An impurity is to be removed from an aqueous solution by adsorption on a selective adsorbent in abatch system. The solution, at $25C$ and containing $0\mathrm{.}1k\frac{g}{m^3}$ of the impurity, will be loaded into a $1m$ diameter baffled cylindrical tank to a $1m$ height agitated by a $0\mathrm{.}33m$ diameter six$-$blade turbine at $30$ rom. All other geometric ratios correspond to curve $1$ in Figure $18-1$ of Perry's Chemical Engineers' Handbook, $9$ th edition. The adsorbent particles are $0\mathrm{.}05cm$ in diameter, have a density of$1,100k\frac{g}{m^3},$ and are completely suspended in the liquid. The diffusivity of the impurity in the solution at $25C$ is $6\mathrm{.}6{10}^{-10}\frac{m^2}{s}.($a$)$ Estimate the mass transfer coefficient for transport of the impurity to the surface of the adsorbent particles. $($b$)$ Predict the time needed to remove $90\%$ of the impurity if $1kg$ of adsorbent particles is added to the tank and the adsorption process is controlled by the external mass transfer resistance. Assume that the impurity concentration at the particle surface is kept at zero because the adsorption is very favorable.