Diffusive particle deposition rates are predicted on the walls of a straight, smooth circular tube in which fully developed turbulent flow exist. As a pre-requisite to calculating particle wall flux, the particle mass balance equation (convective diffusion) is solved, using the method of separation of variables. A sub-layer model is used, in which a thin laminar sub-layer is considered to be imbedded within the turbulent boundary layer. In the region outside the laminar sub-layer, the time-averaged transport of particles to the wall is enhanced because of the turbulence induced eddy-diffusivity of the particles. Inside the laminar sub-layer close to the wall, the effect of turbulence is considered negligible and transport of particles to the surface is dominated by particle Brownian diffusivity. Using method of separation of variables, the problem is reduced to one of solving an ordinary differential equation which belongs to the Sturm-Liouville class of equations. We compute and report the first ten eigenvalues and associated eigenfunctions of the reduced equation along with the relevant constants needed to estimate the evolving particle number density profile and time-averaged particle wall deposition rates. Particle deposition rates have been estimated for Reynolds numbers and particle Schmidt numbers ranges of greatest practical interest. Asymptotic eigenvalues are reported and are seen to be in good agreement with the computed values. Application of the results to estimate the inertial contribution to the particle deposition rates in the eddy diffusion-impaction regime is also illustrated and generalizations of our methods have been discussed.