Earth's atmospheric composition is profoundly altered by the biosphere. All the major constituents (except Ar) are cycled by biology, and the resultant mixture of gases is not in thermodynamic equilibrium. Consequently, atmospheric chemical disequilibrium has been proposed as a method for detecting extraterrestrial biospheres from exoplanet observations. However, disequilibrium can also be maintained by abiotic processes such as photochemistry or volcanic outgassing, and so inferring life from disequilibrium is a question of degree and context. Here, we present the first rigorous calculations of the thermodynamic chemical disequilibrium in the atmospheres of Solar System planets, in which we quantify the difference in Gibbs free energy of an observed atmosphere compared to that of all the atmospheric gases reacted to equilibrium. The purely gas phase disequilibrium in Earth's atmosphere, as measured by this available Gibbs free energy, is not unusual by Solar System standards and smaller than that of Mars. However, Earth's atmosphere is in contact with a surface ocean, which means that gases can react with water, and so a multiphase calculation that includes aqueous species is required. We find that the disequilibrium in Earth's atmosphere-ocean system (in joules per mole of atmosphere) is more than an order of magnitude larger than the disequilibria of all other atmospheres in the Solar System. Disequilibrium in other Solar System atmospheres is driven by abiotic processes, and we identify the key disequilibria in each atmosphere. Earth's thermodynamic disequilibrium is biogenic in origin, and the main contribution is the coexistence of N2, O2 and liquid water instead of more stable nitrate. In comparison, the coexistence of O2 and methane is a small contributor to Earth's atmosphere-ocean disequilibrium. Our metric requires minimal assumptions and could potentially be calculated using observations of exoplanet atmospheres. Our Matlab source code and associated databases for these calculations are available as open source software.