A colloidal suspension contains particles of mass m and density ⇢c suspended in water of density ⇢w in a gravitational field of strength g. (a) Show that the number density n(z) = n0 exp [m(⇢c⇢w)gz ⇢ckBT ] where n0 is the density at the bottom of the vial and z is vertical distance measured from the bottom of the vial. (b) By balancing the di↵usive flux (from the concentration gradient) with the sedimentation flux (from gravity), derive the Stokes-Einstein equation D = kBT/(6⇡⌘a) (Hint: Consider both the di↵usive flux, J1 = Ddn/dz, and the sedimentation flux, J2 = vtn, and balance J1 and J2 across a thin slice of thickness dz).