Phase Projections

In this chapter, the utility of showing mineral assemblages and identifying chemical reactions on composition diagrams has been emphased for 2- and 3-component systems. However, many rocks have minerals that need more than 3-components to describe. How can mineral assemblages be used for petrologic studies when the rock system has more than 3-components? It can be done algebraically, typically with the help of computers, but for most people this is less intuitive and less satisfying than a graphical approach. Fortunately, there is a graphing strategy that can work for some systems with more than 3 chemical components: phase projections.

If one phase is present in most of the assemblages of interest, that phase or mineral composition can be used and counted as one of the system components. By retricting our analysis to only those assemblages that contain the common phase, that component does not need to be included in the composition diagram. If a phase is always present, changing the proportion of the phase component changes only the proportion of that phase, not the equlibrium assemblage. Graphically, the common phase is used as a projection point, reducing by one the number of dimensions needed to show the assemblages on a composition diagram.

As an example, consider the equilibrium assemblages in the 3-component system Mg₂SiO₄-H₂O-SiO₂ shown in Figure 16. In this system, all the minerals have compositions close to the Mg₂SiO₄-SiO₂ line and most of the area of the composition triangle is occupied by H₂O-bearing assemblages. The H₂O phase for the metamorphic conditions of these assemblages is supercritical water. Click on the diagram and mouseover the T-P graph to see the observed equilibrium assemblages and reactions. Projection from water makes sense for this system because water is present in nearly all the assemblages of interest.

To peform the projection from water graphically, draw lines from the projection component (H₂O) through the phases of interest to the composition line defined by the other two components (Mg₂SiO₄-SiO₂). The equilibrium assemblage at any T and P is determined by the bulk chemical composition along that two-component line relative to the projected positions of the minerals. Click on Figure 17 and move the slider to see an animation of the projection.

Figure 18 shows the T-P diagram and equilibrium assemblages after using the H₂O projection. Click on the diagram and mouseover the T-P graph to see the equilibrium assemblages and reactions using the Mg₂SiO₄-SiO₂ composition line. As in a 2-component system, for each reaction one of the phases has a composition that is between the compositions of the other two, and will be on the opposite side of the reaction. However, water is also present and may be used to balance any reaction. The presence of water in the equilibrium assemblages is indicated next to the composition diagram with the label "± H₂O." The ± symbol is used for thermodynamic reasons (fixed activity of water).

Although we do not need to use a projection to show graphically the mineral assemblages for a 3-component system, the example of projecting from H₂O for the Mg₂SiO₄-H₂O-SiO₂ system shows the basic features of a phase projection. A commonly-used and very useful projection (from 3 phases) to reduce a 6-component system to 3-components is shown on the next page.

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