  We consider growing open chemical reaction systems (CRSs), in which
autocatalytic chemical reactions are encapsulated in a finite volume and its
size can change in conjunction with the reactions. The thermodynamics of
growing CRSs is indispensable for understanding biological cells and designing
protocells by clarifying the physical conditions and costs for their growing
states. In this work, we establish a thermodynamic theory of growing CRSs by
extending the Hessian geometric structure of non-growing CRSs. The theory
provides the environmental conditions to determine the fate of the growing
CRSs; growth, shrinking or equilibration. We also identify thermodynamic
constraints; one to restrict the possible states of the growing CRSs and the
other to further limit the region where a nonequilibrium steady growing state
can exist. Moreover, we evaluate the entropy production rate in the steady
growing state. The growing nonequilibrium state has its origin in the
extensivity of thermodynamics, which is different from the conventional
nonequilibrium states with constant volume. These results are derived from
general thermodynamic considerations without assuming any specific
thermodynamic potentials or reaction kinetics; i.e., they are obtained based
solely on the second law of thermodynamics.
