J.R. S. answered 03/30/20
Ph.D. University Professor with 10+ years Tutoring Experience
CsF is the salt of a weak acid, HF. RbI is the salt of a weak acid, HI. Therefore, you would expect CsF to be more soluble, and you are correct in saying that it has to do with the conjugate base. Let's take a look at each salt dissolved in water and then the addition of acid, H+.
CsF ==> Cs+ + F-
F- + H2O ==> HF + OH-
Remember HF is weak so will not ionize much. Addition of H+ will react with F- making more HF and this will increase solubility. Ksp = [Cs+][F-] and if you decrease F-, you decrease Ksp. Another way to look at it, is that addition of H+ will remove OH- from the right side, and according to Le Chatelier, this will push the reaction toward the product side, increasing the solubility.
RbI ==> Rb+ + I-
I- + H+ ==> HI which ionizes completely back to H+ + I- so acid will not affect the solubility.
So, to answer the last part of your question, you identify it by knowing if the salt is from a weak acid or a strong acid, If it's from a weak acid, the conjugate base will be strong and this will increase solubility in acid solution.