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.
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