  The Nitrogen-to-Oxygen (N/O) abundance ratio is an important diagnostic of
galaxy evolution since the ratio is closely tied to the growth of metallicity
and the star formation history in galaxies. Estimates for the N/O ratio are
traditionally accomplished with optical lines that could suffer from extinction
and excitation effects, so the N/O ratio is arguably measured better through
far-infrared (far-IR) fine-structure lines. Here we show that the [N
III]57$\mu$m/[O III]52$\mu$m line ratio, denoted $N3O3$, is a physically robust
probe of N/O. This parameter is insensitive to gas temperature and only weakly
dependent on electron density. Though it has a dependence on the hardness of
the ionizing radiation field, we show that it is well corrected by including
the [Ne III]15.5$\mu$m/[Ne II]12.8$\mu$m line ratio. We verify the method, and
characterize its intrinsic uncertainties by comparing the results to
photoionization models. We then apply our method to a sample of nearby galaxies
using new observations obtained with SOFIA/FIFI-LS in combination with
available Herschel/PACS data, and the results are compared with optical N/O
estimates. We find evidence for a systematic offset between the far-IR and
optically derived N/O ratio. We argue this is likely due to that our far-IR
method is biased towards younger and denser H II regions, while the optical
methods are biased towards older H II regions as well as diffuse ionized gas.
This work provides a local template for studies of ISM abundance in the early
Universe.
