  Thermal rates for the C($^3$P) + O$_2$($^3 \Sigma_g^-$) $\leftrightarrow$
CO($^1 \Sigma^+$)+ O($^1$D)/O($^3$P) reaction are investigated over a wide
temperature range based on quasi classical trajectory (QCT) simulations on
3-dimensional, reactive potential energy surfaces (PESs) for the $^1$A$'$,
$(2)^1$A$'$, $^1$A$''$, $^3$A$'$ and $^3$A$''$ states. The forward rate matches
measurements at 15 K to 295 K whereas the equilibrium constant determined from
the forward and reverse rates are consistent with those derived from
statistical mechanics at high temperature. Vibrational relaxation, O+CO($\nu=1,
2$) $\rightarrow$ O+CO$(\nu=0)$, is found to involve both, non-reactive and
reactive processes. The contact time required for vibrational relaxation to
take place is $\tau \geq 150$ fs for non-reacting and $\tau \geq 330$ fs for
reacting (oxygen atom exchange) trajectories and the two processes are shown to
probe different parts of the global potential energy surface. In agreement with
experiments, low collision energy reactions for the C($^3$P) + O$_2$($^3
\Sigma_g^-$, $v=0$) $\rightarrow$ CO($^1 \Sigma^+$)+ O($^1$D) lead to
CO($^{1}\Sigma^{+}$, $v'=17$) with an onset at $E_{\rm c} \sim 0.15$ eV,
dominated by the $^1$A$'$ surface with contributions from the $^3$A$'$ surface.
Finally, the barrier for the CO$_{\rm A}$($^1\Sigma^+$) + O$_{\rm B}$($^3$P)
$\rightarrow$ CO$_{\rm B}$($^{1}\Sigma^{+}$) + O$_{\rm A}$($^3$P) atom exchange
reaction on the $^3$A$'$ PES yields a barrier of $\sim 7$ kcal/mol (0.300 eV),
consistent with an experimentally reported value of 6.9 kcal/mol (0.299 eV).
