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Oil and water do not mix when detergent is added. What really happens is that (in the usual case) the detergent (which has "oil-like" and "water-like" parts to each molecule) causes the oil to disperse through the water in tiny droplets which have the detergent on the surface of the droplets, making them much more stable than tiny droplets of oil would be in water.

The detergent is usually charged, which makes the droplets repel each other, preventing the tiny oil droplets from reuniting into larger droplets, which is what happens if you vigorously shake an oil water mixture, like salad dressing - that's why oil and vinegar salad dressing has to be used right after preparation.

Oil, water, and detergent is still not a true mixture, since the oil and the water are not really "mixed" in the true sense of the word, but they appear mixed to the eye since the oil particles are too small to be seen with the naked eye.

There is one more requirement for this apparent "mixing" after adding detergent: there has to be a LOT more water or a LOT more oil in the mixture. You can't make an apparent mixture of a 50:50 blend of oil and water by adding detergent, for example.

This means that you can have tiny droplets of water surrounded by detergent in a large amount of oil as well as the more usual case outlined above. That is called a "reverse emulsion" while the more common case above is a "simple emulsion."

It's also the reason why detergent is effective to clean clothes. It "emulsifies" the oils (which soiled clothes are contaminated with) allowing them to be flushed away.

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Q: How detergent changes an oil water suspension into an emulsion?
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