Barium sulfide is produced by reacting barium sulfate ore with coal. The result is barium black ash, which is \(\mathrm{BaS}\) plus insoluble solids. Since \(\mathrm{BaS}\) is soluble in water, it can be washed out with water. In thickeners, insoluble solids in underflow typically carry with them \(1.5 \mathrm{~kg}\) liquid per \(\mathrm{kg}\) insoluble solids. We want to process \(350 \mathrm{~kg} / \mathrm{h}\) of insoluble solids plus its associated underflow liquid, \(1.5 \mathrm{~kg}\) liquid \(/ \mathrm{kg}\) solids, containing \(20.0 \mathrm{wt} \% \mathrm{BaS}\). Use a countercurrent system with \(2075 \mathrm{~kg} / \mathrm{h}\) of water as solvent. The entering water is pure. The outlet underflow liquid is \(0.001 \mathrm{wt} \% \mathrm{BaS}\). Find:

a. The BaS mass fraction in the exiting overflow liquid.

b. The number of equilibrium stages required.