Capturing carbon dioxide from the atmosphere ("air capture") in an industrial process has been proposed as an option for stabilizing global CO2 concentrations. Published analyses suggest these air capture systems may cost a few hundred dollars per tonne of CO2, making it cost competitive with mainstream CO2 mitigation options like renewable energy, nuclear power, and carbon dioxide capture and storage from large CO2 emitting point sources. We investigate the thermodynamic efficiencies of commercial separation systems as well as trace gas removal systems to better understand and constrain the energy requirements and costs of these air capture systems. Our empirical analyses of operating commercial processes suggest that the energetic and financial costs of capturing CO2 from the air are likely to have been underestimated. Specifically, our analysis of existing gas separation systems suggests that, unless air capture significantly outperforms these systems, it is likely to require more than 400 kJ of work per mole of CO2, requiring it to be powered by CO2-neutral power sources in order to be CO2 negative. We estimate that total system costs of an air capture system will be on the order of $1,000 per tonne of CO2, based on experience with as-built large-scale trace gas removal systems.