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there is a simple answer. Anhydrite (powdered) is soluble in hydrochloric acid (HCl), but only with little effect in water. It hydrates over longer period of time < 42°C and turns into gypsum. You need no "hard dissolvents" to decompose this sulfate under laboratory conditions and you can use acid you mentioned above. If you want to carry out differential leaching and save other minerals, the procedure has to be adjusted pH-wise. I do not know the matrix and your goals of study.
The solubility of calcium sulphate in water can be increased several times through the ion pairing effect, i.e. by addition of a soluble salt of a kation that has high affinity to sulphate, (forming an ion pair). For some analytical purposes, we have used pre- cleaned magnesium nitrate solutions. With "cleaning" we mean using ion exchange to remove the trace elements we are interested in (so that all comes from the gypsum/anhydrite). However, for most geochemical analyses, 1 M HNO3 has proven satisfactory. If pH is increased, gypsum will precipitate.
What should be the best experimental schematics for the removal of anhydrite from lithium slag (LS) using HCl or HNO3, without significantly altering the rest of the mineral phases in LS.
For your reference, LS contains Alumino-silicates in the form of spodumene, anorthite, and albite along with calcite.
Fine-grained particles can be dissolved in conc. H2SO4 which is, however, not the method of choice for samples intended to be used for further laboratory trials in sedimentology or metallurgy because it is harming mineralogically the residue. Water has no effect. It is much better to grind your slags and use 10 % hydrous NaCl solution, put it into a container and shake it with a shaking machine for 24 hrs. You can remove both gypsum and anhydrite. The method was elaborated by the late German professor of sedimentology (specialist in salt deposits), Prof. Dr. Correns.