MASS L A W STUDIES, I1
BY S. F. TAYLOR
When to two practically non-miscible liquids a third is added which is consolute with the other two, there are present two liquid phases and a vapor phase. All three components are present in all three phases and with increasing concentration of the third liquid there is a change in the composition of each of the thiee phases. T h e changes in the vapor phase do not form part of the present discussion and will not be referred to again. . T h e changes in the compositions of the two liquid phases are such that these two phases become more nearly alike with increased addition of the third liquid ; eventually become identical and disappear, leaving the trivariant system, solution and vapor. While there are measurements' on record giving the different relative proportions in which two non-miscible liquids and a consolute liquid may be combined in order to form a saturated solution at a given temperature, there are almost no observations showing what solutions form stable pairs. , I n other words we have very little knowledge of the composition of coexistent liquid phases. At the request of Professor Bancroft, I have analyzed six mixtures of benzene, water and alcohol. Before describing the methods and the results it will be well to point out the general form of the relations for this system.2 Let x, refer to the water in the upper liquid phase, x, to the water in the lower liquid phase, y,,y, and z l , z, to benzene and alcohol respectively. It is entirely immaterial what units are selected. I n any case x / z will represent the amount of water with reference to alcohol and x / Y the amount of water with reference to the volume of the solution. If z and I/ are kept constant and made 'Pfeiffer. Zeit. phys. Chem. 9, 444 (1892); Bancroft. Phys. Rev. 3, 2 1 (1895). I' am indebted to Professor Bancroft for the theoretical treatment.
S. F. Taylor
462
equal to unity x / z becomes x and denotes the amount of water in unit quantity of alcohol while x/ I/ becomes x also but denotes the amount of water in unit volume of the solution. Since care will be taken to specify what variable is kept constant in any particular case, there should be no room for confusion. Applying Wenzel’s statement of the Mass Law’ that the strength of the chemical action is proportional to the concentration of the acting substance, we shall have x1”y,/z?+‘ = C,,? for the upper liquid phase and x , ” ~ J ~ / z= , ’ ~C?,, ~ + for ~ the lower liquid phase where rcI and ?z, are
constants but not necessarily integers. In the particular case of water, benzene and alcohol, it has been found experimentally that n, = n, whence it follows that C,,I = C., because we can pass continuously from one liquid ph’ase to the other. Dropping the superfluous subscripts we may write :
By a similar application of the Mass Law we see that the concentrations in the two coexisting liquid phases will be described, when z, = z, = unity, by the equations : x,”’x, = C,
( I ) and
y,””y,= C,
(2)
I n these equations x,, x,, yI and y,, denote the concentrations of water and benzene respectively, in the two phases, referred to constant unit quantity of alcohol while m ,like n, is a constant but not necessarily an integer. From equations I and ( I ) there follows a series of relations. If we express the water and alcohol with reference to a constant unit quantity of benzene we may write : x 1”‘x2 = C’
( 3 ) and z2”’za=C, (4) If such quantities of the two phases are considered that the amount of water in each is always constant and equal to unity we shall have
y,’”= ~ , C, ( 5 ) and z11”z2= C,
(6)
If all three components vary while the volume remains constant we have : Qstwald. Lehrbuch
2,
11, 40.
,
Mass L a w Studies
463
I1 The relations among the constants are shown by the following equations :
C, = r/C, = CJC, = CdC3 c, = /IC, = CJC, = c,c, C' =
c, c,
1
(7) (8)
(9)
Since x, =x,,y,=y, and z,=za when the two liquid phases become identical, we also have the relation : log
c=
12
log nt
c, + log c,
+r
If we use capital letters to denote the amounts of water, benzene and alcohol in a given volurne of the solution at the moment when the two phases become identical we may write :
It should be kept in mind that if equations I and ( I ) hold, the others necessarily follow, being merely algebraical transformations. Equation I has already been shown to describe the variations in composition of each of the liquid phases in the divariaiit ternary system, two solution phases and vapor, so that the primary object of this paper is to test the validity of equation ( I ) and to determine the value of the exponential factor m. Since mixtures, of water, benzene and alcohol can not be analyzed with any comfort gravimetrically, it was thought advisable to make use of an optical method. T h e temperature at which the measurements were to be made was 25' since I had already made a series of determinations for the single phases at that temperature. Solutions were made up containing five cubic centimeters of alcohol and varying but accurately known amounts of benzene and water. These latter quantities were so chosen that the solutions were exactly saturated at 25'. These solutions were warmed to 26' to prevent any danger of precipitation and the index of refraction for that temperature was determined by means of a Pulfrich's refractometer. Laying off the amounts of benzene or of water in the solu-
S. F. TayZor
464
50. 40. 30.
0.33 0.37 0.44
5.00 5.00 5.00
1.48383 1.48I 5 I 1,47597 1.46627 1.44226 1.41758 1.41035 I . 40066 1.38972
0.55
5.00 5.00 5.00 5.00 5.00
2.0
0.79 1.15 1.30 1.52 1.90
5.00
1.8
2.00
5.00
I . 38650
1.0
2.75 3.00 4.00 5.00
5.00 5.00
1.37551 I 37048 1 ' 36432 I . 36072
20. IO.
5.0 4.0
3.0
0.85 0.50
0.33
5.00 5.00
T h e actual experiments were made by putting differen? amounts of the three components into stoppered test tubes, letting the system stand at 25' until the two solution phases were perfectly clear, pipetting off portions of each layer, warming them to 26' and determining the indices of refraction. A few preliminary experiments served to show in what proportions the three components should be taken in order to get enough of each of the layers to make it possi.Jour. Phys. Chem. I , 301 (1897).
Mass Law Stzidies
465
ble to pipette them off with comfort and s t the same time to ensure a sufficient range of concentrations. In Table I1 are given the experimental data as found from the curve. T h e Roman numerals serve to identify the six s y s t e m . In the second column are the cubic centimeters of benzene in an amount of the upper layer suqh that it contains five cubic centimeters of alcohol synthetically. In the third column are the values for water also expressed in cubic centimeters and referred to five cubic centimeters of alcohol as the constant unit. In the fourth and fifth columns are the corresponding values of benzene and water for the lower layer. I n the sixth column are the values for the constant obtained from the equation
1.55 logy,
+ logy, = const
by taking fory,, the values of benzene given in the second column of the table and for y , the values of benzene given in the fourth column of the table. T h e values for solution V I were determined by extrapolation and are therefore not as accurate as those for the other solutions. Upper layer
I I1 1x1 IV V VI
~
I
1
8.5 10.4 16.0 24.0 34.5
51.0
o 86 0.77 0.61
-____
~
~
I
i 1
4.8 3.5 1.8
1.18
1.40 2.00
2. I 2 2 2.120 2.121 2. I 17
2.124 2.124
It will be seen that there is only a very slight variation in \he con. stant ; in fact the change is so slight that there is danger of conveying a false’ idea with regard to the accuracy of the experiments. Fortunately, the data to be given in table VI will offset this to a certain extent. T h e qualitative relations shown in Table I1 are very surprising. I had supposed that in the upper layer the ratio of benzene to water would always exceed unity and that in the lower layer this would be reversed. On this assumption the two phases would become
S. F. Taylor
466
identical when the ratio of benzene to water was unity. If this were true when the quantities of benzene and water were expressed in one set of units, it would no longer be true of another set of units. I t would be rather an interesting problem-though not coming within t h e scope of this paper-to consider whether for some set of units one could establish the proposition tliat the ratio of one of the nonmiscible liquids to the other is necessarily greater than unity in one liquid phase and less than unity in theother and, if this were established, to determine what those units should be. If one expresses the results in cubic Centimeters, the relation does not hold for water and benzene in the presence of ethyl alcohol. In solutions I and I1 the ratio of benzene to water is greater than unity in both solution phases. If the concentration is expressed in reacting weights instead of in cubic centimeters, there is always more water than benzene in the lower layer and more benzene than water in the upper layer-at least so far as these six sets of measurenients go. If it should be shown that for some set of units, the ratio of the two non-miscible liquids becomes unity at the point where two liquid phases cease to be possible, we could write equation ( I I ) in the form
+
by making s = 2 m - 12 r , since X= Y by definition. T h e constant in the sixth column of Table I is not any of the constants in equations 1-12 but we can easily pass from it to them. If the concentrations of benzene and water had been expressed with reference to one cubic centimeter of alcohol, the constant in the table would have been equal to log C., By subtracting 2.55 log 5 we shall have log C,. In table 111 I have tabulated the values of the logarithms of the six constants in equations 1-6 as obtained from my experiments. the unit being the cubic centimeter and setting m = r.55. The actual volume concentrations of the solutions can not be determined from my experiments because we do not know how much expansion or contractiun there is when water; benzene and alcohol are mixed in varying proportions ; but this is not of the slightest importance so far as the quantitative relations which interest US are
Mass L a w Stzidies
TABLEI11
I11 Iv V VI
10.1go--2 10.194-2 10.187-2
0.187-2 10.190-2
0.339
0.849-3 0.853-3 0.851-3 0.859-3 0.845-3 0.845-3
0.660-1 o 662-1 0.66 1-1 0.665-1 0.658-1 0.658-1
0.85173
0.661-1
1.811
1.809
1.810 I . 806
1.813 1.813
1 2.149
1.810
concerned. Professor Bancroft' has shown that the term for the volume disappears when considering one phase and the same proof applies in the same way to two coexistent phases. Let VIbe the volume of the upper liquid phase and V, the volume of the lower liquid phase taken for analysis. Dividing the amount of each of the components, expressed in any units whatsoever, by the volume of the solution gives the concentration per unit volume. Substituting these values in equation I1 we get :
Siiice V, and V2cancel out, we come back to equation 11,thus showing that the actual volume occupied by any given quantity of the solution does not appear in the equations which describe these phenomena. A contraction or expansion on mixing will increase or decrease the volume concentration of each of the coniponents ; but it changes them all in the same ratio and therefore does not change the relative concentrations. If the determinations of water, benzene and alcohol had been expressed in grams, it would be natural to show the way in which the three components vary simultaneously by giving the concentratioii of each component in one hundred grams of the solution. Since m y measurements are given in cubic centimeters I shall tabulate the data, giving the values in cubic ~
IPhys. Rev 3, 116 (rSgg).
S. F. TayZoor
468
centimeters of each component in ten cubic centinieters of the solution synthetically, i n other words the volume concentrations in cubic centimeters per ten cubic centinieters if there were neither expansion nor contraction on mixing. This is done in Table IV, there being added in the eighth column the values of log C' as calculated from equation 11. TABLE IV Upper layer
H,O JC,H,OH~C,H, ~~
1.07 0.29 0.17
1
0.10
o.oG
,
2.31 1.70 1.25 0.89
1
1
1.41
2.04
2.27
1.08
3.22 4.06 4.99
0.59 0.28
4.56 5.05 5.69 5.70 5.35 4.73
0.529-2 0.528-2 0.529-2 0.527-2 0.530-2 0.531-2
T h e changes in the constant, log C', are necessarily of the same order as in the first two tables and merely show that there have beeri no errors in the algebraical transformations. Of more interest are the qualitative changes in the concentration of alcohol in the lower layer. Instead of increasing continuously it passes through a maximum and decreases. This was entirely unexpected and the possibility of such a thing has never even been suggested. This apparently anomalous behavior has, of course, nothing to do with the choice of units. I t would remain just the same if the concentrations were expressed in reacting weights per liter. If we know the total amounts of the water, benzene and alcohol in the two liquid phases, and the concentrations of those two phases, it is possible to calculate the aiiiount of each phase which will be formed from the original mixture and thus check the accuracy of the results. Siiice it would be very difficult to separate the two phases completely and weigh each without loss, the next way would to iiieasure the volumes of the solutions formed. This would be thoroughly unsatisfactory except in the ideal case in which neither contraction nor expansion occurs. What we can do is to turn the problem round and see whether quantities of the two phases could be found such that the total amounts of water, benzene and alcohol in the
'
,
Mass Law Stndies
469
two phases should equal the initial masses or volumes of the three components which had been taken to form the system under consideration. This has been done and the results are given in Table VI. Since absolute agreement was not to be expected it became a question whether to divide the variations up among the three components thus minimizing the apparent error or to solve for the amounts of the two phases necessary to make the calculated quantities of two of the components equal the experimental values, concentrating the error upon the third component. I have adopted the latter course with the first four solutions and the former with the last two. In Table V are given the number of cubic Centimeters of each phase, synthetically, which are assumed in calculating the values in Table VI. Of course the sum of the two phases would add up twenty cubic centimeters if there were no experimental error. Equally of course, that ideal condition is not realized in any single case, In the second, third and fourth columns of Table V I are the values for benzene, water and alcohol in cubic centimeters calculated from the experimental concentrations of the phases and the data in Table V. I n the fifth, sixth and seventh columns are the quantities of benzene, water and alcohol in cubic centimeters which were taken to form the solutions I-VI.
TABLEV ~
~
-
~
_ _ _ ~__.______ - _ _ _ _ _ _____._____ ~ I
1
I 1 13.32 6.73 11 12.7 I 7.5 111 I 10.53 1 9.42
TABLE VI Calculated -
Found
~~~~
~ ~
4.9r
I 5.50
.
-~~
9.45 I 4.90
~
I 5.07
470
S. F. Taylor
T h e relations given in equations 1-6 describe the distribution of one component between two phases when one of the other components is kept constant and equal to unity in both phases. By substituting from equation I in equations 1-6 we can get the expression for the simultaneous concentrations of one component in one phase, and another in the other phase, the concentration of the remaining component being kept constant and equal to unity in both phases. These new equations will all hold absolutely provided equations I and (1)’are absolutely accurate. When z, = z, = unity we have :
When y , =J’, = unity we have :
When x, = x, = unity we have :
In addition there are a series of relations for the distribution of one Component between the two phases when one of the other coniponents is kept constant in one phase and the remaining one in the other phase. They may be deduced from the preceding six equations by substituting for the conditions there imposed the conditions that what are now variables in those equations shall be constant and equal to unity. Doing this we get the following results : When x , =y, ==unity we have : z2/2,111“ = C,”/C
When y, =-x, = unity we have :
C, IC
~ , ~ ~ =/ 2 , ~ ~ ~
When x , = z, = unity we have
;
Y z ” a ’ z Y P ’ J + r-
c/c,”
When x, = z, = unity we have : = I/c,t4~C
n~ii+iii~~ii
Y,
47 =
Mass Law S t d i e s When y, = z, = unity we have :
yl”Lx,?‘fl = qc, When z, =y, = unity we have : =
I/c
(sa.)
( 6aa 1 Of this last batch of equations only the one marked ( z a a ) is of any importance. Letting ?z/m= j and condensing the constants in the equation into a single one the relation may be written in the form : x ~ l ~ ~ ~ < ~ l / & x
2
z:/z, = constad
??+I
6
‘
111.
Here z, is the amount of alcohol in a constant quantity of water and z, the amount of alcohol in a constant quantity of benzene. This is the form in which Nernst should have deduced his Distribution Theorem.’ T h e actual farm differs only slightly from this but the difference is fundamental. Nernst writes the equation as follows : c,”/cI = constant
IV
where c, is the concentration of alcohol in the lower or aqueous layer, expressed in reacting weights per liter of the solution, c, is the concentration of alcohol in the upper or benzene layer expressed in the same units, while n is a constant and an integer, equal to one when the reacting weight of alcohol is the same in the two solvents. Equation I V is deduced on the express condition that the two non-miscible liquids do not become more soluble in each other with increasing addition of the consolute liquid-a condition which is never fulfilled theoretically. Equation I11 is free from this restriction and contains only the aSsumption-made also in deducing equation IV-that the equilibrium between the phases is described by the Mass Law. Equation 111 is therefore applicable to all concentrations, equation I V only to unrealizable ones. For very dilute solutions and for cases where the exponential factor p does not differ much from unity, the two equations I11 and IV will give practically the same results because in very dilute solutions the mass and the volunie concentrations are proportional, while the slight differences between the experimental values and those calculated from equation I V can be attributed to association or dissociation. With increasing ‘Zeit. phys. Chem. 8 ,
IIO
(1891).
S. F. TayZor
472
concentration of alcohol a comparison between the equations becomes superfluous because one describes the 'facts and the other does not. From the data which we already have it is possible to calculate all the constants in equation 111. Since n = 1.85 and m = r.55 njm = r.r9 because the accuracy with which n and m are known does not justify another decimal place in the quotient. T h e constant in the equation is equal to the mth root of C,/C. Since log C = 0.820-2 and log C, = 0.339 the logarithm of the constant will be 0.98. This refers to the concentration of alcohol in cubic centimeters by synthesis in one cubic centimeter of water for one phase and in one cubic centimeter of benzene for the other. If we wish to express the concentrations with respect to five cubic centimeters of water and benzene respectively we must add 0.r9 log 5 t o this value which gives us 1.113 for the logarithm of the constant and 12.96 for the constant itself. T h e experimental value is 12.91 and this discrepancy of less than one-half of one percent would be wiped out if the value of njm had been carried out to one more place of decimals. In the second column of Table VI1 are the concentrations of alcohol expressed in cubic centinieters by synthesis per five cubic centimeters of benzene in the upper layer ;l in the third column are the concentrations of alcohol in the lower phase, expressed in cubic centimeters and referred to five cubic centimeters of water as the unit. In the fourth column are the values of the constant as calculated from these data by the formula ( 5za)'.'9/( 53,) = const
while in the fifth column is the predicted value of this constant.
TABLEVI1 const
I
I
2.94 2.41
I1 I11 IV
'
VI
I
v
1.04 0.73 0.49
JDeduced from Table 11.
21.2
17.9 12.j 8.77 6.59 4.72
12.91 12.88 12.94 12-74 12.91 12.91
12.96 12.96 12.96 12.96 12.96 12.96
Mass L a w Studies
473
In this paper there is given a complete discussion of the equilibrium between two liquid phases at constant temperature for systems made up of two non-miscible liquids and a third consolute with the other two, when the whole isotherm can be presented by a single equation. T h e relations deduced have been tested by means of experiments upon the system, benzene, water and ethyl alcohol. It is also shown that one must use mass concentrations in expressing the distribution of a substance between two liquid phases. '
CowteZZ University
