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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).

untreated_paramediensis_karnik
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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.

knzhou
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    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. – BioPhysicist Aug 04 '18 at 14:09
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    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. – rob Aug 04 '18 at 14:10
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    @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. – rob Aug 04 '18 at 14:12
  • @AaronStevens Good point, I never explicitly stated that, edited. – knzhou Aug 04 '18 at 14:15
  • @rob Another good point, I didn't state that very clearly! Edited. – knzhou Aug 04 '18 at 14:18
  • @rob yes of course :) I just wanted to make sure it was explicitly stated in the answer since this is what the OP was asking about. – BioPhysicist Aug 04 '18 at 15:09
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    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. – BioPhysicist Aug 04 '18 at 15:25
  • @AaronStevens Absolutely, that can be a contribution. In general calculating the entropy of a bucket of water is very hard. In practice somebody else measures it for us and we just go look it up in a chemistry book. But in addition to this, there is the effect that the whole system heats up; the process is exothermic. You have more negative energy due to chemical bonds, which means more positive energy everywhere else by energy conservation. – knzhou Aug 04 '18 at 15:30
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    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. – Walter Mitty Aug 04 '18 at 18:38
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    Although the rest of the answer is excellent, I don't think releasing energy to the surroundings is correct. If that were the case, then salad dressing in an insulated container ought to not separate (or do so very slowly), which is certainly not the case. – DrSheldon Aug 04 '18 at 19:12
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    @AaronStevens: I think you have it spot-on. It's about the number of hydrogen bonds the water molecules can make. Entropy favors water molecules completely surrounded by other water molecules, as this maximizes the number of hydrogen bonds. Water molecules at the surface of oil have fewer hydrogen bonds, so entropy would favor minimizing the number of water molecules at the surface of oil. Hence, the most thermodynamically stable arrangement is isolating the water into one mass. – DrSheldon Aug 04 '18 at 19:22
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    @WalterMitty No, the two are actually exactly the same. Both mean the amount of bits of information you are missing about the system. The entropy in thermodynamics properly includes all information. – knzhou Aug 04 '18 at 19:24
  • @DrSheldon It's fine for an exothermic reaction to happen in isolation; the system just heats up. The energy explanation does suggest that at high temperatures the un-mixing should go slower, but I think the water boils before the rate change becomes important. – rob Aug 04 '18 at 20:08
  • @AaronStevens: You inspired me to write a little Java program that simulates the hydrogen bonding. When the oil and water are fully mixed, 201 bonds form; partially mixed, 212 bonds; and separated, 233 bonds form. Because this question is marked duplicate, I can't post an answer here. However, I did post an answer WITH PICTURES in the original question. Please pop over and take a look! – DrSheldon Aug 04 '18 at 23:14
  • This answer is extremely misleading. It seems to imply that the entropy of the dressing is higher in the unmixed state ("entropy has almost nothing to do with the apparent order or disorder you can see with your naked eye") which is of course incorrect. The entropy of the mixed dressing is higher, as one would expect. However, the internal energy of the mixture is also higher, and the mixture is unstable. As the mixture separates, its entropy decreases and but it also releases energy as heat, so there is no 2nd law violation. DrSheldon is wrong that the unmixed state is higher entropy. – bigjoec May 17 '23 at 04:46