Lecture 23 Phase Equilibrium

Solid-liquid equilibrium Gas - liquid/solid equilibrium Non-ideal systems and phase separation

Ideal solutions - solid-liquid In general for a two component system

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μAS = μA

L μBS = μB

L

Furthermore, assuming that F ≈ G, molar F is:

For ideal solutions

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μiα = (μ 0)i

α + kT ln X iα

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Fα = XA NAvμAα + XB NAvμB

α =

XA NAv (μ0)Aα + XB NAv (μ0)B

α + RT(XA ln XA + XB ln XB )

Ideal solutions - solid-liquid With XA = 1-XB

F as function of composition T > TmA TmB < T < TmA€

Fα = (1− XB )NAv (μ0)Aα + XB NAv (μ0)B

α + RT((1− XB )ln(1− XB ) + XB ln XB )

L

S

XB

FL

S

XB

F

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XBL

€

XBS

Ideal solutions - phase diagram calculations With

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lnXB

S

XBL

=(μ0)B

L − (μ0)BS

kT€

μBS = μB

L

Where

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(μ0)BL − (μ0)B

S =ΔG f

B

NAv

≈ΔF f

B

NAv

=1

NAv

ΔE fB − TΔS f

B[ ]

Heat and entropy change of fusion can be taken form experiment or from statistical mechanics formulas

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(μ0)BL − (μ0)B

S =(u0)B

L − (u0)BS

2− kT[3ln(ν S /ν L ) +1]

Ideal solutions - Phase Diagram

L

XB

T L+

Non-ideal systems - solid-solid phase separation

Two solid phases in equilibrium

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μBα = μB

β

From Eqs 1 and 2, noticing that there is a symmetry about X = 0.5 and that

From the Bragg-Williams approximation

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μBα ,β = (μ 0)B

α ,β + kT ln XBα ,β −

cw

2(1− XB

α ,β )2

(1)

(2)

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(μ 0)Bα = (μ 0)B

β

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T =

cw

2k(1− XB

α )2

lnXB

α

1− XBα

XB

T

+

Non - ideal case: Solid-liquid equilibrium

Liquid ideal, solid not

From equality of chemical potentials €

μBS = (μ 0)B

S + kT ln XBS −

cw

2(1− XB

S )2

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μBL = (μ 0)B

L + kT ln XBL

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kT lnXB

S

XBL

−cw

2(1− XB

S )2 = Δ(μ 0)B

Same for A component

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kT ln1− XB

S

1− XBL

−cw

2(XB

S )2 = Δ(μ 0)A

Solving for XB of solid and liquid gives the phase diagram

Ideal liquid - non-ideal solid phase diagram

L

XB

T

L+

+

L+