thermodynamics - Energy of unmixing

Mixing of two different fluids is associated with an increase of entropy. Conversely, separation of two gases must be associated with a decrease of the entropy of the two fluids.

Is there a minimum requirement imposed by this thermodynamic constraint on the minimum energy needed to separate two fluids?

Answer

This question can be answered, but only with the addition of some extra assumptions. What we know for sure is that the entropy of the universe as a whole cannot decrease, and hence fluids can't be unmixed without increasing the entropy of some other system by an amount equal to $\Delta S_\text{mixing}$. This doesn't immediately tell us anything about work.

However, let us now assume we have to hand a source of mechanical work, and a large heat reservoir at temperature $T$. I'll assume that we'll use the work (somehow) to unmix the fluids, and that any excess heat that gets generated will be dumped in the heat reservoir.

In general the mixed and unmixed states of the fluid system will have different energies. I'll call the difference $\Delta H$, with positive $\Delta H$ meaning that the unmixed state has a higher energy than the mixed one. (I'm guessing that for real miscible fluids this is usually a small positive value, but it needn't necessarily be.)

We'll assume it takes us an amount $W$ of work to unmix the fluids. $W$ has been lost from the work source, and $\Delta H$ has been gained from the fluid system, so the first law says that the energy of the heat bath must increase by $W-\Delta H$.

We can now calculate the total change in entropy. The fluids' entropy has decreased by $\Delta S_\text{mixing}$, and the heat bath's entropy has increased by $(W - \Delta H)/T$, so the total is

$\Delta S_\text{total} = \frac{W - \Delta H}{T} - \Delta S_\text{mixing} \ge 0,$

or

$W \ge T\Delta S_\text{mixing} + \Delta H.$

Changing this to an equality gives you the minimum amount of work required to unmix the fluids. But note that the expression involves $T$, which is the temperature of the heat bath that I assumed to exist. Without the heat bath there would be nowhere for the energy from the work source to go once it's used up.

Also note that $T$ is not necessarily the temperature of the mixed fluids, which could be different from that of the heat bath. I didn't need to make any assumptions about the fluids' temperature in order to work out the above. In practice you'd usually assume the fluids to be in contact with the heat bath, so that the two temperatures are in fact equal. In this case (and only in this case), the minimum work required is equal to the difference in Helmholtz free energy between the mixed and unmixed states.

It's possible for this minimum bound to be negative, meaning that you can actually get work out of the system by unmixing the fluids - but in this case the fluids would be immiscible, so the mixed state would be the unstable one.

Finally, let's make the additional assumption that the fluids are gases and that we have two very special semi-permeable pistons to hand, one at either end of the container. One is permeable to the first gas but completely blocks the second, and the other is permeable to the second gas but blocks the first. We can now very slowly push the two pistons in opposite directions until they meet in the middle, while holding the system in contact with a heat bath. If we do it slowly enough we will have reversibly unmixed the two gases by something similar to reverse osmosis, using an amount of work equal to the minimum $W$ calculated above. (I got that last idea from this brilliant paper on the entropy of mixing by Edwin Jaynes: http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf)