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.

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.

To understand why the concentrations of Pb2+ and SO42- ions remain the same when a saturated solution of PbSO4 evaporates to half of its original volume, we need to consider the principle of equilibrium and the solubility product constant.

When a solid such as PbSO4 is added to water, it can dissolve and dissociate into its component ions, Pb2+ and SO42-. However, as the concentration of these ions in the solution increases, some of them may recombine to form solid PbSO4 through a reverse reaction called precipitation. Eventually, a dynamic equilibrium is established between the dissolved and solid forms of PbSO4.

The equilibrium constant for this reaction is called the solubility product constant (Ksp). For PbSO4, the equilibrium expression is:

PbSO4 ⇌ Pb2+ + SO42-

The Ksp is equal to the product of the concentrations of the Pb2+ and SO42- ions at equilibrium. Mathematically, we can express it as:

Ksp = [Pb2+][SO42-]

Since Ksp is a constant at a given temperature, it means that the product of the concentrations of the Pb2+ and SO42- ions must remain constant as well. This relates to Le Chatelier's principle, which states that if a system at equilibrium is subjected to a stress, it will undergo a change that counteracts the stress to reestablish equilibrium.

Now, let's consider what happens when the saturated solution of PbSO4 is evaporated to half of its original volume. As the solvent (water) evaporates, the volume decreases, but the same amount of solute (PbSO4) is present. This increases the concentration of both Pb2+ and SO42- ions in the remaining solution.

According to Le Chatelier's principle, this increase in concentration will stress the equilibrium and cause the reverse reaction (precipitation) to occur in order to reduce the concentration of Pb2+ and SO42- ions. More solid PbSO4 will be formed until the system reestablishes equilibrium. As a result, the concentrations of the Pb2+ and SO42- ions will remain the same, even though the solution volume has decreased.

In summary, the concentrations of the Pb2+ and SO42- ions in a saturated solution of PbSO4 remain constant when the solution evaporates to half of its original volume due to the equilibrium established between the dissolved and solid forms of PbSO4. The solubility product constant (Ksp) ensures that the product of the ion concentrations remains constant, and Le Chatelier's principle drives the reverse reaction (precipitation) to maintain equilibrium and counteract the increase in ion concentration caused by the decrease in solution volume.