An experiment is designed to measure the partition coefficient, \(K\), associated with the transfer of a pharmaceutical product through a polymer material. The partition coefficient is defined as the ratio of the densities of the species of interest (the pharmaceutical) on either side of an interface. In the experiment, liquid pharmaceutical \(\left(ho_{p}=1100 \mathrm{~kg} / \mathrm{m}^{3}\right.\) ) is injected into a hollow polymer sphere of inner and outer diameters \(D_{i}=5 \mathrm{~mm}\) and \(D_{o}=5.2 \mathrm{~mm}\), respectively. The sphere is exposed to convective conditions for which the density of the pharmaceutical at the outer surface is zero. After one week, the sphere's mass is reduced by \(\Delta m=5.1 \mathrm{mg}\). What is the value of the partition coefficient if the mass diffusivity is \(D_{\mathrm{AB}}=0.2 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\) ?