Let's solve each part step by step, with explanations and final answers: --- ### (a) **Expression for Overall Liquid-Phase Mass Transfer Coefficient (\(K_L\))** According to the two-film theory, the resistance to mass transfer is shared between the gas and liquid phases. The overall liquid-phase mass transfer coefficient, \(K_L\), is related to the individual film coefficients, Henry's law constant (\(H\)), and the enhancement factor (\(E\)), as follows: The total resistance to transfer is the sum of the resistances in the gas and liquid films: \[ \frac{1}{K_L} = \frac{1}{k_L E} + \frac{H}{k_g} \] Here: - \(k_L\) = liquid film mass transfer coefficient - \(E\) = enhancement factor due to reaction in the liquid film - \(H\) = Henry's law constant (molality/partial pressure units) - \(k_g\) = gas film mass transfer coefficient This equation accounts for the effect of a fast pseudo-first-order reaction, which enhances the liquid film mass transfer coefficient by \(E\). --- ### (b) **Concentration Profiles Across Gas-Liquid Interface** For qualitative sketches (as a description): - **Purely Physical Absorption:** The concentration of A in the gas phase decreases linearly through the gas film up to the interface, then decreases linearly through the liquid film to the bulk liquid. The gradient across the liquid film is moderate, showing only physical mass transfer. - **Absorption with Fast Chemical Reaction:** The concentration gradient in the gas film is similar, but within the liquid film, the concentration of A drops steeply near the interface due to rapid consumption by reaction. The gradient is much steeper in the liquid film, showing that the reaction enhances the mass transfer rate by lowering the interfacial concentration of A. **Effect:** The main effect of a fast chemical reaction is to greatly increase the driving force in the liquid film, resulting in a steeper concentration gradient and thus a higher flux of A into the liquid. --- ### (c) **Physical Interpretation of the Hatta Number (\(Ha\))** The Hatta number is defined for a pseudo-first-order reaction as: \[ Ha = \frac{\text{rate of reaction}}{\text{rate of diffusion}} = \frac{k_1 C_{B} \delta_L^2}{D_A} \] or, more commonly, \[ Ha = \frac{\sqrt{k_1 D_A} \, C_{B} \, \delta_L}{D_A} \] **Interpretation:** - If \(Ha \ll 1\): The reaction is slow compared to diffusion, so the overall rate is controlled by the reaction rate. - If \(Ha \gg 1\): The reaction is very fast; as soon as A diffuses into the liquid, it reacts almost instantly. The rate is now controlled by the diffusion of A through the liquid film. - If \(Ha \sim 1\): Both diffusion and reaction contribute significantly to the overall rate. **Conclusion:** The magnitude of the Hatta number determines whether the mass transfer in the system is limited by diffusion (high \(Ha\)) or by chemical reaction (low \(Ha\)) within the liquid film. --- **Final Answers:** (a) \[ \frac{1}{K_L} = \frac{1}{k_L E} + \frac{H}{k_g} \] (b) For purely physical absorption, the concentration profile of solute A is a smooth gradient in both phases. For absorption with a fast reaction, the liquid-side gradient is much steeper, showing a rapid drop near the interface due to enhanced mass transfer by the reaction. (c) The Hatta number quantifies the relative rates of reaction and diffusion. If \(Ha \ll 1\), the process is reaction-controlled; if \(Ha \gg 1\), it is diffusion-controlled; if \(Ha \approx 1\), both processes are significant.