An interesting theoretical calculation is the rate of entropy production due simply to mixing of smokestack gasses (high partial pressure of CO2) with atmosphere (low partial pressure of CO2). In principle this is wasted exergy -- lots of it. One could imagine some combination of semipermeable membranes and turbines to extract power from the mixing process (your mention of membrane separation reminded me of this). To be clear this is sci-fi though at this point.

(I do believe, however, that there exist methods to extract energy from the mixing of salt and fresh water.)

Likewise, this same idea is a strong argument for capturing CO2 from smokestacks instead of from atmosphere: Fundamentally, it takes energy to undo mixing. And if you want to capture the CO2 eventually anyway, then better to just do that than invent a theoretical "heat engine" driven by the difference in CO2 partial pressure, to get energy that you'll just need to spend somewhere else later (and then some) to get the CO2 back out of atmosphere...

...Actually this phrase "heat engine" now makes me think that maybe this isn't even that hard (in theory) using cold temperatures and phase changes to make dry ice. If it's possible to efficiently cycle the gas through these large temperature swings at all, it must require some kind of heat exchangers and regeneration between stages... (Surely if this were realistic people would have figured out how to do it by now, but it's an interesting thought/design experiment in thermodynamics...)