Recent investigations with BACKONE equations of state [S. Calero, Fluid Phase Equilib. 120 (1998) 1-22] revealed the importance of quadrupolar interactions for the thermodynamic behaviour of real fluids. In BACKONE an explicit expression for the quadrupolar contribution to the Helmholtz energy F-Q is used. In order to extend these equations to mixtures, a mixing rule for F-Q is required. Here, an expression for a one-fluid quadrupole moment &, is derived on the basis of perturbation theory using a spherical reference system. This mixing rule is tested first by comparing molecular simulation results for quadrupolar mixtures with those for the equivalent pure Q(x)-fluid for linear molecules of different elongations. Comparisons of pressures, internal energies, and quadrupole energies show in general good to excellent agreement. At high liquid densities, however, discrepancies start to occur at medium elongation which increase with molecular elongation. Excess free energies F-E calculated from the BACKONE-F-Q and the mixing rule show excellent agreement with values obtained from integrating simulation results even at high liquid densities. Finally, average quadrupole-quadrupole energies obtained from BACKONE-F-Q using the mixing rule show again excellent agreement with simulation data.