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Could you drink distilled water with a mineral pill
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I'm a chemist, not a biologist, so I'm a bit fuzzy on stuff like cells. But I can do basic maths (and obviously chemistry) and I have Google.
The difference in sodium between tapwater and distilled water is 40mg per liter. So at most, drinking a liter of distilled water you're going to lose 40mg of sodium more than with tapwater.
Your blood contains 140ish mmol of sodium per liter. At 26 grams per mole, that's 3650mg/liter. The difference between blood and tapwater is already huge, which is why you can easily get hyponatremia from drinking regular tap- or mineralwater (as you showed in your link). Whether it's a difference of 3610mg/l or 3650mg/l doesn't matter at all, the gradient is already very steep with normal water.
The other very obvious bit of evidence are the thousands of people on ocean-going ships who drink water from reverse-osmosis filters, which is basically completely mineral-free too.