  The system of our interest is a dilute binary mixture, in which we consider
that the species have different temperatures as an initial condition. To study
their time evolution, we use the full version of the Boltzmann equation, under
the hypothesis of partial local equilibrium for both species. Neither a
diffusion force nor mass diffusion appears in the system. We also estimate the
time in which the temperatures of the components reach the full local
equilibrium. In solving the Boltzmann equation, we imposed no assumptions on
the collision term. We work out its solution by using the well known
Chapman-Enskog method to first order in the gradients. The time in which the
temperatures relax is obtained following Landau's original idea. The result is
that the relaxation time for the temperatures is much smaller than the
characteristic hydrodynamical times but greater than a collisional time. The
main conclusion is that there is no need to study binary mixtures with
different temperatures when hydrodynamical properties are sought.
