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# Phase Diagrams and Phase Rule

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### Phase Diagrams and Phase Rule

1. 1. Phase Transitions and Phase Rule Dr. Ruchi S. Pandey
2. 2. Thermodynamics and Phase transitions <ul><li>A Phase is a region of space (a thermodynamic system), throughout which all physical and chemical properties of a material are essentially uniform </li></ul><ul><li>Phase changes are governed by laws of thermodynamics </li></ul><ul><li>First Law of Thermodynamics </li></ul><ul><ul><li>Enthalpy – “ Heat transferred between the system and surroundings carried out under constant pressure.” </li></ul></ul><ul><ul><li>H cannot be measured directly, only change in enthalpy  H can be measured (at constant pressure P); </li></ul></ul><ul><li>Second Law of Thermodynamics </li></ul><ul><ul><li>“ In a system, a process that occurs will tend to increase the total Entropy of the universe.” </li></ul></ul><ul><ul><li>q rev is the heat that is transferred when the process is </li></ul></ul><ul><ul><li>carried out reversibly at a constant temperature . </li></ul></ul>Dr. Ruchi S. Pandey
3. 3. Thermodynamics and Phase transitions (2) <ul><li>Spontaneous Process:  S > 0 </li></ul><ul><li>Total entropy change of the universe is given by </li></ul><ul><li>For natural isobaric processes, q rev =  H   S =  H / T </li></ul><ul><ul><li>Also, for energy conservation,  H sys = -  H surr </li></ul></ul><ul><ul><li>Hence , we can write </li></ul></ul><ul><ul><li> </li></ul></ul><ul><ul><li>The quantity -T  S univ is also called the Gibbs Free Energy of the system </li></ul></ul>Dr. Ruchi S. Pandey  G = 0  G > 0  G < 0 exergonic equilibrium endergonic
4. 4. Phase Diagrams <ul><li>Phase </li></ul><ul><ul><li>a form of matter that is uniform throughout in chemical composition and physical state </li></ul></ul><ul><li>Boundaries </li></ul><ul><ul><li>Solid-Liquid ( Fusion ), Liquid-Gas ( Vaporization ), Solid-Gas ( Sublimation ) </li></ul></ul><ul><ul><li>Also Solid-Solid and Liquid-Liquid </li></ul></ul><ul><li>Points </li></ul><ul><ul><li>Critical Point – beyond this a gas cannot be liquefied </li></ul></ul><ul><ul><li>Boiling Point – vapor pressure of the gas = atmospheric pressure </li></ul></ul><ul><ul><li>Melting Point – solid and liquid phase coexist (equilibrium) </li></ul></ul><ul><ul><li>Triple Point - solid, liquid and gas phase coexist </li></ul></ul>Dr. Ruchi S. Pandey
5. 5. Phase Boundaries <ul><li>Chemical Potential ( µ ): For a 1-component system it is the molar Gibbs energy ( G m ) and defines the potential of a system to undergo a physical or a chemical change </li></ul><ul><li>At the phase boundary where phases α and β coexist, </li></ul><ul><li>µ α = µ β (at equilibrium) </li></ul><ul><li>This gives us the Clapeyron Equation: </li></ul><ul><ul><li>Clapeyron Eqn. takes different forms for different phase boundaries </li></ul></ul>Dr. Ruchi S. Pandey <ul><li>Slopes of the boundaries </li></ul><ul><ul><li>Solid-Liquid Boundary </li></ul></ul><ul><ul><li>Liquid-Vapor Boundary </li></ul></ul><ul><ul><li>Solid-Vapor Boundary </li></ul></ul><ul><li>Clausius-Clapeyron Equation </li></ul><ul><ul><li> V = V g -V l/s ~ V g </li></ul></ul><ul><ul><li>V g = RT/P (ideal gas) </li></ul></ul>
6. 6. Gibbs’ Phase Rule <ul><li>P is the number of phases </li></ul><ul><li>C is number of components , i.e. the chemically “independent” constituents, of the system, which can describe the composition of each phase present in the system </li></ul><ul><ul><li>“ independent” means- </li></ul></ul><ul><ul><ul><li>If you have equilibrium balance between reactants and products, the number of components will be reduced by one </li></ul></ul></ul><ul><ul><ul><li>If you have equal amounts (concentrations) of products formed, the number of components will also be reduced by one </li></ul></ul></ul><ul><li>F is the degrees of freedom of the system </li></ul>Dr. Ruchi S. Pandey
7. 7. Phase <ul><ul><li>a form of matter that is uniform throughout in chemical composition and physical state </li></ul></ul><ul><li>Homogeneous phase is uniform throughout in its chemical composition and physical state. (no distinction or boundaries) </li></ul><ul><ul><li>Water, ice, water vapor, sugar dissolved in water, gases in general, etc. </li></ul></ul><ul><li>Heterogeneous phase is composed of more than one phase These phases are distinguished from each other by boundaries. </li></ul><ul><ul><li>A cube of ice in water. (same chemical compositions but different physical states) </li></ul></ul><ul><ul><li>Oil-water mixture. </li></ul></ul><ul><ul><li>The two phases are said to be coexistent . </li></ul></ul>Dr. Ruchi S. Pandey
8. 8. Number of Components <ul><li>NaCl(s) dissolved in water </li></ul><ul><ul><li>Available chemical constituents are four. Na + , Cl - , NaCl and H 2 O </li></ul></ul><ul><ul><li>Because Na + and Cl – have the same amount “equal neutrality” as NaCl, then c = 2 and not 4 </li></ul></ul><ul><li>Decomposition of calcium carbonate </li></ul><ul><ul><li>Available chemical constituents are three. Is it correct to say c = 3 ? </li></ul></ul><ul><ul><li>Because of the equilibrium condition the number of independent components is reduced by one. Thus, c = 2 instead of 3; C = 2, P = 3  F = 2 – 3 + 2 = 1 </li></ul></ul><ul><li>Decomposition of ammonium chloride </li></ul><ul><ul><li>Available chemical constituents are three. Is it correct to say c = 3 ? </li></ul></ul><ul><ul><li>Because of the equilibrium condition the number of independent components is reduced by one. And also because the products formed form a single phase and are formed in equal amounts, the no. of independent components are further reduced by one. </li></ul></ul><ul><ul><li>C = 1, P = 2  F = 1 – 2 + 2 = 1 </li></ul></ul><ul><li>Decomposition of PCl 5 </li></ul><ul><ul><li>Available chemical constituents are three. Is it correct to say c = 3 ? </li></ul></ul><ul><ul><li>Because of the equilibrium condition the number of independent components is reduced by one. C = 2, P = 3  F = 2 – 3 + 2 = 1 </li></ul></ul>Dr. Ruchi S. Pandey
9. 9. Degrees of Freedom <ul><li>Number of intensive variables that can be changed independently without disturbing the number of phases in equilibrium </li></ul><ul><li>Simplistically speaking, there are only three intensive variables which can describe any phase of a chemical system </li></ul><ul><ul><li>Temperature (T), Pressure (P) and Composition/Concentration (  ) </li></ul></ul><ul><li>But what happens to the no. of degrees of freedom or the variance of a system when there are more than one phases? </li></ul><ul><ul><li>To count these, lets assume that we have a heterogeneous system “in equilibrium” consisting of ‘C’ components distributed in ‘P’ phases. </li></ul></ul><ul><ul><li>Lets now derive our phase rule to know the degrees of freedom of such a system. </li></ul></ul>Dr. Ruchi S. Pandey
10. 10. Derivation of the phase rule <ul><li>• In any system the number of intensive variables are: pressure, temperature plus the mole fractions of each component of each phase. </li></ul><ul><li>• Only C-1 mole fractions are needed since </li></ul><ul><li>» Thus for P phase, the number of intensive variables = P(C-1) + 2 </li></ul><ul><li>• At equilibrium the chemical potential of each phase must be equal, </li></ul><ul><li>i.e. μ P1 = μ P2 = μ P3 = μ P4 = μ P5 ….{there are P-1 such equations} </li></ul><ul><li>Since there are C components, equilibrium requires that there are C(P-1) equations linking the chemical potentials in all the phases of all the components </li></ul><ul><li>Now, F = total required variables - total restraining conditions </li></ul><ul><li>F = P(C-1) + 2 - C(P-1) = PC - P + 2 -CP + C = C- P + 2 </li></ul>Dr. Ruchi S. Pandey
11. 11. Phase Diagram of Water Dr. Ruchi S. Pandey <ul><li>A single phase is defined by an area on the phase diagram </li></ul><ul><ul><li>for these regions C=1, P=1  F = 2 </li></ul></ul><ul><ul><li>one can vary either the temperature, or the pressure, or both (within limits) without crossing a phase line. </li></ul></ul><ul><li>Equilibrium of two phases is defined by the black lines in the diagram, also called the phase boundaries </li></ul><ul><ul><li>for these lines C=1, P=2  F = 1 </li></ul></ul><ul><ul><li>If we want to stay on a phase line, we can't change the temperature and pressure independently </li></ul></ul><ul><li>Equilibrium of three phases is the single point O in the diagram </li></ul><ul><ul><li>at this point C=1, P=3  F = 0 </li></ul></ul><ul><li>Metastable state: supercooled water, curve OT </li></ul><ul><ul><li>If the vessel is clean and there is no scope of nucleation, water can be cooled several degrees below its freezing point. </li></ul></ul>
12. 12. Phase Diagram of Water (Experimental) <ul><li>Multiple known structures for solid phase. </li></ul><ul><li>Five more known triple-points, other than the S-L-V point. </li></ul><ul><li>An anomalous liquid! </li></ul><ul><li>Robust Hydrogen bonding </li></ul><ul><li>Negative slope of melting line </li></ul>Dr. Ruchi S. Pandey <ul><li>The Advance of Glaciers </li></ul><ul><li>a natural consequence of negatively sloping melting line </li></ul><ul><li>High pressure causes ice to melt and re-freeze on either side of a bump </li></ul>
13. 13. Phase Diagram of Sulphur Dr. Ruchi S. Pandey <ul><li>Sulfur solid exists in two crystalline forms </li></ul><ul><ul><li>Orthorhombic, S 8 or S(rh) </li></ul></ul><ul><ul><li>Monoclinic, S 4 or S(mo) </li></ul></ul><ul><li>Total no. of phases 4, C=1, F= C-P+2=-1 </li></ul><ul><ul><li>Negative variance is not possible so all 4 phases can never coexist </li></ul></ul><ul><li>Three triple points </li></ul><ul><li>Boiling point at 444.6 o C </li></ul><ul><li>S(rh) changes into S(mo) at 95.6C, only when heated slowly. If heated rapidly, rhombic sulphur passes directly in to the liquid phase </li></ul><ul><li>The metastable triple point occurs at 114C </li></ul>
14. 14. Two components systems Dr. Ruchi S. Pandey <ul><li>Need to know T, P and concentration to describe such systems – the 3D plot looks very complex, so one of the variables is fixed </li></ul><ul><ul><li>P-T graphs (isoplethal), P-C graphs (isothermal), T-C graphs (isobaric) </li></ul></ul><ul><ul><li>Since we are fixing one of the variables the phase rule changes </li></ul></ul><ul><li>Reduced Phase Rule: F = C – P +1 </li></ul><ul><li>Solid-Liquid Phase equilibria: condensed systems at constant pressure (atm. press.) </li></ul><ul><ul><li>Phase diagrams constructed using thermal analysis </li></ul></ul><ul><ul><li>Mixtures of different compositions are first melted much above their melting points and then gradually cooled </li></ul></ul><ul><li>Thermal Analysis </li></ul>
15. 15. Two component solid-liquid equilibria <ul><li>Depending upon the miscibility of the 2 components in the liquid state and nature of the solid that separates on cooling, 2 classes exist; </li></ul><ul><ul><li>I: when the two components are completely miscible in liquid state </li></ul></ul><ul><ul><li>II: when the two components are partially miscible in liquid state </li></ul></ul><ul><li>Components that are miscible in liquid state </li></ul><ul><ul><li>(i) but not miscible in the solid state (Pb-Ag, Bi-Cd syatem) </li></ul></ul><ul><ul><li>(ii) and form a stable compound which melts at a constant temp. to give a liquid with the same composition as that of the solid (FeCl 3 -H2O) </li></ul></ul><ul><ul><li>(iii) and form an unstable compound which melts at a temperature lower than its melting point to give a new solid and a melt which is different from the compound (Na 2 SO 4 -H 2 O) </li></ul></ul>Dr. Ruchi S. Pandey
16. 16. Simple Eutectic System <ul><li>Some important elements of this phase diagram include </li></ul><ul><ul><li>Solidus : boundary below which no liquid phase exists. </li></ul></ul><ul><ul><li>Liquidus : boundary above which there are no solid phases. </li></ul></ul><ul><ul><li>Two solid+liquid fields between the solidus and the liquidus in which one of the two solids plus a liquid is present. </li></ul></ul><ul><ul><li>Eutectic point : a point at which both of the solids and a liquid (three phases) coexist </li></ul></ul><ul><li>general shape: The freezing point of each end-member is depressed by a foreign substance </li></ul>Dr. Ruchi S. Pandey <ul><li>For Ag-Pb system </li></ul><ul><ul><li>Tm(Ag)=961, Tm(Pb)=327 </li></ul></ul><ul><ul><li>Te(Pb(s)-Ag(s)-melt)=303 </li></ul></ul><ul><ul><li>Eutectic composition:2.6% Ag, 97.4% Pb </li></ul></ul><ul><li>Pattison’s process </li></ul><ul><ul><li>Process of raising the proportion of silver in the alloy for its profitable recovery </li></ul></ul>
17. 17. 2 component system where a stable compound with congruent melting point is formed <ul><li>In such a system, compounds are composed of various ratios of the two end members (A & B), or the basic components of the system. </li></ul><ul><li>These end members are assumed to melt congruently. </li></ul><ul><li>The intermediate compound AB2 melts congruently, because at some temperature (the top of the AB2 phase boundary line) it coexists with a liquid of the same composition. </li></ul>Dr. Ruchi S. Pandey
18. 18. <ul><li>Peritectic point - The point on a phase diagram where a reaction takes place between a previously precipitated phase and the liquid to produce a new solid phase. </li></ul><ul><li>The intermediate compound in this diagram (XY2) however is incongruently melting. </li></ul><ul><li>Incongruent melting is the temperature at which one solid phase transforms to another solid phase and a liquid phase both of different chemical compositions than the original composition. </li></ul><ul><li>This can be seen in this diagram as XY2 melts to Y and liquid. </li></ul>Dr. Ruchi S. Pandey 2 component system where a stable compound with incongruent melting point is formed
19. 19. Multi-Component Systems <ul><li>2-component systems </li></ul><ul><ul><li>Liquid-Liquid, e.g. nitrobenzene, hexane etc. </li></ul></ul><ul><ul><li>Liquid-Solid, e.g. water + common-salt </li></ul></ul><ul><ul><li>Solid-Solid, e.g. Tin-Lead solder </li></ul></ul><ul><li>3-component systems </li></ul><ul><ul><li>L-L-L, e.g. water + acetic acid + butanol </li></ul></ul><ul><ul><li>S-S-S, e.g. Steel (see figure) </li></ul></ul>Dr. Ruchi S. Pandey
20. 20. Dr. Ruchi S. Pandey Thank You