A gas that contains CO₂ is contacted with liquid water in an agitated batch absorber. The equilibrium solubility of CO₂ in water is given by Henry’s law (Section 6.4b) C_A = p_A/H_A where C_A (mol/cm³) = concentration of CO₂ in solution P_A (atm) = partial pressure of CO₂ in the gas phase H_A [atm/ (mol/cm³)] = Henry’s law constant. The rate of absorption of CO₂ (i.e., the rate of transfer of CO₂ from the gas to the liquid per unit area of gas—liquid interface) is given by the expression r_A [mol/ (cm²∙s)] = k(C_A – C_A) where C_A = actual concentration of CO₂ in the liquid C^*_A = concentration of CO₂ that would be in equilibrium with the CO₂ in the gas phase (C^*_A = P_A/H_A) k(cm/s) = a mass transfer coefficient. The gas phase is at a total pressure P (atm) and contains y_A (mol CO₂/mol gas), and the liquid phase initially consists of V (cm³) of pure water. The agitation of the Liquid phase is sufficient for its composition to be considered spatially uniform, and the amount of CO₂ absorbed is low enough for P, V, and y_A to be considered constant throughout the process.

(a) Write a differential balance on CO₂ in the liquid phase and integrate it to derive the relation C_A (t) = C^*_A [1 – exp (– kSt/V)] where S (cm²) is the effective contact area between the gas and liquid phases.

(b) Suppose the system pressure is 20.0 atm, the liquid volume is 5.00 liters, the tank diameter is 10.0 cm. the gas contains 30.0 mole% CO₂, the Henry’s law constant is 9230 atm/ (mole/cm³), and the mass transfer coefficient is 0.020 cm/s. Calculate the time required for C_A to reach 0.620 mol/L if the gas-phase properties remain essentially constant.