This dissertation employs the density functional theory (DFT) to investigate the oxidation of carbon monoxide (CO) on the W(111) surface and on the surface of Wn (n=10¡V15) nanoparticles. Since the properties of materials are significantly dependent on material size, we look into the influence of both the size and surface structure of tungsten catalysts on the CO oxidation process. The work contains two parts. Part 1: The adsorption and dissociation of O2 and CO on W(111) surface and Wn (n=10¡V15) nanoparticles. The chemical adsorption of O2 and CO on solid catalysts plays a very important role in heterogeneous catalysis for the CO oxidation reaction. The configurations, adsorption energies, vibration frequencies and electronic structures of adsorbates on W(111) and Wn (n=10¡V15) nanoparticles have been calculated to investigate their surface activity. The results indicate that adsorption of O2 and CO on Wn (n=10¡V15) nanoparticles are more stable compared to on the W(111) surface. The minimum energy pathways and transition states of chemical reaction processes on metal surfaces were also studied by the nudged elastic band (NEB) method. The dissociation barriers of O2 chemisorbed on Wn (n=10¡V15) nanoparticles are smaller those for the W(111) surface. Our results demonstrate that both the surface structure and size of metal significantly influence the adsorption and dissociation properties of adsorbates. Density functional theory-molecular dynamics (DFT-MD) simulation was also adapted to clarify the mechanism of O2 deposition on the W(111) surface. Observations of the variations of energy and bond lengths as a function of time show that the interaction between O2 and W atoms weakens the O¡VO bond, giving rise to the dissociation process. We conclude that the dissociation probability of an O2 molecule is affected by chemisorbed O2 coverage in the vicinity. Part 2: The mechanism of CO oxidation on W(111) and Wn nanoparticles. The oxidation of the CO molecule on transition metals usually follows two reaction pathways, either the Eley-Rideal (ER) mechanism or the Langmuir-Hinshelwood (LH) mechanism. In the ER mechanism, the CO molecule in the gas phase reacts directly with activated O2. The LH mechanism generally involves a few elementary steps, namely the co-adsorption of the O2 and CO molecules, O2 dissociation to form atomic oxygen, diffusion of atomic oxygen, and desorption of CO2. The oxidation of CO on a W10 nanoparticle surface and the W(111) surface are investigated by DFT calculations. Three pathways were studied in this dissertation: (i) CO + O2¡÷CO2 + O, (ii) CO + O2¡÷CO + O + O¡÷CO2 + O and (iii) CO + O¡÷CO2 via both LH and ER mechanisms. The calculated results show that CO oxidation on both the W10 nanoparticle and W(111) surfaces follow the ER rather than the LH mechanism. The CO oxidation on the W10 nanoparticle and W(111) surfaces occurs most easily via pathway (i) as compared to other two.