The connection between the low density Limit of the time correlation equations for the transport coefficients and the solution of the Boltzmann equation to lowest order approximation appear to have been made in essentially two different ways. Either the time correlation function is evaluated by using the time dependent (Linearized) Boltzmann equation, or by utilizing a resummation of an expansion in the reciprocal of a convergence parameter. As well, the connection is often made to only the lowest order solution of the Boltzmann equation, ignoring the possible importance of higher order moments (Sonine polynomials) in the solution of the Boltzmann equation. The present work uses a projection operator method and a subsequent binary collision expansion of the time correlation function to retain all contributions to the transport coefficient from binary collisions. This explicitly avoids an expansion in a divergent parameter and reproduces all Sonine polynomial contributions to the transport coefficient. Gas transport coefficients for a binary mixture are obtained in a similar manner.