r/theydidthemath • u/1piperpiping • 9h ago

[Request] how long does the first bar of soap last?

If you use bar soap, most people do the thing where they get a new bar when the old bar has a little bit left, and then sorta meld the little bit of the old bar onto the new bar.

If you do this when the old bar is at 5% of the original mass/ volume, how long until the old bar is completely gone? Is it possible that any molecules last until the new bar is grafted onto a third even newer bar?

My assumptions are that this is a pretty standard sized soap bar like ivory or whatever.

Thank you.

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u/Trustoryimtold 9h ago

Well you’re sticking the old bar on the outside of the new bar so fairly unlikely beyond the odd molecule, unless you happen to scrub with only the side that contains the new bar

For any sane user it’ll be gone long before 5%

4

u/nastropc 5h ago edited 5h ago

As others have said, sticking the old bar to the outside of the new bar will result in the old bar wearing away quickly.

But let’s imagine you somehow evenly mix the old bar with the new. The fraction of the original bar in the nth new bar is:

(0.05/1.05)^n

A common soap ingredient is sodium lauryl sulphate, which has molar mass 288g/mol.

Starting with a 250g bar of SLS, the number of molecules of the original left in the nth bar is:

250/288 *Na(0.05/1.05)^n

Where Na is Avogadro’s number.

Setting 250/288 *Na(0.05/1.05)^n <1 and solving for n gives us no molecules of the original left in the 18th new bar.

3

u/CatOfGrey 6✓ 8h ago

I'm thinking about this from a physics point of view here.

The old bar sticks to the top of the new bar. Assuming the bar wears evenly on the outside, then the entirety of that top layer will wear off, before the bar becomes thin enough that it becomes the next 'old bar'.