Chinmaya Kumar Behera

 INTRODUCTION TO THERMODYNAMICS A system is some portion of the universe that one might isolate (either physically or mentally) in order to study it. The surroundings are the adjacent portions of the universe outside the system in question. If we consider the system plus the immediate surroundings, energy is conserved in all processes. If, on the other hand, we consider only the system, such as a rock, energy can be lost to, or gained from, the surroundings. Fundamental property of natural systems: Systems naturally tend toward configurations of minimum energy. Such minimum energy configurations, such as the rock on the ground, are referred to as stable. A rock hurled aloft is in an unstable configuration (or state) because it will quickly fall to Earth under the influence of gravity. States, neither stable (lowest energy possible) nor unstable (capable of spontaneous change), are called metastable. Any system that is not undergoing some form of transition is said to be at Equilibrium. It can be either stable equilibrium (such as the rock at the bottom of the slope) or metastable equilibrium (such as the perched rock). GIBBS FREE ENERGY Thermodynamics, as the name implies, deals with the energy of heat and work. Gibbs formulated an energy parameter, Gibbs free energy, that acts as a measure of the energy content of chemical systems. The Gibbs free energy at a specified pressure and temperature can be defined mathematically as: G = H – TS, where G = Gibbs free energy H = enthalpy (or heat content) T = temperature in kelvins S = entropy (most easily perceived as randomness) As a simple example of enthalpy, consider heating water on a stove. The enthalpy of the water increases as it is warmed because you are adding heat to it. When it boils, the steam has a higher enthalpy than the water, even when both are at the same temperature (the boiling point), because you had to add heat to the water to convert it to steam. As for entropy, imagine the ordered structure of a crystal lattice. This is low entropy. Liquids have much less ordered arrangement of atoms and, hence, greater entropy. Gases have even more entropy than liquids because the atoms or molecules are much more widely distributed. “Stable forms of a chemical system are those with the minimum possible Gibbs free energy for the given conditions.” The species on the left side of the reaction are called the reactants, and those on the right are called the products. Of course, the reaction must balance stoichiometrically, in that the number of atoms of each element must be the same on both sides of the reaction. ΔG = ∑(nproductsGproducts – nreactants Greactants ) where ∑ = sum, n = stoichiometric coefficient for each phase in the reaction . If G is negative, then the products have a lower total free energy than the reactants (meaning that they are more stable), and the reaction should run from left to right, as written. THE GIBBS FREE ENERGY FOR A PHASE The most common reference state is to consider pure elements in their natural (stable) form at 25°C (298.15 K) and atmospheric pressure (0.1 MPa), the conditions in a typically overheated laboratory, and assign a Gibbs free energy of zero joules (0 J) to that state. Note that G is an extensive variable of state, in that it is dependent upon the quantity of material in the system (the extent of the system). We can avoid this problem by expressing G in terms of molar free energy, or the number of joules per mole of the substance. For a compound such as quartz, we can measure the heat (enthalpy) change ( ΔH) associated with the reaction of 1 mole Si + 1 mole O2 to 1 mole SiO2 (by a technique called calorimetry). We can also calculate the entropy of quartz based on the assumption that the entropy of any substance is zero at 0 K (based on the third law of thermodynamics), and calculate the change in entropy between 0 K and 298.15 K (as discussed in the following section). From these values of H and S, we can compute the Gibbs free energy for low quartz using Equation (G=H-TS). The result is known as the molar Gibbs free energy of formation (from the elements) and given the symbol , where the subscript stands for formation (from the elements), the superscript refers to the reference state of 298.15 K and 0.1 MPa, and the bar above the G indicates that it is a molar quantity. Variations in the Gibbs Free Energy for a Phase with Pressure and Temperature Once we have the reference state data for geological phases of interest, we can determine the value of the Gibbs free energy (G) of a phase at elevated temperatures and pressures. We can do this by using the following differential equation: dG = VdP – SdT where: G = Gibbs free energy of the phase ,V = volume ,S = entropy G also changes with composition, and parameters for this variation can be added to Equation. for G at different pressures and temperatures. The variation in V with respect to P (called the isothermal compressibility) is sufficiently small for solids that V can be treated as a constant for a fairly large range of pressures, but the volume of liquids and particularly gases will certainly change with pressure, so, for them, calculations that assume a constant V will be in error. S varies appreciably with T for most phases, whether solid, liquid, or gas. The relationship can be expressed as dS = (Cp/T)dT, where Cp is the heat capacity (the amount of heat required to raise 1 mole of the substance 1°C). GIBBS FREE ENERGY FOR A REACTION The side of a reaction equation with lowest G under a given set of conditions is the most stable. The Equilibrium State at any point on the curve that separates the solid and liquid fields, both phases are equally stable, so that they coexist at equilibrium. The curve is thus called the (stable) equilibrium curve. Because both phases are equally stable at equilibrium, they must have the same value of G. Hence ΔG = 0, which must be true anywhere along the equilibrium curve. Equilibrium represents a dynamic state of flux, but the fluxes in this case cancel one another. For our coexisting liquid and solid, the reactions and both happen continuously, but the rates at which they proceed are equal. The amount of each phase will thus remain the same over time if the conditions are unchanged. The dynamic nature of this equilibrium state can be observed macroscopically because the shapes of the crystals will slowly change as they exchange atoms with the liquid. Le Châtelier’s Principle-If a change is imposed on a system at equilibrium, the position of the equilibrium will shift in a direction that tends to reduce that change. Thermodynamic Evaluation of Phase Diagrams There are five interrelated variables in Equation (3): G, S, V, P, and T. We can use these variables to understand various aspects of phase diagrams, such as the slope of an equilibrium curve or why the solid is on the high pressure–low temperature side. Clapeyron Equation It gives the slope of the equilibrium curve. The Phase Rule and One and Two-Component Systems INTRODUCTION: CRYSTALLIZATION BEHAVIOR OF NATURAL MAGMAS General observations about the complex crystallization behavior of natural melts- 1. Cooling melts crystallize from a liquid to a solid over a range of temperature. 2. Several mineral phases crystallize over this temperature range, and the number of minerals tends to increase as temperature decreases. 3. Minerals usually crystallize sequentially, generally with considerable overlap. 4. Minerals that involve solid solution change composition as cooling progresses. 5. The melt composition also changes during crystallization. 6. The minerals that crystallize, as well as the sequence in which they form, depend on the temperature and composition of the melt. 7. Pressure can affect the temperature range at which a melt crystallizes. It may also affect the minerals that crystallize. 8. The nature and pressure of any volatile components (such as H2O or CO2) can also affect the temperature range of crystallization and the mineral sequence. PHASE EQUILIBRIUM AND THE PHASE RULE Basics A system is some portion of the universe that you want to study. The surroundings can be considered the bit of the universe just outside the system. A system may be open (if it can transfer energy and matter to and from the surroundings), closed (only energy, such as heat, may be exchanged with the surroundings), or isolated (neither energy nor matter may be transferred). A phase is defined as a type of physically distinct material in a system that is mechanically separable from the rest. A phase may be a mineral, a liquid, a gas, or an amorphous solid such as glass. A piece of ice is a single phase, whereas ice water consists of two phases (the ice and the water are separable). Two pieces of ice are mechanically separable, but because they are equivalent, they are considered different pieces of the same phase, not two phases. A phase can be complex chemically, but as long as you cannot separate it further by mechanical means, it is a single phase. A component is a chemical constituent, such as Si, H2O, O2, SiO2. For purposes of the phase rule treatment, we shall define the number of components as the minimum number of chemical species required to completely define the system and all of its phases. A pure mineral, such as albite, has a single component (NaAlSi3O8). Minerals that exhibit solid solution, however, are commonly treated as multi-component systems. The proper choice of the number of components for the application of the phase rule is not always easy. For example, calcite may be considered a single-component system (CaCO3). Although this is true at relatively low temperatures if we heat it to the point that it decomposes to solid CaO and gaseous CO2, it would be a two-component system because we would have to use both CaO and CO2 to describe the composition of the solid and gaseous phases. The variables that must be determined to completely define the state of a system can be either extensive or intensive in nature. Extensive variables depend on the quantity of material (the extent) in the system. Mass, volume, number of moles, etc. are all extensive variables. Intensive variables, on the other hand, don’t depend upon the size of the system and are properties of the substances that compose a system. Intensive variables include pressure, temperature, density, etc. If we divide any extensive variable by another one, the extent cancels, and the ratio is an intensive variable. “F=C- φ+2” Where φ is the number of phases in the system, F, the number of degrees of freedom (or the variance) of a system, as the minimum number of intensive variables that need to be specified to completely define the state of the system at equilibrium and C is the number of components. The phase rule applies only to systems in chemical equilibrium. APPLICATION OF THE PHASE RULE TO THE H2O SYSTEM Let’s see how the phase rule works by applying it to a very simple system: the heating of ice on a hot plate. The system is defined by a single component, H2O, so C = 1. If we begin with ice at equilibrium at some temperature below 0°C then our system is completely solid, and φ = 1 as well. The phase rule [Equation (1)] at this point would tell us: F=1-1+2=2 meaning that we must specify only two intensive variables to define the system completely. In the present case, we are free to change two intensive variables as long as the value of the other parameters in the phase rule (C and φ) remain the same. With our ice, we have F = 2 and have chosen P and T as the ones that we shall specify. Alternatively, we could say that we can change P and T independently in our pan of ice (either or both), and still have only ice. Let’s heat the system at constant pressure (turn on the hot plate beneath our pan of ice). We can heat it initially with no change in the parameters of the phase rule (i.e., F remains equal to 2 as long as φ and C are both equal to 1). At each new temperature we can specify T, and the other intensive parameters also have new values (e.g., the ice expands, so the density changes). The phase rule still holds, however, telling us we need to specify two intensive variables if we want to fix the others. Eventually we heat the ice until a new phase appears: the ice begins to melt, and ice and water coexist stably at equilibrium in the pan. Now φ = 2 and F = 1 - 2 + 2 = 1. We need to specify only one intensive variable to completely define the state of the system. Which variable do we choose? Pressure or temperature? The phase rule cannot make this choice for us. ONE-COMPONENT SYSTEMS TWO-COMPONENT (BINARY) SYSTEMS Because C = 2, the variance can be as high as 3 in one-phase systems, requiring three-dimensional diagrams to illustrate properly. Rather than attempt this, we simplify most two-component igneous systems and illustrate their cooling and melting behavior on phase diagrams by fixing pressure and discussing the interactions of temperature and the compositional variables. If we restrict pressure, the phase rule becomes F = C - φ + 1 in the discussions that follow. Because temperature–composition (T-X) diagrams depreciate the importance of pressure in natural systems, we will occasionally discuss pressure effects on the systems in question in this chapter. 5.1 Binary Systems with Complete Solid Solution Example-The plagioclase system, composed of the two components NaAlSi3O8 and CaAl2Si2O8. There might be a tendency to confuse components and phases here. Remember that C (the number of components in the phase rule) is the minimum number of chemical constituents required to constitute the system and all of its phases. It is most convenient to treat this system as the two components CaAl2Si2O8-NaAlSi3O8, corresponding to the composition of the two phases. Ab will be used to indicate the component NaAlSi3O8, and An will indicate CaAl2Si2O8. Because the diagram is a temperature–composition diagram, it might seem appropriate to choose these two. What are the possible compositional variables? They must be intensive, so the choices are the weight (or mole) fraction of any component in any phase. We can define the weight fraction of the An component in the liquid phase: Equilibrium melting is simply the opposite process. The divariant one-phase solid system of composition i in Figure 8 heats up until melting begins. The partially melted system is univariant, and the first liquid formed has composition g. The first liquid to form is not the same as the solid that melts. As heating continues, the compositions of the solid and liquid are constrained to follow the solidus and liquidus, respectively (via a continuous reaction). The liquid moves to composition b as the plagioclase shifts to composition c (the composition of the last plagioclase to melt). Whether the process is crystallization or melting, the solid is always richer in An components (Ca and Al) than the coexisting liquid. Ca is thus more refractory than Na, meaning that it concentrates in the residual solids during melting. Notice how the addition of a second component affects the crystallization relationships of simple one-component systems: 1. There is now a range of temperatures over which a liquid crystallizes (or a solid melts) at a given pressure. 2. Over this temperature range, the compositions of both the liquid melt and the solid mineral phases change. The above discussion considers only equilibrium, crystallization and equilibrium melting, in which the plagioclase that crystallizes or melts remains in chemical equilibrium with the melt. It is also possible to have fractional crystallization or melting. Purely fractional crystallization involves the physical separation of the solid from the melt as soon as it forms. If we remove the plagioclase crystals as they form (perhaps by having them sink or float), the melt can no longer react with the crystals. The melt composition continues to vary along the liquidus as new plagioclase crystallizes along the solidus. Because the crystals are removed from the system, however, the melt composition continuously becomes the new bulk composition, thus shifting inexorably toward albite. As a result, the composition of both the final liquid and the solids that form from it will be more albitic than for equilibrium crystallization and will approach pure albite in efficiently fractionating systems.Fractional crystallization implies that a range of magma types could be created from a single parental type by removing varying amounts of crystals that have formed in a magma chamber. Most natural magmas, once created, are extracted from the melted source rock at some point before melting is completed. This is called partial melting, which may be fractional melting or may involve equilibrium melting until sufficient liquid accumulates to become mobile. Binary Eutectic Systems In a great number of binary systems, the additional component does not enter into a solid solution but changes the melting relationships nonetheless. As an example of a binary system with no solid solution, let’s turn to a system with considerable natural applicability. The system CaMgSi2O6 (Di, diopside)– CaAl2Si2O8 (An, anorthite) is interesting in that it provides a simplified analog of basalt: clinopyroxene and plagioclase. The system is illustrated in Figure 11 as another isobaric (atmospheric pressure) T-X phase diagram. In this type of system, there is a low point on the liquidus, point d, called the eutectic point. Such systems are thus called binary eutectic systems. Because there is no solid solution, there is no solidus (although some petrologists refer to the line g-h as a type of solidus). Let’s discuss equilibrium cooling and crystallization of a liquid with a bulk composition of 70 wt. % An from point a in Figure 11. This T-X phase diagram is also isobaric, so Equation (2) with a single liquid yields F = 2 - 1 + 1 = 2. We can thus specify T and or to completely determine the system. Cooling to 1450°C (point b) results in the initial crystallization of a solid that is pure An (point c). F = 2 - 2 + 1 = 1, just as with the plagioclase system. If we fix only one variable, such as T, all the other properties of the system are fixed (the solid composition is pure anorthite, and the liquid composition can be determined from the position of the liquidus at the temperature specified). As we continue to cool the system, the liquid composition changes along the liquidus from b toward d as the composition of the solid produced remains pure anorthite. Naturally, if anorthite crystallizes from the melt, the composition of the remaining melt must move directly away from An (on the left in Figure 11) as it loses matter of that composition. The crystallization of anorthite from a cooling liquid is another continuous reaction, taking place over a range of temperature. The reaction may be represented by: Liquid1=Solid+Liquid2 We can still apply the lever principle at any temperature to determine the relative amounts of solid and liquid, with the fulcrum at 70% An. At 1274°C, we have a new situation: diopside begins to crystallize along with anorthite. Now we have three coexisting phases, two solids and a liquid, at equilibrium. Ourhorizontal (isothermal) tie-line connects pure diopside at g, with pure anorthite at h, and a liquid at d, the eutectic point minimum on the liquidus. φ = 3, so F = 2 - 3 + 1 = 0. This is a new type of invariant situation, not represented by any specific invariant point on the phase diagram. Because it is invariant, T and the compositional variables for all three phases are fixed (points g, d, and h). The system is completely determined and remains at this temperature as heat is lost and crystallization proceeds (just as with our ice water and boiling water, as discussed above). The amount of liquid decreases, and both diopside and anorthite are produced. Because the amounts (extensive variables) of all three phases change at a constant temperature, it is impossible to determine the relative amounts of them geometrically using the lever principle. The lever principle can be applied, however, to determine the ratio of diopside to anorthite that is being crystallized at any instant from the eutectic liquid. If the liquid composition is the fulcrum (about 42% An), and the solids are pure (0% An and 100% An), the ratio of diopside to anorthite crystallizing at any moment must be 58/42. Removing this ratio keeps the liquid composition from changing from the eutectic as crystallization proceeds. The fact that the compositions of diopside, anorthite, and liquid are collinear is an example of an important relationship that we encounter often in petrology. It is one type of geometric relationship that implies a possible reaction. When three points are collinear, the central one can be created by combining the two outer compositions (in the proportion determined by the lever principle). In the present case the reaction must be: Liquid = Diopside + Anorthite because liquid is in the middle. This type of reaction is a discontinuous reaction because it takes place at a fixed temperature until one phase is consumed. When crystallization is complete, the loss of a phase (liquid, in this case) results in an increase in F from 0 to 1, and thus temperature can once again be lowered, with the two phases diopside and anorthite coexisting at lower temperatures. Because the composition of the two solids is fixed, we have a unique opportunity to determine exactly which of our intensive variables is free to vary; temperature is the only variable left. A discontinuous reaction involves one more phase than a corresponding continuous reaction in the same system, and because this decreases the variance, the compositions of the reacting phases do not vary as the reaction progresses. Only the proportion of the phases changes (usually until one phase is consumed). Such reactions are discontinuous in the sense that the phase assemblage changes at a single temperature due to the reaction. In this case, diopside + liquid gives way to diopside + anorthite as the system is cooled through the reaction temperature (1274°C). Let’s see what happens on the left side of the eutectic point. Cooling a liquid with a composition of 20 wt. % An results in the crystallization of pure diopside first, at 1350°C as the liquidus is encountered at point e in Figure 11. Diopside continues to crystallize as the liquid composition proceeds directly away from diopside toward point d. At point d (1274°C again), anorthite joins diopside and the eutectic liquid in the same invariant situation as above. The system remains at 1274°C as the discontinuous reaction, liquid = Di + An, runs to completion, and the liquid is consumed. In these eutectic systems note that, for any binary bulk composition (not a pure end-member), the final liquid to crystallize must always be at the eutectic composition and temperature. The final cooled product of a binary liquid with no solid solution must contain both anorthite and diopside. To get there, we must have them both coexisting with a melt at some point, and that melt has to be at the eutectic point. Remember that solid-solution systems do not behave this way. In them, crystallization is complete when the composition of the solid becomes equal to the bulk composition, so the final liquid, and the temperature, depends on the bulk composition. Equilibrium melting is the opposite of equilibrium crystallization. Any mixture of diopside and anorthite begins to melt at 1274°C, and the composition of the first melt is always equal to the eutectic composition d. Once melting begins, the system is invariant and will remain at 1274°C until one of the two melting solids is consumed. Which solid is consumed first depends on the bulk composition. If Xbulk is between Di and d, anorthite is consumed first, and the liquid composition will follow the liquidus with increasing temperature toward Di until the liquid composition reaches Xbulk, at which point the last of the remaining diopside crystals will melt. If Xbulk is between An and d, diopside is consumed first, and the liquid will progress up the liquidus toward An. Systems with More Than Two Components REACTION SERIES NL Bowen recognized, as we have, that there are two basic types of reactions that can occur under equilibrium conditions between a melt and the minerals that crystallize from it. The first type, called a continuous reaction series, involves continuous reactions of the type: composition of the melt, the mineral, or both varies across a range of temperature, abound in solid solution series. Examples include Reactions (1, 2, 7, 8, and 9)