  In this paper, the applicability of the entropy method for the trend towards
equilibrium for reaction-diffusion systems arising from first order chemical
reaction networks is studied. In particular, we present a suitable entropy
structure for weakly reversible reaction networks without detail balance
condition.
  We show by deriving an entropy-entropy dissipation estimate that for any
weakly reversible network each solution trajectory converges exponentially fast
to the unique positive equilibrium with computable rates. This convergence is
shown to be true even in cases when the diffusion coefficients all but one
species are zero.
  For non-weakly reversible networks consisting of source, transmission and
target components, it is shown that species belonging to a source or
transmission component decay to zero exponentially fast while species belonging
to a target component converge to the corresponding positive equilibria, which
are determined by the dynamics of the target component and the mass injected
from other components. The results of this work, in some sense, complete the
picture of trend to equilibrium for first order chemical reaction networks.
