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Let's say survivors in the area contaminated by radioactive fallout need to somehow process the water from rivers, rain or pipes. How could they go about it? Is it even possible to 'remove radioactivity' from the water?

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The water itself isn't radioactive, rather particles dissolved in it, because water is a quite good solvent. Therefore, filtration (e.g. with activated carbon) or precipitation can remove such particles. Also a ion exchanger can remove radioactive species. But this can take quite long, especially if the needed amount is large.

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    $\begingroup$ Thermonuclear explosions produce a large amount of tritium. Can't filter that out, because it's likely bound in water. $\endgroup$
    – Karl
    Nov 5 '17 at 14:55
  • $\begingroup$ I'll point out that all surface water has tritium, even before mankind split the atom. $\endgroup$
    – MaxW
    Nov 5 '17 at 14:58
  • $\begingroup$ @Karl OP was not specific about the nature of the radioactive contamination, but the amount of tritium produced by nuclear weapons are generally not a serious threat due to there extreme dilution and short lifetime in body. You also can effectively cure it by drinking normal water. $\endgroup$
    – Greg
    Nov 5 '17 at 15:08
  • $\begingroup$ @Karl - what mechanism produces ‘large’ amounts of tritium? Particularly compared with all the fission byproducts? $\endgroup$
    – Jon Custer
    Nov 5 '17 at 17:58
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    $\begingroup$ The following link shows tritium measured in Ottawa (Canada). It has a large spike in 1964 due to nuclear testing (note that the scale is logarithmic) snobear.colorado.edu/Daniel/isotopes/tritium.html $\endgroup$
    – aventurin
    Nov 5 '17 at 18:48

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