SOME ERRORS INHERENT I N T H E USUAL DETERMINATION OF THE BINARY FREEZING POINT DIAGRAM EVALD L. SKAU1 AND BLAIR SAXTON Department of Chemistru, Yale University, New Haven, Connecticut
Received September 19, 1951
It has long been known that there are marked abnormalities in the freezing points of many binary mixtures near the eutectic composition, but it has never been shown definitely whether they are inherent in the system or whether they are merely apparent abnormalities due to experimental error. If, however, one were to study by the usual approximate method a nearly ideal system for which the accurate data are available, a comparison of the corresponding freezing point-composition diagrams should disclose some interesting facts, since all the differences would be due t o experimental error. Most of the temperature-composition diagrams in the literature have been constructed by the use of the Beckmann method, which consists essentially in cooling the melt below the freezing point, inoculating with the proper kind of crystal, stirring, and noting the maximum temperature to which the thermometer rises. The freezing points so obtained are usually low, owing mainly to two causes: (1) the change in the composition of the liquid due to the solid frozen out, and (2) the thermometer lag? The errors thus involved increase as the eutectic is approached. In many cases the eutectic point itself is not experimentally determined but is obtained by extrapolation of the two branches of the curve. This investigation had for its purpose the study of the nature and the possible magnitudes of the effects due to such inaccuracies on the freezing point-composition diagram, the calculated heats of solution, and on the conclusions drawn as to the ideality of the system. The system P-chlorocrotonic acid-0-chloroisocrotonic acid, for which the accurate freezing point 1 Du Pont Fellow at Yale University, 1924-1925. Present address Trinity College, Hartford, Connecticut. 2 This error may undoubtedly be quite considerable and the fact that in the system here studied it was eliminated, partially counterbalances the fact that our other error was deliberately enhanced. Another cause of low freezing point results is the use of too little undercooling. Since this has been repeatedly discussed (see, for example, Ostwald-Luther, Hand- und Hilfsbuch aur Ausfiihrung physikochemischer Messungen, Leipsig, 1925) and has been eliminated in this work, i t will not be considered here. 183
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EVALD L. SKAU AND BLAIR SAXTON
and heat content data have recently been published Cl), affords an excellent opportunity for a study of this kind. The results obtained show that the eutectic composition as well as the eutectic temperature is affected by the usual inaccuracies of the data. Other causes of error near the eutectic point are pointed out by means of cooling curves. EXPERIMENTAL
The freezing point method used was one which exaggerated the error due to the separation of the solid, but which practically eliminated the error due to the lag in the thermometer registration,
Apparatus The apparatus used was the inner part of the cooling curve apparatus previously described (1). It consisted essentially of a small thin-walled glass tube, 6 mm. in diameter, containing the sample (0.7 to 1.O gram) into which a thermoelement, protected by a thin-walled capillary glass sheath, was inserted through a stopper. This stopper was provided with another small opening through which the undercooled melt could be inoculated by introducing the proper crystals on the end of a glass thread. The thermoelement used was the same as that used in the accurate construction of the diagram for this system.
Procedure The freezing point determinations for this system were made as follows. A known weight of the component A was placed in the freezing point tube and a small amount of component B was then weighed in by difference without removing the thermoelement. After the mixture was made homogeneous by melting completely and stirring, the freezing point was found by allowing the system to cool in the air (without any shield), seeding a t the proper time, stirring, and noting the maximum to which the temperature rose, This was repeated with various degrees of undercooling and the highest maximum obtained was taken as the freezing point for that composition, A little more of substance B was then weighed into the tube and a new determination made, The error in composition is thus cumulative but is estimated to have been at all times less than 0.3 per cent.
Materials The synthesis and purification of the two acids has been described (1). The experimental results The data obtained by the present method are given in table 1. N is the mole fraction in the liquid of the form crystallizing a t the freezing point, t'C., T'K; t' is the true freezing point obtained by interpolation of the
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ERRORS IN BINARY FREEZING POINT DIAGRAM
experimentally correct3 data for the system. These data are plotted in figures 1 and 2. The following facts should be noted: From the temperature-composition diagram, figure 1, it is obvious that the values obtained by the rough method are all lower than those accurately determined. Further, it is apparent that the deviations become increasingly greater as the eutectic is approached and that the eutectic point is shifted by these inaccuracies not only along the temperature axis but also TABLE 1 Experimental freezing point data for the system p-chlorocrotonic acidO-chloroisocrotonic acid
p-chlorocrotonic acid branch 1.000 0.475 0.434 0.389 0.350
1.0000 0.6767 0.6375 0.5900 0.5440
93.6 55.5 52.4 47.0 41.1
2.727 3.043 3.072 3.124 3.183
93.6 59.1 55.4 50.9 46.8
0.0 3.6 3.0 3.9 5.7
60.5 57.9 56.3 55.2 54.2 51.3 49.5 48.1 45.7 44.1 42.8
0.0 0.1 0.3 0.4 0.3 0.6 1.0 1.2 1.6 1.8 1.5
p-chloroisocrotonic acid branch 1.0000 0.9555 0.9271 0.9077 0.889 0.840 0.810 0.788 0.750 0.725 0.706
1. 0000 0.9802 0.9671 0.9579 0.9489 0.9243 0.9085 0.8965 0.8751 0.8603 0.8488
60.5 57.8 56.0 54.8 53.9 50.7 48.5 46.9 44.1 42.3 41.3
2.998 3.022 3.038 3.049 3.058 3.088 3.109 3.125 3.153 3,171 3.181
Eutectic point by extrapolation = 38.9"C.
along the composition axis, the extrapolated temperature being about 38.9"C. instead, of 41.5"C., and the per cent of @-chlorocrotonicacid being about 33.2 instead of 30.8. From the plot of (I
+ log N ) against e, figure 2, it is seen that a T
straight line may be drawn through the inaccurate data in such a way that
* With a maximum error of 0.25%.
near the eutectic point.
T H E JOURNAL OF PHYSICAL CHEMISTRY, VOL. XXXVII, NO. 2
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EVALD L. SKAU AND BLAIR SAXTON
the greatest deviation corresponds to only about 0.5"C. The heats of solution calculated from the slopes of these lines by means of the ideal freezing point-solubility equation are 4970 and 3940, as compared with the values 5220 and 4120 calories per mole calculated from the accurate lines for the 0-chlorocrotonic and the 0-chloroisocrotonic acid respectively. It
Mole fraction (3- s h l o r o c r e t o n i e
acid
FIG.1. FREEZINQ POINT-COMPOSITION DIAGRAM OF THE SYSTEM, 8-CHLOROCROTONIC ACID-P-CHLOROISOCROTONIC ACID Dots represent accurate freezing points; circles represent rough freezing points
is further apparent that the points near the eutectic show a tendency to fall off gradually, especially on the 6-chlorocrotonic acid branch where the eutectic composition is considerably removed from the pure substance. If curved lines be drawn through the points and the heats of solution calculated from their initial slopes, the error would obviously be lessened.
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ERRORS IN BINARY FREEZING POINT DIAGRAM DISCUSSION
There are several possible reasons for the increased deviations near the eutectic point which should be pointed out since, to the best of our knowledge, this situation has never been carefully analyzed. (1) It can be shown that for purely mathematical reasons the error in 2.7
28
28
& 3.0
31
3.2 10
0.9
0.n
0.7
c i + LogN
FIG.2. Log N - 'Oo0 PLOTFOR T
THE
ab
01
SYSTEM,p-CHLOROCROTONIC ACID-R-
CHLOROISOCROTONIC ACID Dots represent values calculated from accurate freezing points; circles represent values calculated from rough freezing points. Upper curves are for p-chlorocrotoniclacid; lower curves are for p-chloroisocrotonic acid.
the mole fraction caused by the freezing of a given fraction of a component of a binary solution increases with the mole fraction of the added component until the two mole fractions are about equal. Let n A and nB be the number of moles of A and B respectively in a binary solution and let a be the number of moles and z = u/nA, the fraction of A crystal-
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lized from this solution in a freezing point determination. Let NA and NB be the mole fractions of A and B, respectively, in solution before crystallization and N'A and N'g be the corresponding quantities after a moles of A have been crystallized. Then ANA = N a
- N,
=
nA
- aNB + nB - a
=
- XNANB =. - xNANB 1
- xN,
(1)
Differentiating, we obtain
--
From the approximate derivative the change in composition reaches a maximum when NA = NB. From the accurate derivative the maximum is reached when N A =
-
Thus when x
X
=
0.01 the maximum
occurs a t NB = 0.4987.4 (2) The general shape of the freezing point-composition diagram adds another source of error. Suppose the error mentioned in (1) be made uniform and small by keeping ANA constant throughout the range of concentration, that is, by decreasing the value of z as we approach the eutectic so as to satisfy equation 1. The error would still usually increase owing to the fact that the change in freezing point for a given change in composition also commonly increases as the eutectic point is approached. This is obvious from the fact that the freezing point-composition lines are ordinarily curved downward. For the ideal solution a t constant pressure this follows from the well-known equation,
where, a t temperature T , NA is the mole fraction in solution, AS, is the molal entropy of fusion, A H A the molal heat of fusion of the component crystallizing, and s is the slope of the freezing point-solubility curve. Differentiating we obtain,
where ACPA is the molal increase in heat capacity of pure A on fusion. Hence whenever 2R
