“Bleach” is a broad term for a lot of chemicals, but the most common household bleach is a chemical called sodium hypochlorite (NaOCl) (or more properly, a very small amount of sodium hypochlorite dissolved in water).
Sodium hypochlorite, per [wiki](https://en.wikipedia.org/wiki/Sodium_hypochlorite#Production), is made mostly by a pretty simple chemical process. Salt (sodium chloride, NaCl) is dissolved in water, which splits the salt into sodium (Na^(+)) and chloride (Cl^(-)) ions. Then an electric current is applied, which splits the water into hydroxide (OH^(-)) and hydrogen (H^(+)) ions.*
The sodium and hydroxide combine to form sodium hydroxide (NaOH), which you know as the chemical called *lye*. And the chloride ions lose their extra electron to the H^+ ions. The net reaction ends up being:
2 NaCL + 2 H2O -> 2 NaOH + CL2 + H2
Or, in words, salt + water -> sodium hydroxide + chlorine gas + hydrogen gas. The hydrogen escapes, and the chloride is collected and passed back through the sodium hydroxide to react with it:
2 NaOH + Cl2 -> 2 NaOCl + H2
This produces more hydrogen, which escapes, and leaves sodium hypochlorite behind.
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The reason bleach works is that sodium hypochlorite is actually a pretty unstable molecule. If you dry it out and pile up a bunch of it, it can explode. But when dissolved in small amounts in water, this instability means it frequently breaks down and releases chlorine. Chlorine is an extremely reactive element, so it easily reacts with other things in the water – like the cell membranes of bacteria you’d like to kill, or the structure of a smelly molecule you’re trying to clean up.
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* Technically this is a lot more complicated than this, H^+ doesn’t just hang around in solution by itself. But this is the way most basic textbooks would present it. In general, chemistry tends to involve a lot of “um, akshully”s.
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