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**Lecture 4: Chemistry of silicate melts and minerals: chemical thermodynamics, melting, mineralogy**

Questions What is Gibbs free energy and how do we use it to understand phase stability in chemical systems? What is a phase diagram and how do we use it to understand the melting of rocks? What minerals dominate igneous rocks in the Earth’s crust, and what does this have to do with their composition and structure? Tools Chemical thermodynamics (i.e., mostly calculus) Ionic radii

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**Chemical thermodynamics**

Thermodynamics is the branch of science that predicts whether a state of some macroscopic system will remain unchanged or will spontaneously evolve to a new state. Kinetics is the branch of science that deals with how long it takes for a system to reach that new state. Mechanics is the branch of science that deals with the motions of small numbers of particles. Thermodynamics is most relevant to the understanding of processes on spatial scales large enough to neglect individual atoms and timescales long enough to neglect kinetics, so that the predictions of thermodynamics describe to good approximation the actual state of nature, rather than the expected state at infinite time. Often, geology and geochemistry deal with very long timescales or very large numbers of atoms, so we use a lot of thermodynamics! All kinetic processes go faster with increasing temperature, and hence the tools of thermodynamics are most useful for predicting the behavior of high-temperature geological phenomena like melting and metamorphism. But even for kinetically limited things (like life), thermodynamics tells which way it is favorable for processes to run.

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**Chemical thermodynamics: Definitions I**

System: the region of interest, of sufficient size that average properties like temperature are well-defined; to be distinguished from the environment (i.e., the rest of the universe) Isolated system: exchanges neither matter nor energy across its boundaries Closed system: may exchange energy across boundaries, but not matter Open system: may exchange matter and energy across boundaries Phase: a physically homogeneous and mechanically separable part of the system, e.g. a vapor, liquid, or mineral A system may be homogeneous (one phase) or heterogeneous (multiple phases). Component: a chemical formula; a basis vector for expressing compositional variations in thermodynamic systems; e.g., H2O, SiO2, Fe, NaCl. Must be independently variable, but we choose the minimum set to span all phases. Avoid at all costs confusing phases (e.g., water or quartz) with components (e.g., H2O or SiO2), even though people often use the same name for both!

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More on Components Choice of components is often arbitrary but number of components is not. Example: System Fe-O can also be described by FeO-Fe2O3. Example: System H2O is a one-component system if the only phases of interest are pure water, ice, and vapor, and if we need not consider electrolysis (i.e. separation into H2 and O2) or acid-base chemistry (i.e. separation into H+ and OH–). Example: System Mg2SiO4-Fe2SiO4 is a two-component system if we are only concerned with olivine and coexisting liquid. But at very high pressure compositions in this system form MgSiO3 perovskite, and we need three components (e.g. MgO, FeO, SiO2) to describe the system; under these conditions the line Mg2SiO4-Fe2SiO4 is a pseudo-binary join. The number of independent compositional variables needed to specify the composition of a system (but not its total mass or size) is one less than the number of components.

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**Chemical thermodynamics: Definitions II**

Equilibrium: a state in which macroscopic physical properties do not change during the period of observation. Microscopic processes are still occurring, but the rate of every process is exactly balanced by the rate of the reverse process. A stable equilibrium is a global minimum in potential energy. Subject to applied constraints, the system cannot achieve lower energy in any way. The system responds to small perturbations by returning to the stable equilibrium state. A metastable equilibrium is a local minimum in potential energy. The equilibrium is stable with respect to small perturbations and does not evolve spontaneously, but it might respond to a large perturbation by evolving away from the metastable equilibrium towards a lower energy state elsewhere. – An unstable equilibrium is a location where the system may not spontaneously evolve, but any small perturbation will cause it to move away from the original state. This is a local maximum in potential energy.

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**Chemical thermodynamics: Definitions III**

Volume (V): the size of a system in units of length3 Temperature (T): a measure of the tendency of a body to exchange microscopic kinetic energy with neighboring bodies. At equilibrium, all parts of a system are at equal temperature. Heat (dq): that which is transferred from hot bodies to cold ones during equilibration. Convention: heat transfer into the system from hot surroundings is positive; heat transfer by the system to cold surroundings is negative. Pressure (P): a measure of the tendency of a body to exchange mechanical energy with neighboring bodies. At equilibrium, all parts of a system are at equal pressure (in the absence of gravitational fields, surface tensions, etc.). Work (dw): the transfer of mechanical energy between objects at different initial pressures. For our purposes work is always given by dw = PdV. So by convention, work is positive when the system expands into low-pressure surroundings; negative when high-pressure surroundings compress the system.

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**Chemical thermodynamics: Definitions IV**

Reversible: an idealized process that proceeds through a sequence of equilibrium states as the parameters (P, V, T, etc.) are varied externally, without any finite deviation from equilibrium. Spontaneous: a real process, where the internal state of a system changes in order to approach equilibrium from an initially disequilibrium state Spontaneous Reversible

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**Chemical thermodynamics: First Law**

It is empirically observed that for any path that brings a closed system from an initial state (P1, V1, T1) to a new state (P2, V2, T2), and back to (P1, V1, T1) that the sum of heat and work transferred across the boundaries of the system is zero. Neither heat nor work is a variable of state; the quantities exchanged around closed paths of both heat and work can be non-zero; only the sum is conserved. Hence it is inappropriate to speak of the amount of heat or work in a system; these quantities are only used for transfers. We can, however, define a variable of state E, the internal energy, whose change for a closed system is given by (4.1) This is the First Law of Thermodynamics. Note: absolute values of E are arbitrary; only its changes dE are significant.

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**Chemical thermodynamics: Second Law**

The thermodynamic temperature scale is defined so that during a reversible cycle among states that returns to the original state, the integral of dq/T is zero. Hence there exists another variable of state S, the entropy, whose change is given by If at any time our closed cycle deviates from reversibility and undergoes a spontaneous change, we find that the integral of dq/T is always positive. So we state another empirical rule: (4.2) (4.3) This is the Second Law of Thermodynamics. If we expand our consideration to the system and its environment, which form an isolated system with dq=0, the second law takes the more familiar form dStotal ≥ 0. In any spontaneous process, total entropy must increase. In a reversible process it is constant.

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**Chemical thermodynamics: Equilibrium**

Combining the first and second laws, (4.4) This provides our first thermodynamic definition of equilibrium and the approach to equilibrium: If in a closed system we fix constant S and constant V, That is, any spontaneous process that occurs at constant S and constant V is associated with a decrease in internal energy E. When E reaches a minimum no further spontaneous changes can occur and all state variables will be constant, so this is a condition for equilibrium…a minimum in E.

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**Chemical thermodynamics: Open systems**

The form of (4.4) is only valid for a closed system (constant mass). If we have an open system that exchanges mass with the environment there are more variables. For a system of one chemical component (i.e., all phases are pure, equal, and constant in composition), we define a new quantity m, the chemical potential, such that for a change in the mass of the system dm, (4.5) Likewise, for a system of n components (independently variable chemical species), each component has a chemical potential mi such that (4.6)

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**Chemical thermodynamics: Partial Derivatives**

Fact from calculus: the total differential of a function of j variables A(x1, x2, …, xj) is related to the partial derivatives with respect to each variable as follows: Comparing this form to (4.6), we see that for reversible changes (4.7)

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**Chemical thermodynamics: E-S-V space**

What is the curvature of the E-surface for a stable phase? Must be concave up! Otherwise at constant total volume we lower E by unmixing into an ever-shrinking low-V, low-P phase and an ever-growing high-V, high-P phase. That is, only a point on a concave-up E surface can be at equilibrium.

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**Chemical thermodynamics: derivative properties**

More definitions: coefficient of isobaric thermal expansion isothermal compressibility isentropic compressibility heat capacity at constant pressure heat capacity at constant volume

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**Chemical thermodynamics: derivative properties**

Note that bS and Cv are related to second derivatives of E, so their sign is fixed by the stability condition on concavity of the E- surface Actually, the E surface needs to be concave up along all directions, i.e. the Hessian Matrix of second derivatives must be positive definite. It can therefore be shown that that bT and Cp are also strictly positive for all stable phases. You cannot have a phase with negative compressibility or heat capacity…it will spontaneously disintegrate! Note that ap can have either sign…it is perfectly acceptable to have a phase with negative thermal expansion.

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**Chemical thermodynamics: other potentials**

We have shown that internal energy E is minimized at equilibrium when the applied constraints are constant S and V. This is almost completely useless…there are hardly any experimental or natural situations where S and V are the independent variables! Why? Because specific S and specific V (=1/r) can differ between coexisting phases at equilibrium, unlike P and T, which must be equal among phases at equilibrium and so (1) are easy to control in the laboratory and (2) must be the independent variables at infinite time. We can get equivalents of 4.4, 4.5, and 4.6 with more useful independent variables that actually apply to natural and realizable experimental settings. We change variables using Legendre Transformations of the form

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**Chemical thermodynamics: other potentials**

First Legendre Transformation: define enthalpy H For a closed system, dE≤ TdS – PdV, so (4.8) For open system of one or many components, respectively: (4.9) (4.10)

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**Chemical thermodynamics: Equilibrium II**

This provides our second thermodynamic definition of equilibrium and the approach to equilibrium: If in a closed system we fix constant S and constant P, Any spontaneous process that occurs at constant S and constant P is associated with a decrease in enthalpy H. When H reaches a minimum no further spontaneous changes can occur and all state variables will be constant, so this is a condition for equilibrium…a minimum in H. This is no longer of strictly theoretical interest: during adiabatic, reversible pressure changes (as in the atmosphere and the Earth’s mantle; in both cases heat flow is negligible compared to advection), S and P are the independent variables, and equilibrium must be found by minimizing H. Note: at constant P, dH=dq, so enthalpy is a direct measure of heat transferred into or out of an isobaric system.

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**Chemical thermodynamics: other potentials**

2nd Legendre Transformation: define Helmholtz Free Energy F For a closed system, dE≤ TdS – PdV, so (4.11) For open system of one or many components, respectively: (4.12) (4.13)

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**Chemical thermodynamics: Equilibrium III**

This provides our third thermodynamic definition of equilibrium and the approach to equilibrium: If in a closed system we fix constant T and constant V, Any spontaneous process that occurs at constant T and constant V is associated with a decrease in Helmholtz free energy F. When F reaches a minimum no further spontaneous changes can occur and all state variables will be constant, so this is a condition for equilibrium…a minimum in F. These constraints are obtainable during isochoric temperature changes, such as in a rigid container like a fluid inclusion in a mineral. Note: at constant T, dF=dw, so Helmholtz free energy is a direct measure of work done on or by an isothermal system.

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**Chemical thermodynamics: other potentials**

Finally, if we do both Legendre transformations, we obtain a definition of Gibbs Free Energy G (Clearly G = H – TS = F + PV) For closed system, (4.14) For open system of one or many components, respectively: (4.15) (4.16)

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**Chemical thermodynamics: Equilibrium IV**

Now we are getting somewhere: If in a closed system we fix constant T and constant P, Any spontaneous process that occurs at constant T and constant P is associated with a decrease in Gibbs free energy G. When G reaches a minimum no further spontaneous changes can occur and all state variables will be constant, so this is a condition for equilibrium…a minimum in G. These constraints are the easiest to understand, the most common in the laboratory, and the most common in geology. From here on we will consider T and P the independent variables and discuss equilibrium as a state of minimum G.

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**Chemical thermodynamics: more on G**

From the definition of dG, we find the partial derivatives of G: (4.17) Thus the second derivatives of G are In G(P,T) space, then, the G surface is concave down, but this does not imply instability. Unmixing to phases at different T and P would violate the conditions of equilibrium and the applied constraints

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Phase diagrams A Phase diagram is a map of the phase or assemblage of phases that are stable in a chemical system at each point in the space of independent parameters (or some subspace, section, or projection thereof). If P and T are the independent variables, this means a phase diagram divides the volume of available conditions into regions where the minimum G is obtained with different phases or assemblages of phases. Begin with a one-component system in which there are three phases: solid, liquid, and vapor solid has the lowest specific entropy, liquid has intermediate specific entropy and vapor has the highest specific entropy (i.e., the entropies of fusion and boiling are positive). Liquid has the smallest specific volume (highest density, as in the case of H2O at low pressure), solid has intermediate specific volume, and vapor has the highest specific volume.

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**Phase diagrams: one component**

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**Phase diagrams: one component**

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**Phase diagrams: two components**

In a one-component system there are two independent variables (e.g., P and T), so a complete phase diagram can be drawn in two dimensions, and stability relations visualized in three dimensions (e.g., G-P-T space). In a two-component system, there are three independent variables: we add one compositional parameter, X, so now the space is four-dimensional. We therefore typically look at two-dimensional sections or projections through phase space to understand such systems. The most common is a map of minimum G assemblages as functions of (T, X) at constant P. We will seek here to understand how the two most common topologies in T-X space are derived by looking at sequences of G-X diagrams at constant P and T.

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**Phase diagrams: two components**

Condition of multicomponent equilibrium (a corollary to minimization of G, etc.): throughout the system at equilibrium, P, T, and mi of all components are equal. If P is larger in any one part of the system, work will be done until volumes adjust to reach constant P at equilibrium. If T is larger in any one part of the system, heat will flow until T is equalized. Likewise, if mi is larger in any one part of the system, mass will diffuse until mi is equalized. Consider a two-component system A-B with the mass of component A present in the system denoted mA and the mass of component B present mB. Total mass m = mA+mB. Define the mass fraction of component A, XA = mA/m. Clearly XB = mB/m = 1 – XA.

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**Phase diagrams: two components**

For system A-B equation (4.16) reduces to It is useful to divide by mass to put this in intensive terms, where a bar over a quantity denotes per unit mass: Now integrate over the whole system at equilibrium (constant T, constant P, and constant mA and mB): So if we draw a plot of G vs. XA at constant T and P, a stable phase is a concave-up curve, otherwise it spontaneously breaks up into two phases to lower G. The chemical potentials are read from the tangent line to the phase at the composition of interest. The intercepts give m of each end-member

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**Phase diagrams: two components, solid solution**

Now let us postulate two phases, solid and liquid, each capable of dissolving the two components in any amount. This is intuitive for a liquid solution, perhaps less so for a solid solution (but think of metallic alloys!). At fixed P and T, the free energy curve of each phase as a function of composition is concave up and the diagram might look like this: The sequence of stable, minimum G assemblages across the diagram is found by locating the common tangent line, which gives the compositions of solid and liquid where they have equal mB and equal mA – they are in equilibrium! Between these points, a mixture of the two phases gives G along the common tangent line, lower than either of the one-phase curves. Outside these points, since a mixture of phases must have positive amounts of each, the lowest G is achieved with one phase alone.

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**Phase diagrams: two components, solid solution**

As we change T at constant P, how does this diagram evolve? Since (∂G/∂T)P = –S, with increasing T, both curves move downwards. The one with the higher entropy (liquid phase in this case) moves down faster, causing the equilibrium points to shift: If we extract the stable, minimum G assemblage, either one phase or two phases, from this sequence of diagrams at each T and combine them, we can generate a T-X plot. This gives away information on the actual values of G, but usually all you need to know is what the minimum G state looks like.

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**Phase diagrams: two components, solid solution**

The resulting T-X diagram might look familiar: This is a map of the stable assemblage for each choice of the three independent variables (P,T,XA), either one phase alone or a mixture of two phases. It is a projection of the minimum envelopes of the sequence of G-XA sections. The blue curve is the liquidus, the locus of minimum temperatures where each bulk composition XA is completely liquid. The green curve is the solidus, the locus of maximum temperatures where each bulk composition is completely solid. In between, the system is partially molten.

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**Phase diagrams: two components, solid solution**

Inside the two-phase region (where the red tie-lines are), the proportion of each phase is given by the lever rule, a statement of conservation of mass. For bulk composition XA where fsolid is the mass fraction in the solid phase, and the composition of the solid is a point on the solidus XAsolid. Likewise for the liquid. Given fsolid =1 - fliquid , we can state the lever rule (4.18)

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**Phase diagrams: two components, eutectic**

Next we consider a different two-component system, this time with two solid phases a and b that tend to have compositions of nearly pure component A and B, respectively. Do not confuse component A, a chemical formula such as SiO2, with phase a, a solid mineral with a definite crystal structure, such as quartz; even though phase a may tend to be very close in composition to pure component A, they are not the same idea. We still have the liquid phase, which can continuously adopt any composition in A-B. Now we might generate a series of G-XA sections at constant P and a range of T like the following.

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**Phase diagrams: two components, eutectic**

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**Phase diagrams: two components, eutectic**

If we assemble the stable sequences from each T into a T-X section... The red line represents a special equilibrium, a eutectic, where the three phases a, b, and liquid coexist (and liquid is intermediate between a and b in composition). It is the lowest temperature at which liquid can exist in this system at this pressure. So this diagram has three kinds of elements: one-phase areas (where temperature and phase composition vary freely), two-phase areas (with a range of temperatures, but fixed phase compositions), and three-phase lines (where temperature and phase compositions are fixed).

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The Phase Rule The Gibbs phase rule is a fundamental relation between the number of components in a chemical system, the number of phases present, and the number of variables that can be independently varied while maintaining equilibrium (the variance, D). Consider a system of C components with f coexisting phases. How many free parameters are there? Total number of parameters: P, T, and C–1 compositional parameters for each phase = (C+1)f Total number of constraints: P must be equal in all phases: f–1 constraints T must be equal in all phases: f–1 constraints m for each component must be equal in all phases: C(f–1) in special cases (critical, singular points, etc.), other constraints Remaining degrees of freedom: (C+1)f – (C+2)(f–1) = C – f + 2

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Minerals Mineralogy is a whole course unto itself! We have time only for the briefest introduction. Definition: A mineral is a naturally occurring, inorganic, solid crystalline material with a defined range of composition. Minerals can be essentially one-component phases (e.g., quartz, basically pure SiO2) or multi-component solid solutions (e.g. olivine, mostly Fe2SiO4-Mg2SiO4). Mineralogical and thermodynamic nomenclature are somewhat different Both mineral groups and specific components are assigned names, sometimes confusingly the same name. The mineral phase olivine is a solid solution between forsterite (Mg2SiO4), fayalite (Fe2SiO4), and some other components. The mineral phase spinel includes components magnetite (Fe3O4), chromite (MgCr2O4), and the component spinel (MgAl2O4).

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Minerals Minerals are periodic structures constructed by packing of ions (either single-atom ions like Na+ or compound ions like carbonate CO32-) Ionic radii and charge balance are the critical factors determining mineral structure Anions (– ions) are big, cations (+ ions) are small, so volume is usually dominated by anions, with cations in interstitial spaces Radius determines whether a cation is likely to be coordinated by 4 (tetrahedral), 6 (octahedral), 8, or 12 anions

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**Classification of Minerals**

Minerals are usually organized by anionic groups: silicates, carbonates, halides, sulfates, phosphates, oxides, etc. Within the silicates, which are all based on arrangements of SiO44- tetrahedra (below ~10 GPa pressure), we classify minerals by the geometry of the network of tetrahedra: Framework silicates: all tetrahedra share four corners with other tetrahedra Layer silicates: every tetrahedron shares three corners with other tetrahedra Double chain silicates: half of the tetrahedra share three, half share two corners Single chain silicates: every tetrahedron shares two corners with other tetrahedra Dimer silicates: each tetrahedron shares one corner with another tetrahedron Isolated tetrahedra silicates: every tetrahedron is isolated Mineral structure is a function of composition, expecially the ratio of octahedral to tetrahedral cations. The above list is in order of increasing fraction of octahedral cations (i.e. things bigger than Al3+).

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**Classification of Minerals**

Framework silicate (quartz, feldspars): all corners shared; no octahedral sites.

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**Classification of Minerals**

Sheet silicate: micas, most clay minerals. The unshared oxygens are all on one side of the layer; these oxygens can help coordinate other cations. The layers are paired together around a layer of octahedrally coordinated cations There are two to six octahedral sites per 8 tetrahedral sites. Example: talc Mg6Si8O20(OH)4.

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**Classification of Minerals**

Chain silicate structures: double chain in amphiboles, single chain in pyroxenes. Again, all the unshared oxygens are on one side of the chain, and these chains pair up around a chain of octahedral sites. There are 7 octahedral sites per 8 tetrahedral sites in amphibole, e.g. anthophyllite Mg7Si8O22(OH)2 There are 8 octahedral sites for each 8 tetrahedral sites in pyroxene. Example: enstatite MgSiO3

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**Classification of Minerals**

Silicate dimer structure: based on Si2O76- groups Example: epidote group Ca2Al3Si3O12(OH). This structure allows about 5 octahedral sites per 3 tetrahedral sites. Isolated tetrahedra: no corner sharing This structure allows 2 octahedral sites for every one tetrahedral site. Example: olivine group (Mg,Fe,Ca,Mn,Ni)2SiO4

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**Major Minerals of Igneous Rocks**

The relationship between mineral structure and ratio of octahedral cations (mostly Fe, Mg, Ca) and tetrahedral cations (mostly Si, Al) allows you to readily understand the minerals that show up in rocks as a function of composition expressed as SiO2 content:

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**Major Minerals of Igneous Rocks: Ultramafic**

The average composition of the Earth’s upper mantle is: SiO2 TiO2 Al2O3 FeO MgO CaO Na2O H2O Others 46% ≤0.5 (Mg+Fe+Ca)/(Si+Al) is between 1 and 2, so the upper mantle is dominated by olivines (isolated tetrahedra structure) and pyroxenes (chain structure). olivine (Mg,Fe)2SiO4 [Mg/(Mg+Fe)~0.9] orthopyroxene (Mg,Fe)2SiO6 clinopyroxene Ca(Mg,Fe)Si2O6 Plus an aluminous mineral that depends on pressure: 0-1 GPa, feldspar (plagioclase) CaAl2Si2O8-NaAlSi3O8 [Ca/(Ca+Na) ~0.9] 1-3 GPa, spinel MgAl2O4 >3 GPa, garnet (Fe,Mg,Ca)3Al2Si3O12 A rock with this mineralogy is a peridotite.

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**Major Minerals of Igneous Rocks: Mafic**

The average composition of the Earth’s oceanic crust is: SiO2 TiO2 Al2O3 FeO MgO CaO Na2O K2O H2O Large enrichments over mantle in TiO2, Al2O3, CaO, Na2O, & K2O; small enrichments in SiO2 and FeO; massive depletion of MgO. (Fe+Mg+Ca)/(Si+Ti+Al) ~ 1, so basalts are dominated by pyroxenes, with alkalis in feldspar Clinopyroxene Ca(Mg,Fe)Si2O6 Feldspar (plagioclase) CaAl2Si2O8-NaAlSi3O8 [Ca/(Ca+Na) ~ ] plus olivine, orthopyroxene, and perhaps a bit of quartz. H2O lives in Amphibole (hornblende) Ca2(Mg,Fe)4Al2Si7O22(OH)2 A volcanic rock with this mineralogy is a basalt. A plutonic rock with this mineralogy is a gabbro.

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**Major Minerals of Igneous Rocks: Felsic**

The average composition of the Earth’s continental crust is: SiO2 TiO2 Al2O3 FeO MgO CaO Na2O K2O H2O 57% Note even larger enrichments over mantle in SiO2, K2O. There are few octahedral cations, so lots of framework silicates (quartz and feldspars to take alkalis). H2O gives micas & amphiboles before alteration Feldspars (plagioclase) CaAl2Si2O8-NaAlSi3O [Ca/(Ca+Na) ~ ] Feldspar (Alkali feldspar) NaAlSi3O8-KAlSi3O8 Quartz SiO2 Mica: biotite KMg3(AlSi3)O10(OH)2 Mica: Muscovite KAl2(AlSi3)O10(OH)2 Volcanic rocks with this composition range from andesite to rhyolite. Plutonic rocks range from diorite to granite.

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**Synthesis: Melting, mineralogy, and differentiation**

Why does partial melting of mantle yield enrichment in partial melt (which goes to form crust) of SiO2, Al2O3, FeO, CaO, Na2O, K2O; leaving a residue enriched in MgO? We can gain insight into this with a few essential phase diagrams. The olivine binary phase loop: an example of continuous solid solution. Mg end-member has higher melting point than Fe end-members. The phase diagram shows that this translates into Mg being more compatible than Fe…the liquid is always enriched in Fe/Mg relative to the residue.

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**Synthesis: Melting, mineralogy, and differentiation**

The Mg2SiO4-SiO2 binary: an example with negligible solid solution and an intermediate phase. The first liquid that appears on melting of a rock consisting of forsterite (olivine) plus enstatite (orthopyroxene) is more SiO2-rich than enstatite. If we turn around and crystallize it, it will make enstatite plus quartz (a model basalt, not a peridotite!). Thus oceanic crust is enriched in SiO2. We can make similar arguments for CaO, Na2O, and K2O, but they require ternary phase diagrams...

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