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In the lab, I'm working in over summer, there is a set of solvent stills.

Most of them use $\ce{CaH2}$ as a drying agent, however ether and THF use Na and K respectively in the presence of benzophenone.

What I don't understand is why the ether still uses Na, whilst the THF uses K. Equally, I'm not sure what the purpose of the benzophenone is — it makes the stills blue, but none of the other solvents (DCM, toluene, triethylamine) have this property.

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Sodium and potassium are both drying agents. Being alkali metals they react (often vigorously) with water.

Sodium wire is often used in diethyl ether stills, whilst potassium is used in tetrahydrofuran stills. In general, potassium is more reactive, and hence a better drying agent.

The reason different desiccants are used is due to temperature. THF boils at a higher temperature, meaning a lump of potassium metal can be used in the still ( the lump of K melts once the THF reaches reflux ). Diethyl ether boils much lower, meaning potassium would not melt, for this reason sodium is often used in the form of a wire (giving a much larger surface area).

In respect to the use of benzophenone (or other indicators such), it generally indicates the presence (or absence) of oxygen in the solution. In the past, it was thought that the purple (blue is actually not great) colour was related to the moisture content, however we now know that the colour is related to the oxygen content. For this reason, using the indicator allows us to determine how oxygen-free the solvent is. A purple THF/ether still in which metal is still present indicates that the solvent is both dry and oxygen free,

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    $\begingroup$ But why are other solvents not distilled using sodium/potassium $\endgroup$ – James Dealon Jun 26 '17 at 0:19
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    $\begingroup$ Sodium/potassium are quite reactive, they cant be used with certain solvents (you mention Et3N for example) as they would simply react $\endgroup$ – NotEvans. Jun 26 '17 at 0:30
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    $\begingroup$ And never, never use halogenated solvents with sodium/potassium. Best case scenario Wurtz, worst case total annihilation. $\endgroup$ – Martin - マーチン Jun 26 '17 at 3:37
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    $\begingroup$ I would think that Na/K also serve to remove any etherperoxides already formed, though, ideally, you would not have any in the first place. $\endgroup$ – TAR86 Jun 26 '17 at 5:11
  • $\begingroup$ Toluene does not need a drying agent as it is dried by azeotropic distillation $\endgroup$ – Waylander Jun 26 '17 at 11:19
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To add to @NotEvans' answer, while I am not able to trace back the original "benzophenone study" he refers to, a (sort of) reprint may be found here, including a small range of molecules that may be generated by extended refluxing over benzophenone.

It may interesting for the OP to see that some of the more frequently used solvents may be dried by pressing across columns packed of $\ce{Al2O3}$, a concept popularised by Grubbs et al. (Organometallics, doi: 10.1021/om9503712). Beside suggested modifications (like here), some groups have it installed instead of the distillations:

enter image description here

(source)

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