Chalk does not dissolve readily. You can bubble CO2 through water and chalk to dissolve it. If you have a carbonator cap and a big plasitc drink bottle, you can fill about 3/4 full, add chalk, pressurise with 30 PSI and shake. If all the chalk does not dissolve, vent, add more pressure and shake again. And so on.
Pickling lime dissolves quickly. The OH ions combine with the H ions to form water. The other thing you add is Ca, which does not impact the flavor. I point this out due to the fact that some use baking soda, but that adds Na.