Wettability gradients are present in many biological systems and have been used to do interesting things like make water run uphill [29]. One way to form such a surface on glass is to immerse the plate in a slowly flowing solution of a silane coupling agent that reacts with \(\mathrm{OH}\) on the glass surface. The reaction can be nearly instantaneous so that the coupling agent concentration at the surface is essentially 0 . We are interested in forming such a surface using \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{SiCl}\) in an inert solvent that has the physical properties of water at \(25^{\circ} \mathrm{C}\). If the free-stream concentration of TMCS is \(1 \times 10^{-4} \mathrm{kmol} / \mathrm{m}^{3}\), the free-stream velocity of the fluid is \(1 \mathrm{~m} / \mathrm{s}\), the plate is \(0.01 \mathrm{~m}\) long with an area of \(2 \times 10^{-4} \mathrm{~m}^{2}\), and the diffusivity of TMCS can be predicted using the Polson equation:

a. How long would we have to expose the plate to the fluid to insure that \(10 \%\) of the available \(\mathrm{OH}\) on the surface was converted at \(x=L\) ? (There are roughly \(2 \mathrm{OH} / \mathrm{nm}^{2}\) on a glass surface).

b. How much of the surface \(\mathrm{OH}\) was converted over the entire plate?

c. What is the surface \(\mathrm{OH}\) concentration at \(x=L / 2\) ?