Question

Why do the concentrations of Pb 2+ and SO4 2- ions in a saturated solution of PbSO4 in contact with solid PbSO4 remain the same when the solution evaporates to half of its original volume?

it has to do with equilibrium.

Answer 1

PbSO4(s) ==> Pb^2+(aq) + SO4^2-(aq)
Ksp = (Pb^2+)(SO4^2-)
As the solvent evaporates, Ksp must stay the same; therefore, PbSO4 must come out of solution to keep the ion product the same.