Molecular nitrogen (N₂) comprises three-quarters of Earth's atmosphere and significant portions of other planetary atmospheres. We report a 19 per mil (‰) excess of ¹⁵N¹⁵N in air relative to a random distribution of nitrogen isotopes, an enrichment that is 10 times larger than what isotopic equilibration in the atmosphere allows. Biological experiments show that the main sources and sinks of N₂ yield much smaller proportions of ¹⁵N¹⁵N in N₂. Electrical discharge experiments, however, establish ¹⁵N¹⁵N excesses of up to +23‰. We argue that ¹⁵N¹⁵N accumulates in the atmosphere because of gas-phase chemistry in the thermosphere (>100 km altitude) on time scales comparable to those of biological cycling. The atmospheric ¹⁵N¹⁵N excess therefore reflects a planetary-scale balance of biogeochemical and atmospheric nitrogen chemistry, one that may also exist on other planets.