# 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?

• I'll close as I don't want to abuse SE as a notebook on my ongoing research into the question ...
– mart
Commented Jan 30, 2013 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) :) Commented Jan 30, 2013 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. Commented Feb 8, 2013 at 15:04
• of course not. even if it's only tangential, maybe you could share this paper?
– mart
Commented Feb 11, 2013 at 9:48
• @mew do you want to share the paper?
– mart
Commented Mar 2, 2015 at 12:35