PARAMETRIC EQUATIONS I K MULTICOMPONEST SYSTEMS
547
PARAMETRIC EQUATIOWS I N T H E TREATMEST OF MULTICOMPONEKT SYSTEMS L . A . DAHL’ National Bureau of Standards, Washington, D.C. Received May 11, 1949 INTRODUCTION
A system of n componknts in an N-component system can be defined analytically by N parametric equations involving n - 1 paramet’ers.zIn each of these equations the percentage of a component is equated to an expression which is linear with respect to the parameter or parameters. This is advantageous, since it affords a means of direct computation of mixtures in the n-component system by assigning values to the parameters. Parametric equations are particularly useful when n is small and :V is large. For example, the parametric equations for a binary system in a system of many components are just as simple as the equations of a binary system in a ternary system, and are as simply derived. Equations of this type are useful in’ planning mixtures for study in phase equilibrium investigations of multicomponent systems. Interpretation of phase equilibrium data requires consideration of nonlinear relations, since curved lines, curved surfaces, etc., appear in phase diagrams and space models representing equilibrium conditions. Holyever, this paper will be concerned only with equations which are linear with respect to the parameters. In a previous paper (1) equations were derived for estimating proportions of phases at equilibrium in mixtures in the quaternary system CaO-5CaO .3A1@4Ca0 ~A1~03.Fez03-2CaC) .SO*, a system of interest in portland cement technology. The equations were derived by the methods described in this paper from the data obtained by Lea and Parker (4,5 ) in their investigation of the system. By substituting quaternary portland cement compositions in the equations the proportions of crystalline phases and liquid are obtained when the liquid is at an invariant point, on a univariant curve, or on an isotherm on a bivariant surface. With the exception of equilibrium at an invariant point, the methods used in deriving the equations were not described and appear here for the first time. TERMINOLOGY AND XOTATIOX
The entire range of compositions which may be formed by varying the proportions of any given set of substances (components) will be referred to as a system. In the broad sense in which the term “system” is used in this paper, the only restriction as to the substances which may be taken as components is t,hat no one of them should be capable of being formed from two or more of the others in any proportions, positive or negative. In considering systems within a system, Research Associate, Iiational Bureau of Standards, Washington, D.C. A parameter may be defined as “an independent variable through functions of which may be expressed other variables.” (Webster) For example, a curve in space may be expressed by three equations, in each of which one of the coordinates, z, y, and z, is equated t o a function of p , a parameter. Equations of this type are parametric equations. 1
548
L. A. DAHL
the latter will be referred to as the primary system and its components as primary components. Systems within the primary system will be referred to as subordinate systems and their components as subordinate components. The number of components in a primary system and the number in a subordinate system will be designated as N and n,respectively. When n = N , the subordinate system is also termed a secondary system (2). The systems to be considered in illustrative examples mill involve various combinations of the oxides CaO, AlpOa, Fe203,Si02, -MgO, SazO, and KeO as primary components. These will be designated as C, A, F, S, M, N , and K, respectively. Percentages of these oxides will be designated by corresponding letters in italics: the symbol A will represent the percentage of A, that is, of A1~03,etc. Compounds will be designated by formulas in which the oxides are treated as elements. For example, the compound CaO.AlZ0a.2SiO2is designated as CAS*. PARAMETERS AND PARAMETRIC EQU.4TION.S
Any curve, surface, or other geometrical figure which may be defined by an equation or equations involving variables representing distances from coordinate axes may also be defined by a series of parametric equations. These equations define each of the variables in terms of one or more independent variables or parameters. To illustrate, let US consider the parabola, X?
+
y2
- 122 - 1 3 -~ 22y + 86 = 0
(1)
The curve may also be represented by the pair of parametric equations,
+ 3p + 4 - 2p + 3 in which p is the parameter. If p 2 + 3 p + 4 and p* - 2 p + 3, respectively, are x
=
p*
?/ = p*
substituted for z and y in equation 1, all terms drop out, leaving the equation 0 = 0. It follows that if any value assigned t o p is substituted in equations 2 and 3, the values of z and y so obtained will satisfy equation 1. The parameter p must have a definite significance, since it is a function of the coordinates of points on the curve. In some instances a variable of known significance may be selected as a parameter, but this is not necessary. In the example given above it would be difficult to ascribe a simple and readily visualized significance to p , and it is sufficient to consider p to be merely an independent variable introduced to define the values of z and y at erery point on the curve. It will be noted that only one parameter appears in equations 2 and 3. This is true for any set of parametric equations defining the coordinates of points on a curved or straight line, regardless of the number of coordinates required for geometric representation. Let us now consider the line which is the intersection of the planes represented by the equations: 32 - 4 y + z + Gz-2v-z-
10 = 0 7 = 0
PARAMETRIC EQTATIOSS I S MULTICOMPOSEST SYSTEMS
549
The 2, y, and z coordinates of any point on the line satisfy both equations. The line may also be defined by the parametric equations, 2d 2 = - + 3 7 3d y=--+5 7 6d z = - + I 7
in which d is the distanre of any given point on the line from the point ( 3 , 5 ,I), which is also on the line. It will be noted that equations 4 and 5 are linear, and that parametric equations 6-8, representing the same line, are also linear. I n general, parametric equations in linear form may be derived to represent any relation capable of being expressed by linear equations in terms of coordinates. On the other hand, if nonlinear equations in terms of coordinates are required to express a given relation; a t least one of the corresponding parametric equations is nonlinear. DEGREES O F FREEDOM
Considering any set of variables, the number of degrees of freedom, F,is the number of variables which may be independently altered. For example, if 2, y, and z are the variables, and there exists no relation between them, F = 3. iln equation involving one or more of the variables reduces the number of degrees of freedom by one, so that F = 2 . That is, values may be assigned to only two of the variables, the value of the third then being fixed. Similarly, two equations further reduce the number of degrees of freedom, so that F = 1. I n treatises on the phase rule and its applications the number of degrees of freedom is usually defined as the number of variables-composition, pressure, and temperature-vhich may be altered independently vithout causing the disappearance of a phase or the appearance of a new phase. This definition is restricted to the variables involved in phase equilibria. I n this study the number of degrees of freedom is considered in a more general sense, as we shall have occasion to deal with composition relations without reference to equilibrium between phases. The number of degrees of freedom possessed by any geometrical figure may be defined as the number of dimensions in that figure. Regardless of the number of dimensions in the space in which a curve lies, a curve has only one dimension and therefore possesses one degree of freedom. A surface has two dimensions and therefore possesses two degrees of freedom. I n dealing with multicomponent systems it is important to bear in mind that a curve represents a univariant condition in any system, regardless of the number of components in the system, or the number of dimensions required for geometrical representation of the system and the additional variables of temperatures and pressure. Considered with reference to composition relations alone-that is, without these additional variables-a system may be said to possess a definite number of degrees of freedom.
550
L. A. DAHL
For example, a binary system, represented geometrically by a straight line, possesses one degree of freedom, whether it is in a ternary or quaternary system, or in a multicomponent system. The number of parameters required to define a given geometrical figure represented by one or more equations is equal to the number of degrees of freedom possessed by the figure. For example, one parameter is required to define a curve, and two parameters to define a surface, regardless of the number of dimensions of the space in which the curve or surface lies. An n-component system is represented geometrically in n - 1 dimensions, since the sum of the percentages of the components is 100 and there are therefore n 1 independent variables. Considered only with reference to composition, the system has n - 1 degrees of freedom, and may therefore be defined by a series of parametric equations involving n - 1 parameters. Equations representing an n-component system are linear.
-
DERIVATION AND PROPERTIES OF PARAMETRIC EQUATIONS
The general method of deriving parametric equations involves four successive steps: (1) determination of the number of parameters required; (2) selection of parameters; (3) setting up the equations with literal constants; and ( 4 ) evaluation of the constants. This procedure may be followed for either linear or nonlinear relations. The parameters may have no readily defined physical significance, or they may represent such variables as temperature, pressure, or the concentrations of primary or subordinate components. When the weight fractions or percentages of subordinate components are taken as parameters, a more direct method may be used. This method will be described first. DIRECT METHOD : PRIMARY COMPONEKTS IX TERMS O F SUBORDINATE COMPOKEKTS
In the investigation of phase equilibria in a multicomponent system it is convenient to study one subordinate system a t a time. For each subordinate system mixtures are planned for heat treatment or other experimental procedures. Weight fractions or percentages of (n - 1) subordinate components may be selected as parameters, with the advantage that limits on values of the parameters are easily formulated. To illustrate the direct method, let us derive equations for the ternary system KAS2-CMS2-C2AS in the quinary system K-C-W.44 (the system K20-Ca0-Mg0-A1203Si02).Let T and s represent weight fractions of KASz and CMS2,respectively, in any mixture of the three subordinate components. Then the weight fraction of C2ASis 1 - r - s. The percentage compositions of the subordinate components are given in table 1. The parametric equation for any given primary component is found by adding the products obtained by multiplying the percentage of that component in each subordinate component by the corresponding weight fraction. That is, K = 29.7% c = 25.90s 40.91(1 - T - s)
M = A = S =
+ 32.24~ + 37.18(1 37.9% + 55.48s + 21.91(1 18.62s
T T
- s) - S)
551
PARAMETRIC EQUATIONS I S MULTICOMPOXENT SYSTEMS
Simplifying,
K = 29.7% C = -4O.9lr
-
15.01s 18.62s A = -4.94r - 37.18s S = 16.07r 33.57s
M =
+
+ 40.91 + 37.18 + 21.91
Any assigned values of r and s substituted in equations 9-13 will give a composition in the system KAS~-CMSZ-CZASor in an extension of that system. Compositions will be restricted to that system if neither r nor s is negative s does not exceed 1. and T Upon adding equations 9-13, it is found that the terms involving parameters drop out, leaving the relation:
+
K
+ C + M + A + S = 100.00
~-
per
per cent
K C
SI A S
~
I ~
32 24 37 98
25 90 18 62
I
55 48
I
40 91 37 18 21 91
WEIGET PPACIION
I
I
per cent
ccn1
29 78
T
I
s
l
1 - T - 8
This constitutes a check on the accuracy of the computations required in deriving the equations. Equations in which the parameters are weight fractions or percentages of subordinate components are useful in many problems requiring the computation of mixtures in a given subordinate system. Their application to such problems is simplified through the fact that limits on values of the parameters are readily seen. The range of values which may be assigned to the parameters are not mutually independent, since their sum must not exceed 1.00 when weight fractions are used, or 100 when percentages are used. For instance, in this example, if r = 0.2, s may have values from 0.0 to 0.8. When variables other than subordinate components are taken as parameters the ranges of values of parameters are similarly dependent upon one another, but they are not so simply expressed nor so easily found. SUBORDINATE COMPONEh-TS I&- TERMS O F PRIMARY COMPONESTS
In the foregoing example the equation for each primary component is expressed in terms of the parameters T and s. To obtain equations for the weight
552
L. A. DAHL
fractions or percentages of the subordinate components in terms of two of the primary components, any two of these equations may be solved for T and s. The pair of equations selected should be those defining the primary components which are intended t o appear in the final equations. Equations in terms of K and iM are obtained directly, since K = 29.7% and M = 18.62s (equations 9 and 11). That is, Per cent KASZ = lOOr = 3.3580K Per cent CMS, = 100s = 5.3706;M By difference, Per cent CZAS = 100
- 3.3580K
- 5.3706iM
Equations in terms of C and 6 will be needed in a problem to be considered presently. These may be derived from equations 10 and 13, which are repeated below, with terms transposed. 40.91r 16.07r
+ 15.01s = -C + 40.91 + 33.57s = S - 21.91
(10)
(13)
Solving for r and s, Per cent KAS, Per cent CMS,
= =
+
lOOr = -1.32586 - 2.9662C 150.35 lOOr = 3.61396 1.4194C - 137.24
+
(14) (16)
By difference, Per cent C,AS = -2.28778
+ 1.5458C + 86.89
(16)
GENERAL METHOD
The general method, involving the successive steps indicated on page 560, may be applied in any case, including those previously discussed. The variables selected as parameterk may be composition variables, or they may be other variables, such as temperature and pressure. To illustrate the general method of deriving parametric equations, let us derive equations for the binary system CAS,-C,AS in the ternary system C-A-S (the system Ca0-L41~03-Si02).I n table 2 the percentage compositions of the subordinate components are given. Only one parameter is required for a binary system, and primary component S is selected. The percentages of S are indicated in the table as values of the parameter for each of the subordinate components. The literal equations for C and A are in the form indicated below for C, C=uS+b (17) in which a and b are constants to be evaluated. Substituting in equation 17 the values of C and S in the two subordinate components, the following equations are obtained:
+ +
43.18~ 6 = 20.17 2 1 . 9 1 ~ 6 = 40.91
553
PARAMETRIC EQUATIOKS I N MULTICOMPONEKT SYSTEMS
Solving for a and b, and substituting in equation 17: C = -0.97518 62.27
+
By the same procedure for A :
A
=
-0.02493
+ 37.73
Substitution of any assigned value of 3 in equations 18 and 19 will lead to a composition in the binary system CASi-CAS or in an extension of that system. TABLE 2 SUBORDINATE COMPONENTS PBIMABY COMPONENTS
I
C
CtAS
per cent
per cent
20.17 36,65
i
A
I
CAS
~
40.91 37.18 PABAK€I%B
S
I
43 18
p~
21 91
TABLE 3 SUBORDINATE COMPONENTS PBIMARY C O Y P O W N T S
KASi 1
K. . . . . . . . . . . . . . . . . .
1
hl . . . . . . . . . . . . . . . . . . . . A ....................
1
CiAS cent
0.00 0.00 37.18
29.78 18.62 0.00
32.24
~
~
$6,
1 9 C
CMSi
pcr cent
PABAMEIEPS
'i ::
55 48 ~
25 90
~
21 91 40 91
Since the percentages of B in C A S and CAS, are 21.91 and 43.18, respectively, compositions may be restricted to the system CAS,-C,AS by limiting S to values from 21.91 to 43.18. DETERMINATIOK O F LIMITS OK PARAMETERS
I t was mentioned previously (page 551) that when two or more parameters are required the ranges of values of the parameters are not mutually independent. As a basis for discussing 1 he determination of limits on the values which may be assigned to parameters, we shall consider a ternary subordinate system, which requires two parameters. We shall again derive equations for the ternary system KAS&MS,-CzAS in the quinary system K-C-M-AS, this time selecting S and C as parameters. The data are given in table 3.
554
L. A. DAHL
The parametric equations have the form indicated below for K .
K
= aS
+ bC + c
in which a, b, and c are constants to be evaluated. The procedure is the same as in the preceding example. The equations derived by that procedure are given below:
+ +
K = -0.39488 - 0.8830C 44.78 M = 0.67285 0.2643C - 25.56 A = -1.27803 - 0.3813C 80.78
+
Limits can be assigned to one of the parameters by observing the maximum and minimum values shown in table 3 for that parameter. Since C has the wider range of values, from 0.00 to 40.91, these may be taken as limits on the values of C. The limits on S must now be defined in terms of C. The limits on S must restrict all compositions to the system KAS2-CMS2-C2AS.They are therefore determined from equations of the subordinate components of this system in terms of S and C. These equations were obtained in a previous example involving the same primary and subordinate systems (page 552) and are repeated below.
+ +
Per cent KAS, = -1.32585 - 2.9652C 150.35 Per cent CMS, = 3.61358 1.4194C - 137.24 Per cent C2AS = -2.28778 1.5458C 86.89
+ +
(14) (15) (16)
The percentages of the subordinate components are positive or zero for all compositions in the subordinate system, To meet this condition in equation 14,
1.32585 > 150.35
or,
- 2.9652C
- 2.2365C
(23)
S Q 37.98 - 0.3928C S > 37.98 + 0.6757C
(24) (25)
S > 113.40 Similarly, from equations 15 and 16:
For any given value of C there is only one minimum and one maximum value of S. Inequalities 23 and 25, which express maximum values of S , must therefore apply to different ranges of values of C. By equating the right-hand members of these inequalities, and solving for C, we obtain C = 25.90. For this value of C both inequalities indicate the same maximum value of S . By trial of a single value of C less than 25.90 it is found that inequality 25 gives a smaller quantity for the maximum value of S than does inequality 23. Inequality 25 therefore applies when C is less than 25.90, while inequality 23 applies for larger values of C. The limits on C and S may he stated as follows.
PARAMETRIC EQUATIONS I N MULTICOMPONENT SYSTEMS
555
C from 0.00 to 25.90: S C 37.98 - 0.3928C S > 37.98 0.6757C
+
C from 25.90 to 40.91 :
S C 37.98
- 0.3928C
S > 113.40 - 2.2365C
PARAMETERS OTHER THAN COMPOSITION
In the foregoing study we have derived linear equations in which each of the parameters represents the quantity of a component. Other variables may be selected as parameters. The analytical treatment of systems with reference to phase equilibria involves temperature or pressure, or both, as variables. Relations between these variables and composition are seldom linear, so that when such relations are involved the use of linear equations must be restricted to cases in which the relations are approximately linear. For example, a curve may be so short that it can be taken as a straight line without appreciable error. Or, if a curve as a whole may not be treated as a straight line, it may be possible to divide the curve into segments, treating each segment as a straight line. This mode of attack is frequently advantageous because of the simplicity of linear equations. The derivation of linear equations involving parameters other than composition variables will be illustrated in the case of a segment of the boundary between the C3S and CZS primary phase regions in Rankin and Wright's phase diagram of the ternary system CaO-A1203-Si02 (6). This boundary is curved, but segments representing intervals of 100°C. are approximately straight lines. Furthermore, points at 100' intervals on the curve are nearly equally spaced, which is evidence that 100" segments of the corresponding curve in the space model of the system are approximately straight lines. Parametric equations will be derived for the segment between 1500" and 16OOOC. Let us designate the compositions at the 1500" and 1600°C. points as X and Y , respectively. The data required for deriving parametric equations, with T (temperature) as a parameter, are given in table 4. The form of the equation is that shown below for C.
C=aT+b Proceeding by the general method, the following equations are obtained.
+ +
C = 0.0332' A = - 0.0692' S = 0.036T
Total
+ 9.1 + 135.8 -
44.9
100.0
556
L. A. DAHL METHODS O F APPLICATION
The examples which have been considered in describing the derivation and properties of parametric equations suggest some applications to the analytical treatment of multicomponent systems. More complex problems may be solved when a knowledge of the derivation and properties of parametric equations is accompanied by experience in performing algebraic operations. To demonstrate the manner in which the application of the equations may be extended, two examples will be considered in this section, one dealing with the planning of mixtures for use in an investigation of phase equilibria, the other with the analytical treatment of phase equilibria data. In a previous paper (2) the derivation and properties of intrinsic equations were described. It was pointed out that the difficulty of applying intrinsic equations increases with N - n, while this situation is reversed in the case of paraTABLE 4 SUBORDINATE COldPOhZZTS'
PRIXAUPY COYPOhTNTS
I
Y
per rent
per ccnl
c...................................
58.6
61.9
A ....................................
32.3 9.1
25.4
s..................................
1
I
T
12.7 PARAMETER
1500
I
1600
* Estimated from phase diagram metric equations. To illustrate, relations in a quaternary system within a quinary system may be expressed by a single intrinsic equation in simple form, and for many purposes this type of equation is to be preferred. On the other hand, a binary system within a quinary system is much more easily treated by means of parametric equations. PLANNING MIXTURES I S T H E INVESTIGATIOX O F A QUIKARY SYSTEM
Composition relations in a quinary system can not be represented in a phase diagram or space model, since four dimensions are required. h plan followed by Eubank (3) in an investigation of a portion of the system S-C-A-F-S (the system Na20-CaO-Al2O8-Fe2O~-SiO2) was to study phase equilibria in a series of sections through the quinary system, represented by triangular diagrams, in each of which the percentages of two components were arbitrarily fixed. By a systematic choice of the percentages of the two components, the series of diagrams can be arranged in columns with reference to one of the components and in rows with reference to the other, for convenient study and comparison. The planning of mixtures in the various sections investigated by Eubank will be con-
557
PARAMETRIC EQUATIONS I N MULTICOMPONENT SYSTEMS
sidered as an example of the application of parametric equations. His problem was to investigate the effect of the addition of Ka20, in the form of NCs& to compositions in a portion of the system CaO-C,A3-C2F-C2S. This portion of the system is defined by a tetrahedron with vertices as indicated in table 5. The compound NCsA3 and the compositions at the vertices of the tetrahedron define a quinary system. The problem that we shall consider is the calculation of mixtures in the plane (triangular) sections through the quinary system, in which two components vary from plane to plane but are constant in each plane. The required data are shown in table 6. TABLE 5 VXRTEX COMPOXENTS
I
1
1
I
2
I
3
4 ~
par ccnl
prr cent
C
c*s c 5A3
per a n I
pcr ccnl
20
0
0
0
15
15 59 26
35
15 39 46
C&
39 26
TABLE 6 SECONDAPY COMPONENTS PRIXARY COMPONEXTS
SYMBOL
v
1
per ccnl
w
X
par ccnl
par cenl
pcr ccn1
0 35 39 26 0
0 15 39 46 0
B D
20
0
15
E
39 26
15 59 26
G H
I
Y
Z per Cenl
0 0 0 0 100
WEIGET FRACTION
-
I--v-w ~
v
!
w
~ - x - y x
This investigation is confined to a portion of the system C-C2S-C&-C2F?JC8A3,and the components of this system are prepared in quantity for use in making mixtures. It is therefore convenient to treat these compounds as primary components. As a preliminary step we shall derive equations for the system V-W-X-Y-Z in the system B-D-E-G-H by the direct method previously described (page 550). The equations so derived are as follows: Per cent C = B = 20v (26) Per cent C2S = D = 15v 15, 352 15y (27) Per cent C.& = E = 39v 59w 392 39y (28) Per cent C2F = G = 26v 26w 262 46y (29) Per cent SCeAs = H = -1OOv -100w -1001: -1OOy 100 (30)
+ + +
+ + +
+ + +
+
~
y
558
L. A. DAHL
Now let us arbitrarily select CA3 and NCaA3 as the components to be constant in each triangular diagram representing a plane section through the quinary system. Equations 28 and 30 give the percentages of these components but involve four parameters, whereas only two parameters are required for a ternary subordinate system. Let us choose z and y as the parameters to be retained, and let E1 and HI represent the constant values of E and H, respectively, in any given diagram. Transposing terms in equations 28 and 30, and substituting E1 and HI, respectively, for E and H, we obtain the equations:
+
39~ 5 9 = ~ E1 1000 1 0 0 ~= -Hi
+
- 392 - 39y - lOOz - 1 0 0 ~+ 100
Solving for v and w:
+
= - O.O5Ei - O.0295Hi - z - y 2.95 (31) w 0.05Ei 0.0195Hi - 1.95 (32) Since the parameters v and w are to be eliminated through the substitutions which follow, we must now consider the requirement that neither of these variables can be less than zero. From the above equations we obtain the following inequalities expressing this condition: V
+
z
+ y > 0.0295(100 - Hi) - 0.05E1
(318) (32a) Substituting in equations 26, 27, and 29 the values of v and w in equations 31 and 32, we obtain the equations,
E1 C 0.39(100 - Hi)
Per cent CaO = B = 0.59(100 Per cent C2S = D = 0.15(100 Per cent C2F = G = 0.26(100
- HI) - El - 20(2 + y) - HI) + 20z - HI) + 20y
(33) (34) (35)
By substituting for El and HI the percentages of CsA3 and NC8A3 to be constant in a particular diagram representing a section through the quinary system, parametric equations are obtained for calculating mixtures in that section. For example, let E1 = 45 and H1 = 8. These values satisfy inequality 32a. Substituting in equations 33-35, Per cent CaO = B = 9.28 - 20(r Per cent C2S = D = 13.80 202 Per cent C2F = G = 23.92 20y
+ +
The value of B in equation 33a will be negative if z
+ y)
(334 (34%) (35a)
+ y is greater than 9.28/20,
or 0.464. We may therefore assign limits to the parameters
2 and y by stating that 1: and y must be zero or positive, and that their sum must not exceed 0.464. Within these limits, compositions obtained by substituting assigned values of z and y in the equations will be restricted t o the secondary system V-W-X-
Y-z
.
In drawing a series of triangular diagrams, each with constant percentages of CaA3and NCsA3, it is necessary to determine the composition at each vertex in
559
PARAMETRIC EQUATIONS I N MULTICOMPONENT SYSTEMS
terms of the three remaining primary components, as a basis for extimating the composition at any point in the diagram. This may be done by assigning limiting values to x and y. In this case, when one of these variables is equal to zero, the maximum value of the other variable is 0.464. We therefore have three pairs of values of x and y representing the vertices, as indicated in table 7. ESTIMATIOK O F PROPORTIONS O F PHASES
The interpretation of phase equilibrium data represented by a phase diagram involves determination of the phases present under any given condition, and estimation of the proportions of those phases. In the case of systems represented in a plane diagram, these operations are performed graphically. Parametric equations provide a convenient means of performing the same operations analytically, as will be shown in a specific example. Before considering the example illustrating the application of parametric equations, we shall discuss a rule of procedure which is important in applying analytical methods to the interpretation of phase equilibrium data. Let us eon-
x
CrS
CaO
Y
[
GF per Cent
1 2 3
* By
0 00 000 0 464
0 00
'
0464 0 00
1
1
000
0 00
1
13 80 23 08
23 92 33 20 23 92
substitution in equations 33a-36a.
sider an invariant point L at which three solid phases, A, B, and C, are capable of coexisting with the liquid at temperature T I .During slow fusion of a mixture in the system A-B-C the temperature remains constant at temperature T 1 while the quantity of liquid increases and the proportions of solid phases change. The quantity of liquid reaches a maximum, at which point one of the solid phases disappears, or two or more disappear simultaneously. The maximum quantity of liquid which can form at that temperature is present when one of the solid phases disappears, or when two or more disappear simultaneously. Further formation of liquid L is impossible, since it would leave a solid residue which can not be composed of solid phases with which liquid L can coexist. The problem of determining the maximum amount of liquid at an invariant point is therefore one of determining the quantity of liquid present when one of the solid phases disappears. A convenient method of accomplishing this is to express the compositions of the liquid and the mixture under study in terms of the solid phases capable of coexisting with the liquid. Let us suppose that the percentages of A, B, and C in this expression of composition of the liquid are A,, B,, and C1, respectively. Then the percentages of A, B, and C in the mixture are reduced O.OIA1, 0.01B1, and O.OlC1, respectively, by each per cent of liquid formed. I t is then a
560
L. A. DAHL
simple matter to determine the percentage of liquid which mill lead to the disappearance of a solid phase, and to determine the percentages of the solid phases then present. The rule to be followed in estimating the proportions of solid phases and liquid in a mixture in a state of equilibrium may be stated as follows: Compositions of the mixture and the liquid should be expressed in terms of the solid phases capable of coexisting with the liquid and, when necessary, such additional substances as are needed to make a total of N substances. The significance of the latter part of this rule will be clear when it is presently applied. The foregoing rule will be applied in the solution of a problem pertaining to mixtures in the system CAS2-CAS-CS in the ternary system Ca0-A1203-Si02, using data obtained by Rankin and Wright (6) in their investigation of the system. When mixtures in the system CAS2-C2AS-CS are slowly fused, the first stable liquid is formed at 1265”C., at the invariant point 38 per cent CaO, 20 per cent A1203, 42 per cent SiOz. This point will be designated as X. Liquid of composition X is capable of coexisting with solid CAS2, GAS, and CS at 1265°C. The first two phases are essentially pure compounds, while the CS phase probably contains a small quantity of other constituents in solid solution. However, since the actual composition of the CS phase which is capable of coexisting with liquid X is not known, it will be treated here as a pure compound. Following the above rule, the composition of the liquid is converted into terms of CAS2, GAS, and CS, obtaining the “potential composition,” 30.30 per cent CASZ, 23.92 per cent CZAS, 45.78 per cent CS. Let B1, D1,and GI represent the potential percentages of CAS,, GAS, and CS, respectively, in any given mixture. Then for any state of equilibrium at 1265OC. the percentages of the solid phases may be found from the equations, Per cent solid CAS2 = Bl - 30.30m Per cent solid C2AS = D 1- 23.92m Per cent solid CS = G1 - 45.78m in which m is the weight fraction of liquid X in the mixture. .411 three of the solid phases may be present at 1265”C., but as heat is supplied the quantity of liquid X increases, and each of the solid phases decreases in quantity, until one of the solid phases disappears. The quantity of liquid has then reached the maximum which may be present at that temperature. Further introduction of heat causes the liquid to leave point X, and the temperature rises. To determine the maximum quantity of liquid at the invariant point temperature, we equate the right-hand members of the equations to zero and solve each of the resulting equations for m. The lowest value of m so obtained is the maximum weight fraction of liquid at the invariant point temperature. Upon substituting this value of m in equations 36-38 the percentages of solid phases present when the liquid leaves point X is obtained. The course of the liquid phase upon leaving X is also learned by this procedure. For example, if equation 36 gives the lowest value of m, CASz is the disappearing phase, and the liquid leaves point X to follow the univariant curve for CzAS and CS. Similarly, if equations 36 and 37 give identical values of m, lower than that obtained from equation 38, CAS, and G A S dis-
561
PARAMETRIC EQUATIOKS I X MULTICOMPONEXT SYSTEMS
appear simultaneously, and the liquid passes into the CS primary phase region, in a direction tolyard the composition point for CS. From the foregoing considerations it may be seen that equations 36-38 provide a means of classifying mixtures in the system CAS,-GAS-CS, with reference to the course which the liquid will follow upon leaving point X . To describe the method of estimating proportions of phases when the liquid is on a univariant curve we shall select the range of compositions for which equation 38 gives the lowest value of m For mixtures in this range, m = G1/45.78 when the liquid starts to follow the univariant curve for CASz and C,AS. This curve appears as a straight line in the phase diagram of the system. It extends from point X , which we have already considered, to the invariant point for CAS,, GAS, and A1203, which we shall designate as Y . The univariant curve may then TABLE 8 CObIPOhFNTS
1
1 I
i !
CAS2 CzAS
* Invariant point
Yt
OXIDE COMPOSITION
CaO AlzOs SlOZ
cs
I
X' SYMBOL
I
G
per cent
per cenl
38. 20. 42.
29.2 39.0 31.8 POTENTIAL COMPOSITION
30 30 23 92 45 78
11
54 42 51 24 - 5 66
for CASz, C A S , and CS (6).
t Invariant point for CAS*, G A S , and ALO8 (6). be designated as the line X Y . The data required for estimating proportions of phases when the liquid is on the line X Y are shown in table 8. At present lye are concerned with equilibria in which the only solid phases are CAS, and CzAS. The potential compositions of the liquid phase and any mixture under consideration must therefore be expressed in terms of CAS,, C2AS, and a third substance, since the system is ternary. The third substance may be selected arbitrarily, without influencing estimates of the proportions of phases. However, whenever it is possible, it is convenient to select components-of the system in which mixtures under consideration lie. We have therefore selected CS as the third component in the potential compositions given in table 8. Potential compositions are expressed to a higher degree of accuracy than the oxide compositions merely to insure consistency in the results. This applies also to the computations which follow. For the parameter in equations for the line X Y we may select B , D,or G. However, later operations in which the parametric equations are used are somewhat simplified by selecting G, as will presently be seen. Following the general
562
L. A. DAHL
method described on page 552 the following parametric equations for the line XY are obtained: Per cent CAS2 = B = - 0.46896 Per cent G A S = D = - 0.5311G G Per cent CS = G =
+ 51.77 + 48.23
(39) (40) (41)
Let B1, D1,and G1 represent the potential percentages of CAS,, GAS, and CS in any given mixture. If m is the weight fraction of liquid, the percentages of the solid phases are given by the following equations, in which the coefficients of m are the values of B, D,and G in equations 39-41. Per cent solid CAS2 = B1 - (51.77 Per cent solid GAS = D 1- (48.23 Per cent solid CS = GI - Gm
- 0.4689G)m - 0.5311G)m
(42) (43) (44)
However, solid CS cannot exist in equilibrium with liquid on the line XY,so that GI - Gm = 0, or G = Gl/m. Substituting Gl/m for G in equations 42 and 43, and simplifying, we obtain equations 45 and 46 (below) for all equilibrium states in which the liquid phase is on the line XY.
+ 0.4689G1 + 0.5311G1 -
Per cent solid CAS, = Bl Per cent solid GAS = D1 Per cent liquid =
51.77m 48.23m 100.00m
(45) (46) (47)
I t was shown previously (page 561) that m = G1/45.78 when the liquid leaves point X to follow the line XY.This is therefore the minimum value of m which may be substituted in equations 45-47. As the value of m increases, the percentages of CAS2 and GAS decrease, until at some point one of these phases disappears. At this point the liquid leaves the line XY and passes into the primary phase region for the solid phase which remains. The maximum value which may be assigned to m is therefore found after substituting the composition of a mixture in equations 45 and 46, by choosing the lower of the two values obtained when the right-hand members of these equations are equated to zero, and solved for m. Interest is usually confined to the proportions of phases when the minimum and maximum weight fractions of liquid are substituted in equations similar to equations 4 5 4 7 . ESTIMATION O F PHASES I N T H E NEIGHBORHOOD O F A PERITECTIC POINT
In the foregoing example invariant point X is a eutectic point, and its POtential composition can be expressed in positive percentages of the solid phases with which liquid X may coexist. That is, the invariant point is in the system CAS,-GAS-CS. The portions of the univariant curves involved in estimating proportions of phases in mixtures in that system are also located in the system. On the other hand, a peritectic point is outside of the system whose components are the solid phases capable of coexisting with the liquid. Univariant curves leading from a peritectic point in a direction of higher temperatures are also outside of the system in the neighborhood of the peritectic point. Both positive
PARAMETRIC EQUATIONS I N MULTICOYPOXEST SYSTEMS
563
and negative values therefore appear in the potential compositions of such liquids. The fact that negative values appear does not interfere with the application of the method which has been described for equilibria at eutectic point X and for points on the univariant curve for CAS1 and C2ilS. This may be illustrated by considering a hypothetical case in which the invariant point is a peritectic. Let us suppose that the peritectic point P represents a liquid capable of coexisting with the solid phases A, B, and C, and that its potential composition, expressed in terms of the solid phases, is A = 65, B = - 25, C = 60. Then if AI, Bl, and C1 are the potential percentages of A, B, and C, respectively, in any given mixture, and m is the weight fraction of liquid P, Per cent solid A = A , - 65m Per cent solid B = B, 25, Per cent solid C = C1 - 60m
+
These equations are similar to equations 36-38, except that the coefficient of m in the second equation is positive. From this it is seen that the quantity of solid B increases as the quantity of liquid increases, and that this phase consequently does not disappear. There are only two univariant curves leaving point P in a direction of higher temperatures, the curves for A and B and for B and C. The value of m when one of the solid phases, A or C, disappears is determined in the same manner as in the case of a eutectic point. Estimation of the proportions of phases when the liquid is on one of the univariant curves is accomplished in the same manner as in the preceding example. I t is not necessary to give separate treatment to the portions of the univariant curves inside and outside of the system A-B-C, since the signs appearing in the potential composition at any point on these curves take care of the situation. KONLISEAR REL4TIOSS
Estimation of proportions of phases when the liquid phase is on a univariant curve has been illustrated by an example in which the curve appears as a straight line in the phase diagram. Curved lines may be represented by nonlinear parametric equations. However, the use of nonlinear equations leads to difficulties in both the derivation of equations and application to individual mixtures. I t is much more simple to treat the curve as being composed of a series of straight lines. Usually the degree of accuracy with which the location of a curve has been established is not so great that nonlinear equations can be considered to he much more accurate than a successive series of linear equations representing segments of the curve treated as straight lines. There may be instances in which curvature is so great that this mode of treatment is not satisfactory, but it is usually the most practical method of dealing with nonlinear relations. SUMMARY
The derivation and use of parametric equations in problems encountered in the investigation of phase equilibria are described. These problems are of two kinds: ( I ) the design of mixtures to be subjected to heat treatment or other ex-
564
QIUSEPPE CILENTO
perimental operations in the course of an investigation, and (2) analytical interpretation of the data obtained. Advantages of parametric equations are their simplicity when dealing with a system of a small number of components within a multicomponent system, and their form, which permits direct computation of the quantity of each component when values are assigned to the parameters. The form of the equations also simplifies the operations required in deriving equations for estimating percentages of phases under specified conditions of equilibrium. REFERENCES (1) DAHL,L. A , : Rock Products 41-42 (1938-39). (2) DAHL,L . 8 . :J. Phys. & Colloid Chem. 62, 698 (1948). (3) E U B A N KW. , R . : Dissertation, The Johns Hopkins University, 1947. (4) LEA, F . M., AKD PARKER, T. W . : Phil. Trans. Roy. SOC.(London) A234 (So. 731), 1 (1934). (5) LEA, F. M., AND PARKER, T. W . : Dept. Sci. Ind. Research, Building Research Tech. Paper No. 16, 52 pp. (1936). (6) RANKIN, G. A , , A N D WRIGHT,F. E . : Am. J. Sci. 39, 1 (1915).
SOME BINARY SYSTEMS OF AXALOGOUS ORGANIC CHALCOGEN DERIVATIVES' GIUSEPPE CILENTOZ
Departamento de Qulmica, Faculdade de Filosojia, Cidncias e Letras, Universidade de Sbo Paulo, Sbo Paulo, Brasil Recezved J u n e 87, 1949
In this paper are reported solidus-liquidus binary systems between o-phthaloyl derivatives (Ia: X = 0, S, Se, KH) and also between triaryl phosphates (11: R = H, CHs; X = 0, S, Se). They constitute part of a systematic research undertaken by H. Rheinboldt et al. in the isomorphogenic relationships of organic oxygen, sulfur, selenium, and tellurium compounds. 0 x
II
(\ 2
X
II
0 Ia
@ \
0 Ib
(RaO-)3P+X
I1
P a r t of the thesis submitted by Giuseppe Cilento in partial fulfillment of the requirements for the degree of Doctor of Science a t the Universidade de SLoPaulo, Brasil, October, 1946. 2 Present address: Fundapao Andrea e Virginia Matarazzo, Faculdade de Medicina, Universidade de SLo Paulo, Caixa Postal 100-B, Silo Paulo, Brasil. 1
