In part I of the present work a constitutive theory is established for a gas diffusing, as a dilute solution, in a linear thermoelastic solid. The theory is compatible with the laws of thermomechanics i.e., mass, momentum and energy balance, as well as with the principles of positive entropy production and material objectivity. The assumptions of small concentrations of the gas and the absence of viscous effects simplify the analysis considerably. This allows a thermomechanical model for the solid to be assumed, independent of the presence of the gas. The constitutive model adopted for the solid is the linear uncoupled theory of thermoelasticity. In this level of generality a transport theory is developed for the gas by using the conservation laws of mass, linear momentum and internal energy. Some uncommon identities for the thermodynamic quantities of the gas are also established, based on the derived functional dependence for the internal energy, entropy, free energy and stress tensor of the gas. Thus a Gibb's type equation is obtained. In addition a comparison with classical irreversible thermodynamics, previous intuitive theories and experiments is made.