On the Clausius formulation of the second law in stationary chemical networks through the theorems of the alternative

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.