  The non-steady-state formation of small clusters and the growth of grains
accompanied by chemical reactions are formulated under the consideration that
the collision of key gas species (key molecule) controls the kinetics of dust
formation process. The formula allows us to evaluate the size distribution and
condensation efficiency of dust formed in astrophysical environments. We apply
the formulation to the formation of C and MgSiO3 grains in the ejecta of
supernovae, as an example, to investigate how the non-steady effect influences
the formation process, condensation efficiency f_{con}, and average radius
a_{ave} of newly formed grains in comparison with the results calculated with
the steady-state nucleation rate. We show that the steady-state nucleation rate
is a good approximation if the collision timescale of key molecule tau_{coll}
is much smaller than the timescale tau_{sat} with which the supersaturation
ratio increases; otherwise the effect of the non-steady state becomes
remarkable, leading to a lower f_{con} and a larger a_{ave}. Examining the
results of calculations, we reveal that the steady-state nucleation rate is
applicable if the cooling gas satisfies Lambda = tau_{sat}/tau_{coll} > 30
during the formation of dust, and find that f_{con} and a_{ave} are uniquely
determined by Lambda_{on} at the onset time t_{on} of dust formation. The
approximation formulae for f_{con} and a_{ave} as a function of Lambda_{on}
could be useful in estimating the mass and typical size of newly formed grains
from observed or model-predicted physical properties not only in supernova
ejecta but also in mass-loss winds from evolved stars.
