A person applies an insect repellent onto an exposed area of \(A=0.1 \mathrm{~m}^{2}\) of their body. The mass of spray used is \(m=2\) grams, and the spray contains \(15 \%\) (by mass) active ingredient. The inactive ingredient quickly evaporates from the skin surface.

(a) If the spray is applied uniformly and the density of the dried active ingredient is \(ho=2000 \mathrm{~kg} / \mathrm{m}^{3}\), determine the initial thickness of the film of active ingredient on the skin surface. The temperature, molecular weight, and saturation pressure of the active ingredient are \(32^{\circ} \mathrm{C}, 152 \mathrm{~kg} / \mathrm{kmol}\), and \(1.2 \times 10^{-5} \mathrm{bar}\), respectively.

(b) If the convection mass transfer coefficient associated with sublimation of the active ingredient to the air is \(\bar{h}_{m}=5 \times 10^{-3} \mathrm{~m} / \mathrm{s}\), the partition coefficient associated with the ingredient-skin interface is \(K=0.05\), and the mass diffusivity of the active ingredient in the skin is \(D_{\mathrm{AB}}=1 \times 10^{-13} \mathrm{~m}^{2} / \mathrm{s}\), determine how long the insect repellent remains effective. The partition coefficient is the ratio of the ingredient density in the skin to the ingredient density outside the skin.

(c) If the spray is reformulated so that the partition coefficient becomes very small, how long does the insect repellent remain effective?