# When a general base associates in water, with K_b "<<" 10^(-5), what is the algebraic expression for x as a function of K_b and the equilibrium concentration of the base?

Mar 9, 2017

As usual, write out the reaction.

${\text{B"(aq) + "H"_2"O"(l) rightleftharpoons "OH"^(-)(aq) + "BH}}^{+} \left(a q\right)$

By constructing an ICE table, you would obtain:

${K}_{b} = \left(\left[\text{OH"^(-)]["BH"^(+)])/(["B"]) = x^2/(["B}\right] - x\right)$

As ${K}_{b}$ $\text{<<}$ ${10}^{- 5}$, we can of course use the small $x$ approximation, and thus, $\left[\text{B"] - x ~~ ["B}\right]$. Thus:

$\textcolor{b l u e}{x \approx \sqrt{{K}_{b} \left[\text{B}\right]}}$

And you can always use this expression for a weak base equilibrium in which ${K}_{b}$ is on the order of ${10}^{- 5}$ or less. What does $x$ represent, and what is the answer?

Did you check your answer by trying to see if the $\text{pH}$ was more than $7$?