An estimate of the degree of CO-depletion ($f_D$) provides information on the physical conditions occurring in the innermost and densest regions of molecular clouds. A key parameter in these studies is the size of the depletion radius, i.e. the radius within which the C-bearing species, and in particular CO, are largely frozen onto dust grains. A strong depletion state (i.e. $f_D>10$, as assumed in our models) is highly favoured in the innermost regions of dark clouds, where the temperature is $<20$ K and the number density of molecular hydrogen exceeds a few $times$10$^{4}$ cm$^{-3}$. In this work, we estimate the size of the depleted region by studying the Infrared Dark Cloud (IRDC) G351.77-0.51. Continuum observations performed with the $Herschel$ $Space$ $Observatory$ and the $LArge$ $APEX$ $BOlometer$ $CAmera$, together with APEX C$^{18}$O and C$^{17}$O J=2$rightarrow$1 line observations, allowed us to recover the large-scale beam- and line-of-sight-averaged depletion map of the cloud. We built a simple model to investigate the depletion in the inner regions of the clumps in the filament and the filament itself. The model suggests that the depletion radius ranges from 0.02 to 0.15 pc, comparable with the typical filament width (i.e.$sim$0.1 pc). At these radii, the number density of H$_2$ reaches values between 0.2 and 5.5$times$10$^{5}$ cm$^{-3}$. These results provide information on the approximate spatial scales on which different chemical processes operate in high-mass star-forming regions and also suggest caution when using CO for kinematical studies in IRDCs.
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