  Motivated by the work of Cooper & Showman, we revisit the chemical relaxation
method, which seeks to enhance the computational efficiency of
chemical-kinetics calculations by replacing the chemical network with a handful
of independent source/sink terms. Chemical relaxation solves the evolution of
the system and can treat disequilibrium chemistry, as the source/sink terms are
driven towards chemical equilibrium on a prescribed chemical timescale, but it
has surprisingly never been validated. First, we generalize the treatment by
forgoing the use of a single chemical timescale, instead developing a pathway
analysis tool that allows us to identify the rate-limiting reaction as a
function of temperature and pressure. For the interconversion between methane
and carbon monoxide and between ammonia, and molecular nitrogen, we identify
the key rate-limiting reactions for conditions relevant to currently
characterizable exo-atmospheres (500-3000 K, 0.1 mbar to 1 kbar). Second, we
extend chemical relaxation to include carbon dioxide and water. Third, we
examine the role of metallicity and carbon-to-oxygen ratio in chemical
relaxation. Fourth, we apply our pathway analysis tool to diagnose the
differences between our chemical network and that of Moses and Venot. Finally,
we validate the chemical relaxation method against full chemical kinetics
calculations in one dimension. For WASP-18b-, HD 189733b- and GJ 1214-b-like
atmospheres, we show that chemical relaxation is mostly accurate to within an
order of magnitude, a factor of 2 and $\sim 10\%$, respectively. The level of
accuracy attained allows for the chemical relaxation method to be included in
three-dimensional general circulation models.
