The mantle structures observed by seismic tomography can only be linked with convection models by assuming some relationships between temperature, density and velocity. These relationships are complex and non linear even if the whole mantle has a uniform composition. For example, the density variations are not only related to the depth dependent thermal expansivity and incompressibility, but also to the distribution of the mineralogical phases that are themselves evolving with temperature and pressure. In this paper, we present a stoichiometric iterative method to compute the equilibrium mineralogy of mantle assemblages by Gibbs energy minimization. The numerical code can handle arbitrary elemental composition in the system MgO, FeO, CaO, Al$_2$O$_3$ and SiO$_2$ and reaches the thermodynamic equilibrium by choosing the abundances of 31 minerals belonging to 14 possible phases. The code can deal with complex chemical activities for minerals belonging to solid state solutions. We illustrate our approach by computing the phase diagrams of various compositions with geodynamical interest (pyrolite, harzburgite and oceanic basalt). Our simulations are in reasonable agreement with high pressure and high temperature experiments. We predict that subducted oceanic crust remains significantly denser than normal mantle even near the core mantle boundary. We then provide synthetic tomographic models of slabs. We show that properties computed at thermodynamic equilibrium are significantly different from those computed at fixed mineralogy. We quantify the three potential contributions of the seismic anomalies (intrinsic thermal effect, changes in mineralogy induced by temperature variations, changes in the bulk composition) and show that they are of comparable magnitudes. Although the accuracy of our results is limited by the uncertainties on the thermodynamic parameters and equations of states of each individual mineral, future geodynamical models will need to include these mineralogical aspects to interpret the tomographic results as well as to explain the geochemical observations.
