chapter 9 molar phase diagrams. molar phase diagram : what happens if the potentials in a phase...
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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
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