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We investigate the fast relaxation of translational and internal temperatures in nonequilibrium gas models derived from the kinetic theory. We study the Chapman-Enskog expansion in the fast relaxation limit and establish that the difference between the translational and equilibrium temperatures becomes asymptotically proportional to the divergence of the velocity field which yields the volume viscosity term of the limiting one-temperature equilibrium fluid model. We establish existence theorems and prove local in time error estimates between the out of equilibrium solution and the one-temperature equilibrium fluid solution for well prepared initial data and justify the apparition of the volume viscosity terms. Numerical simulations are finally presented of the impact of volume viscosity during a shock/hydrogen bubble interaction.