Problem 1 (Filtration): You are performing ultrafiltration of an enzyme solution at constant volume to remove low molecular weight species. Constant volume is maintained by the addition of media to the feed to balance the liquid volume lost through the membrane. The flow tubes for this system are 0.1 cm in diameter and 1 m long. The protein has a diffusion coefficient of 10^-6 cm²/s, the solution has the viscosity and density of water at standard temperature and pressure. The system operates at a bulk stream velocity of 100 cm/s. (A) At this velocity, determine the polarization modulus for a transmembrane flux of 50 L×m^-2×h^-1. (B) Estimate the polarization modulus if an enzyme with twice the molecular weight is being purified.

Problem 2 (Crystallization): You have an antibiotic dissolved in water at 2 M that you would like to crystallize (in bulk, not at an interface) at 20°C. At this concentration and temperature the antibiotic is supersaturated. (A) Calculate the height of the free energy barrier, assuming a crystal-water interfacial energy of 0.1 J/m² and bulk free energy difference between dissolved organic and organic crystals of -0.1×10⁹ J/m³. Put this answer in units of k_BT. (B) Calculate the rate of nuclei formation assuming the rate at which molecules attach to a nucleus is 10¹¹/s, Z = 10^-3. HINT: Use class notes and the example provided on the Wikipedia page on CNT. (C) How long will you need to wait in order for a single crystal nuclei to form? What alternatives would you explore to make crystallization happen faster?

Problem 3 (Precipitation): You wish to precipitate a protein contained in 100 L of aqueous solution at 20°C in a tank at a concentration of 0.1 g/L. The protein has a molecular weight of 600,000 and a diffusion coefficient of 5×10^-7 cm²/s at 20°C. Precipitate particles have a density of 1.4 g/cm³. The solution is stirred during the addition of a precipitating agent using a 50 W motor. Calculate the “molecular weight” of the average nuclei (protein cluster) at the end of the initial mixing period.