We have developed and studied methods to characterize the time-varying composition of liquid microdroplets, under controlled changes to environmental conditions, using Raman tweezers. This work has focussed on measurements of inorganic salts, such as nitrate and sulfate anions, which comprise a major fraction of the dissolved solutes in atmospheric aerosols. The experimental Raman intensities for the anions of inorganic salts in optically tweezed droplets were found to be in good agreement with theoretical estimates of photon scattering. The detection limit for sodium nitrate salt in a single particle was found to be ~4 pg. The mass of an inorganic salt in the droplet can be estimated from the Raman intensity of the anion bands using a calibration curve which is independent of droplet volume. The volume of the droplet, and concentration of solute, can be found directly from the spacing of morphology dependent resonances in the Raman band of water, or indirectly from the integrated-intensity of the Raman band for the solvent. The later strategy eliminates the uncertainty in the collection efficiency of Raman-scattered light related to varying particle sizes.