11. H. LIETZKE ASD R. W. STOUGHTON
508
Vol. 60
THE CALCULATION OF ACTIVITY COEFFICIENTS FROM OSMOTIC COEFFICIENT DATA’ BY M. H. LIETZKEAND R. W. STOUGHTON Chemistry Division, Oak Ridge h7ational Laboratory, Oak Ridge, Tennessee Received September 88,I961
A non-linear least squares method of fitting osmotic coefficient data in order to calculate activity coefficients is described. With this technique, the osmotic and activity coefficients of electrolytes can be expressed within the precision of the measurements over wide ranges of concentration. The method is greatly to be preferred over graphical integration if a high speed computer is available to perform the calculations.
Several different semi-empirical equations have been considered for representing the logarithm of an activity coefficient over a wide range of concentration. Each equation involves a one parameter Debye-Huckel term plus two or three higher terms, each of which is the product of a parameter and a simple function of the ionic strength I . When any one of the activity coefficient equations is substituted into the Gibbs-Duhem relation and integrated analytically, a corresponding equation may be obtained for the osmotic coefficient in which each of the parameters retains its ideiitity. Thus the parameters may be evaluated using a non-lineal least squares method from osmotic coefficient data and used to calculate activity coefficients us. concentration, or vice verso. Since it is relatively difficult to get accurate osmotic coefficient data at low concentrations and since an integration from zero concentration is required to evaluate activity coefficients therefrom, the possibility of obviating some of the difficulty usually encountered in such graphical integrations has been investigated by the use of analytical methods using an IBM 7090 computer. When a single-parameter Debye-Huckel expression for the activity coefficient of an electrolyte
is differentiated and substituted into
:SD”
$=1+-
mdlnyt
and the integration performed analytically, then the following expression for the osmotic coefficient 4 is obtained +=I--
’
A81
[(I
+ AI’h) 2 In (1
+ AI1/z)- 1 + AI%]
(3)
For use at higher concentrations linear, quadratic and cubic terms may be added as 9 = 1
- s
[(I
+ A I V ~ -) 2 In (1 + A Z V ~-)
The corresponding equation for the activity coefficient then becomes (1) This document is based on work performed for the U. 5. Atomic Energy Commission at the Oak Ridge National Laboratory, Oak Ridge, Tennessee, operated b y Union Carbide Corporation.
(5)
In equation 4 the parameters A , B, C and D for any particular electrolyte may be evaluated by the method of least squares2 and the corresponding activity coefficients may be computed immediately using equation 5. I n order to determine how many parameters were needed to give a good fit with equation 4, the osmotic coefficients of NaCl (as listed by Robinson and Stokes3) were fitted both with and without a cubic term. At both 25 and 100’ the inclusion of the cubic term produced a significantly better fit: the variance of the fit a t 25’ was 3.1 X without the cubic term and 1.2 X 10-7 with the cubic term; at 100’ the variance of the fit was 1.2 X without the cubic term and 7.3 X lo-’ with the cubic term. Hence all the calculations presented in this paper were performed with equation 4 as written. I13 Table I are given the parameters describing the concentration dependence of the osmotic coefficients of a variety of electrolytes as well as the variances of fit. Most of the data were taken from Robinson and stoke^.^ However, the values for LiC1,4 LiBr,6 LiI,4 KCl,4 KBr,6 RbC1,4 BeS0d6 and COzS046were computed from the original isopiestic molalities, while the values a t 99.6’ were computed from the isopiestic molalities reported by Patterson, Gilpatrick and Soldano.’ I n all cases the value of S (the Debye-Huckel limiting slope) was taken as 1.17202 a t 25’, 1.4107 a t 99.6”, and 1.4122 at 100’ for a 1 : l electrolyte. When the coefficients obtained by fitting osmotic coefficient data with equation 4 are used in equation 5 it is not necessary to normalize any of the activity coefficients to a particular value a t 0.1 m since the integration of equation 2 was performed analytically from m = 0. Thus the activity coefficients reflect better the differences between various salts (particularly the 2 :2 sulfates). Calculations with 25” data and equations 4 and 5 have shown that osmotic coefficients in the range 1-3 m can be used to calculate values of both osmotic and activity (2) H. Margenau and G. Murphy, “The Mathematics of Physics and Chemistry, ’ D. Van Nostrand Co., Inc., New York, N. Y.,1956. (3) R. A. Robinson and R. W. Stokes, “Electrolyte Solutions,” Academic Press, Inc., New York, N. Y.,1955. (4) R. A. Robinson and D. A. Sinclair, J . Am. Chem. Soc., 86, 1830 (1934). (5) R. A. Robinson, zbid., 87, 1161 (1935). (6) R. A. Robinson, J . Cham. Soc., 4543 (1952). (7) C. S. Patterson, L. 0. Gilpatriok and B. A. Soldano, ibid., 2730 (1960).
ACTIVITYCOEFFICIENTS FROM OSMOTIC COEFFICIENT DATA
March, 1962
PARAMETERS Electrolyte
NaCl LiCl LiBr LiI KCl KBr RbCl CaClz MgCL BaClz BaBrz T\’azS04 BeSOl
uozso,
MgS0.i MnS04 NiS04 cuso4 ZnSOr CdS04
LiCl KC1 csc1 BaClz Na2S04 MgSO4
uo2so4
DESCRIBING THE
CONCENTRATION
B
A
1.45397 1.48996 1.24741 2.12556 1 ,30752 1.29231 1.30240 1.61291 1.60067 1 .59925 1.62543 1 ,24072 1.23250 0.998119 1 ,37486 1.28920 1.31677 1.14652 I.27839 1 20516
I.36704 1.87392 I.40727 1,17244 2.18824 1.58747 1.11309
x
10%
2.23565 10.2909 14,7940 13.8683 - 0.359188 0.994831 - 2.05431 4.56577 6.63253 1.23161 3.81918 -. 6.58044 - 1,54261 0.399490 -. 5.42492 -. 5.47447 -. 7.20761 -. 2.25375 -. 5.56227 5.08061
-
4.62413
- 2.83699 - 1.45738 6.81811
- 11.0123
- 6.80762 -’
2.11182
TABLE I DEPENDENCE O F THE
OSMOTIC COEFFICIENTS O F A VARIETY O F
TROLYTES x 108
c
D x
25O 9.30838 6.04782 - 0.61962 - 0.261474 7.17091 4.34095 13.9445 8.57310 9.00272 8.81121 8.02100 7.26282 5.10411 2.27077 8.42636 7.36518 10.2081 -. 1.13297 7.60615 7.00382 99.6’ 20.0758 19.5070 10.1090 - 7.60404 15.2961 6.51886 2.52142
509
104
- 5.36209
Ofit*
X 1084
ELEC-
Range,b m
5,40012 12.8635 40.7768 - 5.67500 - 3.50742 - 15.6796 - 2.73800 - 2.54526 - 6.92099 - 5.19535 - 1.94540 - 0.945238 - 0.563181 - 1,89929 - 1.56926 2.45406 6.24346 - 1.25180 - 1.69009
0.12 5.91 6.09 19.39 1.46 0.24 74.82 4.03 15.01 0.84 1.90 4.41 37.08 11.14 3.84 17.58 3.34 3.42 5.67 4.52
0.006-6.0m .111-3.159 .166-3.325 ,120-3 ,152 ,1084.810 .loo-5.000 .4234.962 ,002-6.00 .loo-5.00 .loo-1 .80 .loo-2.00 ,100-4.00 ,102-4.286 ,113-6.371 .loo--3.00 .loo-4.00 .loo-2.50 .loo-1.40 .loo-3.50 .loo-3.50
-25.0075 -21.7686 - 8.77824 5.47538 - 7.04151 - 1.15772 - 0.580834
8.03 6.64 18.53 17.97 0.55 10.90 40.99
0.950-3.845 0.993-4.742 1.018-4.924 0.714-2.199 0.891-.3.178 1.001-4,753 2.083-5.041
- 7.13179
0.73
-
looo a
NaCl 1,55510 3.64784 6.43661 Variance of fit. b Range of concentrations used in making the fit.
coeficienl s at lower concentrations with good accuracy. Thus activity Coefficients may readily be calculated over a wide range of concentrations frorn relatively few osmotic coefficient data. Moreover, the use of equations 4 and 5 is greatly to be preferred over graphical integration if a highspeed computer is available to perform the nonlinear least squares fits. Care should be exercised, however, in using the equations (particularly if a cubic term is included) for extrapolation to concentrations higher than those used in making the fit since jictitious points of inflection may be encountered in this region. The method also will fail if an electrolyte exhibits sufficient ion association so that values of the osmotic coefficients fall below the limiting slope at low concentrations. The method of computation described in this paper represents a considerable extension over the method recently described by Guggenheim and Stokes.8 In their paper Guggenheim and Stokes demonstrated that a two-parameter equation was valid for CaC12 solutions to m = 0.4. They then assumed that the two-parameter expressions were valid for other typical 2 : 1 and 1: 2 electrolytes and chose the two parameters in each case to give the best fit of the isopiestic values a t molalities (8) E. A. Guggenheim and R. H. Stokes, Trans. Faradaz, Soc., 64, 1646 (1958).
0.05-4.0
0.1, 0.2, 0.3 and 0.4. With the coefficients so obtained the activity coefficient of each salt was computed at 0.1 m. In the present work the osmotic coefficients are fitted with equation4 and those parameters are immediately chosen for any electrolyte which satisfy the statistical criterion that x ( + o b s - +c)2 be minimized. It is interesting to i
compare the values of the activity coefficients a t 0.1 m computed in the present work with those reported by Guggenheim and Stokes for several electrolytes. These values are presented in Table 11. TABLE I1 ACTIVITYCOEFFICIENTOF SEVERAL ELECTROLYTES AT 0.1m
VALUESOF
THE
y
hlgClz
CaCI2 BaC12 BaBrz NazS04
(old)*
0.529 .518 .500 .513 .445
y (G. and
S.)8 y (present oalcd.)
0.528 .518 508 .517 .452 I
0.525 .520 .509 .519 .448
Acknowledgments.-The authors wish to express their appreciation to Drs. George Scatchard and G. E. Boyd for helpful suggestions and encouragement.
