Problem: The pH of a 0.10 M solution of a weak base is 9.82. What is the Kb for this base?

We are being asked to calculate the K_b for a 0.1 M weak base (HB).

Step 1. Equilibrium reaction:        HB_(aq) + H₂O_(l) ⇌  BH₂⁺_(aq) + OH^-_(aq)

$\boxed{{\mathbf{K}}_{\mathbf{b}}\mathbf{=}\frac{\mathbf{products}}{\mathbf{reactants}}\mathbf{=}\frac{\left[{\mathrm{BH}}_2\right]\left[{\mathrm{OH}}^{-}\right]}{\left[\mathrm{HB}\right]}}$

$$\begin{gathered} {\mathbf{K}}_{\mathbf{b}}\mathbf{=}\frac{\left[\mathbf{x}\right]\left[\mathbf{x}\right]}{\left[\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{-}\mathbf{x}\right]} \\ \boxed{{\mathbf{K}}_{\mathbf{b}}\mathbf{=}\frac{{\mathbf{x}}^{\mathbf{2}}}{\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{-}\mathbf{x}}} \end{gathered}$$