The Chapman--Enskog solution to the Enskog kinetic equation of polydisperse granular mixtures is revisited to determine the first-order contributions $varpi_i$ to the partial temperatures. As expected, these quantities (which were neglected in previous attempts) are given in terms of the solution to a set of coupled integro-differential equations analogous to those for elastic collisions. The solubility condition for this set of equations is confirmed and the coefficients $varpi_i$ are calculated by using the leading terms in a Sonine polynomial expansion. These coefficients are given as explicit functions of the sizes, masses, composition, density, and coefficients of restitution of the mixture. Within the context of small gradients, the results apply for arbitrary degree of inelasticity and are not restricted to specific values of the parameters of the mixture. In the case of elastic collisions, previous expressions of $varpi_i$ for ordinary binary mixtures are recovered. Finally, the impact of the first-order coefficients $varpi_i$ on the bulk viscosity $eta_text{b}$ and the first-order contribution $zeta^{(1)}$ to the cooling rate is assessed. It is shown that the effect of $varpi_i$ on $eta_text{b}$ and $zeta^{(1)}$ is not negligible, specially for disparate mass ratios and strong inelasticity.
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