  We study a connection between chemical thermodynamics and information
geometry. We clarify a relation between the Gibbs free energy of an ideal
dilute solution and an information-geometric quantity called an $f$-divergence.
From this relation, we derive information-geometric inequalities that give a
speed limit for a changing rate of the Gibbs free energy and a general bound of
chemical fluctuations. These information-geometric inequalities can be regarded
as generalizations of the Cram\'{e}r--Rao inequality for chemical reaction
networks described by rate equations, where unnormalized concentration
distributions are of importance rather than probability distributions. They
hold true for damped oscillatory reaction networks and systems where the total
concentration is not conserved so that the distribution cannot be normalized.
We also formulate a trade-off relation between speed and time on a manifold of
concentration distribution by using the geometrical structure induced by the
$f$-divergence. Our results apply to both closed and open chemical reaction
networks, thus they are widely useful for thermodynamic analysis of chemical
systems from the viewpoint of information geometry.
