THE VISCOSITY O F BINARY MIXTURES BY EUGENE C. BINGHAM
In 1905 Jones and Binghaml advanced the hypothesis that in binary mixtures fluidities are normally additive. The prevailing assumption had been that viscosities are additive, but it was seen that if fluidities are additive viscosities cannot be. There was theoretical justification2 for the newer hypothesis and the existing experimental data were in its for whereas it had been repeatedly noted that “the viscosity of a mixture of miscible and chemically indifferent liquids is rarely, if ever, under all conditions, a linear function of the composition”4 it was found that the fluidity of such a mixture is much more nearly a linear function of the composition. Binary mixtures do not afford the best opportunities for testing the validity of the above hypothesis, since so many liquids are associated and do not fulfil the requirement of being “chemically indifferent” to each other; so that, on mixing, either the association may break down causing a rise in the fluidity or a more or less feeble chemical combination may take place causing a drop in the fluidity. During the past nine years the author and his co-worked have been engaged in accumulating other lines of evidence and in working out some of the consequences of the hypothesis so long as the truth of the fundamental hypothesis should remain unquestioned. Recently, however, James Kendal€6working in the Nobel Institute of Physical Chemistry under the direction Am. Chem. Jour., 34, 481 (1905). Lees [Phil. Mag., [ 6 ] I, 128 ( I ~ o I ) ] considered the fluidity formula for mixtures but discarded it because he found an empirical formula which better accorded with his data. Jour. Am. Chem. SOC.,33, 1257 (1911). Am. Chem. Jour., 35, 195 (1906). Thorpe and Rodger: Jour. Chem. SOC., 71,374 (1897). Bibliography and summary of results, Phys. Rev., 35, 407 (1912). 6 Meddelanden K. Vetenskapsakademien Nobelinstitut, 2, No. 2 5 (1913).
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of Arrhenius has reached the conclusion that ‘ ‘ the logarithmic viscosity (or fluidity) of a solution is the characteristic additive property, and not these quantities themselves.” Kendall reached this conclusion by obtaining the viscosity data for as many “normal” mixtures as possible and then comparing the observed values with those calculated on the various hypotheses that ( I ) viscosities are additive when volume concentrations are employed, ( 2 ) logarithmic viscosities are additive when volume concentrations are employed, (3) fluidities are additive when volume concentrations are employed, (4)fluidities are additive when weight concentrations are employed, and ( 5 ) logarithmic viscosities (or fluidities) are additive when molecular concentrations are employed. He finds that the last gives better agreement and on this fact he bases the conclusion that the logarithmic viscosity (or fluidity) of a solution is the characteristic additive property. It is highly important that it be soon decided once for all whether viscosity, fluidity, or the logarithm of the fluidity (or viscosity) is to be regarded as the characteristic additive property in a given case. Viscosity, as a property of matter, is of fundamental importance in many lines of investigation. It is absolutely necessary that we have a clear understanding of the nature of viscosity before we can hope to obtain a satisfactory understanding of conductivity, migrations of ions, rate of diffusion, rate of crystallization, and of the nature of colloidal solutions; and we may reasonably expect to find viscosity related in an important manner to volume, vaporpressure, association, hydration, and other properties. Discussion of this subject therefore seems desirable. Admitting a slight numerical superiority for the empirical logarithmic formula, is one justified in making an inference on account of. it in regard t o a fundamental physical relationship? As in the expansion of gases, may it not be that the ideal case is never realized and that, moreover, the exceptions are for the most part in one direction? Whatever the answer, the importance of the law which applies strictly to the ideal case only, like Boyle’s law, is not lessened thereby.
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There is a further objection to Kendall’s conclusion, which is more serious. He has applied his test to so-called “ normal ” curves only, and following Dunstan’sl classification, Kendall regards all viscosity curves as normal which do not have a maximum or minimum. This classification places all mixtures in three categories which are quite sharp but also quite arbitrary, since, as has already been pointed out,2 the occurrence of a maximum or minimum is an accidental circumstance, depending very largely on the nearness to equality in the viscosities of the components of the mixtures. If the viscosities of the components are exactly equal, the feeblest sort of chemical combination will be sufficient to produce a maximum; and on the other hand, a large amount of chemical combination may be insufficient to produce a maximum if the viscosities of the components are very ~ n e q u a l . ~ We shall now attempt to prove that the method of selecting the data used in the comparison described above would have tended to produce the result noted by Kendall even if fluidities are normally additive. In Fig. I AFB is drawn to represent a linear fluidity curve, abscissas representing concentrations and ordinates fluidities. If this were the normal fluidity curve, supposed to be given by unassociated and chemically indifferent liquids, the normal viscosity curve obtained by plotting the reciprocals of the fluidities would be ufb, which is part of an equilateral hyperbola whose asymptotes are CD and CE, the point C being the point where the fluidity curve cuts the X-axis. On the other hand, if the viscosities are additive, the normal viscosity curve is of the form uVb, so it is seen that the viscosity of any mixture is smaller, if the fluidities are additive, than would be expected, Jour. Chem. SOC.,91, 83 (1907). Phys. Rev., [ z ] I, 108 (1913). Kendall himself notes that Dunstan’s classification is not altogether satisfactory, for “even where the curve of the viscosity appears to be of the normal type, changes of state may have occurred in the mixture which are not sufficient to bring about a maximum or a minimum point.” For an attempt t o obtain a more natural classification see Zeit. phys. Chem., 83, 660 (1913).
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Eugene C. gingham
if viscosities were additive. It can be provedl mathematically that this must invariably be the case except when the fluidities of the components are identical, in which case the difference becomes zero. Now if logarithmic viscosities (or fluidities) are additive, the normal viscosity curve will be alb, which is intermediate between the other two curves ; and it can be shown that if logarithmic fluidities are additive, the viscosity of a mixture must always be greater than would be expected on the hypothesis that fluidities are additive and less than would be expected on the hypothesis that viscosities are additive.
Fig. I
Were liquids usually chemically indifferent and unassociated, there would be no difficulty in deciding which one of these hypotheses is correct. Unfortunately for our purpose, in the greater number of mixtures there is a change of temperature and volume on mixing and the cause of these changes also affects the viscosity and other properties. When there is a considerable contraction and heat evolution on mixing, there seems to be an invariable increase in the viscosity, and on the other hand, when there is expansion and heat absorption on mixing there is a lowering of the viscosity. In Jour. Am. Chem. SOC.,33, 1261 (1911); cf. Phys. Rev., 35,411(1912).
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taking the viscosity curves which show neither maximum nor minimum as his material for comparison, Kendall practically took the curve alrb as his normal curve, since it lies half way between the two extremes, where a maximum or minimum occurs and yet the object of the investigation was to discover which curve should be considered normal. Having taken as normal those curves which approximate to alrb instead of to afb or a h , the hypothesis that logarithmic fluidities are additive is placed a t a disadvantage and the hypothesis that fluidities are additive is placed at a still greater disadvantage. It would have been at least as logical to have taken as normal those$uidity curves which do not give a maximum or minimum and the result would certainly have been different, although of no value, because the case would have been prejudged in the choice of the data. The proper definition of a normal curve, to be as free as possible from objection, is the curve given by substances which mix without volume or heat change. This definition does not prescribe whether viscosities, fluidities, or other functions of the viscosity are to be compared and the question whether weight, volume, or molecular concentrations are to be used is likewise left open. If fluidities are really additive, Kendall’s choice of data gave preference to mixtures in which there is heat evolution and contraction since the corresponding increase in viscosity would correct the sagging of the curve afb. Thus while this method of choosing the data established a prejudice in favor of the hypothesis that viscosities are additive, yet the data are against that hypothesis, and the utmost that was accomplished was the shifting of the average curve, so that it lies between those expected on the two extreme hypotheses and somewhere near the intermediate curve given by the hypothesis that logarithmic viscosities are additive. From what has been said it seems probable that Kendall has considered a greater number of mixtures where contractioii and heat evolution take place than of the opposite kind; certainly examples of this kind are more common in the literature. Contraction and heat evolution seem t o be quite
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common in water mixtures, for example, and it is quite possible that these effects are due to a feeble chemical combination which is more pronounced and common than the opposite effect of dissociation. I have elsewherel attempted to show that the conditions in ordinary mixtures are quite analogous to those in electrical conduction when the resistances are in parallel, i. e., the fluidities, like the conductivities, are addim2cp2where ml and m2 represent the fractive or cp = mlql tions of the total volume occupied by each of the components, having fluidities of cpl and cp2, respectively. When under certain conditions the mixture is not homogeneous as in certain emulsions, the conditions are analogous to those in electrical conduction when the resistances are in series, i. e., the viscosities are additive or 7 = mlql n z 2 ~ 2 . Thus the fact is accounted for2 that when a homogeneous mixture is cooled below its critical solution temperature, the fluidity decreases quite unexpectedly. It is not apparent that so simple an explanation could be obtained if logarithmic fluidities are the characteristic additive property. When conductors are arranged so as to be in parallel and in series simultaneously, a checkerboard arrangement results. Lees3 has shown that for such an arrangement of viscous substances the logarithm of the resulting fluidity is equal to the logarithms of the components multiplied by their respective fractions of the total volume, or
+
+
log
cp =
ml log
cp1
+
PI42
log
cp2.
The author believes* that ordinarily this condition is a fleeting one and that after a time a homogeneous mixture results in which the fluidities are strictly additive. But in any case, as Lees has clearly pointed out, volume percentages must be employed and not molecular percentages Phys. Rev., 35, 407 (1912). Jour. Am. Chem. SOC.,33, 1257 (1911). 3 Phil. Mag., [6]I, 128 (1901). Phys. Rev., 35, 409 (1912).
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as Kendall has proposed. Kendall states1 “ A molecular formula has not been proposed. This is certainly strange, since from first principles the viscosity is a function of the molecular concentration, ‘ viscosity being essentially the frictional resistance encountered by the molecules of the solution in moving over one another.’ ” It is undoubtedly true that the size of the molecules determines the fluidity of each component, but it is difficult to see how the size of the molecules enters a second time into the calculation when two or more components are combined in a mixture. Finally there are various other lines of evidence in favor of the hypothesis that fluidities are additive, which at least have not yet been shown to be eqaally in favor of the hypothesis that logarithmic fluidities are additive. For example, suspensions of finely divided solids in liquids in which they are insoluble give fluidities2 which are a linear function of the volume concentration. Moreover these curves converge toward a definite zero of fluidity. The viscosity and the logarithmic fluidity curves are not linear and they do not meet except at infinity. A. Batschinski3 has very recently discovered that the volume of a liquid is a linear function of its fluidity u = a bp where a and b are constants characteristic of the given liquid. This would be a very surprising relation, if fluidities were not additive. Making use of the hypothesis that fluidities are additive it has been found4 possible to calculate the association of associated substances and to obtain values which are in excellent agreement with values calculated by other methods. It is probable that dissociation and combination play an unexpectedly common and important r81e in mixtures and solutions. It has been repeatedly pointed out that there is almost always some heat and volume change when liquids
+
Page 3. Am. Chem. Jour., 46, 278 (1911); cf. Phys. Rev., 35, 419 (1912). Ann. Soc. d’encour. sc. exper. et de leurs applic. Suppl., 3, 1913. Zeit. phys. Chem., 66, 28 (1909);cf. Phys. Rev., [ 2 ] I, 106 (1913).
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are mixed, which has been taken as an evidence of chemical action. Under these circumstances, it is well-nigh useless to attempt to find an empirical formula to fit all cases. It seems more logical to first discover what is the characteristic additive property. All cases which then do not fit the accepted formula-perhaps very few will fit it-will be explained on the basis of dissociation or combination or both. If these chemical changes are subject to the familiar chemical laws of Definite and Multiple Proportions and Mass Action, then it may prove a comparatively simple matter from the fundamental fluidity (or other) formula and the affinity constants to calculate the fluidity of any mixture at any temperature, knowing the fluidities of the components. Work by the author and his co-workers along this line has already yielded some very promising resu1ts.l The formation of a so-called ” hydrate” or other feeble combination is not to be lightly assumed as a pure matter of convenience. The trouble in the past has been not in a lack of evidence but that the evidence has been confusing. When two liquids were mixed, the contraction in volume and heat evolution were taken as evidences of chemical combination, and the composition of the hydrate was taken to be that of the mixture in which the contraction and heat evolution were a maximum. The maximum in the viscosity curve was likewise taken to indicate the composition of a hydrate, but this hydrate was different and moreover its composition seemed to vary with the temperature. Naturally when the testimony of the different properties upon this important question was so greatly a t variance, chemists were forced to regard the whole question as open. It is a satisfaction therefore to be able to state that in the cases which have been carefully investigated, the maximum deviation from the linear fluidity-volume concentration curve is found in the same mixture which shows the greatest heat and volume changes on mixing. In the case of 1
Zeit. phys. Chem., 83, 662 (1913).
?'he Viscosity of Binary Mixtures
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ether and chloroform,' for example, this mixture also corresponds exactly to the formula C4Hla0.CHC13,and furthermore the vapor-pressure curves of these mixtures give evidence of the formation of the same complex. Thus the way seems open for a considerable extension of our conceptions in regard to chemical combination and the difficult problem of hydration in solution. Richmond College Richmond, Vu. June 30, 1913 .~
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1
LOC.cit.
