From a mathematical model that describes a complex chemical kinetic system of
N species and M elementrary reactions in a rapidly stirred vessel of size
V as a Markov process, we show that a macroscopic chemical thermodynamics
emerges as Vββ. The theory is applicable to linear and
nonlinear reactions, closed systems reaching chemical equilibrium, or open,
driven systems approaching to nonequilibrium steady states. A generalized
mesoscopic free energy gives rise to a macroscopic chemical energy function
\varphi^{ss}(\vx) where \vx=(x_1,\cdots,x_N) are the concentrations of the
N chemical species. The macroscopic chemical dynamics \vx(t) satisfies two
emergent laws: (1) (\rd/\rd t)\varphi^{ss}[\vx(t)]\le 0, and (2)(\rd/\rd
t)\varphi^{ss}[\vx(t)]=\text{cmf}(\vx)-\sigma(\vx) where entropy production
rate Οβ₯0 represents the sink for the chemical energy, and chemical
motive force cmfβ₯0 is non-zero if the system is driven under a
sustained nonequilibrium chemostat. For systems with detailed balance
cmf=0, and if one assumes the law of mass action,\varphi^{ss}(\vx)
is precisely the Gibbs' function βi=1Nβxiβ[ΞΌioβ+lnxiβ]
for ideal solutions. For a class of kinetic systems called complex balanced,
which include many nonlinear systems as well as many simple open, driven
chemical systems, the \varphi^{ss}(\vx), with global minimum at \vx^*, has
the generic form βi=1Nβxiβ[ln(xiβ/xiββ)βxiβ+xiββ],which has
been known in chemical kinetic literature.Macroscopic emergent "laws" are
independent of the details of the underlying kinetics. This theory provides a
concrete example from chemistry showing how a dynamic macroscopic law can
emerge from the kinetics at a level below.Comment: 8 page