Chapter 9

Molar phase diagrams

• molar phase diagram :

what happens if the potentials in a phase diagram are replaced by their conjugate molar quantities?

i.d. potential axis replaced by axis - a single phase field : the same general shape in a molar

phase diagram - a two-phase region : phases in equil with each other will no

longer fall on the same

- the same pt same values of T,-P, i in both phases, however the molar quantities are different between phases they are connected by a or konode (in Germany) in a molar phase diagram

• why separated ?

at equil in the T,-P, i diagram, on the surface of G-D,

T=T, -P=-P but Sm≠Sm

, Vm≠Vm

in G-D,

2

21

2111

cjj

m

c

iimm dYXdzdPVdTSd

In considering two phases, and in equil with each

other, the system is then moved away from equil by changing the value of one potential, Yj

jjm

jm dYXXd )()( 1111

supposing α is the phase favored by increased Yj value

0)(

0)(

1111

11

jjm

jm dY

dXX

d

곧 만큼의 차이 (difference) 가 에 나타난다 (shows up in the molar phase

diagram).

jm

jm XX 11

• Fig. 9.1

T Sm

(const P, tie-line perpendicular to -P axis)

-P Vm (const T, tie-line perpendicular to T axis)

likewise, if B XB(=NB/NA) then the same happens

one-phase fields separate and leave room for a field

• Fig. 8.11

4-phase field tetrahedral 3-phases field prismatic2-phase field 3D volume1-phase field 3D volume

T,-P, B (A obtained from G-D)

potential phase diagram:

a 4-phase field a point

S,V, xB(zB)

molar phase diagram:

• Fig. 9.4

The topology of potential phase diagrams is very simple and each geometrical element is a .

A phase diagram with only molar axes have a relatively simple topology. All the phase fields have the same as the diagram itself.

For the unary system all the phase fields have 2D and for the binary system they have 3D.

• phase boundaries - in a potential phase diagram, all the geometrical

elements of pt, line, plane being phase fields - but in a molar phase diagram, pts, lines and

sometimes planes playing the roles of phase ( 상경계 )

- in a 2D molar phase diagram, 4 kinds of phase regions

meet each other at each pt

α+β+γ, 3-phase 2D 요소α+β, β+γ, α+γ 2-phase 2D 요소α, β, γ 1-phase 2D 요소

same dimensionality

• in a 3D molar diagram a four-phase region : each corner of a tetrahedron

being connected to a one-phase field for example, at the pt related with single phase (Fig. 9.4)

4-phase : 1 region → 3-phase : 3 regions → 2-phases : 3 regions → 1-phase : 1 region →

∴ two molar axes three molar axes insert Fig. 9.8 insert Fig. 9.9

8 regions,8 phase fields

Fig. 8.5 (a) Elementary unit of a phase diagram with two molaraxes. (b) Topological equivalence.

Fig. 8.6 Elemetary unit of a phase diagram with three molar axes.

• in a 2D molar diagram, Masing (1949) → the # of phases in the phase fields changes by one unit in crossing a linear phase boundary

• generalization by Palatnik and Landau (1964) D+ + D- = r - b

D+ : # of phase that appear D- : # of phase that disappear r : # of axes in the molar diagram b : dimensionality of the phase boundary

crossed

• special phase boundaries liquid / liquid + solid liquid + solid / solid solid / solid1+ solid 2

(MPL boundary rule)

• sections of molar phase diagrams

- for practical reasons one likes to reduce the # of axes in a complete molar phase diagram

- sectioning at a constant value of a molar variable : isoplethal section or

- MPL boundary rule applicable to the sections

∵ r – b not changing by sectioning (r – b : r and b decreases by 1, respectively, each

sectioning, in the same way for ns sectioning)

ex) 3D (1 sectioning) → 2D r=3→2 phase boundary : volume b=3→2 phase boundary : plane b=2→1 phase boundary : line b=1→0

ⓐ ⓑ

ⓒ

in the elementary unit of a molar phase diagram sectioned a sufficient # of times to make it 2D

case of , Dⓐ + + D- = r – b = 2 (2D) - 1 (line) = 1 ∴ e-1 e or e-2 case of , Dⓑ + + D- = r – b = 2 (2D) - 1 (line) = 1 ∴ e e+1 or e-1 e-2 e-1 or e-3 case of , Dⓒ + + D- = r – b = 2 (2D) – 0 (pt) = 2 ∴ e+1 D+ = 2, D- = 0 e-1 D+ = 1, D- = 1 e-3 D+ = 0, D- = 2

which one is true in this case ? see the next slide!

•Fig. 9.11

• Schreinemakers’ rule (1912) - at const T & P, for isobarothermal sections of

ternary diagrams, extrapolations of one-phase field boundaries must either both fall inside three-phase fields or one inside each of two-phase fields

- Hillert (1985) generalized this in molar phase diagram

Schreinemacher under const T & P ∴ Gibbs free energy (G) be used Hillert generalized molar diagram ∴ based upon U (internal energy)

)( 1

ii

c

ii dNVdPSdTdGdNPdVTdSdU

j

k

Nkj

k

j

NjkNjk

i

c

ia

aa

NNU

N

NNU

NNNU

dNdXYdU

l

ll

2

2

1

treating T & -P like (chemical)potentials

same value & same sign

from Maxwell relation

- in a more general case, we shall denote k and j components

- if one of the two boundaries extrapolates outside the a-k-j triangle

see thin line above, Nj increases, closer to k → k increases → therefore

Fig. 9.14 b

- if the k boundary extrapolates into the triangle, then

0 ,0

k

j

j

k

NN

Fig 9.14 a

0

j

k

N

0

k

j

N

in the same way

- this rule applicable to equipotential sections (at const T & P) as well as mixed phase diagram

(the rule being satisfied at all the intersections in this diagram)

- it may be used as a convenient guide when other information is lacking

Fig. 9.15

• Fig. 9.17

- by now, we know that the phase field opposite to the starting one has the same number of phases, e-1 in this case

- e-3 & e+1 not possible according to Schreinemacher’s rule