In this paper, a new, continuous formulation of the adjoint equations is derived for a steady, continuum, inviscid gas mixture in thermochemical non-equilibrium for force-based objective functions. These adjoint equations, when solved in conjunction with the governing equations, provide sensitivity information that can be used in a gradient-based optimization framework for shape design. The governing and adjoint equations are implemented in an unstructured, three-dimensional CFD solver, enabling efficient, optimal design of complex geometries with aerothermodynamic considerations in the presence of high-enthalpy, chemically reactive gas mixtures. Surface gradients calculated using the adjoint formulation provided herein are validated against finite-difference gradients for blunt-body and simple wing configurations.