  The kinetics and vibrational relaxation of the ${\rm N}(^4S) +{\rm
  O}_2(X^3\Sigma^-_g) \leftrightarrow {\rm O}(^3P) + {\rm NO}(X^2\Pi)$ reaction
is investigated over a wide temperature range based on quasiclassical
trajectory simulations on 3-dimensional potential energy surfaces (PESs) for
the lowest three electronic states. Reference energies at the multi reference
configuration interaction level are represented as a reproducing kernel and the
topology of the PESs is rationalized by analyzing the CASSCF wavefunction of
the relevant states. The forward rate matches one measurement at 1575 K and is
somewhat lower than the high-temperature measurement at 2880 K whereas for the
reverse rate the computations are in good agreement for temperatures between
3000 and 4100 K. The temperature-dependent equilibrium rates are consistent
with results from JANAF and CEA results. Vibrational relaxation rates for O +
NO($\nu=1$) $\rightarrow$ O + NO($\nu=0$) are consistent with a wide range of
experiments. This process is dominated by the dynamics on the $^2$A$'$ and
$^4$A$'$ surfaces which both contribute similarly up to temperatures $T \sim
3000$ K, and it is found that vibrationally relaxing and non-relaxing
trajectories probe different parts of the potential energy surface. The total
cross section depending on the final vibrational state monotonically decreases
which is consistent with early experiments and previous simulations but at
variance with other recent experiments which reported an oscillatory cross
section.
