Microdrop platforms in microfluidic technologies provide developments for the study of a variety of systems by means of micrometer sized fluids. Concentrating the solutes within an aqueous microdrop via slight dissolution of water into a surrounding organic phase is a process which has important applications in biological systems. In our previous work we described the thermodynamic equilibrium of this concentrating process for two types of solutes, those with limited and unlimited solubility, and investigated the role of temperature, the amount of organic phase, and the initial concentration of the solutes. It was found that, during the concentration process which is accompanied by microdrop shrinkage, for specific initial conditions the system may encounter more than one equilibrium state. In this work, by means of free energy analysis based on Gibbsian surface thermodynamics, we investigate the stability behavior of the system to provide more detailed information so that these systems can be designed more accurately.