The thermodynamic evaluation of reaction feasibility, based on the standard Gibbs energies of reaction, faces difficulties when, instead of isolated reactions, we are examining whole pathways. For pathways, we seek not only to decide whether they are feasible but also to pinpoint the pathway segment that causes thermodynamic difficulties. The obstructing pathway-segment may either be a single reaction (localized bottleneck) or a sequence of reactions (distributed bottleneck) which cannot take place simultaneously. We present a duality theory that converts the primal problem of selecting concentrations of species to make a pathway feasible to its dual problem of selecting linear combinations of reactions that make the pathway infeasible. The dual problem leads to an algorithm that can determine the thermodynamic feasibility of any chemical reaction system. The method involves the analysis of individual reactions and the selective construction of larger subpathways; it uncovers localized and distributed thermodynamic bottlenecks of a pathway. The method is applicable to reaction systems of any origin or topology, provided that concentrations of species are restricted to positive intervals.