Lecture 12: Phase diagrams

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Lecture 12 Phase diagrams

• PHYS 430/603 material
• Laszlo Takacs
• UMBC Department of Physics

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Phases and the phase rule

• Phase Uniform agglomeration of material - no
  boundaries inside
• Gibbs phase rule f n - p 2
• f degrees of freedom, i.e. number of freely
  selectable thermodynamic parameters. Typically
  temperature, pressure, concentrations.
• n number of chemical components.
• p number of phases in equilibrium
• When phases are in equilibrium, atoms of any
  component can be transferred between phases.
  Equilibrium requires equal chemical potentials,
  mathematically equations that reduce the number
  of freely selectable parameters.

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The phase diagram of UF6 as a function of p and
T n 1, thus f 3 - p
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Pressure-temperature phase diagram of single
component systems sulfur and silicon dioxide.n
1, thus f 3 - p
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The T-p phase diagram of FeUsually processes
are carried out at atmospheric pressure, thus we
will usually neglect the pressure dependence.
Extreme high pressure does cause phase changes.
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• n 2, thus f 4 - p
• The phase diagram of a two-component system is
  usually shown for a fixed - ambient - pressure.
  The remaining variables are temperature and the
  concentration of either component. Notice
• allotropic phases (Ti),
• solid solutions,
• compound phases,
• two-phase regions,
• transformations,
• etc.

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The lever rule on the amount of each phase in a
two-phase region

• Sample is 1 mole total, contains
• (1 - c) mole of component A
• c mole of component B
• We can also consider the sample as
• m? mole of phase ? with c? plus
• m? mole of phase ? with c?
• Total of component A in the sample
• 1 - c m? (1 - c?) m? (1 - c?)
• Total of component B in the sample
• c m? c? m? c?
• The sum of these two equations gives
• 1 m? m? as it should be.
• From the last two equations we get
• m? (c? - c) / (c? - c?)
• m? (c - c?) / (c? - c?)
• m? / m? (c? - c) / (c - c?)

?
?
c? c c?
c cB
A
B
Notice that c? is not the same as cA, even if
phase ? is a solid solution based on A.
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What determines the phase diagram?

• Equilibrium is given by the minimum of the Gibbs
  potential, G. If more than one phase are
  possible, the phase with the lowest G is the
  equilibrium state.
• G H - TS At low temperature enthalpy dominates.
• Entropy becomes increasingly important at
  higher T.
• Ideal solution Two chemically similar components
• G H - T (Sv SM) GM - TSM
• Assume
• GM changes linearly from A to B.
• Configurational entropy SM -Nk c ln c (1-c)
  ln (1-c)
• (same idea as with vacancies)
• The slope of SM goes to /- 8 when c --gt 0 or 1

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G(T,c) of an ideal solution

• GM is a simple weighted average of the Gibbs
  potential of the components and ?Sm is the
  configurational entropy.
• Notice that the entropy term becomes more
  important with increasing temperature, increasing
  the curvature of the G(c) curve.

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• Comparing G for the ideal solution and the
  similar G for the liquid phase allows the
  construction of a hypothetical phase diagram.
• Notice that S is lower for low, L is lower for
  high temperature.
• Notice the changing curvature.
• In intermediate states, neither L nor S gives the
  smallest possible G. The state is a mixture of
  the two common tangent gives the concentrations,
  the lever rule gives the relative amounts.

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Why the common tangent?

• For an average concentration of c, we have x
  fraction with c1 and (1 - x) with c2.
• c x c1 (1 - x) c2, thus
• x (c2 - c) / (c2 - c1) 1 - x (c - c1) / (c2
  - c1)
• The Gibbs potential for the mixture
• G G1 x G2 (1 - x)
• G1 (c2 - c) / (c2 - c1) G2 (c - c1) / (c2
  - c1)
• This is a linear equation that describes a
  straight line between the end points. We get the
  lowest G when finding the lowest straight line
  still having end points on the G(c) curve - that
  is the common tangent.

G1
G2
c1 c c2
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The Ag-Au system is close to an ideal solution.
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Regular solutions

• Beside mixing entropy, differences in nearest
  neighbor bond energy provide a variation of the
  enthalpy term HAA, HBB, HAB.
• Suppose N atoms (A and B together), z nearest
  neighbors, c is concentration of B.
• NAA 1/2 N (1-c) z(1-c) 1/2 (total A)
  (mean A neighbor)
• NBB 1/2 N c z c
• NAB N z c (1-c)
• Hm NAAHAA NBBHBB NABHAB 1/2 Nz (1-c)HAA
  cHBB 2c(1-c)H0
• where H0 HAB - 1/2(HAA HBB)
• The last term in Hm has a maximum for H0 gt 0,
  competes with the minimum caused by
  configurational entropy. The slope of entropy is
  /- 8 at the borders, the slope of Hm is finite.
• Can result in limited solubility.
• G Hm - TSm

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• Gibbs free energy for a regular solution
• H0 lt 0 makes Gm sharply peaked - ordering,
  compound formation
• H0 gt 0 can result in two minima. For c1 lt c lt c2,
  the lowest energy state is a mixture - phase
  separation, limited mutual solubility.
• If HAA HBB and c ltlt 1, the solubility limit
  from dG/dc 0 is
• c1 exp(-zH0/kT)