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By the second law of thermodynamics, entropy tends to increase when the system is let on itself. And if entropy is a measure of disorder, how come mixing oil in water and letting the system reach equilibrium, ends up with the oil and water well separated? I see no disorder whatsoever, while in reality the entropy increased compared to the initial state (oil and water seemingly well mixed by e.g. shaking the container).

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This is because entropy has almost nothing to do with the apparent order or disorder you can see with your naked eye. That's just a pop science simplification.

Compare the entropy of a dictionary and an identically sized book full of random gibberish. You might think the latter has a higher entropy, because the content is disordered. But the entropy of the words in the book is not even a million millionth of the total entropy, which overwhelmingly comes from thermal motion of the molecules in the paper. (Compare the number of characters in the book to the number of molecules, on the order of $10^{23}$.) If you hold the dictionary for even a second, the heat from your hand will make the entropy of the dictionary higher.

In the case of oil and water, it is energetically favorable for the oil to be separate from the water, because these molecules bind more strongly to themselves than to each other. The extra energy released is now available to thermal motion of the oil and water molecules, or to the surrounding air molecules, increasing the entropy of the universe. This overwhelms the decrease of entropy associated with clumping the oil together.

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    $\begingroup$ So you are saying that the final state has less entropy than the mixed state? If this is so, then entropy had to increase somewhere else. Where is this entropy increase? This is what the OP is asking about. I am not saying you're wrong, I just think this should be explicitly stated since this is the heart of the question. $\endgroup$ Aug 4, 2018 at 14:09
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    $\begingroup$ Nice answer. Note that it's mostly irrelevant that the oil and the water have different densities ("energetically favorable for the oil to be on top"), because water will happily dissolve low-density ethanol or high-density table salt. The energetic favorability comes from the water-water interactions and/or oil-oil interactions being stronger than water-oil interactions. When the mixture is stirred, energy is released when the water molecules find each other. $\endgroup$
    – rob
    Aug 4, 2018 at 14:10
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    $\begingroup$ @AaronStevens The entropy of the universe increases when the water and oil un-mix, because the energy released by the un-mixing has many many ways to interact with the environment. $\endgroup$
    – rob
    Aug 4, 2018 at 14:12
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    $\begingroup$ Can't you also say there is more entropy in the un-mixed case because the polar water molecules have more configurations to make hydrogen bonds with their neighbors? Your new addition makes it seem like when oil and water separate the entire system heats up, and I am not sure this is the case. $\endgroup$ Aug 4, 2018 at 15:25
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    $\begingroup$ The words in the dictionary have entropy in the Information Theory sense of the word. The motion of the molecules have entropy in the Physics sense of the word. While these two meanings are intimately connected, they shouldn't be confused. $\endgroup$ Aug 4, 2018 at 18:38

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