Do liquids have a higher entropy before or after being mixed together?

Question

Suppose I mix oil and water by stirring the liquid, then which state has the higher entropy, the mixed or the unmixed state?

Is this similar to:
When mixing coffee and milk, which state has the higher entropy, the mixed or the unmixed state?

Answer 1

The entropy of a mole of a mixture is $$S=RT \sum_i x_i \log(x_i)$$ where $R$ is the universal gas constant, $T$ is the temperature and $x_i$ is the molar fraction of substance $i$; $\sum_i x_i =1$. So generally a mixture will have a higher entropy (and it will take energy to purify it into the component substances).

However, the change in free energy of a system $\Delta H = \Delta U-T\Delta S$ matters for whether a change can happen spontaneously: this happens if $\Delta H<0$ ($U$ is the internal energy). Normally a big increase in entropy ($\Delta S \gg 0$) is enough to cause this and can even overcome an increase in internal energy (as is the case of dissolving ammonium nitrate in water, which is endothermic and cools the mixture).

But if the increase in internal energy would be big enough then even a big entropy gain may not be enough to outweigh it. In the case of oil and water there is a surface energy along the interface between the substances that makes $\Delta U$ big and proportional to the interface surface area - hence converting oil and water into an even mixture is not favoured. Milk and water have far less surface energy so they mix easily. Add a surfactant to the oil and it can also be turned into an emulsion.

Answer 2

Generally, the entropy will go up when you mix materials. Each bit can now be in more places, so there are more possible states of the final product. In the example of the coffee and milk, that mixture is complete and irreversible, so the entropy has gone up.

But there are lots of special cases:

• If the materials are the sane, I.e. water and water, entropy doesn’t change: the situation hasn’t changed. Unless you also stirred vigorously, in which case you added energy making more motion states available, so entropy went up.

• if the materials react with themselves or each other, that can increase or decrease thermal energy, which can cause the resulting entropy to increase a lot (combustion, I.e. hypergolic rocket fuels) or almost not at all (oil and water, which self-separate).

Edit: Given the back-and-forth in the comments, here's an image to clarify the point:

From right to left are four initial states at the top, and final states at the bottom. Imagine there's initially a barrier between the blue and yellow liquids, which is then removed and they're allowed to mix.

In the left-most case, they mix entirely; that's an irreversible process, and clearly entropy goes up.

In the right-most case, which is meant to represent oil on top and water on the bottom, nothing changes. The barrier can be put back; the situation is completely reversible. There need be no increase of entropy in that case even when the materials are mixed. (The two states in the middle are intermediate, in that they're not obviously reversible but oil-water separate still greatly reduced the entropy gain over the left-hand cast; that kind of situation has a greatly reduced entropy rise over the complete-mixing case)

It's been argued in the comments that when you mechanically mix something, you must add energy, and that adds entropy. That can happen, but it need not: If the components are in a constant-temperature environment, for example, the entropy of the mixture (remember the original question is asking about the entropy of the liquids, not of the universe) is unchanged by adding mechanical energy, because the energy of the mixture is unchanged after that mechanical energy turns to heat and exits in the the environment.

Bottom line: Most mixing greatly increases entropy because it's irreversible. The more irreversible, the more the entropy increase.