We report high resolution observations of the $^{12}$CO$(1rightarrow0)$ and $^{13}$CO$(1rightarrow0)$ molecular lines in the Carina Nebula and the Gum 31 region obtained with the 22-m Mopra telescope as part of the The Mopra Southern Galactic Plane CO Survey. We cover 8 deg$^2$ from $l = 285^{circ}$ to 290$^{circ}$, and from $b = -1.5^{circ}$ to +0.5$^{circ}$. The molecular gas column density distributions from both tracers have a similar range of values. By fitting a grey-body function to the observed infrared spectral energy distribution from Herschel maps, we derive gas column densities and dust temperatures. The gas column density has values in the range from $6.3times 10^{20}$ to $1.4times 10^{23}$ cm$^{-2}$, while the dust temperature has values in the range from 17 to 43 K. The gas column density derived from the dust emission is approximately described by a log-normal function for a limited range of column densities. A high-column density tail is clearly evident for the gas column density distribution, which appears to be a common feature in regions with active star formation. There are regional variations in the fraction of the mass recovered by the CO emission lines with respect to the total mass traced by the dust emission. These variations may be related to changes in the radiation field strength, variation of the atomic to molecular gas fraction across the observed region, differences in the CO molecule abundance with respect to H$_{2}$, and evolutionary stage differences of the molecular clouds that compose the Carina Nebula-Gum 31 complex.
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