Hand washing removes bacteria with mechanical means. The bacteria get washed off your hands, not killed. There is no resistance to build up against this. Hand sanitizer kills with alcohol and that mode of killing is not easily protected against. No DNA modification can protect against this (ie, mutate)
Mutation like you’re thinking about only occurs when there’s a transitional change, and not a sudden one.
Both Soap & Hand sanitizer we use, are too strong for any bacteria to have any chance of survival, and since they are dead, they can’t create offspring anymore.
If you’d instead would use heavily diluted hand sanitizer, and lessen the degree of dilution over time, the bacteria on your hands will indeed mutate to the point of being resistant to pure hand sanitizer.
[A short video about bacteria mutating to resist antibiotics](https://youtu.be/plVk4NVIUh8)
Soap is an example of an amphiphilic molecule. It has parts that are attracted to water and parts that are attracted to oil. That is how soap removes dirt – the oil-loving part dissolves in the dirts and oils and the water loving-part dissolves in water. As a result the dirts and oils get dissolved into the water and rinse off of you.
Cell membranes, including bacteria, are [bi-lipid](https://en.wikipedia.org/wiki/Lipid_bilayer) which is also amphiphilic, except they are arranged to separate the cell contents from the outside world.
The soap interferes with the bacteria’s cell membrane, damaging it. When the bateria’s cell membrane is damaged enough, that cell can no longer carry on its functions and dies.
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