The integral momentum and tracer equations for the mean motion with the turbulence contribution in momentum and tracer fluxes are integrated on the centreline of either plane or round buoyant jets, using suitable assumptions for the spreading coefficients and a closing function, and unified first- and second-order solutions are derived in the entire buoyancy range for mean axial velocities and mean concentrations. Comparisons to experimental data in the literature validate the model and show that second-order solutions deviate less than first-order solutions. Both types are used in conjunction with the integral continuity and kinetic energy equations for the mean motion to determine the variation of the local Richardson and Froude numbers, dispersion ratio, bulk dilution, dilution ratio, entrainment coefficient and mean velocity, kinetic energy flux and its gradient for the mean motion; and the variations of these quantities are evaluated using reported experimental or theoretical data. Finally, the variation of the product of kinetic energy flux and the local Richardson number is examined and a universal constant for both plane and round buoyant jets is revealed, leading to a unified definition of the local Richardson number, which is independent of the flow and mixing geometry and could be useful. Simple computational programming and good overall agreement make the proposed model a very promising tool for laboratory and field studies, outfall design and validation of numerical models.