The diffusion of a protein into tissue is studied at short times by measuring the uptake of the labeled protein into the tissue of thickness L. Initially there is no labeled protein in the tissue layer. At time zero, the surface at x=0 has a (constant) concentration of C0. (A) Assuming the tissue can be modeled as a semi-infinite medium, derive an expression for the protein concentration profile as a function of both time and position. (B) Derive an expression for determining the protein uptake M which can be defined as: M(t)=0tJ(t,x=0)dt Given the following values for the system: M(t=600s)=8.741013molC0=1109cm3molL=0.015cmS=1cm2 (where S is the area normal to the direction of protein movement), (C) determine the value for the diffusion coefficient. (D) Use the diffusion coefficient found in part C and the given time window (0t600s) and tissue thickness L to validate the assumption of a semi-infinite medium. Determine the approximate time when the model's assumption may no longer be valid (defined when the concentration of the protein at the far edge of the tissue is 1% of C0 ). (Ans: part C=D=1109scm2; part D= t3.91hr)