Charge Balance and power law?

I've come across a statement in a paper* that's poorly explained, and I can't work out the reasoning behind it. I suspect this is embarrassingly basic, but could anyone please explain the theory behind the following (from the bottom of the second paragraph of the discussion)?

For a reaction involving B(OH)4- and CO32-, a B(OH)4- can balance the charge of only 1/2 a CO32-, thus a dependence [of the B/Ca ratio in synthetic aragonite] on [B(OH)4-]/[CO32-]0.5 would be expected.

I'm unsure why a power law applies here.

* Holcomb, M.; DeCarlo, T. M.; Gaetani, G. A.; McCulloch, M. Factors affecting B/Ca ratios in synthetic aragonite. Chem. Geol. 2016, 437, 67–76. DOI: 10.1016/j.chemgeo.2016.05.007.

• Your quoted statement would make sense in discussing the dependence of the equilibrium constant of a reaction involving $2\ce{B(OH)_4^-}$ on one side and $\ce{CO_3^{2-}}$ on the other, but I'm unsure if this is relevant. – a-cyclohexane-molecule Jul 17 '17 at 21:55
• Thanks @a-cyclohexane-molecule. Could you point me towards some reading so I can understand why this is this case? – oscarbranson Jul 17 '17 at 23:58
• I don't have a good resource on hand, but a Google search regarding equilibrium constant should be productive. Essentially, we define the equilibrium constant for the reaction $\ce{aA + bB \to cC + dD}$ as $$K = \frac{[A]^a[B]^b}{[C]^c[D]^d}.$$ Specializing to this reaction indeed shows the desired proportionality. Again, do note that I'm not certain that this is what the authors of the paper are getting at. – a-cyclohexane-molecule Jul 18 '17 at 0:05
• I think you're exactly right. They discuss the stoichiometry of the reaction as $B(OH)_4 + 0.5 CO_3^{2-}$, so this makes sense. Thanks for the pointer! – oscarbranson Jul 18 '17 at 4:21