Here, we present measurements of inorganic and organic particles using a novel instrument that simultaneously measures both acid and base compounds via temperature-programmed desorption: a Filter Inlet for Gases and AEROsols (FIGAERO) combined with a Bipolar Time-of-Flight (BTOF) mass spectrometer. We measured particles from three different sources: atomized salt solutions, aerosols formed in the CERN CLOUD chamber, and ambient aerosols from a site in New York. We find that acids and bases do not desorb at temperatures characteristic of the volatilities of their pure compounds, but rather they desorb at higher temperatures that depend on particle composition related to the entropy of the ionic condensed state, i.e. the lattice energy. For monoprotic acids, the desorption is rate-limited by a proton transfer reaction, evidenced by simultaneous desorption of the neutral acid-base pair. For ammonium sulfate ((NH<sub>4</sub>)<sub>2</sub>SO<sub>4</sub>), we observe a step-wise release of ammonia and increase of the sulfate-to-ammonia ratio: first to triammonium sulfate hydrogen sulfate ((NH<sub>4</sub>)<sub>3</sub>H(SO<sub>4</sub>)<sub>2</sub>; letovicite), and then to ammonium hydrogen sulfate ((NH<sub>4</sub>)HSO<sub>4</sub>)). We find that organic acids and alkyl organosulfates also desorb at temperatures higher than those expected for their pure compounds, indicating reactive uptake. This suggests that a large fraction of organic compounds in particles may be light, ionic species, rather than fragments of low-volatility oligomers. Our studies show that the FIGAERO-BTOF is a highly sensitive and versatile instrument for elucidating condensed-phase aerosol chemistry.