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Non-equilibrium vibrational dissociation of CO$_2$ at lowtranslational-rotational temperatures $T$ is investigated computationally forconditions of microwave induced plasmas. Semi-analytic treatment of vibrationalrelaxation in CO$_2$ in shock tube and acoustic experiments is summarized. Astate-to-state vibrational kinetics model applied for the simulations isbenchmarked and adjusted to the relaxation times obtained in gas dynamicexperiments. The governing parameter $Q/n^2_0$ has been introduced, where $Q$is the specific volumetric power coupled in plasma and $n_0$ is the initialnumber density of CO$_2$. The modelling results indicate a rapid increase ofthe rate of the primary dissociation process CO$_2$ + M $\to$ CO + O + M when$Q/n^2_0$ exceeds some critical value. A simple analytic calculation of $\left(Q/n^2_0 \right)_{crit}$ is proposed which agrees well with the numericalresults. At $T$=300 K the estimated $\left( Q/n^2_0 \right)_{crit} \eqsim6\cdot10^{-40}$~W$\cdot$m$^3$.

Vibrational relaxation and triggering of the non-equilibrium vibrational decomposition of CO$_2$ in gas discharges