Theoretical studies of carbon isotopic fractionation in reactions of C with C$_{2}$: dynamics, kinetics, and isotopologue equilibria

Abstract

Our current understanding of interstellar carbon fractionation hinges on the interpretation of astrochemical kinetic models. Yet, the various reactions included carry large uncertainties in their (estimated) rate coefficients, notably those involving C with C$_{2}$. In this work, we provide theoretical thermal rate coefficients as a function of the temperature for all possible gas-phase isotope-exchange reactions of C+C$_{2}(X^{1}\Sigma_{g}^{+},a^{3}\Pi_{u})$. For this, we employ the quasi-classical trajectory method, with the previously obtained potential energy surfaces of C$_{3}$ dictating the forces between the colliding partners. The calculated rate coefficients show a positive temperature dependence and are markedly different from previous theoretical estimates. While the forward reactions are fast and inherently exothermic owing to the lower zero-point energy content of the products, the reverse processes have temperature thresholds. For each reaction considered, analytic three-parameter Arrhenius-Kooij formulas are provided that readily interpolate/extrapolate the associated forward and backward rates. These forms can further be introduced in astrochemical networks. Apart from the proper kinetic attributes, we also provide equilibrium constants for these processes, thence confirming their prominence in the overall C fractionation chemistry. In this respect, the $^{13}$C + $^{12}$C$_{2}(X^{1}{\Sigma}^{+}_{g})$ and $^{13}$C + $^{12}$C$_{2}(a^{3}{\Pi}_{u})$ reactions are found to be particularly conspicuous, notably at the typical temperatures of dense molecular clouds. For these reactions and considering both equilibrium and time-dependent chemistry, theoretical $^{12}$C/$^{13}$C ratios as a function of the gas kinetic temperature are also derived and shown to be consistent with available model chemistry and observational data on C$_{2}$