R. A. Pasternak a"d A. P. Bmdvl
Standford Research Institute Menlo Park, California
1
1
Random and Systematic Errors in the Determination of Association C O N S ~ U N ~ S
During a preliminary study of associatiou equilibria in solutiou of carboxylic acids and of amides, carried ont with an improved version of the thermoelectric ~smometerz,~ of Brady et al.,',5 the authors became concerned with the large magnification factors connecting the uncertaiuties in the experimental data with that of the derived equilibrium constant, aud with the large systematic error in the constant produced by even a small vapor pressure of any participatiug species. Since these effects have ueither been considered uor discussed thoroughly in the literature, a short presentation of the arguments appears appropriate. Uncertainty Due lo Random Experimental Errors
The discussion will he restrict,ed to the dimerisation 2N, s S1 and Raoult's law ill he assumed t o hold for each species (similar co~isiderations~ o u l dapply to more complex cases). The experimental data are: N
+
Ni 2X1; tho total mole frsction expressed in terms of monomer jj=-. , nlensured "osmotic" total mole frsction at oquilib=
PO
rinm
' The aut,horsare indebted to Meehrolab, Ine., Mountain View, California for partial support of this work. HILL, A. V., Proe. Roy. Soe. (London), AlZ7, 9 (1930). a BALDE~. E. J . . Riodrmamiea., 46 (19391. . . ' BRADY,'A. P . , H u ~ ; H . ,A N D MCBAIN,J. W., J. P I I ~Chem., . 55, 304 (1051). The present instrument has been described at the "Thirteenth Annual Conference on Electrical Techniques in Medicine and Biology" Bethesda, Maryland, October, 1960.
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Journal o f Chemical Education
For solute species with negligible vapor pressure fl R, S2,and the equilibrium constant is given by:
+
Differentiation, introduction of X algebraic manipulation leads to
=
=
.TIN, and
the ahsolute values of the differentials being take11 since the errors in N and iV are independent. In Table 1, columns 2 and 3, the factors by which the relative uucertainties 6N/N and 6D/N in the measurcments are multiplied to give the relative uncertainty iu I< are shomn as functions of X. Even a t X = 0.8, at about the minimum value of the factors, an experimental uncertainty of each in N and Rleads to a maximum uncertainty in K of about llyoand a standard uncertainty of 8.3%, for a singlc determination. Table 1
For X = 0.55, equivalent to a degree of association of 0.9, which is not unusual for such studies, the same experimental uncertainty represent a standard nncert,aint,yof about 40% in K: Systematic Error Due lo Volatility of Solute
Some of the major methods for determining the activity of the solvent in a solution, i.e., the ehullioscopic, isopiestic, and thermoelectric methods, are hased on the direct or indirect measurement of total vapor pressure over the solution relative to the pure solvent or to a solntion of known vapor pressure. It is in general assumed that the solutes have no vapor pressure of their own; however, a solute ~vitheven a small partial pressure introduces a quite significant systematic error. If each solute species, mole fraction AT,, has in its pure state a vapor pressure pi, then
In the case of dimerization, it might reasonably be assumed that the vapor pressure of the dimer can be neglected; this is, however, not necessarily so for the monomer. In this case, equation ( 3 ) can be rearranged to give the apparent "osomotic" mole fraction
where A
=
p,/po.
The true ecpilihrium constant is
and, if K' is the constant calculated from equation (1) n-hich neglects A,
In the last column of Table 1 the factor (3 - 2 X ) / ( 1 - X ) is given as a function of X. Its value is quite large, especially at low degrees of association, and even at virtually complete association it has a limiting value of four. For the calculation of the actual error it remains to estimate A. The value of 4 for any solute can be estimated in first approximation from its boiling point by assuming Trouton's rule (taking the entropy of vaporization at the boiling point as 21 cal (OK)-I) and using the Clausius-Clapeyron equation
where Toand T, are the boiling points of solvent and solute and T is the temperature of measurement. In estimating T, it should be borne in mind that in the present case it refers to a hypotlietTcal pure monomer. It appears justified to assign to such a monomer a boiling point about that of a structurally similar compound of the same molecular weight, vhich, ho!~-ever, is not able to polymerize. For example, in the case of the evtensively investigated benzoic acid a~sociation,~ an appropriate monomer would he phenyl formate, bp 446°K. I n Tahle 2 the values of 4 , using equation (6) and assuming benzene as a solvent, are given for different temperatures. Combining these results with those shown in the last column of Table 1, it is seen that significant Tahle 2
systematic errors occur in the experimental constant, the magnitude of which depends pronouncedly on the degree of association and on temperature. At the boiling point of benzene, and for a degree of association of 0.5 (X = 0.75),the error amounts to over 30%. I t is likely that these effects have not been detected hecause the large association constant of benzoic acid requires studying system a t relatively high degrees of association where the systematic errors do not change rapidly with concentration. It is also obvious from equation (3) that even for quite significant values of A the partial pressure of a volatile solute at low concentration is very small, and that therefore transfer of solute through the vapor phase is undoubtedly extremely slow. To arrive at the above estimate of errors caused by the partial pressure of the monomer, a number of assumptions were necessary. However, in principle both the true equilibrium constant Zi and A can be derived directly from precise experimental data for N and fl obtained over a wide range of X . The only necessary condition is that the partial pressure of the monomer obey Henry's Ian.. Equation (4) can he rewritten in the form
A plot of the experiment fl/N versus (2fl
- N)Z/N would be a straight line with an intercept of (1 - 4 ) and a slope of K/ (1 - 2A).
-
-
See for example: ALLEN,G., A N D CALIXN, E. F., Trans. Farada!, Soe., 49, 895 (1953); and WALL,I. T.,A N D BANES,F. W , .I. Am. Chem. Soc., 67, 898 (194.5).
Volume 40, Number 5, May 1963
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