John D. Worley and David R. Fussaro
universitv of Cincinnati cincinndti. Ohio 45221
Calculation of Activity Coefficients
11
I
from Spectroscopic Data A
The determination of activity coeficients for mixtures of nonelectrolytes is usually carried out by measuring the partial pressures of the components and their mole fractions in both the vapor and liquid phase. The experimental analysis may be simplified through the use of the Gihbs-Duhem equation (1-3). This reduces the experimental problem to the somewhat less formidable task of measuring only the mole fractions of all components in the liquid phase and their respective partial pressures. By restricting the system to a mixture of one nonvolatile and one volatile component of low solubilities in water, the activity coefficient of the volatile component may be determined as a function of its mole fraction from spectroscopic measurements. The essence of this experiment is that it allows the substitution of solubilities and optical absorbance~for partial pressures. The former parameters are easily measured with an ultraviolet spectrophotometer and do not require the construction of any special apparatus beyond the standard equipment available in an undergraduate physical chemistry laboratory. Experimental
Equipment necessary for carrying out this experiment are an ultraviolet spectrophotometer, several 250-cc laboratory bottles with caps, several sample bottles with a minimum volume of 10 cc and of suitable diameter for fitting inside the laboratory bottles, a constant temperature bath, a 6-in. hypodermic needle and syringe, and matched absorbance cells of 1 cm path length. The isopiestic solute transfer technique, which forms the basis for this experiment, was developed by Chris-
Optical absorbance of p-rylene in oqueous phase versus mole fraction p-rylene in hydrocarbon phase. Solid line corresponds to the curve expected if p-xyiene-dodecone mixtures obeyed Raouit's tow, i.e., y = 1 .O. = 0.884. Points correspond to average d u e s from two or more 0.5-C. obtorbonce measurements. Temperotvre = 32.0
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lourno1 of Chemical Education
chemistry experiment
tian, Affsprung, and co-workers and has been previously described by these investigators (4-6). I n our studies, the only essential differencewas that the aqueous and nonaqueous phases were interchanged in the vessels where the equilibrium process was carried out. Samples were equilibrated with mixtures of p-xylene and dodecane of known mole fraction in a constant temperature water bath at 32 + O.l0C for at least a period of 45 hr. Subsequent studies have shown that equilibrium is usually established within a period of approximately eight hours, but to assure ourselves that we had all systems a t equilibrium we let the samples equilibrate with the p-xylene-dodecane mixtures for a t least 48 hr. Absorbance measurements were made a t h = 274 mp in 1-cm cells. Solutions were thermostated by mounting them in a temperature-regulated cell holder. The temperature of the solutions in the cells was believed constant to Zt0.5"C. Analyses were carried out using a Beckman DR-1A recording spectrophotometer. Dodecane was the product of Phillips Petroleum Co. and was believed 99 mole yopure. The p-xylene was obtained from the Eastman ICodak Co. and had a melting point range of 12-13'C. No further purification of reagents was attempted. Mixtures of p-xylene aud dodecane at known mole fraction were prepared by weighiug. Aqueous samples were prepared using distilled water. Absorbance measurements were made against a reference of distilled water. Results and Discussion
The results of this experiment aresummarized by the figure which shows a graph of the optical absorbance of p-xylene in the aqueous phase versus the mole fraction of p-xylene in the hydrocarbon phase a t 32°C. The absorbance appears to be a nonlinear function of the mole fraction of p-xylene. We have interpreted these results in terms of deviations from Raoult's law in the hydrocarbon phase and have calculated activity coefficients for p-xylene from them. In carrying out our calculations we have made the following assumptions: (a) Beer's law is valid for pxylene in water over its entire concentration range, (b) the absorbance of the p-xylene is a linear function of the p-xylene activity over the nonaqueous phase. The first assumption is supported by the work of Bohon and Claussen (7). The second assumption seems reasonable since the absorbance of benzene is apparently linear with respect to activity (8). Deviations from linearity would be expected i f the solute showed a tendency to self associate in the aqueous phase. In view
of the fact that p-xylene is considerably less soluble in water than is benzene, it appears even less likely pxylene will associate with itself. Beer's law and a modified version of Henry's law are given by the equations A;:, = e;;,1c:,
(1)
where superscripts refer to the phase and subscripts to the component (5). At:, is the absorbance, ,::c is the p-xyleue absorptivity, 1 is the path length of the cell, C:;, is the molar concentratiou of p-xylene in the aqueous phase, a::, is the p-xylene activity, and k is a constant of proportionality related to the Henry's constant and the vapor pressure of pure xylene a t 3Z°C. To describe the activity of the p-xylene above the hydrocarbon phase we use the equation a:;
=
y!;:XZF
(3)
where 7:;: is the activity coefficient for p-xylene and X g ; is the p-xylene mole fraction. At equilibrium the activity of p-xylene must be the same in both the aqueous and non-aqueous phases. We may write a:
-- a:;,
=
kC:;, = y:;fX:;f
(4)
Solviug eqn. (1) for CZ,, substituting in eqn. (4) and rearranging gives
If X:$
+
1.0 then 7::;
-
Calculated Activity Coefficients for Various Mole Fractions o f Xylene a t 3Z°C.
1.0 and
which allows the constant, &/k, to he determined from the absorbance of an aqueous solution saturated with pure p-xylene a t a fixed tempcrature. Suhstitution of eqn. (6) into eqn. (5) and solving for 7:;: gives
which permits the activity coefficients of p-xylene to be calculated from spectroscopic data and a knowledge of the mole fraction of p-xylene in the hydrocarbon phase. Our absorbance values (see the table) at unit activity of p-xylene agree quite well with those in the literature (7). I n calculating 7:;; we have used e,":,l/k = A:$ = 0.884 + 0.005. We estimate our error in measuring the absorbance of the samples to be *0.002 absorbance units. The points at Xt$ = 0.905 show larger deviations from the mean than the other data points. We have taken these points to determine the maximum error introduced in the calculation of the activity coefficients. We estimate an error range of j ~ 3 . 7 7in~ the
a his work. b Internolated valuea from Bohon and Clauaren
(7).
7:;: at 1%; = 0.905. Unfortunately we have been unable to find activity data on the p-xylene-dodecane system, which were determined by an independent method, so that we might compare findings. It is notable that the direction of deviation for the system studied here is the same as the chemically similar system, benzene-hexane, studied by Christian, et al. (9). I\loreover, the activity coefficients for p-xylene and benzene in their respective solvents are qualitatively similar. An experiment such as the one proposed here has the advantage of exposing the student to a number of facets of solution thermodynamics without an undue expenditure of time. Experimentally it would seem possible to extend studies such as this one to other systems of interest. It might be pointed out that the restriction of systems to one volatile and one nonvolatile component is more a convenience than a necessity. All that is really necessary is for the two components to be mutually exclusive for purposes of analysis, and to have low solubilities in water. The disadvantages of this method for evaluating activity coefficients are that the solubilities of the hydrocarbons in water are very sensitive to small temperature changes, and the hydrocarbons have a strong tendency to "flash" whcn the sample is being taken for analysis. These disadvantages can be overcome by a good constant temperature water bath and by some practice at manipulating the aqueous samples. Literature Cihd
(1) (2) (3) (4) (5) (6)
(7) (8) (9)
SCATCHARD, G., Ann. Rev. Chem., 2,269 (1953). BARKER, J. A,, A&. J . Chem., 6 , 207 (19.53). CHRISTIAN, S. D., J. CHEM. EDUC.,39,521 (1962). CHRISTIAN, S. D., AFFSPRUNG, H. E., JOHNSON, J. R., A N D WORLEY. J. D.. J. CHEM.EDUC.. .40.421 . (1963). , . CHRISTIAN, S. D., AFFSPHUNG, 1%.E., A N D JOHNSON, J. R., J . Chem. Soc., 1896 (1963). CHRISTIAN, S. D., AFFSPRUNG, H. E., A N D JOHNSON, J. R., J.Chem. Sac., 1 (1965). BOHON, R. L., AND CLAUSSEN, W. F.,J . Am. Chem. Soc., 73, 1571 (1951). WORLEY, J. D., Can. J. Chem., 45,2465 (1967). CHRISTIAN. S. D.. NEPARKO. E.. AND AFFSPRUNG. H. E..
Volume 45, Number 8, August 1968
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