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MME213 Phase Diagrams and Transformations Lecture 5 A. K. M. Bazlur Rashid Professor, Dept. of Materials and Metallurgical Eng. Bangladesh Univ. of Eng. and Tech., Dhaka-1000 Thermodynamics and Phase Diagram 2 – Single Component Systems © Copyright Protected Lecture Outcome (LO) At the end of this lecture, students should be able to 1. analyse the dependence of Gibbs free energy on temperature and pressure, 2. differentiate various types order of phase transformations. 3. examine some important unary systems used in materials science and engineering. 2/24 Single-Component Phase Diagram β A single component system is the one containing a pure element or one type of molecule that does not dissociate over the range of temperature of interest. pure copper rod alumina powder water 3/24 β The phase diagrams of single component systems are generally plotted using (P, T) co-ordinate systems. Gibbs Phase Rule F = C-P+2 • For pure copper, the number of phase a system can exist are solid, liquid and gas. • For silica, however, besides liquid and gas, the system can exist in more than one solid state. • If a system exhibits more than one phase, the relative amounts of each co-existing phase cannot be determined using this unary phase diagram. Phase Diagram for Pure Copper Phase Diagram for Water Phase Diagram for Silica 4/24 An elementary principle of chemistry states that every liquid or solid tends to be in equilibrium with its vapor. Piston Then, what is happened to the vapour phase of solid and liquid? Can there be a completely liquid region in the one component diagram? Will not vapor exist in equilibrium with the liquid? P Cylinder Metal vapour Liquid metal (a) P • The explanation lies in the manner in which the one-component system is determined. • Only the metal being investigated is contained in the cylinder that exerts pressure on the system; even air is excluded. • If the external pressure is equal to the vapor pressure of the liquid metal at the given temperature, both liquid and vapor exist in equilibrium in the cylinder (Fig. a). Liquid metal • However, if the external pressure is greater than the vapor pressure of the liquid metal, the piston is forced down, the vapor condenses, and only the liquid phase remains (Fig. b) (b) 5/24 β For unary systems, the two-phase equilibria curves in (P, T) space is described mathematically by the function P = P (T). β The Clapeyron equation is a differential form of this equation. ππ βπ βπ» = = ππ βπ πβπ β For any pair of coexisting phases in the unary system, integration of the Clapeyron equation yields a mathematical expression for the corresponding phase boundary on the phase diagram. β However, to predict the phases that are stable, or mixtures that are in equilibrium at different temperatures, it is necessary to determine the variation of Gibbs free energy, G with temperature, T. 6/24 Gibbs Free Energy as a Function of Temperature How CP is related with H and S? ππ = βπ»π = πΆπ ππ πΆπ = πΏπ ππ = π ππ» ππ ππ = π βπ»π = π»π = ΰΆ± πΆπ ππ 298 π 0 By convention, H of pure material in its most stable state at 298 K is taken as zero. By convention, S of homogeneous material in its most stable state at 0 K is taken as zero. Slope = CP T 0 298 T, K Entropy, S H Slope = ππ ππ = π πΆπ π Temperature 7/24 Variation of G with T πΊ = πΊ(π, π) Enthalpy / Gibbs free energy CP πΆπ ππ − ππΌππ π π πΆπ βππ = ππ = ΰΆ± ππ π ππ» = πΆπ ππ + π 1 − ππΌ ππ ππΊ = −πππ + πππ ππΊ ππ = −π ; π ππΊ ππ =π π At constant pressures, π2 βπΊ = − ΰΆ± πππ ; π1 Slope = ππΊ ππ H Slope = CP 0 −TS Slope = -S G = −π π G decreases with increasing T (at constant P) at a rate given by –S. Temperature Variation Gibbs free energy, G, with T 8/24 G H (liquid) d H (solid) c Latent heat, L b Tm a G (solid) G (liquid) f Liquid stable T Variation of enthalpy, H, and free energy, G, with T for solid and liquid phases of a pure metal. (L = latent heat of melting, Tm = equilibrium melting temperature) G H (liquid) d H (solid) c Latent heat, L b • However, SL > SS and thus GL decreases more rapidly with increasing T than that for the solid. • At Tm, GL = GS and both the solid and liquid phases can coexist in equilibrium. • Beyond Tm, the liquid phase has the lowest free energy and is therefore the equilibrium state of the system. 9/24 To summarise, Which is larger, HL or HS ? • HL > HS at all temperature Which is larger, SL or SS ? Tm a • Therefore, at low temperatures, GL > GS. • For temperatures up to Tm, the solid phase has the lowest free energy and is therefore the equilibrium state of the system. e Liquid stable • At all temperatures, the liquid phase has a higher H (or U) than the solid phase. • SL > SS at all temperature e Liquid stable G (solid) f G (liquid) Liquid stable T Variation of enthalpy, H, and free energy, G, with T for solid and liquid phases of a pure metal. (L = latent heat of melting, Tm = equilibrium melting temperature) Which is larger, GL or GS ? • GL > GS at temperatures below Tm • GS > GL at temperatures above Tm This is because Gibbs free energy of liquid decreases more rapidly with increasing temperature than that of the solid. Effect of Pressure on Gibbs Free Energy β The temperature Tm at which both solid and liquid phases coexist is called the equilibrium melting temperature. β But this equilibrium melting temperature Tm only applies at a specific pressure (1 atm). β At other pressures, this equilibrium temperature will differ. 11/24 Consider the P-T diagram for pure iron. β Effect of pressure on equilibrium temperatures: ο Increasing pressure raises L β g-iron equilibrium transformation temperature. ο Increasing pressure lowers a-iron β g-iron equilibrium transformation temperature. ο Increasing pressure lowers a-iron β e-iron equilibrium transformation temperature. Effect of pressure on the equilibrium phase diagram for pure iron 12/24 β The reason for these changes derives from the equation ππΊ = −πππ + πππ β At constant temperature, the free energy of a phase increases with pressure such that ππΊ ππ =π π β If the two phases in equilibrium have different molar volumes, their respective free energies will not change by the same amount at a given temperature and equilibrium will, therefore, be disturbed by changes in pressure. β The only way to maintain equilibrium at different pressures is by varying the temperature. 13/24 β Consider a single-component (unary) system having two phases, a and b, are in equilibrium. β The free energy of the phases as a function of temperature and pressure can be written as From classical thermodynamics, the conditions for πΌ ↔ π½ phase equilibrium : π πΌ = ππ½ β ππ πΌ = πππ½ = ππ π πΌ = ππ½ β ππ πΌ = πππ½ = ππ ππΌ = π π½ β πππΌ = ππ π½ = ππ ππΊ πΌ = π πΌ ππ − π πΌ ππ ππΊπ½ = ππ½ ππ − ππ½ ππ β For unary systems, it can be shows that G = m β Then, for πΌ ↔ π½ equilibrium πΊ πΌ = πΊπ½ ππ ππ = ππ β ππΊ πΌ = ππΊπ½ ππΊ ′ = −π ′ ππ + π ′ ππ + πππ π= ππΊ′ ππ = π,π π ππΊ ππ =πΊ π,π ππ½ − π πΌ βπ πΌ→π½ = ππ½ − π πΌ βπ πΌ→π½ 14/24 ππ ππ = ππ ππ½ − π πΌ βπ πΌ→π½ βπ» πΌ→πΊ = = πππ βπ πΌ→πΊ ππ½ − π πΌ βπ πΌ→π½ πΌ ↔ πΎ Equilibrium βπ = π πΎ(πΉπΆπΆ) − π πΌ(π΅πΆπΆ) = −π£π βπ» = π» πΎ(π»π) − π» πΌ(πΏπ) = +π£π ππ ππ FCC (0.74) = ππ βπ» <0 πππ βπ πΏ ↔ πΏ Equilibrium BCC (0.68) βπ = π πΏ − π πΏ = +π£π βπ» = π» πΏ − π» πΏ = +π£π ππ βπ» = >0 ππ ππ πππ βπ Unary phase diagram for pure iron 15/24 β Thus, to determine a unary phase diagram, integration of the following two equations are required to determine P = P (T) curve of two-phase equilibria of the system. For equilibrium between two condensed-phases • S ↔ L equilibrium ππ ππ • α ↔ β equilibrium For equilibrium between condensed phase and gas phase • S ↔ G equilibrium • L ↔ G equilibrium ππ ππ ππ ππ βπ πΌ→π½ = βπ πΌ→π½ Clapeyron Equation βπ» πΌ→πΊ πβπ» πΌ→πΊ = = πππ βπ πΌ→πΊ π π 2 Clausius - Clapeyron Equation 16/24 Order of Phase Transformations β Phase transitions are driven by the minimization of the free enthalpy of the system: if at a certain temperature, the entropy contribution of the Gibbs enthalpy outweighs the enthalpy contribution in ΔG = ΔH − TΔS the high-temperature phase will become thermodynamically stable. Unfortunately, the term order is used for two different concepts in relationship to phase transitions. β The precise nature of this change, i.e. how smoothly or abruptly it occurs, is different for different types of phase transition. On the one hand, each phase transition involves an ordered (low-temperature) and a disordered (high-temperature) phase . β To describe this, phase transitions are classified into first-order and second-order transitions. On the other hand, the order of the transition (in the mathematical sense of the word) determines the severity of the changes as described above. • The order referred to here is the order of the differential of the Gibbs energy for which a step is observed at the phase transition. 17/24 First Order Transition Example: Melting, vaporisation, sublimation, etc. Ice ↔ Water Water ↔ Steam Liquid copper ↔ Solid copper Characteristics of First-Order Transition • Large heat energy absorbed or liberated during the transition In all these transitions, T and P remain constant while S and V change dG = -SdT + VdP S = -(πG/πT)P V = (πG/ πP)T CP = T (πS/ πT)P = T(π 2G/ πT2)P a = (1/V) (πV/ πT)P b = -(1/V) (πV/ πP)T • Gibbs free energy changes continuously and at the point of transition, G remains constant • First-order derivatives (V, S) and the second-order derivatives (a, b, CP) of Gibbs free energy changes discontinuously as a function of temperature 18/24 Second Order Transition Example: Liquid He-I ↔ Liquid He-II (at Lambda point, TL = 2.19 K) Ferromagnetic materials ↔ Paramagnetic materials (at Curie point. For Fe, TC = 1043 K) Characteristics • No heat energy absorbed or liberated during the transition • Continuous Gibbs free energy • Smooth transition in first derivatives of Gibbs free energy but discontinuity in second derivatives • No change in entropy and volume • Discontinuity appears in specific heat • Entropy and volume doesn’t change discontinuously 19/24 FIRST-ORDER PHASE TRANSITION ππΊ ππ −π = G ππΊ ππ π= π πΆπ = π π π,π To ο₯ DV DS = L/T T T ππ ππ T T SECOND-ORDER PHASE TRANSITION G π S DS = 0 T T CP DV=0 T T 20/24 • The Gibbs free energy, G, of both phases involved in the transition is a smooth function of temperature. At the transition point, the curves of both phases intersect - on crossing the transition point, the other phase becomes the thermodynamically stable one. • As a result, there is a kink in the free energy of the system (under equilibrium conditions) at the transition point of a first-order transition. In a second-order transition, the free energy of both phases are identical over a limited temperature range before diverging either side of the transition. Both curves have the same tangent at the transition point. • For first-order transitions, the kink in G corresponds to a step in its first derivatives at the transition point. This is a result of the high latent heat associated with the transition. • In the case of second-order transitions, there is no latent heat and therefore no step in the entropy at the transition. However, the slope of the curve changes abruptly, producing a kink similar to that in G itself for first-order transitions. • For a first-order transition, the heat capacity therefore goes to infinity when the transition point is approached from either side. For a second-order transition, the kink in S merely results in a step in its derivative. 21/24 Examples of Unary Phase Diagrams • Three allotropic forms (d, g and a) with different crystal structures and different properties. • a-iron (BCC), also called Ferrite, exists from the lowest temperatures up to 910°C, and it's magnetic up to its Curie temperature of 768°C. 770 K 11 GPa • g-iron (FCC), also called Austenite, exists from 910-1401°C. • Above 1401 °C, there is d-iron (BCC). At normal pressure, the d-iron melts at 1538°C. Unary phase diagram of pure iron • At very high P, there exists another solid, called e-iron (HCP). Although not technologically useful, is important for geologists as it is found in Earth’s core. 22/24 • Silica is polymorphic material. • The polymorphic forms of silica crystals are: 1. At room temperature, the stable form of silica is a-quartz. 2. At 573°C, a-quartz will change to β-quartz. 3. On cooling, β-quartz will revert to a-quartz. 4. At 870°C, stable β-quartz will change to β-Tridymite. 5. At 1470°C, β-Tridymite will change to β- Cristobalite, which melts at 1713°C. (a) (b) Unary phase diagram of silica 6. At high temperature, metastable β-quartz will change to β-Cristobalite or melt to liquid silica. (a) Phase diagram for silica system SiO2 at higher pressures (b) Stability relations in the silica system at atmospheric pressure 23/24 Next Class Lecture 6 Thermodynamics and Phase Diagrams 3 – Binary Solutions