Why is hard water, when initially basic, turn acidic when softened with sodium ion-exchange resins?

Hard water is basic, this is true. Soft water may or may not be basic after the ion exchange resin bed is done. It depends on how the system is set up.

If you put a polishing bottle after the bed, then you can make Deionized water which is acidic at a pH of 5.5. This happens when the purified water, hungry for solute, dissolves CO2 out of the atmosphere to create carbonic acid H2CO3 (aq). More.

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