A Diabolically Difficult Chemistry Puzzle

Originally posted 2022-07-24

Obligatory disclaimer: all opinions are mine and not of my employer

In 2012, I was a mentor at the USNCO Study Camp, which prepares and selects the top students in the U.S. to send to the International Chemistry Olympiad. I wrote a question for the final exam which only 2 of the 20 students completed. Both students admitted to brute-forcing the system of equation in 5 unknowns. I found it hilarious at the time that I’d stumped the entire camp with a seemingly trivial problem. The students were not as amused! Without further ado:

Balance the following equation.

\[\ce{BH4- + BF3 -> B11H14- + BF4- + H2(g)}\]

Source: https://doi.org/10.1016/S0022-328X(03)00460-1

This 11-boron, 14-hydrogen species has \(C_{5v}\) symmetry (dodecahedron with missing vertex), and its parent species \(\ce{B11H11^2-}\) is apparently fluxional, rapidly interconverting with no one boron identifiable. If you’re wondering how I stumbled across such an obscure reaction, I decided at some point that boron was probably a simple enough element that I could learn everything there was to know about boron if I’d really tried. Boy, was I wrong! It turns out a lot of research went into obscure borohydride species as potential energy-dense rocket fuels during the Cold War.

A hint and solution for this equation comes after the spoiler break.

v v v v v v

Hint

This is a redox equation. Find the two half-reactions.

v v v v v v

Solution

The borohydride and fluoroborate species all have boron in their (+3) oxidation state, while the B11 species has a 15/11 oxidation state, which is about (+1.36) on average. The hydrogen, on the other hand, goes from hydride at the (-1) oxidation state to hydrogen gas at the (0) oxidation state. We’ll have to find the right ratios that lead to net electron transfer balance.

In a traditional redox reaction, typically what happens is that you first insert electrons to form the correct half-reaction, and then use \(\ce{H2O}\), \(\ce{H+}\) or \(\ce{OH-}\) species to balance oxygens, hydrogens, and charges. As a simple example, for the reduction of Cr(+6) to Cr(+3):

\[\ce{CrO4^2- -> Cr^3+}\]

\[\ce{CrO4^2- + 3e- -> Cr^3+}\]

\[\ce{CrO4^2- + 3e- -> Cr^3+ + 4H2O}\]

\[\ce{CrO4^2- + 3e- + 8H+ -> Cr^3+ + 4H2O}\]

How do these steps apply to this problem? This isn’t an aqueous reaction - borohydrides are pyrophoric in air! Instead of using \(\ce{H+}\) to balance our charges, we’ll use \(\ce{H-}\).

\[\ce{11BH4- + 20e- -> B11H14- + 30H-}\]

\[\ce{2H- -> 2e- + H2}\]

Balanced reaction:

\[\ce{11BH4- -> B11H14- + 10H- + 10H2}\]

Free hydride is not in the final equation - this is where the fluoroborate species come in. You could see the role of \(\ce{BF3}\) as a hydride sponge.

\[\ce{4BF3 + 4H- -> 3BF4- + BH4-}\]

Balancing and cancelling the hydrides, the final balanced equation is

\[\ce{17BH4- + 20BF3 -> 2B11H14- + 15BF4- + 20H2}\]