  In this article the Gordan theorem is applied to the thermodynamics of a
chemical reaction network at steady state. From a theoretical viewpoint it is
equivalent to the Clausius formulation of the second law for the out of
equilibrium steady states of chemical networks, i.e. it states that the
exclusion (presence) of closed reactions loops makes possible (impossible) the
definition of a thermodynamic potential and vice versa. On the computational
side, it reveals that calculating reactions free energy and searching
infeasible loops in flux states are dual problems whose solutions are
alternatively inconsistent. The relevance of this result for applications is
discussed with an example in the field of constraints-based modeling of
cellular metabolism where it leads to efficient and scalable methods to afford
the energy balance analysis.
