A41iALYTIC.lL METHODS I N PHASE RULE PROBLERIS BI GEORGE W MORE?
It frequently happens in phase rule work that an analytical method of treating some problems is desirable, even tho the problem may be capable of solution by graphical means I n other cases graphical methods are either laborious or impossible, and recourse must be had to extended computations. Examples of these cases will be treated in the follolling description of a general analytical method especially adapted to phase rule work This method has previously been applied to working out general theorems,' but it is also of convenience in practical work
-1 problem uhich frequently arises in the study of three-component systems is that of the locus of points fulfilling a desired relation, such as passing thru t n o points in the component triangle T i hen triangular coordinates are used, as I S generally the case, care must be taken in applying trigononietrical considerations because of the special method of measurement The sine of 60°, as taken in triangular coordinates, ii: 3 j , instead of I 247 The usual equation of analytical geometry for a line passing thru t u o points
is, however, applicable, and will be found of great convenience. Simplified, this leads to an equation of the form
Ax
+ By
=
C
(2)
using x and y as expressed variables, z being determined by the relation x y z = I . The same end result may be obtained by several methods.
+ +
As an illustration of the use of this equation, suppose it is desired to make mixtures on the join between the compounds &O.2SiO2 and K20.3Ca0. 6Si02 in the ternary system K20-CaO-Si02. Letting t,hese phases be P' and P", respectively, y = CaO, z = S O 2 , and substituting the proper values for these t w o points, equation I becomes y-o ~-
-
,269j-0
z = ,5606
z-.5606 .5 j26--.j606
+ ,0445 y.
Substitution of even values of y = weight fraction C'aO gives corresponding values of z = weight fraction Si& along this join, as illustrated in Table I. Morey and Williamson: J. Am. Chem. Soc., 40, 59 (191j)
1746
G E O R G E T. MOREY
TABLE I Composition of mixtures between K20.2Si02and K20.3Ca0.6Pi02 y = CaO
x
z = SiOn 0 . j606
0
0.1
‘5651
0 . 2
.5695 . 5 j26
0.2697
=
KlO
0.4394 ,3349 ,230 j
,1577
Values of x are calculated from x = I - y - z. I n a four-component system it is frequently advisable to study various three-component systems before considering the phase relations in the system as a whole. I n this case, it is convenient to have an analytical expression for a plane passing t,hru 3 pointsin the tetrahedron representingthe iyhole system; and to express compositions in the resulting triangle, not in ternis of the three compounds themselves, but in terms of some three of the four components. This usually facilitates computation of the mixtures, and avoids the graphical distortion resulting from the use of compounds as components. Equation I may be regarded as a solution of the general equation 2 , thru the point,s P’ and P”, In a four-component system, with component weight fractions represented by x, y, z, and w, where x+y+z+w = I , there will be an equation of the type Ax + By cz = (3)
+
n
Substitut,ing the values for the 3 points I”, I”’, P”’, wyc have three linear equations, from which the values of the coefficients can be obtained by elimination. This can best be done by means of the determinant’ notation, when equation 3 becomes: I
y‘
2’
’
I
K‘
I Z‘
y’
I
This determinant, which may be w i t t e n
gives the equation of a plane cutting the component tetrahedron and passing thru the points P’, P”, and P’”. Particular notice should be taken of the cyclic arrangement of the quantities; and of the fact that in each determinant coefficient the row corresponding in order to the letter of which that determinant is the coefficient is replaced by a row of unities. As an example of its application, consider the system albite-orthoclase-silica in the four coni1 This notation is merely a shorthand method of solving a series of linear equations by operating on the coefficients in a prescribed manner. The same end result can be obtained by the usual processes of elimination, and the above equation thus verified. The use of determinants is well lvorth while in the present case, and a few minutes application to the section on Determinants in J . K. Melior’s “Higher Mathematics for Students of Chemistry and Physics” will give sufficient fnmilirtrit,v for all auulications in connection with phase rule st,udirn.
A S A L Y T I C A L METHODS I N P H A S E RULE PROBLEMS
I747
ponent system Na20-KzO-A12O3-SiO2. Calling the oxide components x,y,w, and z, calling P’, P”, P”’ albite, orthoclase, and SiO,, respectively, in the order given, the equation becomes:’
If
0.6874 0.6476
O.ab,,
0.1182
x f
I
0
I
0.1692
I
0
I
0.6874 0.6476
I
0
1
0
I
y+
I
2=
This becomes
1:
: I
0.1
0
0.6874 0.7915
1
1
‘
1
x+
I
I
0.1182
I
0.6874
I
0.7915
0
0
p+
I748
G E O R G E W . hlOREY
Elimination of x between this and the equation of the triangle (5) gives z = 0.68j4
+ 1.0403 y
(7)
by means of which the compositions of the mixtures given in Table I1 have been calculated.
TABLE I1 Coinposition of mixtures determined by equation 7 KzO (Y)
Na20 (X)
81203 f 1-X-S-Z)
0.6874
0 . 1 IS2
0,1944
0.01
,6978
0.03
,7186
,1064 ,0827
0.05
.i3i4
.oj 9 I
'1958 1987 , 2 0 15
0.07
. j602
,0355
0.10
,7915
0
,2013 ,208j
0
A five-component system offers almost inwperable difficulties in the way of complete portrayal, but parts of several five-component systems have been worked out. The extension of the above method of treatment is obvious. Equation 2 takes the form, AX
+ By + C Z + Dw
=
E
(8)
in which x, y, z, w represent the weight fractions of any four of the five components. Equation 4 takes the form,
(Di,,,) x -t iDxizn) Y
+
(DX,1w)
z
+
(JJx,,i)
w =
D x p w
(9)
This symbolizes a tetrahedral section of the 5-diinensional figure representing the entire system, and requires four points for its evaluation. Elimination of one of the composition variables thru two such equations results in the equation of a plane, which will represent the triangular intersection of two tetrahedrons. I s an illustration consider the ternary system kaliophyllite (Kp), KzO. A1203.2Si(fa-diopside (Di), C"O.1IgO.z SiOy-silica, in the five-component system Ei20-Ca0-11g0-i\1203-Si0~. tising K20 and CaO as auxiliary points, obtain the equation of the tetrahedrons K20-lip-Di-silica and CaO-Kp-DiS O ? , which viill take the forms,
+ z + (Dt5zI)w ( D F p w ) x + (DL.\\) v + (D9,iW) e + (D;yz1) vi (D!yzw)
x+
(D$iSw)9
(&yiB)
=
DkYzw and D&za
(10)
(11)
in which Dk and Dc are written for the determinants involving K2O and CaO, respectively, and x, y, z, IT are the weight fractions of E;.&, C'aO, A 1 2 0 3 , SiOz,
AS.%LTTICAL M E T H O D S I S PHASE RULE PROBLEXS
I749
respectively, leaving the weight fraction MgO to be calculated from the relation: weight fraction 1IgO = I-x-y-z-w. This gives the equation for the triangle Kp-Di-SiO,:
x
+ 0.8259 y + 0.4802
TV =
0.4802
(12)
As in the preceding case, a line represents the intersection of two planes, and its equation is obt,ained by elimination of one of the composition variables by the aid of another equation. For example, the locus of mixtures on the join between leucite (Lu) and diopside may be obtained by passing a plane thru these two point,s, giving an equation involving, as before, x, y, z. Eliminating z between this equation and (9) gives the equation of the desired locus. In the preceding discussion, the equations have been given in the form of t,he determinants involving the weight fractions of the components, but in the elimination between two equations involving determinant coefficients the determinants have been evaluated before multiplication, instead of first carrying out the cross multiplication. For example, to get equation 9, z is eliminated from ( 7 ) and (8). This leads to the equation:
which can be further operated on by carrying out the cross multiplications. But this is difficult and the resulting products are inconrcnient to evaluate, while as a rule the original coefficients are simple, and usually can be evaluated, or at least reduced t o a difference of two products, by inspection. Also, it is frequently possible t o avoid an actual evaluation of an equation by employing an obvious artifice. For example, in the triangle albite-orthoclaseSiO,: on the Or-SiOz side, x = 0 ; lines parallel to the various sides are given by x = const., y = const., etc.; thru SiO, and any point on the Or-Ab join the ratio K : S a is constant, etc. Also, in the triangle kaliophyllite-diopsideSiOs, the equation of the join orthoclase-diopside will be determined by the by means of which condition that x:z = (K?O:A1203).,th,,l,,, = 94.2:101.94, x can be climinated, leaving an equation betmen CaO and SiO2. The above may be generalized as follows. In a system of n + I components, in which the weight fractions of the various components are reprc. -n), the equation of an n-fold is sented by x,y,z,. . .n, ( I - x - y - z . . represented by
in which D l y z ,, n, , Dxlz. .D, etc. represent the determinants composed of the n columns I , x,z. . . n ; x1 r 1 z l . . . n ; etc. and t h e n rowsgivingthevaluesofthese weight fractions for the n phases determining the n-fold. The equation of an n - I foldisbestobtained byelimination of one of the quantities, s,y,z. . . n be-
I750
GEORGE W. MOREY
tween the equation of z different n-folds, etc. I n forming these equations it is always advisable to use as auxiliary points phases of simple composition, such as one of the components itself.
summary It is often convenient to make use of analytical methods in connection with graphical ones in the treatment of phase rule problems. Such methods are outlined and their application illustrated. Geophysical LabOTatOTy, Carnegie Institution of Vashington, February, 1950.
