A theoretical approach is presented to express the concentrations of CO2 and NaOH along with the absorption rate of CO2 in a CO2–aqueous elastic xanthan gum (XG) solution containing NaOH. In this study, we obtain approximate analytical solutions for the system of nonlinear differential equations under steady-state conditions by using the homotopy analysis method (HAM) and Padé approximation technique (PA). The homotopy analysis method is used for the analytical treatment of the equations and the Padé approximation for increasing the convergence region of the HAM analytical solution. The results reveal that the HAM with the enhancement of the Padé approximation is a very effective, convenient, and quite accurate method for such types of partial differential equations. Furthermore, in this work the numerical simulation of the problem is also reported using the Scilab/Matlab program. An agreement between analytical and numerical results is noted.