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Let say I have, a dishwashing soap, plenty of cold water (5 °C) and only have 2 L of hot water (60 °C) to clean a t-shirt with oil stains. Should I use the 2 L to wash the t-shirt and then the cold water to rinse the shirt or should I wash the t-shirt with only 1 L of hot water and then use the second liter for a first rinse and then the cold water to finish rinsing?

If the soap has already "captured" the fat molecule (in the micelles), I shouldn't need the hot water to rinse the micelles, right?

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    $\begingroup$ Every sensible washing machine rinses with cold water. $\endgroup$ – Karl Aug 9 '18 at 21:54
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Depends on carry-over & solubility

Generally, when you have two cleaning fluids, one of which is less effective than the other (the "rinse"), you need to use the good solvent to get the contaminant concentration below the solubility limit of the poor solvent, before switching solvents. Otherwise, you will need a heck of a lot of the poor solvent (albeit that in this case it is only cold water.) How do you do that with only 2 L of good solvent (soapy hot water)?

This sort of question is formally studied in both industrial cleaning plants, and decontamination processes. Determining the ideal process is not exactly rocket science but it is fairly complicated.

It depends on the solubility of the contaminant in the washing fluid -- usually, the dynamic solubility, since processes like these can't sit around waiting for a long soak. However, even if the contaminant is perfectly miscible in the cleaning fluid, it still also depends on "carry-over": the fraction of cleaning fluid that remains on or in the cleaned object between rinses.

In the case where the contaminant is miscible and the carry-over fraction is very small, then you actually get the best results (theoretically, at least) from a very large number of very small rinses. But that won't apply in your case: a wet t-shirt is capable of holding quite a large fraction of 2 litres.

For more practical size of problem like yours, the optimal solution is challenging to find analytically, but straightforward to find numerically. For example, you can set up a simple simulation in (say) MS Excel and run the Solver to get lowest residual concentration against an input parameter of "fraction of hot water used for each wash."

Oh, and it simplifies things considerably to know: you get best result when each titre is the same. That is, you don't need to consider using, say, 2/3 of the hot water on a first wash and 1/3 on a second. If you have n washes, just use 1/n each time.

Incidentally, for many practical parameters, the solution is so sensitive to carry-over fraction that often the most economical approach is to minimise carry-over; in your case, give the t-shirt a really thorough wringing out between rinses!

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