  We establish a Hessian geometric structure in chemical thermodynamics which
describes chemical reaction networks (CRNs) with equilibrium states. In our
setup, the ideal gas assumption and mass action kinetics are not required. The
existence and uniqueness condition of the equilibrium state is derived by using
the Legendre duality inherent to the Hessian structure. The entropy production
during a relaxation to the equilibrium state can be evaluated by the Bregman
divergence. Furthermore, the equilibrium state is characterized by four
distinct minimization problems of the divergence, which are obtained from the
generalized Pythagorean theorem originating in the dual flatness. For the ideal
gas case, we confirm that our existence and uniqueness condition implies
Birch's theorem, and that the entropy production represented by the divergence
coincides with the generalized Kullback-Leibler divergence. In addition, under
mass action kinetics, our general framework reproduces the local detailed
balance condition.
