The calculation of two-temperature transport coefficients in an argon-hydrogen plasma at atmospheric pressure is performed using a new theory of two-temperature transport properties recently presented. The latter takes into account the coupling between electrons and heavy species, coupling neglected in the already existing theories of Devoto and Bonnefoi. Transport coefficients are calculated at two-temperatures, the kinetic temperature of electrons T(e) being different from that of heavy species T(h). This paper is divided into two parts. The first one is related to elastic processes and its aim is to compare the results obtained with this new theory for viscosity mu, translational thermal conductivities K(e)(tr) and k(h)(tr) and electrical conductivity sigma with the previous results of Bonnefoi. The composition is calculated with the modified equilibrium constant of van de Sanden et al. and the most recent interaction potential are discussed. As it could be expected the electron translational thermal conductivity and the electrical conductivity calculated when taking into account or not the coupling between electrons and heavy species show non-negligible discrepancies. Besides this comparison, the results also show the drastic influence of the non-equilibrium parameter theta =T(e)/T(h) on the values of sigma,mu, k(e)(tr), and k(h)(tr).