Far-infrared molecular emission is an important tool used to understand the excitation mechanisms of the gas in the interstellar medium (ISM) of star-forming galaxies. In the present work, we model the emission from rotational transitions with critical densities n greater than or similar to 10(4) cm(-3). We include 4 3 < J <= 15-14 transitions of CO and (CO)-C-13, 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 ISM 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 similar to 10 cm(-3) and a dispersion sigma = 2.1 in the log of the density, most of the emission of molecular lines, even of gas with critical densities >10(4) cm(-3), 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 (CO)-C-13 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, with negligible contribution from denser gas. Specifically, the Mach number, M, of the model galaxy is similar to 10. 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 greater than or similar to 30 in order to obtain significant (> 10%) emission from n > 10(4) cm(-3) gas. Such Mach numbers are expected in star-forming galaxies, luminous infrared galaxies (LIRGS), and ultra-luminous infrared galaxies (ULIRGS). This method provides a way to constrain the global PDF of the ISM of galaxies from observations of molecular line emission. As an example, 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.
