Refractometric Determination of Stability Constants J. O'M. Bockris and P. P. S. Saluja, J. Phys. Chem., 76,2298 (1972). A. I. Vogel, "A Textbook of Practical Organic Chemistry", Longmans. Green and Co., London, 1948. "Handbook of Chemistry and Physics", 48th ed, The Chemical Rubber Co., Cleveland, Ohio, 1967-1968. I. M. Kolthoff and E. B. Sandell, "Textbook of Quantitative Inorganic Analysis", 3rd ed, MacMillan, New York, N.Y., 1952. See paragraph at end of text regarding supplementary material. J. C. Philip and A. Bramley, J. Chem. SOC.,107,377 (1915). J. C. Philip and A. Bramley, J. Chem. SOC.,107, 1831 (1915). S.Glasstone and A. Pound, J. Chem. SOC.,127,2660(1925). S . Glasstone, D. A. Dimond, and E. C. Jones, J. Chem. SOC., 129,2935 (1926). F. A. Long and W. F. McDevit, Chem. Rev., 51, 119 (1952). N. Schlessinger and W. Kubasowa. Z. Phys. Chem. A, 142,25 (1929). A. F. Scott, J. Phys. Chem., 35,3379 (1931). B. Lunden, Z.Phys. Chem. (Leipzig), 192,345 (1943). C. H. Spink, Ph.D. Thesis, Pennsylvania State University, University Park, Pa., 1962. R. E. Gibson, J. Am. Chem. SOC.,57,284 (1935). D. S.Allam and W. H. Lee, J. Chem. SOC.A, 5 (1966). Yu. S.Manucharov, I. G. Mikhailov, and V. A. Shutilov, Vestn. Leningr. Univ., Ser. Fiz. Khim., 19 (16),No. 3,65 (1964). E. B. Freyer, J. Am. Chem. SOC.,53, 1313 (1931). W. Geffcken, Z.Phys. Chem. A, 155, l(1931). F. Vaslow, J. Phys. Chem., 70,2286 (1966). M. S. Stakhanova and V. A. Vasilev, Russ. J. Phys. Chem., 37, 839 (1963). "International Critical Tables of Numerical Data. Physics, Chemistry
821 and Technology", 1st ed, McGraw-Hill, New York, N.Y., 1929. Yu. A. Epikhin and M. S. Stakhanova, Russ. J. Phys. Chem., 41, 1157
(1967). Yu. A. Epikhin, M. S. Stakhanova, and M. Kh. Karapet'yants, Russ. J. Phys. Chem., 40,201 (1966). A. Passynski, Acta Physicochim. URSS, 8,385 (1938). J. C. Hindman, J. Chem. Phys., 36, 1000 (1962). M. A. Paul, J. Am. Chem. SOC.,74,5724(1952). N. C. Den0 and C. H. Spink, J. Phys. Chem., 67, 1347 (1963). 0. D. Bonner, Rec. Chem. Prog., 32, l(1971) n-PeOAc is sufficiently insoluble in aqueous solutions that the excess of standard base added to the aliquots of the aqueous phase for analysis was of about the same magnitude as the concentration of HF in the concentrated KF solutions.50 Hence unequivocal determinations were not possible. "Stability Constants", Chem. SOC.,Spec. Publ., No. 17 (1964). (a) There is some doubt as to whether the F- ion is larger than the Na+ ion.52(b) The relative strengths of cationic and anionic binding of water is also unresolved although some a ~ t h o r sfeel ~ ~that , ~ cationic ~ bonding is stronger. J. E. Desnoyers and C. Joiicoeur in "Modern Aspects of Electrochemistry", Vol. 5, J. O'M. Bockris and E. E. Conway, Ed., Plenum Press, New York, N.Y., 1969. G. Engel and H. G. Hertz, Ber. Bunsenges. Phys. Chem., 72,808 (1968). P T. McTigue and A. R. Watkins, Aust. J. Chem., 18, 1943 (1965). J. E. Prue, A. J. Reed, and G. Romeo, Trans. faraday SOC.,67, 420
(1971). A. Holtzer and M. F. Everson, J. Phys. Chem., 73,26 (1969). G. Nemethy and H. A. Scheraga, J. Chem. Phys., 36,3382(1962).
Refractometric Determination of Stability Constants Lajos Barcza Institute of Inorganic and Analytical Chemistry, L. Eotvos University, 1443 Budapest, Hungary
(Received September 22, 1975)
The theoretical and practical possibilities of refractometry have been discussed in view of the determination of equilibrium constants. The method developed is very quick and simple (if some requirements are fulfilled) and applicable to the determination of a (rather low) stability constants range which cannot be investigated easily by any other methods. The method has been demonstrated on an extensively studied system (on the formation of HgCl3- and HgC1d2- species) and the measured and computed results are practically the same as the results determined by other methods.
Introduction
I t is well known that the phenomenon of refraction is connected after all to the polarizability of electron clouds. As the latter is changed by the formation of complex compounds,l it can be supposed that refractometry is a very good tool for investigating complex equilibria. In the literature, several papers can be found (partly summarized in ref 1-4) using refractometry for the investigation of formation stoichiometries, but only two deal with the problem of the determination of stability constant^.^,^ The method used for finding stoichiometric ratios has been, first of all, the method of continuous variation, which can often give inconsistent result^.^ Its quantitative solution had been used by Japanese authors5 and by Giles and his co-workers, together with the solute excess method6 known and criticized in spectrophotometry a long time. The main point of the criticisms against the application of refractometry in physical-chemical research is that no general relation is known for the connection between the refractive index (or its square or molar refraction, etc.) and
the concentration.2,8 However, this fact only strengthens the first assumption; it means that refraction is influenced also by weak, secondary, and long distance interactions and these can be investigated by refractometry, too, only the experimental and computing techniques must be chosen and controlled very carefully. Chemical Background
In analytical chemistry, the empirical relationship for some relatively diluted solutions are used and p r ~ v e d ~ , ~ very often: n = no
+ kc
(1)
where n is the refractive index measured; no is that of the solvent; h is an empirical factor; c is the concentration of the solute expressed in volume or weight percent, or even in molarity. The h refractive factors or increments are fairly constant up to 0.5 M and often higher (1-2 M) concentrations in most aqueous and nonaqueous solutions having either electrolytes or nonelectrolytes as solutes. The Journal of Physical Chemistry, Vol. 80, No. 8, 1976
Lajos Barcza
822
However, there are some inconsistencies, because it can be pointed out3 (even theoretically) that the refractive indices are additive, first of all, when the concentrations are expressed in molar volumes or at least in concentration units related to the volume. In eq 1, the concentration of the solute as molarity is given in such a scale, but the concentration of the solvent seems to be constant. In concentration range mentioned, it cannot be true, that is, the constant k must be an apparent constant. When the concentration of the solvent is expressed also in molarity, eq 1 can be rewritten as n = no* co
+ n1*c1
(2)
where c1 is the concentration of the solute; co is that of the solvent, both expressed in molarity; and nL*can be considered (as the molar absorbance coefficient) as the molar refractive coefficient. There is no difficulty in finding the connection between eq 1 and 2, only some requirements must be postulated. (i) The kind of interactions between the molecules of the solvent and solute must not be changed in the concentration range investigated; that is, a solvation factor fls must be constant. (ii) The structure of the solvent is not influenced. (iii) There is no change in the degree of the dissociation or association of the solute. (iv) The density of the solution ought to be varied nearly linearly by the concentration as
d = do
+ fldcl
(3)
where d and do are the densities of the solution and the pure solvent; fld is a constant. (Equation 3 is equivalent to eq 1 not only formally but theoretically, too. I t means that its validity depends also on requirements (i)-(iii).) It is essential, that eq 3 must be valid within f0.1% for the concentration range under investigation, which is fairly true for most solutions up to 0.5 M (even to 1-2 M) concentrations. These requirements can be substituted into eq 2 and it can be rewritten as n=
no* lOOOdo
Mo
+
c1 (n1*
no* +(1000fld - MI)- n0*f1.) MO
(4)
where Mi is the molecular weight. It can be easily seen that the first part of the right-hand side is really equal to the refractive index of the pure solvent; and the multiplier of c1 defines exactly the meaning and the validity range of the so-called refractive increment k , or better, that of the apparent molar refractive coefficient ( n L Msee , eq 7). Its value can be measured for different simple substances easily: .IM
= ( n - no)/cl
(5)
The refractive index of a multicomponent solution can be characterized by a set of such equations, in the general form, as n = no
+ znpci
compound^.^ Another difference between the molar absorbance and refractive coefficients is that the latter is valid only for a rather narrow concentration range since it can be determined in relatively concentrated solutions. However, the possibilities can be extended (first of all for aqueous solutions) using a rather high constant ionic strength. The structure of the solvent, the conditions of the solvation, and roughly the density, too, would be practically influenced and kept constant by the inert electrolyte; eq 5 would be valid in more concentrated solutions of different species and thermodynamic constants could be directly calculated from the measured data.
An Experimental Example Although both of the quantitative methods m e n t i ~ n e d ~ , ~ dealt with the investigation of hydrogen bridged complexes, the ability of the present method is not demonstrated on a similar system (in spite of fact that it is very favorable for such ones, too), but on a more complicated still better known system. The equilibria between mercury(I1) chloride and chloride ions
+
HgC12 C1HgC13K3 = [HgCh-]/[HgC12] [Cl-]
(6)
where the definition of the apparent molar refractive coefficients is (7)
The equilibria can be followed and the equilibrium conThe Journal of Physical Chemistry, Vol. 80, No. 8, 1976
stants can be computed using eq 6. First of all, the apparent molar refractive coefficients have to be determined (and controlled) for every reactants separately. When they react, the refractive indices measured in their mixtures cannot be described by the type of eq 6 composed simply.as a set of eq 5 measured separately. The differences between the measured and calculated values are directly caused by new species having new and different apparent molar refractive coefficient(s). Using the principle of electroneutrality (as is usual in coordination chemistry) and a computer, the real eq 6 can be solved for the stability constant(s1 by a rather simple iteration procedure. (The computing is really so simple, that no details are necessary and can be managed in most of cases even by desk computer. Some practical problems will be discussed later.) Although there are similarities between the molar absorbance and refractive coefficients, a typical difference is that all of the components are refractometrically active. The interactions can be detected by refractometry when the apparent molar refractive coefficients of reactants and products differ on a measurable scale. Sometimes these differences are large enough for the detecting of rather weak complexes (such as the formation of a complex between propionic aldehyde and dimethylformamidelO),but there is the possibility that the apparent molar refractive coefficient of the product is nearly the sum of those of the reactants. In these cases no interactions can be detected, although rather stable complexes have been found in the system by other methods.ll This phenomenon seems to connect with substances forming strong hydrogen bridges with the solvent when the further interaction is of the same type. It follows that the lack of measurable change in refractive indices is no proof against the formation of complex
(8)
and
+
HgClz 2C1- * HgC14'K3K4 = [HgC14"-]/[HgC12] [C1-12
(9)
had been investigated by several methods and several au-
823
Refractometric Determination of Stability Constants
',3470
1,3425
1,3420 1,3450
1.3415
:3433 1,34104
0 1
~
!
~
:
~
94
02
!
46 04
0,6
0,e
I
-c--
~
1 F; NaCl 3 MUaNC3
0,8 02
Figure 1. Refractive indices of solutions containing 1,000 M Na+, a M CI-, and 1.000 - a M Nos- (solid line, calculated values using the computed apparent molar refractive coefficients).
GC4
qca
O,l2
0,'6
q2OM HgCI,
(1000MNaCI)
Figure 3. Refractive indices of HgCIz solutions in 1.OOO M NaCl (broken line, tangent of first points for calculating of approximate n M ~ gvalue; ~ p solid line, calculated one using the computed nM and Kvalues).
"I 1,360 -/x
n
/x
/
x/x
.xr
1,3440 --
/x
/x
/x
1,3L2G--
/"
/' /"
1,3655
t 3 1
02 08
04 CE
06
04
CB 02
1 M NaCl
0 M NoNO, 0,2000 M HgCI,
Figure 4. Refractive indices of 0.2000 M HgCIz solutions containing 1.000 M Na+, a M CI-, and 1.000 - a M NOS- (solid line, calculated values using the computed constants).
thors;12 the values of the equilibrium constants measured (in 0.5-3 M Na(C104), at 25 "C) are very consistent to each other (log K3 = 0.75-0.95; log K3K4 = 1.85-2.13). These equilibria have been reinvestigated by the present method in 1M Na(N03) and a t 25 "C. All of the reagents were of highest purity and the following stock solutions were prepared at 25.00 "C: 1.000 M NaN03, 1.000 M NaC1,0.2000 M HgClp in 1.000 M NaN03, 0.2000 M HgClz in 1.000 M NaC1. During the investigations, one of the solutions was pipetted into an exactly thermostated vessel (the temperature variation did not exceed the limit of f0.05 "C) and its refractive index was measured several times with a Zeiss type dipping refractometer. Then a small volume of the second solution (also measured accurately and exactly thermostated a t 25.00 "C) was mixed with the solution in the vessel and the refractive index was measured again after some minutes. This procedure was repeated several times, as the whole procedure and its inverse. In the first series, sodium nitrate was mixed with sodium chloride. I t means that the sodium concentration was kept constant (1.000 M), while the nitrate was decreased from 1.000 M and the chloride was increased to 1.000 M concentration. As can be seen in Figure 1, the rz vs. c relation is 0 3 n M ~ , c lcan be directly calculated linear; both n M ~ , ~ and using eq 6. In the second step, a series of HgCl2 solutions in 1.000 M NaN03 was produced and measured in a similar way (Figure 2). Mercury(I1) chloride does not dissociate or hydro-
lyze to a degree12 measurable by refractometry, so its apparent molar refractive coefficient can be calculated, too. In spite of that, there are no linear relations when NaCl and HgC12-NaC1 (chloride concentration constant, Figure 3) or HgC12-NaN03 and HgClpNaCl (mercuric chloride concentration constant, Figure 4) solutions are mixed. These results can be explained if the new species formed have different rzM values and parallel with their formation the concentrations of mercury(I1) and chloride ions (see eq ~ ~ Z can be cal8-9) decreased. An approximate ~ M H ~ C value culated from the slope of the first part of the curve on Figure 3 and a less accurate n M ~ g ~ value i s - can be supposed as a mean value of n M ~ g ~ land r z -n M ~ g ~These l z . data are the initial ones in the iterative computer calculation and will be refined in the successive steps together with the stability constants seeking the best fit. For the iterations, eq 5, 6, 8, and 9 and the equations of mass balance
+ [HgC13-] + [HgC1k2-] [CI-IT = [CI-] + [HgCla-] + 2[HgC14'-]
[HgC12]~ = [HgClz]
(10) (11)
were used. The guessed pairs of K3 and K4 values were 10-10, 5-20, and 20-5 (and without any results 10-0 and 0-100). The best fit was always found in a few cycles with the values K3 = 7.83 f 0.09 or log K3 = 0.894 and K3K4 = 106.5 f 0.6 or log K3K4 = 2.025. The Journal of Physical Chemistry, Vol. 80, No. 8, 1976
J4rgen Lyngaae-J4rgensen
824
Discussion The figures and the final results show that the recent refractometric method gives identical results with other methods. (Further examples for its application will be published later.) The conditions of the method are also demonstrated. (i) The type of species formed in system under investigation must be known or a t least precisely guessed and then proved by calculations similar to other methods, e.g., to potentiometry. (ii) The linearity of eq 5 must be proved separately for all of the reactants. (iii) T o increase accuracy, a number of different concentrations must be made and measured a t constant temperature. However, the method developed is very advantageous, because (i) it is very simple and quick; (ii) less stable complexes of different type can be investigated for which we do not have too many or simple methods; (iii) even the stabili-
ty constants of hydrogen bridged complexes can be measured in aqueous solutions, also.
References and Notes (1)S. S.Batsanov, "Refractometry," Consultants Bureau, New York, N.Y., 1961. (2) B. W. Joffe, Jen. Rev., 17, 170 (1972). (3)B. V. loffe, "Refr. met. khim." (izd. Khim.), Leningrad, 1974.
(4)G. C. Pimentel and A. L. McClellan, "The Hydrogen Bond," W. H. Freeman, San Francisco, Calif., 1960,p 55. (5) 2. Yoshida and E. Osawa, Bull. Chem. SOC.Jpn., 38, 140 (1965). (6)C. H. Giles, J. Gallangher, A. McIntosh, and S. N. Nakhwa, J. SOC.Dyers Colour 88,360 (1972). (7)M. M. Jones and K. K. Innes, J. Phys. Chem., 62, 1005 (1958). (8)B. V. loffe, Usp. Khim., 29, 137 (1960). (9)S. 2. Lewin and N. Bauer, "Treatise on Analytical Chemistry." Part I, Vol. 6, I. M. Koithoff and Ph. J. Elving, Ed., Interscience, New York, N.Y., 1965,p 3895. (IO)F. M. Arshid, C. H., Giles, and S. K. Jain, J. Chem. SOC.,559 (1956). (11) H. Fritzsche, Z.Chem., 6,39 (1966). (12)L. G. Sillen and A. E. Martell, Ed., "Stability Constants," The Chemical Society, London, 1964 and 1971.
Melting Point of Crystallites in Dilute Solutions of Polymers. Poly(viny1 Chloride) in Tetrahydrofuran Jergen Lyngaae-Jsrgensen instituttet for Kemiindustri, The Technical University of Denmark, Bygning 227,Lyngby, Denmark (Received July 3 1, 1975) Publication costs assisted by Statens Teknisk- Videnskabelige Forskningsrad
A model is derived which permits the prediction of the melting point of crystallites in dilute polymer solutions based on the melting point of crystallites in the pure polymer, the molecular weight, and the interaction parameter. The model prediction is in accordance with experimental data for poly(viny1 chloride) in tetrahydrofuran. Furthermore, an expression for the chemical potential of repetition units in the range between the theory for extremely dilute solutions and the Flory-Huggins theory for solutions with uniform segment distribution has been shown to be a reasonable approximation.
Introduction The fact that poly(viny1 chloride), PVC, molecules form aggregates in dilute solutions was realized by Dotyl in 1947. Since then extensive investigations of the phenomenon have shown that aggregates can be formed in all known solvents2 but in greater amounts in poor solvent^.^,^ Hengstenberg4 concluded that the aggregates were held together by crystallites. In accordance with Hengstenberg's work, it is found that the amount of aggregates is a function of the degree of crystallinity, temperature, solvent power, and solvent concentration.j-I5 The rate of recrystallization in dilute solutions depends on the tacticity of the ample,^ but is normally a very slow process a t high dilution and low degree of tacticity.7J0J1 The aggregates found in dilute solutions of tetrahydrofuran (THF) are built of a number of single molecules bound together by a crystalline nucleus.16 The characterization of PVC molecules by light scattering etc. presupposes a solution consisting of single molecules. The purpose of this work was therefore to develop and experimentally verify a model for the prediction of the The Journal of Physical Chemistry, Voi. 80, No. 8, 1976
melting point of crystallites in dilute solutions as a function of the melting point of the crystallites in pure PVC, the solvent concentration, and the solvent power. The model was tested with tetrahydrofuran since it is a solvent often used in characterization work.
Theoretical An expression is sought for solutions of PVC molecules in the concentration range where light-scattering, gel permeation chromatography, etc. is performed. The model is derived as follows. A t the melting temperature equilibrium between repetition units in crystallites and repetition units in solution exists; stated otherwise this means that p,,c = p,, where wLUcand pu are the chemical potential of a repetition unit in the last crystallite and a repetition unit in solution, respectively. wuc and pu are expressed as functions of temperature, molecular weight, the melting temperature of crystallites in pure polymer, the interaction parameter, and volume fraction of polymer. From the condition pUc = ku one can then isolate the melting temperature as a function of other variables. The derivation of an expression for wu is performed by
