If I start with pure water and add $\ce{LiOH}$ the conductivity of solution increases. However, if I then dissolve boric acid in the same solution the conductivity falls. I would have thought adding more cations or anions to a system would always increase conductivity, but in this case it does not. Why is this?
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TL;DR: The addition of boric acid effectively switches the conducting anionic solute from hydroxide to borate, which has a lesser contribution to the conductivity due to its larger size and corresponding lower diffusivity. Most solute systems will not behave this way.
In sufficiently dilute solution, the contribution $\kappa_i$ of each dissolved ion to the overall conductivity is proportional to the concentration, diffusivity, and squared charge of that ion:
$$ \kappa_i \propto z_i^2D_iC_i $$
And, the total conductivity is just the sum of these contributions:
$$ \kappa = \sum_i{\kappa_i} $$
In this case, you're starting with $\ce{LiOH}$, which dissociates more or less completely into aqueous $\ce{Li^+}$ and $\ce{OH^-}$, and the conductivity rises as you add more of it. Once you start adding boric acid, though, you're no longer just dissociating ions into the solution.
Boric acid is a funny creature. Even though it looks like it might be a triprotic Arrhenius acid if we were to write it as $\ce{H3BO3}$, in reality it's a monoprotic Brønsted acid that either exchanges a hydroxide ion with the water solvent:
$$ \ce{B(OH)3 + 2H2O <=> B(OH)4^- + H3O^+} $$
or, at sufficiently high $\mathrm{pH}$ $(\mathrm pK_\mathrm a = 9.2)$, it just takes up a hydroxide ion from solution:
$$ \ce{B(OH)3 + OH^- <=> B(OH)4^-} $$
So, when added to a solution of $\ce{LiOH}$, boric acid just substitutes dissolved $\ce{OH^-}$ for dissolved $\ce{B(OH)4^-}$, with no net addition of anions to the solution! Thus, the $C_i$ contribution to $\kappa$ doesn't really change much. As well, both $\ce{OH-}$ and $\ce{B(OH)4^-}$ are monovalent anions, so there's no effect on any of the $z_i^2$ terms either.
The overall decrease in $\kappa$ comes from the change in diffusivity that results from the swap of hydroxide for borate. The diffusivity of an anion is roughly proportional to its ionic volume—bigger ions experience more "drag" from the surrounding solvent molecules, so they diffuse (move around) more slowly. I couldn't find an ionic radius for borate on a quick search, but just from their compositions, it seems sensible to assume that the borate anion is considerably larger than the hydroxide anion, and thus its diffusivity should be correspondingly less.
