Water flows through a thin tube, the walls of which are lightly coated with benzoic acid \(\left(\mathrm{C}_{7} \mathrm{H}_{6} \mathrm{O}_{2}\right)\). The water flows slowly, at \(298 \mathrm{~K}\) and \(0.1 \mathrm{~cm} / \mathrm{s}\). The pipe is \(1 \mathrm{~cm}\) in diameter. Under these conditions, equation (2-98) applies.
(a) Show that a material balance on a length of pipe \(Z\) leads to

where \(v\) is the average fluid velocity, and \(c_{A}{ }^{*}\) is the equilibrium solubility concentration.

(b) What is the average concentration of benzoic acid in the water after a length of pipe of \(2 \mathrm{~m}\). The solubility of benzoic acid in water at \(298 \mathrm{~K}\) is \(0.003 \mathrm{~g} / \mathrm{cm}^{3}\), and the mass diffusivity is \(1.0 \times 10^{-5} \mathrm{~cm}^{2} / \mathrm{s}\) (Cussler, 1997).

Data From Equation 2-98:-

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-kaz CAout C1-exp (2-106)