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# Phase Diagrams and Phase Rule

## by Ruchi Pandey, Teacher at Delhi on Feb 04, 2011

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## Phase Diagrams and Phase RulePresentation Transcript

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