Preface: Buffer solution (acid-base buffer). I am provided with a weak base, which I will designate B. $\mathrm{p}K_\mathrm{a}$ for $\ce{B}$'s conjugate acid, which I will designate $\ce{BH}$, is $8.1$, and its mole weight (sic) is $121.1$. I'm assuming the latter is the molar mass, though I don't know how that helps me solve this problem.
Problem: $\pu{0.1 M}$ ($\pu{mol/L}$) of $\ce{B}$ was dissolved in $\pu{1.0 L}$ water, then titrated to $\mathrm{pH}$ $8.1$ and $7.5$. I am to assume no change in volume.
The last statement sounds like an odd assumption to me, how do you titrate something without [significantly] altering the volume? I always assumed titration is only done by adding a liquid to another liquid, is this not correct?
I am to find the concentration of $\ce{BH}$ at $\mathrm{pH}$ $8.1$ and $7.5$.
This question looked really easy to me at the first glance, but after a few hours I realized I had no idea on how to go about this.
I've tried calculating the equilibrium concentrations and going from there. This just ends up in $x \neq x$ in my calculations (where $x$ is the concentration of some substance), which obviously doesn't hold.
I've used the acid dissociation-, base dissociation- and Henderson–Hasselbalch equations, but nothing seems to do the trick. I might not be the brightest person to have grazed the surface of the planet, but I'm confident that I've got a firm grasp on elementary arithmetic.
The answer is provided, and should be $\pu{0.05 M}$ at $\mathrm{pH} = 8.1$.
I would highly appreciate somebody showing or explaining the thought process to me.