Context. M 33 is a gas rich spiral galaxy of the Local Group. Its vicinity allows us to study its interstellar medium (ISM) on linear scales corresponding to the sizes of individual giant molecular clouds.
Aims: We investigate the relationship between the two major gas cooling lines and the total infrared (TIR) dust continuum.
Methods: We mapped the emission of gas and dust in M 33 using the far-infrared lines of [C II] and [O I](63 μm) and the total infrared continuum. The line maps were observed with the PACS spectrometer on board the Herschel Space Observatory. These maps have 50 pc resolution and form a ∼370 pc wide stripe along its major axis covering the sites of bright H II regions, but also more quiescent arm and inter-arm regions from the southern arm at 2 kpc galacto-centric distance to the south out to 5.7 kpc distance to the north. Full-galaxy maps of the continuum emission at 24 μm from Spitzer/MIPS, and at 70 μm, 100 μm, and 160 μm from Herschel/PACS were combined to obtain a map of the TIR.
Results: TIR and [C II] intensities are correlated over more than two orders of magnitude. The range of TIR translates to a range of far ultraviolet (FUV) emission of G0, obs ∼ 2 to 200 in units of the average Galactic radiation field. The binned [C II]/TIR ratio drops with rising TIR, with large, but decreasing scatter. The contribution of the cold neutral medium to the [C II] emission, as estimated from VLA H I data, is on average only 10%. Fits of modified black bodies to the continuum emission were used to estimate dust mass surface densities and total gas column densities. A correction for possible foreground absorption by cold gas was applied to the [O I] data before comparing it with models of photon dominated regions. Most of the ratios of [C II]/[O I] and ([C II]+[O I])/TIR are consistent with two model solutions. The median ratios are consistent with one solution at n ∼ 2 × 102 cm-3, G0 ∼ 60, and a second low-FUV solution at n ∼ 104 cm-3, G0 ∼ 1.5.
Conclusions: The bulk of the gas along the lines-of-sight is represented by a low-density, high-FUV phase with low beam filling factors ∼1. A fraction of the gas may, however, be represented by the second solution.

Maps of TIR, [C II], [O I] shown in Figs. 2 and 3 are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/cat/J/A+A/639/A61

Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.