A solution containing hydrogen peroxide with a mass fraction xp0 (kg H2 O2/kg solution) is added to a storage tank at a steady rate m0 (kg/h). During this process the liquid level reaches a corroded spot in the tank wall and a leak develops. As the filling continues, the leak rate m1 (kg/h) becomes progressively worse. Moreover, once it is in the tank the peroxide begins to decompose at a rate rd (kg/h) = kMp where Mp (kg) is the mass of peroxide in the tank. The tank contents are well mixed, so that the peroxide concentration is the same at all positions. At a time t = 0 the liquid level reaches the corroded spot. Let M0 and Mp0 be the total liquid mass and mass of peroxide, respectively, in the tank at that moment, and let M(r) be the total mass of liquid in the tank at any time thereafter.

(a) Show that the leakage rate of hydrogen peroxide at any time is m1 Mp/M.

(b) Write differential balances on the total tank contents and on the peroxide in the tank, and pro vide initial conditions. Your solution should involve only the quantities m0, m1, xp0, k, M. M0, Mp, Mp0, and t.