Atmospheric chemical disequilibrium has been proposed as a method for detecting extraterrestrial biospheres from exoplanet observations. Chemical disequilibrium is potentially a generalized biosignature since it makes no assumptions about particular biogenic gases or metabolisms. Here, we present the first rigorous calculations of the thermodynamic chemical disequilibrium in Solar System atmospheres, in which we quantify the available Gibbs energy: the Gibbs free energy of an observed atmosphere minus that of atmospheric gases reacted to equilibrium. The purely gas phase disequilibrium in Earth's atmosphere is mostly attributable to O2 and CH4. The available Gibbs energy is not unusual compared to other Solar System atmospheres and smaller than that of Mars. However, Earth's fluid envelope contains an ocean, allowing gases to react with water and requiring a multiphase calculation with aqueous species. The disequilibrium in Earth's atmosphere-ocean system (in joules per mole of atmosphere) ranges from ∼20 to 2 × 10(6) times larger than the disequilibria of other atmospheres in the Solar System, where Mars is second to Earth. Only on Earth is the chemical disequilibrium energy comparable to the thermal energy per mole of atmosphere (excluding comparison to Titan with lakes, where quantification is precluded because the mean lake composition is unknown). Earth's disequilibrium is biogenic, mainly caused by the coexistence of N2, O2, and liquid water instead of more stable nitrate. In comparison, the O2-CH4 disequilibrium is minor, although kinetics requires a large CH4 flux into the atmosphere. We identify abiotic processes that cause disequilibrium in the other atmospheres. Our metric requires minimal assumptions and could potentially be calculated from observations of exoplanet atmospheres. However, further work is needed to establish whether thermodynamic disequilibrium is a practical exoplanet biosignature, requiring an assessment of false positives, noisy observations, and other detection challenges. Our Matlab code and databases for these calculations are available, open source.