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We present a pilot survey of 21 lines of sight towards UCHII regions in theH2CO 1-1 (6cm) and 2-2 (2cm) transitions, using the H2CO centimeter lines as amolecular gas densitometer. Using Arecibo and Green Bank beam-matchedobservations, we measure the density of 51 detected H2CO line pairs and presentupper limits on density for an additional 24 detected 1-1 lines. We analyze thesystematic uncertainties in the H2CO densitometer, achieving H2 densitymeasurements with accuracies ~ 0.1-0.3 dex. The densities measured are notcorrelated with distance, implying that it is possible to make accurate densitymeasurements throughout the galaxy without a distance bias. We confirm thatultracompact HII regions are associated with gas at densities n(H2)~10^5cm^-3.The densities measured in line-of-sight molecular clouds suggest that theyconsist of low volume filling factor (f~10^-2) gas at high (n(H2)>10^4cm^-3)density, which is inconsistent with purely supersonic turbulence and requireshigh-density clumping greater than typically observed in gravoturbulentsimulations. We observe complex line morphologies that indicate densityvariations with velocity around UCHII regions, and we classify a subset of theUCHII molecular envelopes as collapsing or expanding. We compare thesemeasurements to Bolocam Galactic Plane Survey 1.1mm observations, and note thatmost UCHII regions have 1.1mm emission consisting of significant (5-70%)free-free emission and are therefore not necessarily dominated by opticallythin dust emission. A comparison of our data with the Mangum et al. starburstsample shows that the area filling factor of dense (n(H2)~10^5cm^-3) moleculargas in typical starburst galaxies is <~0.01, but in extreme starburst galaxieslike Arp 220, is ~0.1, suggesting that Arp 220 is physically similar to anoversized UCHII region.

Galactic H2CO Densitometry I: Pilot survey of Ultracompact HII regions and methodology