Four proteins are dissolved in solution and should be fractionated. The constants in the Cohn equation

for the proteins are given in the Table below long with the concentration of each protein in solution.

Assume that the Cohn equation constants are valid when other proteins are present.

$($a$)$ Determine the maximum percentage recovery of protein $4$ that can be obtained if it is desired to

obtain a precipitate containing $98\%$ protein $4.$

$($b$)$ Determine the maximum percentage recovery of protein $3$ that can be obtained if it is desired to

obtain a mother liquor containing $97\%$ protein $3.$