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Some applications of isoactivity lines
SOME
APPLICATIONS MATS
OF
ISOACTIVITY
LINES*
HILLERTt
A connection between phase boundaries and isoactivity lines in isothermal sections of ternary is shown. Thereby laws relating to these diagrams can be proved in a very simple way. The connection between the slope of tie-lines and the change in activity along a phase boundary strated qualitatively and a quantitative formula is derived. CERTAINES
APPLICATIONS
DES
LIGNES
On indique l’existence d’un rapport entre les limites des phases isothermes de diagrammes ternaires. De la, les lois se rapportant tres facilement. Le rapport existant entre la pente des lignes de conjugaison limite de phase, est dCmontr6 qualitativement; on deduit aussi EINIGE
ANWENDUNGEN
VON
diagrams is demon-
D’ISOACTIVITI?
et les lignes d’isoactivid dans des sections a ces diagrammes peuvent etre demontrees et la variation de l’activite une formule quantitative.
le long d’une
ISOAKTIVITATSKURVEN
Es wird eine Beziehung zwischen den Phasengrenzen und den Isoaktivitatskurven in den isothermen Bereichen der ternlren Diagramme gezeigt. Auf Grund dessen lassen sich die diese Diagramme betreffenden Gesetzmassigkeiten sehr einfach beweisen. Die Beziehung zwischen der Steigung der Konoden und der Aktivitltsanderung langs der Phasengrenze wird qualitativ erllutert, und es wird eine quantitative Formel abgeleitet.
have been drawn in two different ways in Figs. 1 and 2. It is evident that Fig. 2 shows an impossible arrangement, since a straight line from the component is cut three times by an isoactivity line. The direction of tielines can thus give some information about isoactivity lines in one-phase regions. By means of this a better understanding has been gained of some phenomena in alloyed iron-carbon-systems.3 Some other uses of isoactivity lines will be demonstrated in this paper.
INTRODUCTION
An isoactivitv line in an isothermal section of a ternary diagram-is the locus of all points which represent a certain chemical activity of a component. Such lines afford a very suitable means of presenting the results of activity measurements and have been used for this purpose by, among others, R. P. Smith’ and J. Chipman.2 Isoactivity lines also have some general properties which might be of practical value. It has been shown3*4 that an isoactivity line for a component cannot intersect a straight line from the component more than once. Figures 1 and 2 show a simple ternary diagram with two one-phase regions and one two-phase region. Tie-lines have been drawn in the two-phase region. The component represented by the right corner is considered. It is self-evident that each tie-line is a part of an isoactivity line, since different compositions along it consists of phases of the same compositions, only in different amounts. The continuations of these isoactivity lines in the one-phase regions
FIG. 2. Impossible arrangement of isoactivity the right corner component. I. CONNECTION AND
FIG. 1. Possible arrangement the right corner
of isoactivity component.
METALLURGICA,
VOL.
3, JANUARY
lines for
1955
LINES
Figure 3 shows a two-phase region. Three tie-lines have been drawn and the middle one has been extended to the right. Consider a component P, arbitrarily chosen on this extension. Each tie-line must be part of an isoactivity line for this component (as well as for all other components). Suppose the continuations of these isoactivity lines in the one-phase region form angles with the phase boundary ab as Fig. 3a demonstrates. Such a straight line can then be drawn from the component, that intersects one of the isoactivity lines twice (in d and e). This is forbidden, as already mentioned in
* Received May 28, 1954; in revised form June 22., 1954. assachusetts Institute of Technology, Cambridge, Massach:s:ts. ACTA
BETWEEN ISOACTIVITY PHASE BOUNDARIES
lines for
34
HILLERT:
ISOACTIVITY
LINES
35
the introduction. The only way of avoiding this error is to let the isoactivity line through g be tangent to the phase boundary at this point as Fig. 3b demonstrates. II. APPEARANCE
OF TERNARY
FIG. 3. Isoactivity lines (i) for a component P, chosen on the extension of a tie-line. (a) Impossible case. (b) Possible case.
the two tie-lines from the corner. The isoactivity lines for A and B (i, and in) in the one-phase region must be tangent to the phase boundaries at the corner. The right part of Fig. 4 can thus be constructed from the left part. If the B content is increased in the sample without formation of a new phase, then the A activity is decreased, since isoactivity lines for .4 further away from .4 than in are reached. Thus, aa, --<0 dnB
and
6 G
DIAGRAMS
The laws relating to ternary diagrams have been extensively treated by Schreinemakers.5 Intersections between two phase boundaries in isothermal sections of ternary diagrams were later treated by Lipson and Wilson6 in a simpler way. Section I in this paper now offers a possibility of an even simpler way of proving the laws relating to isothermal sections. Figure 4 demonstrates this for a specific case. Consider a sample with the composition represented by the upper corner of the three-phase region. Two components, il and B, are chosen on the extensions of
8 ln aA -----0. anB
The B-activity is, however, increased if the A content is increased, since then isoactivity lines for B closer to B than iR are reached. Thus (a lnae)/(&A)>O. This is impossible since the two derivatives must be equal : ah
aB 1 d2G a h a,J =_. ---= RT dnna%B he ’ an,
The left part of Fig. 4 shows therefore an impossible case. The extensions of the two phase boundaries must lie either both within the three-phase region [(a lnae)/(&zn) and (a lnaA)/(&ze) are both negative] or one on each side [(a hB)/(anA) and (a haA)/(a%e) are both positive]. Figure 5 illustrates the last case.
n !4 c9
I
A :
\
X8
FIG. 4. Impossible appearance of ternary diagram. III. CHANGE
OF ACTIVITY ALONG BOUNDARIES
PHASE
It was mentioned in the introduction that the direction of the tie-lines in a two-phase region gives some information about the direction of the isoactivity lines in the one-phase regions. At the same time, the direction of tie-lines also gives information about the change of activity along the phase boundariesP The activity will thus decrease upwards along the phase boundaries in Fig. 1 since each point on the phase boundaries lies on an isoactivity line further away from the component (in this case the right corner) than any lower point. This change in activity can in fact be computed quantitatively. Let (x, y, z) and (x’, y’, z’) be the composition (in mole fraction) of two points, one on each phase boundary and both on the same tie-line. The activities al, a2 and aa of the three components must be identical in the two points. The Gibbs-Duhem equation gives :
a,
xd lnai+yd
lnaz+zd
lnaa=O,
(1)
x’d lnar+y’d
lnag+z’d
lna3=0,
(2)
can be eliminated: (xz’-x’z)d
In ai+ (yz’-y’z)d d In al --= d In a2
-
1n a2=0,
(3)
yz’- y’z xz’- 2’2
(4)
Notice that z=l-x-y and z’=l--x’- - y’. The coordinates for one of the points can be substituted by the slope of the tie-line, which is
if x and y are considered as rectangular coordinates. (It is always possible to choose a right-angled triangle for a ternary representation.) The result, which is of general
FIG. 5. Possible appearance of ternary diagram.
ACTA
36
validity,
METALLURGICA,
__= d In a?
-
k+y-kx
1-y+kx’
3,
19.55
X-axis,‘i.e., yO=O. Let ulo be the activity at this ‘point.
will be: d In al
VOL.
of component
1
(6)
In :=
-[k(l-x)+y],
.1°
It is easy to show geometrically that k+ykx is equal to zero if component 1 lies on the extension of the tie-line with the slope k. Equation (6) then gives d lnal =0 and the activity of component 1 does therefore not change in the direction of the phase boundary. This is a rigorous proof of the statement in Section I that phase boundary and isoactivity lines are tangent at the end points of the tie-line, if the component has been chosen on its extension. Equation (6) can be simplified by some approximations. Consider only dilute solutions, i.e., 1 -y+ kel, and assume Henry’s law holds for component 2 at the point (x, y, z), i.e. d lnaz=dy/y. Equation (6) then simplifies to d In al (7) d?,= -[ ;(l-X)+l], which can be integrated easily if k/y is a constant [related to the partition coefficient K for component 2 between the two phases by k/y= (K- 1)/(x’-XX)]. This might often be a good approximation. We integrate Eq. (7) from the end-point of the phase boundary on the
where al is the activity at an arbitrary point on the phase boundary, k is the slope of the tie-line from this point and y is its mole fraction of component 2. If the mole fraction of component 1 varies along the phase boundary, x should be some mean value. ACKNOWLEDGMENT
The author wishes to thank Professor E. Rudberg, Director of the Swedish Institute for Metal Research, for many valuable discussions during the course of this work and Professor C. Wagner of M.I.T. for his assistance in simplifying the derivations in Section III. REFERENCES 1. R. P. Smith, J. Am. Chem. Sot. 70, 2724 (1948). J. Chipman, Disc. Faraday Sot. 4, 23 (1948). 3”: M. Hillert, Jernkontorets Annaler 136, 25 (1952). 4. E. Rudberg, Jernkontorets Annaler 136, 91 (1952). 5. F. A. Schreinemakers, “Die tern&en Gleichgewichte,” Braunschweig, 1911, 1913. (Part III of Roozeboom, “Die Heterogenen Gleichgewichte.“) 6. H. Lipson and J. C. Wilson, J. Iron and Steel Inst. 142, 107 (1940).
APPLICATIONS MATS
OF
ISOACTIVITY
LINES*
HILLERTt
A connection between phase boundaries and isoactivity lines in isothermal sections of ternary is shown. Thereby laws relating to these diagrams can be proved in a very simple way. The connection between the slope of tie-lines and the change in activity along a phase boundary strated qualitatively and a quantitative formula is derived. CERTAINES
APPLICATIONS
DES
LIGNES
On indique l’existence d’un rapport entre les limites des phases isothermes de diagrammes ternaires. De la, les lois se rapportant tres facilement. Le rapport existant entre la pente des lignes de conjugaison limite de phase, est dCmontr6 qualitativement; on deduit aussi EINIGE
ANWENDUNGEN
VON
diagrams is demon-
D’ISOACTIVITI?
et les lignes d’isoactivid dans des sections a ces diagrammes peuvent etre demontrees et la variation de l’activite une formule quantitative.
le long d’une
ISOAKTIVITATSKURVEN
Es wird eine Beziehung zwischen den Phasengrenzen und den Isoaktivitatskurven in den isothermen Bereichen der ternlren Diagramme gezeigt. Auf Grund dessen lassen sich die diese Diagramme betreffenden Gesetzmassigkeiten sehr einfach beweisen. Die Beziehung zwischen der Steigung der Konoden und der Aktivitltsanderung langs der Phasengrenze wird qualitativ erllutert, und es wird eine quantitative Formel abgeleitet.
have been drawn in two different ways in Figs. 1 and 2. It is evident that Fig. 2 shows an impossible arrangement, since a straight line from the component is cut three times by an isoactivity line. The direction of tielines can thus give some information about isoactivity lines in one-phase regions. By means of this a better understanding has been gained of some phenomena in alloyed iron-carbon-systems.3 Some other uses of isoactivity lines will be demonstrated in this paper.
INTRODUCTION
An isoactivitv line in an isothermal section of a ternary diagram-is the locus of all points which represent a certain chemical activity of a component. Such lines afford a very suitable means of presenting the results of activity measurements and have been used for this purpose by, among others, R. P. Smith’ and J. Chipman.2 Isoactivity lines also have some general properties which might be of practical value. It has been shown3*4 that an isoactivity line for a component cannot intersect a straight line from the component more than once. Figures 1 and 2 show a simple ternary diagram with two one-phase regions and one two-phase region. Tie-lines have been drawn in the two-phase region. The component represented by the right corner is considered. It is self-evident that each tie-line is a part of an isoactivity line, since different compositions along it consists of phases of the same compositions, only in different amounts. The continuations of these isoactivity lines in the one-phase regions
FIG. 2. Impossible arrangement of isoactivity the right corner component. I. CONNECTION AND
FIG. 1. Possible arrangement the right corner
of isoactivity component.
METALLURGICA,
VOL.
3, JANUARY
lines for
1955
LINES
Figure 3 shows a two-phase region. Three tie-lines have been drawn and the middle one has been extended to the right. Consider a component P, arbitrarily chosen on this extension. Each tie-line must be part of an isoactivity line for this component (as well as for all other components). Suppose the continuations of these isoactivity lines in the one-phase region form angles with the phase boundary ab as Fig. 3a demonstrates. Such a straight line can then be drawn from the component, that intersects one of the isoactivity lines twice (in d and e). This is forbidden, as already mentioned in
* Received May 28, 1954; in revised form June 22., 1954. assachusetts Institute of Technology, Cambridge, Massach:s:ts. ACTA
BETWEEN ISOACTIVITY PHASE BOUNDARIES
lines for
34
HILLERT:
ISOACTIVITY
LINES
35
the introduction. The only way of avoiding this error is to let the isoactivity line through g be tangent to the phase boundary at this point as Fig. 3b demonstrates. II. APPEARANCE
OF TERNARY
FIG. 3. Isoactivity lines (i) for a component P, chosen on the extension of a tie-line. (a) Impossible case. (b) Possible case.
the two tie-lines from the corner. The isoactivity lines for A and B (i, and in) in the one-phase region must be tangent to the phase boundaries at the corner. The right part of Fig. 4 can thus be constructed from the left part. If the B content is increased in the sample without formation of a new phase, then the A activity is decreased, since isoactivity lines for .4 further away from .4 than in are reached. Thus, aa, --<0 dnB
and
6 G
DIAGRAMS
The laws relating to ternary diagrams have been extensively treated by Schreinemakers.5 Intersections between two phase boundaries in isothermal sections of ternary diagrams were later treated by Lipson and Wilson6 in a simpler way. Section I in this paper now offers a possibility of an even simpler way of proving the laws relating to isothermal sections. Figure 4 demonstrates this for a specific case. Consider a sample with the composition represented by the upper corner of the three-phase region. Two components, il and B, are chosen on the extensions of
8 ln aA -----0. anB
The B-activity is, however, increased if the A content is increased, since then isoactivity lines for B closer to B than iR are reached. Thus (a lnae)/(&A)>O. This is impossible since the two derivatives must be equal : ah
aB 1 d2G a h a,J =_. ---= RT dnna%B he ’ an,
The left part of Fig. 4 shows therefore an impossible case. The extensions of the two phase boundaries must lie either both within the three-phase region [(a lnae)/(&zn) and (a lnaA)/(&ze) are both negative] or one on each side [(a hB)/(anA) and (a haA)/(a%e) are both positive]. Figure 5 illustrates the last case.
n !4 c9
I
A :
\
X8
FIG. 4. Impossible appearance of ternary diagram. III. CHANGE
OF ACTIVITY ALONG BOUNDARIES
PHASE
It was mentioned in the introduction that the direction of the tie-lines in a two-phase region gives some information about the direction of the isoactivity lines in the one-phase regions. At the same time, the direction of tie-lines also gives information about the change of activity along the phase boundariesP The activity will thus decrease upwards along the phase boundaries in Fig. 1 since each point on the phase boundaries lies on an isoactivity line further away from the component (in this case the right corner) than any lower point. This change in activity can in fact be computed quantitatively. Let (x, y, z) and (x’, y’, z’) be the composition (in mole fraction) of two points, one on each phase boundary and both on the same tie-line. The activities al, a2 and aa of the three components must be identical in the two points. The Gibbs-Duhem equation gives :
a,
xd lnai+yd
lnaz+zd
lnaa=O,
(1)
x’d lnar+y’d
lnag+z’d
lna3=0,
(2)
can be eliminated: (xz’-x’z)d
In ai+ (yz’-y’z)d d In al --= d In a2
-
1n a2=0,
(3)
yz’- y’z xz’- 2’2
(4)
Notice that z=l-x-y and z’=l--x’- - y’. The coordinates for one of the points can be substituted by the slope of the tie-line, which is
if x and y are considered as rectangular coordinates. (It is always possible to choose a right-angled triangle for a ternary representation.) The result, which is of general
FIG. 5. Possible appearance of ternary diagram.
ACTA
36
validity,
METALLURGICA,
__= d In a?
-
k+y-kx
1-y+kx’
3,
19.55
X-axis,‘i.e., yO=O. Let ulo be the activity at this ‘point.
will be: d In al
VOL.
of component
1
(6)
In :=
-[k(l-x)+y],
.1°
It is easy to show geometrically that k+ykx is equal to zero if component 1 lies on the extension of the tie-line with the slope k. Equation (6) then gives d lnal =0 and the activity of component 1 does therefore not change in the direction of the phase boundary. This is a rigorous proof of the statement in Section I that phase boundary and isoactivity lines are tangent at the end points of the tie-line, if the component has been chosen on its extension. Equation (6) can be simplified by some approximations. Consider only dilute solutions, i.e., 1 -y+ kel, and assume Henry’s law holds for component 2 at the point (x, y, z), i.e. d lnaz=dy/y. Equation (6) then simplifies to d In al (7) d?,= -[ ;(l-X)+l], which can be integrated easily if k/y is a constant [related to the partition coefficient K for component 2 between the two phases by k/y= (K- 1)/(x’-XX)]. This might often be a good approximation. We integrate Eq. (7) from the end-point of the phase boundary on the
where al is the activity at an arbitrary point on the phase boundary, k is the slope of the tie-line from this point and y is its mole fraction of component 2. If the mole fraction of component 1 varies along the phase boundary, x should be some mean value. ACKNOWLEDGMENT
The author wishes to thank Professor E. Rudberg, Director of the Swedish Institute for Metal Research, for many valuable discussions during the course of this work and Professor C. Wagner of M.I.T. for his assistance in simplifying the derivations in Section III. REFERENCES 1. R. P. Smith, J. Am. Chem. Sot. 70, 2724 (1948). J. Chipman, Disc. Faraday Sot. 4, 23 (1948). 3”: M. Hillert, Jernkontorets Annaler 136, 25 (1952). 4. E. Rudberg, Jernkontorets Annaler 136, 91 (1952). 5. F. A. Schreinemakers, “Die tern&en Gleichgewichte,” Braunschweig, 1911, 1913. (Part III of Roozeboom, “Die Heterogenen Gleichgewichte.“) 6. H. Lipson and J. C. Wilson, J. Iron and Steel Inst. 142, 107 (1940).