Phase Equilibria Lectures GIK Institute Pakistan.

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The Gibbs free energy of a system is defined by the equation G=H-TS Thermodynamic Function Enthalpy is a measure of the heat content of the system and is given H=E+PV The internal energy arises from the total kinetic and potential energies of the atoms within the system. Kinetic energy can arise from atomic vibration in solids or liquids and from translational and rotational energies for the atoms and molecules within a liquid or gas whereas potential energy arises from the interactions, or bonds. between the atoms within the system.
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Phase Equilibria Lectures GIK Institute Pakistan.

Transcript of Phase Equilibria Lectures GIK Institute Pakistan.

  • The Gibbs free energy of a system is defined by the equation

    G=H-TS

    Thermodynamic Function

    Enthalpy is a measure of the heat content of the system and is given

    H=E+PV

    The internal energy arises from the total kinetic and potential energies of the atoms within the system.

    Kinetic energy can arise from atomic vibration in solids or liquids and from translational and rotational energies for the atoms and molecules within a liquid or gas whereas potential energy arises from the interactions, or bonds. between the atoms within the system.

  • If a transformation or reaction occurs the heat that is absorbed or evolved will depend on

    the change in the internal energy of the system.

    However it will also depend on changes in the volume of the system and the term PV

    takes this into account, so at constant pressure the heat absorbed or evolved is given by

    the change in H

    Thermodynamic Function

    When dealing with condensed phases (solids

    and liquids) the PV term is usually very small in

    comparison to E, that is H = E.

    G includes entropy (S) which is a measure of the randomness/disorder of the system.

  • A system is said to be in equilibrium when it is in the most stable state. i.e. shows no

    desire to change. An important consequence of the laws of classical thermodynamics is

    that at constant temperature and pressure a closed system (i.e. one of fixed mass and

    composition) will be in stable equilibrium if it has the lowest possible value of the Gibbs

    free energy, or in mathematical terms

    dG=0

    G, that the state with the highest stability will be that with the best compromise

    between low enthalpy and high entropy.

    Equilibrium and Free Energy

  • Thus at low temperatures solid phases are most stable since they have the strongest atomic binding and therefore the lowest internal energy (enthalpy). At high temperatures however the - TS term dominates and phases with more freedom of atom movement, liquids and gases, become most stable.

    Equilibrium and Free Energy

    dG=0

  • Graphite and diamond at room temperature and pressure are examples of stable and

    metastable equilibrium states. Given time. therefore. all diamond under these conditions

    will transform to graphite.

    Equilibrium and Free Energy

    Any transformation that results in a decrease in Gibbs free energy is possible. Therefore

    a necessary criterion for any phase transformation is

    G=G2-G1

  • Intensive & Extensive Properties

    All thermodynamics function can be divided into two types of properties:

    Intensive and Extensive Properties

    Intensive Properties

    Independent of size of system

    Example : T and P

    Extensive Properties Are directly proportion to the quantity of material in a system Example: E,H,V,G and S

  • Single Component System

    Phase changes can be induced in a single component system by changes in temperature

    at a fixed pressure, say 1 atm.

    A single component system could be one containing a pure element or one type of

    molecule that does not dissociate over the range of temperature of interest.

    To predict the phases that are stable or mixtures that are in equilibrium at different

    temperatures ,It is necessary to be able to calculate the variation of G with T

  • Gibbs Free Energy as a Function of Temperature

    Variation of Cp (Sp. heat) with temperature

    Variation of Enthalpy(H) with temperature Variation of Entropy (S) with temperature

  • Gibbs Free Energy as a Function of Temperature

  • Variation of enthalpy and free energy for liquid and solid

    phases of pure metal .L is the latent heat of melting and Tm is

    the equilibrium melting temperature

    At low temperatures GL > GS.

    However, the liquid phase has a

    higher entropy than the solid phase

    and the Gibbs free energy of the

    liquid therefore decreases more

    rapidly with increasing temperature

    than that of the solid.

    Enthalpy & Free energy for liquid & solid phases of Pure Metal

    Tm is the melting Temperature.

  • Free energy for liquid & solid phases of Pure Metal

    The phase with the lowest free energy at

    a given temperature will be the most

    stable.

    It shows that below the melting

    temperature the solid phase is most

    stable, and above this temperature the

    liquid phase is stable.

    At the melting temperature, where the

    two curves cross, the solid and liquid

    phases are in equilibrium.

  • Free energy and isomorphous diagram

    Molar quantities: g = G/mole h = H/mole s = S/mole

    Xi = mole fraction of i = Ni / Ni

    If the (molar) gibbs free energy of pure A is gA, and that of pure B is gB, then the (molar) gibbs free energy for the combination of pure components is

    g (pure, combined) = gAXA + gBXB

  • Free energy and isomorphous diagram

    As a function of composition, the

    gibbs free energy for the combination

    of pure A and pure B is a straight line

    connecting gA and gB

  • Free energy and isomorphous diagram

    Now, lets remove the imaginary partition and let

    the A and B atoms mix. There should be some

    change in g due to this mixing is:

    gmix = hmix - Tsmix

    The enthalpy term, hmix, represents the

    nature of the chemical bonding, or the extent

    to which A prefers B, or A prefers A as a

    neighbor.

  • Free energy and isomorphous diagram

    gmix = hmix - Tsmix

    The entropy term, smix, signifies the increase in

    disorder in the system as we let the A and B atoms

    mix. It is independent of the nature of the chemical

    bonding.

    Isomorphous, binary system as an ideal solution.

    That is, A couldnt care less whether it is sitting next

    to another A to a B atom. Like Ni-Cu system. Under

    these circumstances,

    hmix = 0.

  • Free energy and isomorphous diagram

    Let us qualitatively examine the entropy of mixing. If

    we have a system of pure A, and let the A atoms mix

    with one another, there is no increase in entropy

    because they were mixed to begin with. The same

    holds true for a system of pure B.

    If we have just one atom of B in a mole of A, removing

    the invisible partition hardly changes the amount of

    disorder. But, as we get to a 50:50 composition of A:B,

    the increase in entropy as we allow the system to mix

    is enormous. Therefore, smix is:

  • Free energy and isomorphous diagram

    From that we can easily get the change in gibbs

    free energy due to mixing:

    gmix = hmix - Tsmix

    hmix =0

    gmix = - Tsmix

  • Free energy and isomorphous diagram

    sum g(pure, combined) and

    gmix to get the total gibbs

    free energy of the solution

    as a function of composition:

  • At this point we have the general shape of the

    g(XB) curves for phases in a binary system. This

    overall shape holds for both the solid and liquid

    phases, so long as both make ideal (or close to

    ideal) solutions.

    Now compare gsol(XB) with gliq(XB) for various

    temperatures, and examine how these correlate

    to the phase diagram.

    Free energy and isomorphous diagram

  • Free energy and isomorphous diagram

    (1) High temperature:

    At temperatures above the melting points of both pure A and pure B, the liquid is the stable phase for all compositions.

    Therefore, gsol(XB) > gliq(XB) and the

    gibbs free energy curves look like:

    (2) Low temperature:

    At temperatures below the melting points of both pure A and pure B, the solid is the stable phase for all compositions. Therefore, gsol(XB) < gliq(XB) and the gibbs free energy curves look like:

  • Free energy and isomorphous diagram

    (3) Intermediate Temperature:

    As the temperature is brought down from high

    to low, the gsol(XB) starts to move below that

    of gliq(XB). The minima in the two curves, in

    general, do not occur at the same point, so for

    some compositions, gsol(XB) > gliq(XB), while

    for others gsol(XB) < gliq(XB). Therefore, at

    temperatures between the melting points of

    pure A and pure B, the solid and liquid curves

    look like:

  • Free energy and isomorphous diagram

    Lets say we start out with a liquid of

    composition XBO and cool it to To.

    The gibbs free energy of the liquid would be

    given by point (1). The system realizes it

    could lower its gibbs free energy by

    transforming to a solid.

    The gibbs free energy of that solid would be

    given by point (2) on the g(XB) diagram.

    But, how low can you go?

  • Free energy and isomorphous diagram

    The system can, in fact, lower its free energy

    even further by splitting up into a solid of

    composition XBS and a liquid of composition

    XBL.

    The gibbs free energy of the solid is given by

    point (4) on the g(XB) diagram and that of

    the liquid by point (5) on the same diagram.

  • Free energy and isomorphous diagram

    A system of overall composition XBo (at To). Then,

    we only have solid phase present, and the gibbs

    free energy is given by point (4) If system has of

    overall composition XBL, then we only have liquid

    phase present and the total gibbs free energy is

    given by point (5).

    If a system of a composition exactly in the

    middle of XBS and XBL , then half the moles of

    our system would be in the solid phase and the

    other half in the liquid phase.

    The gibbs free energy would be given by a point

    half-way between points (4) and (5), sitting

    precisely on the line that connects them.

  • Free energy and isomorphous diagram

  • Isomorphous system with Free energy

    We have 2 phases liquid and solid. Lets consider Gibbs free energy curves for the two phases at

    different T

    T1 is above the equilibrium melting

    temperatures of both pure components:

    T1 > Tm(A) > Tm(B) the liquid phase will be the stable phase for any

    composition

  • Isomorphous system with Free energy

    Decreasing the temperature below T1 will have two effects:

    1. GA Liquid and G B

    liquid will increase more rapidly than GA solid and G B

    Solid

    2. The curvature of the G(XB) curves will decrease.

    Eventually we will reach T2 melting point of pure component A, where

    GA Liquid = G A

    Solid

  • Isomorphous system with Free energy

    For even lower temperature T3 < T2 = Tm(A) the Gibbs free energy curves for the

    liquid and solid phases will cross.

    The common tangent construction

    can be used to show that for

    compositions near cross-over of G

    solid and G liquid, the total Gibbs free

    energy can be minimized by

    separation into two phases.

  • As temperature decreases below T3 GA Liquid and G B

    liquid continue

    to increase more rapidly than GA solid and G B

    Solid

    Therefore, the intersection of the Gibbs free

    energy curves, as well as points X1 and X2 are

    shifting to the right, until, at T4 = Tm(B) the

    curves will intersect at X1 = X2 = 1

    At T4 and below this temperature the Gibbs free

    energy of the solid phase is lower than the G of

    the liquid phase in the whole range of

    compositions the solid phase is the only stable

    phase.

    Isomorphous system with Free energy

  • Based on the Gibbs free energy curves we can now

    construct a phase diagram for a binary isomorphous

    systems

    Isomorphous system with Free energy

  • Binary Solutions with Miscibility Gap

    Lets consider a system in which the liquid phase

    is approximately ideal, but for the solid phase we

    have Hmix > 0

    i:e. the A and B atoms 'dislike' each other.

  • Binary Solutions with Miscibility Gap

    Therefore at low temperatures (T3) the free energy

    curve for the solid assumes a negative curvature in

    the middle.

  • Binary Solutions with Miscibility Gap