The concerns towards the rise in carbon dioxide (CO₂) concentrations in atmosphere and its impact on climate change have been increasing since past decades. While one of the primary sources of CO₂ emissions are power or manufacturing plants (large point sources) that burn fossil fuels to generate electricity, the cumulative emissions from vehicles and humans/animals (small but distributed point sources) can be significant. This necessitates the development of carbon capture technologies not only from flue gases but also directly from the atmosphere. Most of these technologies are based on CO₂ adsorption in a fixed-bed reactor. The current work presents the modeling and experimental studies on the topic, where we developed reduced-order model to describe the transport and adsorption of CO₂ in a packed-bed reactor and perform the lab-scale experiments to investigate adsorption characteristics. The reduced-order model (ROM) is expressed in multiple concentration modes while capturing the small-scale physics using mass-transfer coefficients. We illustrate that ROM developed here are more accurate than traditional pseudo-homogeneous model and can be utilized for real time optimization and parametric/bifurcation studies. We also show the use of ROM in solving inverse problems to analyze the experimental data in obtaining transport/reaction parameters. Finally, we provide the physical interpretation of the various terms used in the model, perform sensitivity analysis and discuss its use in design/scale-up of the reactor.