APPROXIMATE RULES FOR SOLUTE ACTIVITIES
313
3. If the curves of h/(N2/N1) and ~ ' / ( N ~ / N us. ~ ) (N:/N~) can be extrapolated to infinite dilution with no more than a finit? crror, the values of the holutr activity coefficients (Q/N2) and (a2/m) obtained from the extrapolated curves will he infinitely more accurate a t infinite dilution ( i t . , thc error is reduced by a factor of m) than could be obtained from the expcrimental data alone. The extrapolation can be made to within ) finite a finite error provided the functions ~ / ( N z / N ' ) and ~ ' ( N z / N ~have limits. 4. For an ideal solution ~ ' / ( N ~ / N should ~) be everywhere zero, while ~ / ( N Z / N ~has ) a definite form as a function of ( N ~ / N ~with ) the limiting value of 1/2 and the limiting slope -1/3. Different types of deviations from ideality and their effects on the limiting values and limiting slopes of the ~ / ( N ~ / N and ~ ) ~ ' / ( N ~ / N us. ~ ) (N~/N') curves are discussed. REFERENCES (1) cf. HILDEBRAND, J. H. : Solubility of Non-electrolytes. ilmerican Chemical Society Monograph No. 17. Reinhold Publishing Corporation, New York
(1936). (2) LEWISA N D RANDALL: Thermodynamics and the Free Energy of Chemical Sub-
stances. McGraw-Hill Book Company, New York (1923). ( 3 ) VANLAAR:Z. physik. Chem. 72, 723 (1910);83, 599 (1913).
APPROXIMATE RULES FOR SOLUTE ACTIVITIES I N DILUTE S o L w r I o x s OF NON-ELECTROLYTES. GENERALIZED FORM OF BURY'S RULE' MERLE RANDALL, W. F. LIBBY,
AND
BRUCE LONGTIN2
Department of Chemistry, University of California, Berkeley, California Received June l d , 1959
A recent article (3) described the general nature of the function h/r (used in calculating solute activities from solvent activities) for various types of deviation from the ideal solution laws. The behavior and significance of this and related functions in the limit of high dilution, together with a number of other problems, may be made still clearer by the use of series expansions of the various functions involved. 1 Clerical assistance of the Works Progress Administration 0. P. No. 665-08-3-144 is gratefully acknowledged. * Shell Fellow in Chemistry, 1938-39.
314
M. RANDALL, W. F. LIBBY ANB g. LONGTIN POWER SERIES EXPANSION BY METHOD OF RESIDUES
Let y denote any empirical function of x which is capablc of expansion into a power series in z. Let the series expansion be y = A0
+ A1z + A222 + . . . + A,zn + . . .
(1)
The essential requirement is that all of the derivatives of y exist and be finite a t z = 0. It is desired to obtain the coefficients A o , AI, AP, . . . A,, . . . etc., directly from the experimental data. This may be accomplished by a relatively simple device. Let us define the series of functions,
Y’ Y”
= (y =
- Ao)/z
(Y’
- Al)/Z
+ A ~ +x A3z2 + . . . = A2 + A3z + . . .
= A1
Y”’ = (Y” - A z ) / z = A )
+ . . . , etc.
(2) (3)
(4)
Each function is obtained from the preceding function as the difference between that function and the limit which it approaches at x = 0, divided by the value of z. The limit approached by each of these functions as x approaches zero is one of the desired coefficients Ao, AI, Az, . . . , A,, . . . . This procedure is the basis of the special graphical methods developed by Lewis and Randall (2) for the calculation of solute activities. Thus the functions In ( a l / r ) and h / r are the first and second derived functions (Y’ and Y”) which would be used in obtaining the series expansion of In al in terms of the mol ratio r. Likewise the functions elm and - A ( j / m ) are those which would be used in obtaining a series expansion of the freezing point lowering, e, in terms of the molality m, as is explained later. The application of this method is further illustrated in figures 1 and 2 by means of a set of experimental data. From the data it is estimated that y approaches the limit A . (figure 1). The value of (y - AD)corresponding to any one point is given by the vertical distance from the point to the horizontal line y = Ao. The value of the function Y’ corresponding to this point is given as the ratio of this distance to the abscissa (z coordinate) of the point. If the experimental data should fall on a straight line through the point (x = 0, y = A o ) ,then Y’ would have the same numerical value for all of the experimental points. The values of Y’ obtained from the experimental data are plotted in figure 2. From these values the function Y” may be obtained by a repetition of the above process. It is to be noted that the points scatter widely in the neighborhood of x = 0. This occurs because the experimental errors in y are magnified by dividing by z. The estimated limit A 1 was based chiefly on extrapolation from the higher values of z, where the experimental errors are not so greatly magnified.
APPROXIMATE RIJLES FOR SOLUTE ACTIVITIES
315
As the process is repeated to obtain the higher order coefficients, the experimental errors will become seriously magnified up to successively higher values of x. The estimated values of the coefficients will be based chiefly on the experimental data a t higher and higher values of x, until finally the data scatter too widely t o allow any reliable estimate of the values of higher coefficients. This merely means that with this number of terms in the series and the estimated coefficients, equation 1 has been made to fit the data within the limits of error. The addition of further terms might alter the shape of the resulting curve slightly but would not cause it to fit the data better. The proper choice of the coefficient A0 assures that the curve represented by equation 1 will approach the correct limit at x = 0. If there
FIG.1
FIG.2 FIG.1. Direct plot of a set of data
FIG.2. Plot of first derived function of the set of data of figure 1
were only the one additional term A lx, the curve would be a straight .line whose slope is so chosen as to fit the experimental data near z = 0. The higher terms are present to insure that the curve will fit the data a t higher values of x as well. Thus it is only natural that their coefficients should be determined chiefly by the data a t higher values of x. When the curve has been made to fit the data up to the highest value of x, the information which the data can give is exhausted, and it is futile to include higher terms in the series. It is always likely that a small error E will be made in estimating the limit A0 approached by y. The function Y r rwill then have the apparent value (y - A0 .t E ) / % instead of the correct value (y - A o ) / z . Thus an error (.tt / z ) is produced in Y ’ . While this error is infinite a t x = 0, i t becomes smaller-until negligible-at higher values of x. Thus if the region in the neighborhood of z = 0 is disregarded, the errora made in
316
M. RANDALL, W. F. LIBBY AND B . LONGTIN
estimating A o , Al, etc. are not cumulative in the estimation of the higher coefficients. The series expansion may be expressed analytically as a Maclaurin's series. From this standpoint the coefficients AI, Az, . . . , A,, . . . can be expressed in terms of the limiting values of the derivatives dy/dx, dzy/dx2, . . . , dny/dxn, . . . at x = 0. Thus we obtain the limiting laws Lim Lim x+o Y' = A1 = x+o dY /&
(5)
Thus the coefficient A1 gives the limiting slope of the curve of y us. x, while the coefficient A2 depends upon the limiting curvature, and the higher coefficients depend upon the way in which the curvature changes. One might have started with the derived function Y' rather than the Thus we function y, obtaining Y" as the 3 r d function derived from .'1 may also establish the rules Lim Y,, - Lirn dY'/dx x+o x+o Lim Y),( x+o
1 Lim d("-l)Y//dx -~ (n - l ) ! 2 4 0
(9)
and many similar sets based on the higher functions Y", ctc. as the first of the set. THE
h/m
RULEB
The function In al (the natural logarithm of the solvent activity, al) may be considered as a function of the mol ratio r (Le.' the ratio N ~ / Nof~ mol fraction of the solute to that of the solvent), which may in many cases be expressed by a power series in r. To find the coefficients of the series we define the functions
;":,[
y = In a1
Y' = (y - O)/r = (l/r) Inal Y"
=
h / r = (Y'
+ I ) / r = (l/r)[l +
y = 01 (10)
r+o Y' = - 1 1 [Lim (l/r) In all, etc.
(11) (12)
The function Y" is the function h/r used in calculating solute activities.
317
APPROXIMATE RULES FOR SOLUTE ACTIVITIES
An interesting rule may be obtained by combining equations 11 and 12 with equation 9. Thus for very dilute solutions Lim Lim Lim r-0 ( h l r ) = r+o d(l/r) In al/dr = r-tO dh/dr The final member of equation 13 appears, since h differs from (l/r) In a1 only by the term unity, which contributes nothing to the derivative. Equation 13 shows that the limiting value of h/r is the limiting slope of a curve of ( l / r ) In a1 us. r. The accuracy to which the limiting slope of this curve may be estimated directly is certainly no greater than the accuracy of the data a t high dilution. However, it was shown in the previous article (3) that the limiting value of ( h / r ) could under some circumstances be estimated far more accurately than the data alone would permit. Thus the plot of (h/r) us. r furnishes a more sensitive means of estimating the limiting slope of the curve of (l/r) In al us. r. Lewis and Randall (2) have shown that the so1ut.e activity, a2, is to be obtained by the differential equation h/r -d In (a2/r)/dr = dh/dr (14)
+
If In (a2/r) can be expressed by a power series in r and has the value
In yo a t infinite dilution, equation 5 shows that d In (a2/r)/dr has the same limit as"(l/r) In (azlryo). Remembering equations 13 and 14, one may therefore write
This is an exact statement of the approximate rule of Lewis and Randall that In ( a ~ / r y ois) equal to -2h. For aqueous solutions, in which the activity is taken equal to the molality (m = 5 5 . 5 1 ~ )a t infinite dilution, it may be written in the form Lim 1 Lim - In (az/m) = - 2 (16) m+O m m-0 (h/m) It is a simple matter to determine the limits within which equation 16 will give the activity coefficient (a2/m)accurately. To a first approximation ( h / r ) may be expressed as ( h / r ) o Br, where B is the limiting slope of the curve of ( h / r ) us. r. To this approximation dh/dr = (h/r)J 2Br, since h is given as (h/r)or Br2. Thus integration of equation 14 gives (to this degree of approximation)
+
+
+
+
In (azlrro) = 2(h/r)or (3/2)Br2 (17) and for aqueous solutions In (ar/m) = 2 ( l ~ / m ) ~ m (3/2)B'm2 (18) where B' is the limiting slope (B/(55.51)2) of the curve of ( h / m ) vs, m.
+
318
M. RANDALL. W. E’. LIBBY AND B. LONGTIN
Equations 17 and 18 differ from equations 15 and 16 only by the presence of the terms containing the coefficient B. The limiting laws of equations 15 and 16 will give the activity coefficient accurately if the term (3/2)Br2 is negligible. In general they will hold to within an accuracy of 0.01 per cent in (azlryo) up to the concentration a t which (3/2)Br2 has the value In 1.0001 = It was previously shown (3) that for an ideal solution B has the value -1/3. In this case the highest value of r a t which this accuracy will obtain is 0.014, corresponding to a molality of 0.80. For cadmium and tin amalgams, with values of B = 24.4 and -0.75, respectively, the accuracy of 0.01 per cent in (uzlryo) will hold up to r = 1.65 X (0.092 M ) and 0.009 (0.53 M ) , respectively. These last two are cases in which the deviations from ideality are fairly large. In general it may be said that the Lewis and Randall rule, namely, In az/ryo = -2h, should hold quite accurately up to a t least 0.1 M except in extreme cases. On the other hand, equations 17 and 18 should hold accurately up t o the neighborhood of 5 M (r = O.l), as is indicated by the data for cadmium and tin amalgams (3). In the region in which equation 18 holds, the activity coefficient (azlrn) is given as exp [2(h/m)om
+ (3/2)B’m21
This may be expanded in series to give az/m = 1
+ 2(h/m)0m + [(3/2)B’ + 2 ( h / m ) h 2
(19)
to the order of approximation of m2. An exactly similar equation may be written for (uz/ryo),with T and B substituted in the right-hand member for m and B’. Noting that r can be expanded as (NZ N: ... N; . . .), this latter equation may be rearranged to give
+ +
+
Uz/yoNz = 1
+ [I
+ 2(h/r)o]Nz + [I + 2(h/r)o + 2(h/r)i + (3/2)Bhi
+
(20)
Equations 19 and 20 may be used to calculate the activity directly from the limiting value and limiting slope of the plot of h/r us. r, and are valid up to mgderately high concentrations. FREEZING POINT LOWERING
The freezing point lowering, 8, produced by the presence of solute to the extent of m molal, is a function of the molality, which ordinarily is capable of being expressed by a power series in m. By definition it approaches zero as limit a t infinite dilution. In determining the coefficients of the series expansion the functions 0’= e/m, 0” = (e/m - A)/m, etc. (where
APPROXIMATE RULE&
FOR SOLUTE ACTIVITIES
319
A = 1.858 is the limiting value of elm) would be defined. The function j/m d e h e d by Lewis and Randall (2) is -@"/A. Equation 6 thus shows that the limiting value of j/m a t infinite dilution is given as
This limiting value depends on the curvature of the freezing point us. molality curve. Thus it is evident that the limiting value of (j/m) may be made uncertain by a common tendency to extrapolate to infinite dilution along a straight line. Similarly equations 7 and 8 show that the limiting slope of the (j/m) us. m plot is given by the equation
The limiting slope depends upon the rate of change of curvature of the freezing point lowering curve a t infinite dilution, and is made even more uncertain by the tendency toward linear extrapolation. Since both e and elm are functions of m, elm is also a function of 8. Bury (1) has assumed that (elm) can be expanded in a power series in 8, with the limiting value A a t infinite dilution. Thus by equation 5 the limiting value of [(elm) - A]/e is equal to the limiting slope, d(e/m)/de of a plot of (elm) us. 0. Since 0 approaches Am in the limit, the limiting Bury's value of [(elm) - A]/e is also equal to the limiting value of -j/m. rule states that
This rule can be given several generalized forms. If elm is considered as being expressed by a power series in m, then equation 8 gives
since 0'' differs from elm only by the additive constant A. This particular forin is analogous to equation 13. From the considerations established there, it is evident that the plot of j/m us. m or 0 together with Bury's rule furnishes a more sensitive and more accurate method of determining the slopes d(e/m)/de and d(e/m)/dm than the plots of elm itself. This is a reversal of Bury's proposal. Bury's rule is valuable in giving better values of e/m rather than better values of j / m .
520
M. RANDALL, W. F. LIBBY AND B. LONCTIN
In aqueous solutions the activity of water is given to a high degree of approximation in terms of the freezing point lowering as (2) In al = - (1/55.51) [@/A)
+ 2.85 x
W4e2
+ 2.3032
f 8.8 X 10-76'3]
(25)
The coefficient of the term in e3 is a t present quite doubtful, and this term is ordinarily omitted. The function x is that defined by Lewis and Randalla to take into account the difference in In a1 between the given temperature and the freezing point of the solution. It depends on the heats of mixing, is everywhere zero for ideal solutions, and vanishes a t least as rapidly as m2in other cases. From equation 25 and the definition of h/m, it follows that h/m = j/m
- 2.85
- 8.8
X
X lo-' (e/m)2e
+ (55.51 X 2.303)x/m2
(27)
+ 55.51x0
(28)
In the limit of infinite dilution Lim Lim (j/m) (him) = m+O m+O
- 2.85 X 10-'A2
where X o is the finite limit approached by 2.303x/m2. Equation 28 gives the value of (h/m)o needed in equation 19, directly in terms ?f the freezing point data. Equation 19 also requires the limiting value, B', of the slope, d(h/m)/dm, of the curve of h/m us. m. Since by equation 5
and
these limiting values may be obtained by subtracting equation 28 from equation 27, dividing by m, and passing to the limit. Thus Lim Lim B' = m+O d(h/m)/dm = m+O d(jlm)/dm Lim [(elm)' - X']/m - 8.8 X 10 - 2.85 x m+O (elm)'
:20
-,
+ 55.51 iFo[2.303x/m2- Xo]/m a
(31)
Reference 2, page 289: (26)
APPROXIMATE
RULES FOR SOLUTE ACTIVITIES
Again equation 5 shows that Lim m-+O has the value
- ~'Ilm
which by equation 24 becomes -2A2(j/m)o. limit As, while we will write X1 for m+O
32 1
The term in (e/m)3 has the
[2.303x/m2 - Xd/m
Equation 31 takes the final form
B' = Lim d(j/m)/dm m-+O
+ 5.7 X 10-'A2(j/m)0 + 8.8 X 10-'Aa + 55.51x1
(32)
The values of (h/m)o and B' obtained from freezing points by means of equations 28 and 32 can be used in equation 19 to calculate solute activities with the same accuracy as values obtained from other types of data. BUMMARY
1. A simple general method of residues is given for obtaining the coefficients in a Maclaurin's series directly from experimental data. The functions (h/m) and (j/m) used in calculating solute activities are shown to form special cases of the application of this method. 2: Equations are given whereby the solute activity may be calculated accurately for solutions up to about 5 M from a knowledge of the limiting values of the function (h/m) and the slope of a curve of (h/m) us. m (likewise (j/m) and the slope of a curve of (j/m) us. m) a t infinite dilution. 3. It is shown that Bury's rule and analogous rules may be useful in obtaining improved values of freezing point lowerings and solvent activities from extrapolated curves of j/m and h/m vs. m, respectively. This follows since in some cases it is possible to extrapolate (j/m) and (h/m) with an accuracy greater than that of the experimental data, REFERENCES ( 1 ) BURY:J. Am. Chem. SOC.48, 3123 (1926).
(2) LEWISAND RANDALL: Thermodynamics and the Free Energy of Chemical Substances. McGraw-Hill Book Company, New York (i923). (3) RANDALL AND LONQTIN: J. Phys. Chem. 44, 306 (1940).
