Inverse Gas Chromatography Derivation

Introduction

Chromatography is something mostly chemists have familiarity with. So what is it? Chromatography can be cheaply described as the practice of spatially separating mixtures of compounds by their prolonged exposure to a medium, or stationary phase, with which chemical interactions occur. The general idea is that a chemical mixture is allowed to flow through some medium with which some of its components will have more or less affinity too. As the mixture travels through the column, the compounds which spend the least time adsorbed to the stationary phase will exit, or "elute", from the column first. Those which have a higher affinity will take longer to elute. So, that's the jist - we create, buy, or modify, a stationary phase, physical conditions, and solvent mixtures to separate chemicals for purification, or subsequent analysis.

To a lot of chemists, what I've stated is common knowledge. Albeit - explained very briefly and crudely. But, how many people have heard of inverse chromatography? In my experience, not a whole lot...

Inverse chromatography is the opposite of normal chromatography. Instead of separating a mixture, we elute pure "probe" compounds through/over a material which we intend to study. For purposes of this blog post we will concern ourselves principally with inverse gas chromatography because gasses are much easier to deal with then say - liquids. Yes - so our stationary phase is now our analyte, and our probes are typically alkanes of fixed length.

Why would you want to do this though? It doesn't purify anything? Well, if we're clever we can determine thermodynamic information about the material we intend to study! Why alkanes? Well, you'll need to read on to know, but basically, Doris and Grey concocted a clever scheme of indirectly measuring the dispersive energy of the analytes surface using this technique.

Surface chemistry sure is neat but its typically challenging (cough expensive) to study. This is bad news for people in coatings, paints, pharmaceuticals, etc. But inverse gas chromatography is a kind of playful and relatively cheap means to acquire some surface information about an analyte.

Unfortunately when I was reading about it years ago, no one was explaining in detail how in principle it worked, or how it was derived. Intuition is easy, but questions like "why infinitely dilute/henrys law conditions?", "what if the temperature is changed?", or "why alkanes as probes?" popped into my curious little head. If you're in the same boat - no worries, I found some school notes where I wrote down what seems to be a full derivation for how it works. If that interests you - read on, if not, have a good one.

Derivation

The Abstract Part

Interfacial internal energy between the probe and stationary phase can be written as

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Now, consider interfaces have no volume, therefore

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after integration we can state,

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Which yields the following total differential

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Which can be equated with the first equation to give,

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Therefore,

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Now we can state our system is isothermal, thus,

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and,

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For simplicity sake let us presume there is 1 analyte and 1 probe(i = 2)

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Now let us define surface excess by the number of components over the surface area,

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Therefore...

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One component in the system is the surface, and thus cannot have surface excess,

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Remembering our General Chemistry course we can define Gibbs energy in terms of the fugacity of the gas at isothermal conditions, recall,

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And on substitution with the above equation yields,

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remembering that: inline_formula not implemented

thus,

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which on rearrangement yields,

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Now remember the ideal gas law,

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by substituting the above we can arrive at the following

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Now, recall that surface pressure under Henry's law conditions (infinite dilution) is

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therefore,

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Considering the free energy of the equilibrium at the surface of the material,

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and finally,

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The Concrete Part

Now we have an expression with 3 unknowns. How could we possibly deal with that? Dorris and Gray had a great idea. If they passed probe molecules over the surface of their analyte, and each experiment added one more methyl moiety then the following will hold:

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it follows that....

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and,

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and we can abstract away the pressures of the probes to a constant c,

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And further by remembering, inline_formula not implemented

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The mathematics now describe an experiment: Calculate the retention volumes of various n-Alkanes differing by only one chain length.

Remember the equation for retention volume is the following

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So by plotting the retention volumes vs the number of carbons in a chain we may obtain the dispersive free energy associated with adding one methylene unit to the probe which interacts with our analytes surface. That is - the slope of the line represents the free energy per unit chain length.

Then, we can use the Fowkes Relation to estimate the work of adhesion to determine the dispersive surface energy of the analyte as follows. I also derived this but it seems superflouos here,

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where inline_formula not implementedcan be referenced via equations, tables, etc. So, there you have it. An experiment where retention volume, and number of carbons of a probe can find you (with some constants) the surface dispersive energy of your analyte. As promised?

Apparently I used references when I concocted this many years ago, so those are below,

References

Physics and Chemistry of Interfaces Second Revised and Enlarged Edition. Hans-Jurgen Butt, Karlheinz Graf, Michael Kappl. 2006. Wiley-VCH Verlag GmbH & Co. KGaA. Weinheim. ISBN: 3-527-40629-8

Mohammadi-Jam, S; Waters, K. Advances in Colloid and Interface Science. 2014, 212, 21–44.

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