It depends on why its unsafe to drink in the first place. Other people have covered microbes and heat, so I’ll mention the cold. If you’re drinking sea or salt water, you can begin to freeze it so that it separates into salt rich water and salt deficient ice, then pour out the water and melt the ice.
Bacterial cells are mostly made of proteins. Many proteins are fine with getting cold, but not fine with getting hot; high temperatures cause the protein molecules to unfold from their normal shapes, or even break apart, meaning they don’t work properly anymore. If this happens to enough of the proteins in a cell, the cell dies. This is why heat kills bacteria but cold doesn’t (unless it’s really really super cold).
There are common super tiny living things called Giardia that are kind of like turtles. When they see a dangerous environment they hide away in their shell that they call a cyst. When you freeze the water the little Giardia just hang out in their shell which is insulated like a cozy cabin. But when you boil water it gets too hot in the shell so the Giardia have to leave their cyst/ Shell where they are exposed and then killed.
Heat kills things because high temperatures triggers chemical reactions which irreversibly destroy some important things. For us humans, this starts happening when our skin reaches around ~60°C. Some microorganisms have found ways to survive even higher temperatures, but ultimately a temperature above 120°C will kill (nearly) all living things, with the exception of some very rare extremophile “archaea” found in thermal springs.
Cold does the opposite. It stops chemical reactions. The way that cold can kill is that ice crystals can grow and expand in ways that pierces cell membrane. That’s why most vegetables will turn to mush if frozen in a household freezer. But bacteria have found a way to survive this: They can turn themselves into a dormant form, where they are almost completely dried out. In this form, these so called spores are not vulnerable to ice formation, so they can survive even extremely low temperatures.
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