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Impurity diffusion in Zr is potentially important for many applications of Zr alloys, and in particular for their use of nuclear reactor cladding. However, significant uncertainty presently exists about which elements are vacancy vs. interstitial diffusers, which can inhibit understanding and prediction of their behavior under different temperature, irradiation, and alloying conditions. Therefore, first-principles calculations based on density functional theory (DFT) have been employed to predict the temperature-dependent dilute impurity diffusion coefficients for 14 substitutional alloying elements in hexagonal closed packed (HCP) Zr. Vacancy-mediated diffusion was modeled with the eight-frequency model. Interstitial contributions to diffusion are estimated from interstitial formation and select migration energies. Formation energies for each impurity in nine high-symmetry interstitial sites were determined, including significant effects of thermal expansion. The dominant diffusion mechanism of each solute in HCP Zr was identified in terms of the calculated vacancy-mediated activation energy, lower and upper bounds of interstitial activation energy, and the formation entropy, suggesting a rough relation with the metallic radii of solutes. It is predicted that Cr, Cu, V, Zn, Mo, W, Au, Ag, Al, Nb, Ta and Ti all diffuse predominantly by an interstitial mechanism, while Hf, Zr, and Sn are likely to be