<div>In the two-balloon experiment, two rubber balloons are connected and allowed to exchange gas. Owing to the non-monotonic relationship between the radius of the balloon and the pressure of gas inside of it, the two-balloon system presents multi- and in-stabilities.</div><div><br></div><div>Herein, we consider a two-adsorbent system, where two different adsorbents are allowed to exchange gas. We show that, for rigid adsorbents, the thermodynamic equilibrium state is unique.</div><div><br></div><div>Then, we consider an adsorbent-balloon system, where an adsorbent exchanges gas with a rubber balloon. This system can exhibit multiple states at thermodynamic equilibrium-- two (meta)stable and one unstable. The size of the balloon, pressure of gas in the balloon, and partitioning of gas between the adsorbent and the balloon differ among the equilibrium states. Temperature changes and the addition/removal of gas into/from the adsorbent-balloon system can induce catastrophe bifurcations and show hysteresis. Furthermore, the adsorbent-balloon system exhibits a critical temperature where, when approached from below, the discrepancy of balloon size between the two (meta)stable states decreases and, beyond, bistability is impossible.</div><div><br></div><div>Practically, our findings preclude multiple partitions of adsorbed gas in rigid mixed-linker metal-organic frameworks and may inspire new soft actuator and sensor designs.</div>
Thermodynamic Equilibrium in Relativity: Four-temperature, Killing Vectors and Lie Derivatives
