1944
ERNESTGRUNWALD AND JAMES F. HALEY,JR.
Acknowledgment. The technical assistance of Mr.
P. J. D’Arcy and Mr. C. J. Halpin in all phases of the experimental work is gratefully acknowledged. We
also wish to thank Mr. A. Westwell of the National Research Council Computation Centre for programming the Barker calculation.
Acid Dissociation Constant of Trifluoroacetic Acid in Water Measured by Differential Refractometry1
by Ernest Grunwald and James F. Haley, Jr. Department of Chemistry, Brandeis University, Waltham, Massachusetts 06164
(Received January 16, 1068)
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The refraction per cubic centimeter of a dilute solution (p = (nz - l)/(nz 2)) differs from that of pure where J , RJ and VJ denote apparent molar refraction and volume and solvent by (p - po = F ( R J - ( ~ ~ V J ) C
cJ denotes molar concentration of the J t h solute species, and the summation extends over all solutes. A(R - poV) for chemical reactions in solution can be large, especially when ionic charge is generated, so that equilibrium constants can be deduced. Medium effects on RJ are small compared to those on VJ and can often be neglected. Refraction measurements on aqueous solutions of acetic acid, sodium trifluoroacetate, and cyanoacetic acid up to moderate concentrations show that the effects of solute-solute interaction here are small, except for those interionic effects that vary colligatively with the square root of the ionic strength. Refraction measurements on aqueous solutions of trifluoroacetic acid at 22’ lead to the following parameters for acid dissociation: KAO = 1.1 i: 0.3 M and AVO = -2.5 1.5 cmS. Measurements were made with a Rayleigh interferometer by an improved optical technique.
The refractive index, like any specific property, can be analyzed to yield equilibrium constants in dilute liquid solution. Although refractometry has not been used extensively for this purpose, it offers certain advantages. Differences in refractive index, even very small ones, can be measured with classical precision by interferometry.2.3 The (differential) change of refractive index per molar change in the concentration of the Jth solute, 6n/6cJ,can often be predicted with fair accuracy, and the quantity A(snl6c) is relatively large for many reactions. Refractometry is especially useful for the measurement of fairly large dissociation constants, on the order of unity, for which most other methods become difficult. We shall describe the refractometric determination of the dissociation constant of trifluoroacetic acid in water. This is a strong acid (KAO = 1.1M a t 22O), and the degree of dissociation, a, differs significantly from unity only a t acidities that are higher than any attainable in dilute aqueous solution. Recent measurements of a by nmr and Raman spectroscopy have involved trifluoroacetic acid concentrations above 0.5 M.4 We find that by refractometry, K A O can be obtained with better than 30% accuracy from data in the concentration range 0.0-0.6 M . The Journal of Physical Chemistry
Theory The Lorenz-Lorentz equation for the specific refraction of a mixture5 reduces, in the dilute solution limit, to eq 1. In that equation and throughout this p/P =
[(I - CWJ)ROl/MO J
+ C(WJRJ/MJ) J
paper, symbols without a subscript refer to a solution; symbols with a zero subscript refer to the pure solvent; and those with a J subscript to the Jth solute, and the (1) Work supported by the Petroleum Research Fund of the American Chemical Society. Grateful acknowledgment is made to the donors of that fund. One of the authors, J. F. H., is indebted to the American Cancer Society for the award of a summer stipend under Grant No. IN-29. (2) N. Bauer and K. Fajans in “Physical Methods of Organic Chemistry,” A. Weissberger, Ed., Interscience Publishers, Inc., New York, N. Y., 1949, Chapter 20. (3) W. Geffcken and A. Kruis, 2. Phys. Chem., B23, 176 (1933); A. Kruis, ibid., B34, 13 (1936). (4) (a) G. C. Hood, 0.Redlich, and C. A. Reilly, J . Chem. Phys., 23, 2229 (1966); (b) M. M. Kreevoy and C. A. Mead, Discussions Faraday SOC.,39, 166 (1966); (c) A. K. Covington, ibid., 39, 176 (1966). (6) H. A. Lorentz, Ann, Phys., 9, 641 (1880); L. V. Lorenr, ibid., 11,70 (1880).
ACIDDISSOCIATION CONSTANT OF TRIFLUOROACETIC ACIDIN WATER summation extends over all solute species. W is the weight fraction; R is the apparent molar refraction; M is the molar weight; p is the density; and cp is given by eq 2, where n is the refractive index. cp = (n2
- l)/(n2 + 2)
(2)
Beginning with eq 1, Kohnere and Geffcken’ have derived eq 3 for the special case of a single solute, K 1ooo(Y, -
(00)
E
[RK -
(3)
POVKIcK
whose apparent molar volume is VK. In principle, RK and VK are concentration dependent. However, in sufficiently dilute solution RK and V K become constant and cp - cpo becomes proportional to C K . ~ To relate cp - cpo to the experimental quantity n no, it is convenient to expand p(n) in a Taylor series about the point cpo = cp(no). The result is
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Because In - no]is small, the series converges rapidly. For water, the coefficient of (n - no)2is one-third that of n - no. Moreover, for most solutes in water, n - no < 0.01, even at 1 M concentration. Thus in our calculations, we need retain only the linear term. On that basis, eq 3 reduces to
1945
water. For the un-ionized acid, RHA,,= 13.00 cma/mol, while $ 2 2 ~=~ 2.36 ~ cm3/mol, according to our measurements. R increases slightly on acid dissociation; AR = RHt R A ~- R H A=~ 0.69 Thus AR/ REAc = 0.05, while AR/QHA,,= 0.29, nearly six times larger. I n addition, AV = VH+ VA,,- - V H A is ~ negative, -9.2 cm3,10 because the ionic products bring about a marked electrostriction of the aqueous solvent. Thus cpoAV = -1.90 em3, AQ = 2.59 ema, and Q H t QA,= 4.95 cm3, more than twice as large as &A,,! It follows that ((p - cpo)/c is a strong function of the degree of dissociation.
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Effect of Solute-Solute Interaction In principle, RK, VK,and hence OK are functions of composition. For reasons that will become evident shortly, we shall assume that the concentration dependence of QK can be treated analogously to that of . V X . ~ O J ~ Our measurements extended up to 0.6 M concentrations, and the solutes were nonelectrolytes and univalent ions. Under these conditions we may represent VKfor nonelectrolytes in good approximation as a linear function of molar concentrations, as in eq 8, where the coefficient / 3 j is ~ specific and the summation includes J = K. For univalent electrolytes, VL may be represented by eq 9, which contains an additional term, SvZ/,, derivable from interionic attraction theory,1°-12 that is a colligative function of the ionic strength p. VK = VKO
6000n0
+ 212‘~
(no2
- no) =
(RK - (POVK)CK ( 5 ) VL
Equation 5 is easily extended to the more general case of a dilute solution containing several solute species. We shall assume that at the concentrations to be considered, specific solute-solute interaction can be neglected. The result is eq 6, where a single symbol, QJ, represents RJ (POVJ.
-
RJ
+
J
~v
