation is derived that resembles a Gibbs-Duhem relation equating convected entropies, from which two distinguishable forms of entropy are proven to exist for such systems; this questions prevailing developments which presume a singular or characteristic entropic form that couple all work and heat ows, such as in Onsager ΓΏrst-order thermodynamics. In particular, a closed path undergone by the system does not return the environment to the initial state for one of these entropic forms. The Biot assertion that the entropy contribution due to diathermal heat transfer does not form a state function is therefore contradicted and a local entropy is shown to exist. Several other new composite state functions for work and heat ow are shown to exist for open systems. From Gibbs' results, it is suggested that the ensuing chemical potentials used routinely may possibly ignore heat e ects. The functions developed here are suitable for application since the functions are proven to exist, rather than presumed to exist.