Far-infrared molecular emission is an important tool used to understand the excitation mechanisms of the gas in the inter-stellar medium of star-forming galaxies. In the present work, we model the emission from rotational transitions with critical densities n >~ 10^4 cm-3. We include 4-3 < J <= 15-14 transitions of CO and 13CO, in addition to J <= 7-6 transitions of HCN, HNC, and HCO+ on galactic scales. We do this by re-sampling high density gas in a hydrodynamic model of a gas-rich disk galaxy, assuming that the density field of the interstellar medium of the model galaxy follows the probability density function (PDF) inferred from the resolved low density scales. We find that in a narrow gas density PDF, with a mean density of ~10 cm-3 and a dispersion sigma = 2.1 in the log of the density, most of the emission of molecular lines, emanates from the 10-1000 cm-3 part of the PDF. We construct synthetic emission maps for the central 2 kpc of the galaxy and fit the line ratios of CO and 13CO up to J = 15-14, as well as HCN, HNC, and HCO+ up to J = 7-6, using one photo-dissociation region (PDR) model. We attribute the goodness of the one component fits for our model galaxy to the fact that the distribution of the luminosity, as a function of density, is peaked at gas densities between 10 and 1000 cm-3. We explore the impact of different log-normal density PDFs on the distribution of the line-luminosity as a function of density, and we show that it is necessary to have a broad dispersion, corresponding to Mach numbers >~ 30 in order to obtain significant emission from n > 10^4 cm-3 gas. Such Mach numbers are expected in star-forming galaxies, LIRGS, and ULIRGS. By fitting line ratios of HCN(1-0), HNC(1-0), and HCO+(1-0) for a sample of LIRGS and ULIRGS using mechanically heated PDRs, we constrain the Mach number of these galaxies to 29 < M < 77.
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