# Amount of energy to separate Gases - relationship to concentration

I want to understand the efficiencies of separating mixed gases, and for that I want to understand the thermodynamic limit case. Looking at the wikipedia page for entropy of mixing, I find the following formula: $$\Delta S_{mix}=-nR \sum_{i}^rx_i\ln x_i$$ with $r$ being the number of phases, $x_i$

With gases, the Gibbs free energy is entropy x temperature, no enthalpy.

Now, the change of entropy with mixing/separating is highest for a two phase mixture with both components at equal concentrations. Now I'm confused: I know that removing small traces from a mixture requires more effort/energy per mass/mole than a bulk. Thermodynamics tell me that this is not necessarily so.

While the specific energy is higher to remove trace amounts, the Gibbs energy still poses an upper bound to the energy necessary to separate gases assuming a 100% efficient process.

Now I assume that the relatively larger effort for trace amounts stems from engineering difficulties (PSA needs higher pressure, centrifuges spin faster, more passes through membranes...) but there is no law of thermodynamics stating: Thou shalt not easily remove trace amounts from mixtures of gases.

Or did I miss something?

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I'll close as I don't want to abuse SE as a notebook on my ongoing research into the question ... – mart Jan 30 '13 at 9:27
It seems OK to me. This post is useful to others, it's fine if left open (and who knows, you may get a much better answer that improves your understanding) :) – Manishearth Jan 30 '13 at 11:17
Great question! I hope you don't mind me using it in my next paper with the answer and some extra notes. – Mew Feb 8 '13 at 15:04
of course not. even if it's only tangential, maybe you could share this paper? – mart Feb 11 '13 at 9:48
@mew do you want to share the paper? – mart Mar 2 '15 at 12:35