Problem $1(35$ points$)$ You were asked to precipitate an antibody from the supernatant of a CHO cells culture. The antibody has a molecular weight of $121$kDa, a precipitate density of $1\mathrm{.}3\frac{g}{c}{m}^3,$ a hydrodynamic diameter of $9\mathrm{.}1nm$ and a diffusion coefficient of $3\mathrm{.}5{10}^{-7}c\frac{m^2}{sec}.$ It will be precipitated by a sudden $pH$ change in a $200$ liters tank stirred with a $0\mathrm{.}2hp$ motor. The feed concentration is $0\mathrm{.}2m\frac{g}{m}L$ and a collision effectiveness factor of $0\mathrm{.}05sec.$

$($a$)$ Estimate the time during which precipitate growth will be controlled by diffusion.

$($b$)$ The concentration of particles at the end of that time.

$($c$)$ The time required to grow $100m$ particles.

Assume that nucleation is fast, that flocculation is negligible and that the properties of the solution are those of water.