Continuum-based theories, such as Navier–Stokes (NS) equations, have been considered inappropriate for flows under nonequilibrium conditions. In part, it is due to the lack of rotational degrees-of-freedom in the Maxwell–Boltzmann distribution. The Boltzmann–Curtiss formulation describes gases allowing both rotational and translational degrees-of-freedom and forms morphing continuum theory (MCT). The first-order solution to Boltzmann–Curtiss equation yields a stress tensor that contains a coupling coefficient that is dependent on the particles number density, the temperature, and the total relaxation time. A new bulk viscosity model derived from the Boltzmann–Curtiss distribution is employed for shock structure and temperature profile under translational and rotational nonequilibrium. Numerical simulations of argon and nitrogen shock profiles are performed in the Mach number range of 1.2–9. The current study, when comparing with experimental measurements and direct simulation Monte Carlo (DSMC) method, shows a significant improvement in the density profile, normal stresses, and shock thickness at nonequilibrium conditions than NS equations. The results indicate that equations derived from the Boltzmann–Curtiss distribution are valid for a wide range of nonequilibrium conditions than those from the Maxwell–Boltzmann distribution.