F. J. C. ROSSOTTI AND HAZEL ROSSOTTI
926
Vol. 63
GRAPHICAL METHODS OF DETERMINING SELF-ASSOCIATION COXSTANTS. I. SYSTEJIS COXTA41NINGFEW SPECIES BY F. J. C. ROSSOTTI ASD HAZEL ROSSOTTI Department of Chemistry, The L‘niuersity of Edinburgh, Scotland Reeezved September 23, 1960
New graphical methods of calculating self-association constants for systems containing a monomer and up t o three oligomers are proposed and compared with existing methods.
Methods’-* for determining equilibrium constants of self-association reactions’ @ j - B, (1) have lagged far behind those for treating binary complex formation. This paper and its sequel describe a more comprehensive treatment of the computation of stoichiometric self-association constants defined by p IB91 P -
bs Q
Q
where BQis the highest oligomer formed in the concentration range studied. The sum of the concentrations of each species is given by Q
Q
[&I
Pqbq
=
(4)
1
1
dS= -B _ db
(7)
b
follows from equations 3 and 4, regardless of the nature of the species present. Calculation of b from X(B).-The value of b may be obtained from measurements of B and S using Bjerrum’s integrated form’ of equation 7
bq
where b is the concentration of free monomer. Emphasis is placed on making the fullest use of the experimental data, and on obtaining realistic limits of error for the constants. Concentration Variables. The total analytical concentration of B is given
8 =
only one of the variables b or S as a function of B, the relat,ionship* , 4
where S1is the value of S corresponding to a known value, bl, of the monomer concentration. It is often difficult to determine a suitable point, 81, bl, for systems in which appreciable association occurs a t the lowest concentrations studied. We recommend two different types of method: (i) In sufficiently dilute solutions, it is likely that the dimer will be the only complex formed. Then from equations 2, 3 and 4 b=2S-B
and log ( B - S ) = log
(9)
+ 2 log (25 - B )
(10)
Thus if the plot of log ( B - X) against log ( 2 s B ) is a straight line of slope 2, the value of b a t any point on the line is given by equation 9. Alternatively, if equation 9 is valid
If one or both of the functions B(b) or S(b) can be determined, the power series 3 or 4 may in prinB -_1- + 2 b (11) ciple be solved for the (Q - 1) unknown paramS l + b eters & provided that the activity coefficients of all species remain effectively constant over the where b = Pz(2S - B ) . A smooth curve may then concentration range studied. Stepwise associa- be drawn through the experimental points BS-l. log (2s - B), using the template BS-l (log b), tion constants calculated by means of equation 11. The value of b for any point lying on the curve may then be obtained using equation 9. (ii) If complexes for the react’ioiis higher than the dimer are present in the most dilute solutions studied, the curves B(B,’S) and S ( B / S ) Bpi B Bg (6) may be extrapolated until they meet a t the point may then be calculated. Experimental methods where B / S = 1 and B = X = b. for determining values of b and,&’ in solut’ion have If neither procedure (i) nor (ii) gives a satisbeen described e l s e ~ h e r e . ~The value of S in factory value of bl, an approximate value may be gaseous systems may be calculated using the ideal used to calculate preliminary association constants, gas law. Although it is usually possible to measure which are then used to refine the value of bl. (1) J. Bjerriim, Kem. kfaanedsblad, 24, 21 (1943). Calculation of S from b(B).-The value of S niay (2) AT. Davies and H. E. Hallam, J . Chem. Edue., 33, 322 (1956). he similarly obtained from measurements of B and (3) H. Dunken, Z . p h y s z k . Chem., 45B, 201 (1940). b, using Kreuzer’s integrated form4 of equation 7 (4) J. Kreuzer, ibid.. 53B,213 (1943).
+
( 5 ) F. J. C. Rossotti and H. Roasotti, “The Determination of Stability Constants.” McGraw-Hill Book Co., New York, N. y., in press, Chap. 16. (6) N. E. White and hI. Iiilpatrick, J- P h y s . Chem., 59, 1044 fl95.5).
(7) li. L. IVolf, H. 1)iinkv11 and I (Q - 1) sets of values B, b or S, b are available, m ! / ( Q- l)!(m - Q l)!equations mould have to be solved in order to use all the data. Calculations of this type may readily by performed by a high-speed electronic computer, but are very tedious to carry out with a desk calculator. Moreover, since the equations will probably be ill-conditioned on account of small experimental errors, it may be difficult to choose the "best" set of constants, and to estimate realist limits of error. Successive Extrapolation.-In the absence of an electronic computer, self-association constants are most conveniently obtained by graphical methods. For example, data B, b may be analyzed by plotting the function
+
F2
=
B-b - ' b2 - 2/32
+ 383b +
Q q/3*bQ-2 (15) 4
aga,inst b. The value of 2/32 is given by the intercept' and that of 3p3as the limiting slope as b tends to zero. Higher values of PP may be similarly obtained by t'he successive extrapolation method, which is analogous to t8hatintroduced by Leden'" for determining stability const-ant,sof binary mononuclear complexes. In general, values of t/3t and (t 1)/31+1 may be obtained as the intercept and limiting slope of the plot of
+
F L = Bb-'
-
t-I
~ / 3 & - ~= tpt
+ ( t + l)Bt+i b +
1
Q
@W-t (16)
927
may result in gross errors in the higher constants. However, if measurements have been made a t high concentrations, and the formula of the highest complex is known, functions of the type
may be plotted against b-I. The value of PQ is obtained from the intercept, and that of pQ-l from the limiting slope as b-' tends to zero. Values of the lower constants may also be obtained by successive extrapolation of polynomials in b-l. The two sets of constants obtained from polynomials in b and b-l may be refined by successive approximations to ensure that the functions B(b) and s ( b ) , calculated by substitution of the stability constants into equations 3 and 4, give an adequate description of the experimental data. The main advantage of successive extrapolation methods is their independence of any assumption about the nature of the complexes. However, they can only be used successfully for experimental data of high precision. Moreover, if Q > 3, the whole range of experimental data cannot be considered simultaneously, nor can limits of error in the constaiits readily be obtained. Curve-fitting Methods. The precision of the measurements as carried out often is insufficient to justify the calculation of more than three independent parameters. Association constants may therefore be conveniently obtained by curve-fitting methods of the type introduced by SillBn. The procedures recommended below mainly use logarithmic plots which allow the functions B(b)and S(b) to be considered simultaneously over the whole concentration range studied. They give realistic estimates of the limits of error in the constants obtained, and an automatic check on the agreement between the calculated functions, and the experimental lots. Computational errors may be introduced by (i) the smoothing of experimental data and choice of residual integral in calculating b or 8, and (ii) the use of log-log plots which appear to reduce small discrepancies between experimental and calculated functions. The validity of t hc association constants should therefore always be checked by substituting them into equations 3 and 4 to ensure that good agreement between the obserT7ed and calculated functions B ( S ) is obtained over the whole concentration range studied. The Degree of Association of the Complexes.Some indication of the type of species present may be obtained by calculating the degree of association
t+z
against' b. St'einerl' used the limiting slopes of functions Ft(b) to obtain association constants from data B, b. White and Kilpat'rick6have used a similar procedure, based on equation 4, to obt'ain values of Pt from data S , b. Small errors in the computat,ion of the lower constants by these methods will accumulate, and (9) G. Preuner and W.Schupp, Z. phvsik. Chem., 6 8 , 129 (1909). (10) I. Leden, ibid., 188A, 160 (1941); "Potentiornetrisk Undrrsohning av nigra. Kadiiiirinisalter? Iioinplexitet," L u n d , 1943. ( 1 1 ) R. E'. Steiner, Arch. Biochem. B i o p h y s . , 39, 333 (1952); 44, 120 (1953); 49, 400 (1951).
of the complexes. An integral value of i j = (2 over the whole concentration range shon-s that there is a unique oligomer. BQ. If i j mries with concentration, a number of complexes are formed, and (2 2 plllaX. For exaniple, if iilllaX = 3 2 and (dii ( 1 2 ) L 0 Sill6n. .Icta Cheni. S c a n d , 10, 188 (1936)
F. J. C. ROSSOTTI A S D HAZEL ROSSOTTI
928
Vol. 65
Treatment of a Single Cornpiex, Bg.-Several graphical methods may be used to obtain the association constant of a single oligomer, BQ. (i) If the functions B(b) or S(b) are used, no previous knowledge of Q is required. From equations 3 and 4 log ( B - b) = log &PQ + Q log b ( 19) and log ( S
- 6)
= log PQ
+ Q log b
( 80)
The plots of log ( B - b) and log (S - b ) against log b are therefore straight lines of slope 2 and of intercept, log QPQ and log PQ, respectively. (ii) Alternative rearrangement of equations 3 and 4 gives B log - = log (1 b
+ QPpbQ-l) = log ( I + QbQ-‘)
(21)
and log
s- = log (1 + PQbQ-1) = log (1 4bQ-’) b
(22)
where the normalized (dimensionless) variable b is given by -1
0 Log b.
1
log b = log b
1 + Q-___ log PQ - 1
(23)
The experimental values of log Bb-l and log Sb-’ are best plotted on the same diagram as functions of log b. Sets of normalized curves log Bb-’ (log b)Q and log Sb-l (log b)Qmay be calculated for different values of Q using equations 21 and 22, (see Fig. 1). Pairs of theoretical curves for the same value of Q are plotted on the same scale as the experimental curves, and superimposed on them so that the ordinates of the two graphs are coincident (see Fig. 2). The correct value of Q is that which gives normalized curves of the same shape and separation as the experimental plots. The association constant may be obtained by solving equation 23, using corresponding values of b and b in the position of best fit. The limits of error in @Q are obtained from the permissible displacement of the normalized curves along the log b axis. Values of Q and @Q may be obtained similarly from plots of B/b or S / b against log b. (iii) Combination of equations 3 and 4 gives the linear relationship log ( B - S) = Q log (QS - B ) log PQ ( Q - 1) log ( Q - 1) (24) The non-logarithmic form of equation 24 has been used by Dunken3J and others to obtain values of @Q -1.2 -0.8 -0.4 by one-point calculation. The value of Q must be Log b. found from equation 18, or by trial and error. Fig. 2.-Benzenesulfonanilide in naphthalene15 a t 80”. Treatment of the System B Be BQ.--The Xormalized curves log Bb-1 (log b)&and log Sb-1 (log b)Qfor Q = 2 superimposed on the experimental data log Bb-l data B, S, b can often be described in terms of the (log b ) (open circles) and log Sb-1 (log b) (full circles) in formation of a dimer, and of one other oligomer BQ the position corresponding to log pz = 0.28. usually a trimer. For systems of this type, equadB)Jma,is low, it is likely that Q = 4 or 5 . If V tions 3 and 4 may be written rises sharply with concentration to give high values (25) of ijmax,an extended series of “rnultimer~”’~ may be formed.14 and
Fig. 1.-(a) log B/b and (b) log S/b as functions of log for several values of Q (equations 21 and 22).
+
+
+
(13) hl. Dnvi(15 i n “IIy(lrogen Rondinr.” rd. D. I%ailziand H. W . Thompson. Pergnrnon Press, London. 1959, p . 860. (14) t’. J. C. Rossotti and H. Rossotti. J . I’hys. Chcm.. 6 5 , Y 3 0
(1961): D. hl. W. Anderson, J. L. Duncan and F. J. 6. Rossotti, J . Chem. Soc., in press (1961).
f l a ) K. Sheele and A. Hartman. Kolloid-Z., 131, 196 (1933).
GRAPHIC METHODOF DETERMINING SELF-ASSOCIATION CONSTASTS
June, 1961
929
Substitution of the normalized variables log F = log F log cp = log
- log Pz
* - log
(27)
0.6
(28)
p2
and log b = log b
+log -Q-2
BO B2
-2.2
(29)
into equations 25 and 26 gives log F = log ( 2
+ QbQ-2)
and log Q, = log (1
+ bQ-2)
(30)
4 0.4 00
2
-2.0
(31) and log @ are
The experimental values of log F plotted on the same diagram as functions of log b. Pairs of theoretical curves log F (log b)Q and log Q, (log b)Q are calculated for a number of likely values of &. These are superimposed on the experimental plot in the position of best fit with the axes of the two graphs parallel (see Fig. 3). The correct value of Q is that used to calculate the pair of normalized curves of the same shape and separation as the experimental functions. The values of pZ and PQ may be obtained by solving equations 27 and 29, or 28 and 29, using corresponding values of the logarithms of b and b, and of either F and F or and Q,, in the position of best fit. The limits of error in log p2, may be obtained from the permissible vertical movement of the normalized curves across the experiment data and those in log pO/p2 from the permissible horizontal movement. Analogous curve-fitting methods may be developed for analysing systems containing monomer and any two complexes of unknown stability, provided that the formula of one of them is known. Treatment of the System B Bz B3 B4.If monomer, dimer, trimer and tetramer are present, equations 3 and 4 may be written as
Log b
0
-2.5
-3.0
I
I
1
-0.5 0 Log b. Fig. 3.-Pyrazole in carbon tetrachloridel6 a t 18'. Normalized curves log F (log b)Q and log CP (log b)Q for Q = 3 superimposed on the experimental data log F (log b ) (open circles) and log (log b ) (full circles) in the position corresponding to log PZ = 1.67 and log Pa = 3.89.
+ + +
F
=
whence log F = log F log CP = log
B -b-b2 - 2/32
+ 3P3b + 4P4b2
- log Pz = log (2 + 3Rb + 4b2) - log Pz = log (1 + Rb + b2)
(32)
(34) (35)
01
I
I
I
I
where log b = log 2,
+ -21 log Pz
(36)
and R=-
Pa (PZP4) 'la
( 37)
Thus the shapes of the functions log F (log b) and log @ (log b) are determined by the parameter R, and their positions on the ordinate and abscissa depend on the values of Pz, and on the ratio p4/pz, respectively. The experimental functions log F (log b) and log @ (log b) are plotted on the same graph, and compared with pairs of normalized -2 -1 0 1 2 curves log F (log b)R and log Q, (log b)a, calcuLog b. lated for a number of values of R, using equations 34 and 35 (see Fig. 4). The correct value of R is Fig. 4.-(a) log F and (b) log Q, as functions of log b for several values of R (equations 34 and 35). that used to calculate normalized curves of the same shape and separation as the experimental functions. The values of p2 and P4 may be obtained by solving equations 34 and 36, or 35 and 36 (16) D. M. \V. Anderson, J. L. Duncan and F. J. C. Rossotti, J . Chem. Soc., 140 (1961). using corresponding values of the logarithms of b and
F. J. C. ROSSOTTI AND HAZEL ROSSOTTI
930
b, and of F and F or @I and CP in the position of best fit. The value of p3 may then be calculated from the appropriate value of R by means of equation 37. The limits of error in the association constants may be obtained from the permissible variations in R, and in the position of acceptable fit. A similar treatment may be devised for any system containing three complexes of known formula but of unknown stability. Analysis of Weight Average Molecular Weights. -The weight average molecular weights, RwJ of solutions of high polymers may sometimes be obtained from measurements of turbidity or of sedimentation velocity. Steiner" has shown that, since Q Mi 1vw=
log b--
1
Q
-
Mi
qaP9bq 1
B
(38)
dB.21 1
the value of b may be obtained using the expression
=
B
where MI is the molecular weight of the monomer. The free monomer concentration must have a known value, bl, a t one total concentration, B,. The function B(b) may then be treated as described above. If required, the function S(b) may be obtained by a second integration, using equation 12. Alternatively, equation 38 may be rearranged to
and curve-fitting methods may be applied to the power series W(b). For example, if Q = 3 w-1
Q ?[B91
- log b2
Vol. 65
4P2
+ 9Pab
The values of P 2 and p3 may be obtained by a procedure analogous to that described for the functions F(b) and @(b), uiing equations 25 and 26. The three sets of data (Mw, b ) , ( B , b ) and (8, b ) may therefore be treated simultaneously.
GRAPHICAL METHODS O F DETERMINING SELF-ASSOCIATION COSSTANTS. 11. SYSTEMS CONTAINING MANY SPECIES BY F. J. C. ROSSOTTI AND HAZEL ROSSOTTI Department of Chemistry, The University of Edinburgh, Scotland Received September 23, 1960
The number of independent self-association constants which may be determined for systems containing a large number of multimers is limited by the precision of the experimental data. New graphical methods are proposed for determii ing approximate association constants on the assumptions that the constants are related to one, two or three determinable parameters. The methods are illustrated with reference t o 2-n-butylbenzimidazole, N-methylformamide, X-propylacetamide and N-methyltrichloroacetamide in benzene, and to aqueous butyric acid.
In spit'e of numerous of equilibria in systems which are extensively associated, few attempts have been made to obtain a large number of independent association const'ants. Although values' of PP have been obtained by successive extrapolation* and successive approximations,8 the precision of the data rarely permits the determination of more than two or three independent association constants. Kow, if several constant's were inter-related, the number of independent parameters to be determined would be reduced. Many authors have assumed that all the st'epwise association constants K , are identical2l8-l7or that (I) E. N. Lassettre J . A m . Chem. Soc., 59, 1383 (1937); Chem. Revs., 20, 259 (1937). (2) K. L. Wolf and R. Wolff, Angew. Chem., 61, 191 (1949). (3) R. E. Conniok and W. H. Reas, J . A m . Chem. Sac., 73, 1171 (1951). (4) N. E. White and M. Kilpatriok, J . Phys. Chem., 59. 1044 (1955). ( 5 ) M. Davies and H. E. Hallam, J . Chem. Educ., 33, 322 (1956). (6) F. J. C. Rossotti and H. Rossotti, "The Determination of Stability Constants," McGraw-Hill Book Co., New York, N. y . , in press, Chap. 16. 17) F. J. C. Rossotti and H. Rossottti, J . Phys. Chem., 65, 926 (1961), where symbols are defined and equations with numbers preceded by Roman I will be found. (8) K. L. Wolf, H. Dunken and K. Morkel, Z . p h y s i k . Chem., 46B, 287 (1950). (9) H. Kemyter and R. Rleoke, ibid., 49B, 229 (1940).
the first one or two stepwise association constants differ from the rest12,18-21 Other one-parameter relationships between st'epmiseassociation constants have been given by Connick and Reasj3and a twoparameter relationship was suggested by Lassettre' in 1937. The treatment of simple syst'ems by curve-fit,t.ing7 is now ext,ended to show how experimental data for more highly associated systems may be described in terms of one, t'mo, or three independent parameters. If D (equation 1-18) t8endsto high values and dD/db increases sharply, we assume that a very large series of multimers is formed and test various simple hypotheses concerning the intcrrelationship of the association constants. (10) J. Kreueer and R. Mecke, ibid.,49B, 309 (1941). i l l ) J. Kreueer, ibid., 53B, 213 (1943). (12) J. Bjerrum. X e m . Maanedsblad, 24, 21 (1943). (13) R. Ginell, J . Coll. Sci., 3, 1 (1948); 5, 99 (1950). (14) K. L. Wolf and G. Metzger, Ann., 563, 157 (1949). (15) Y . Douoet and S. Bugnon, J . chim. phys., 54, 155 (1957). (16) E. Thilo and G. Kruger, Z. Elektrochem., 61, 24 (1957). (17) P. J. Flory, J . Chem. Phys., 17, 425 (1944): 14, 49 (1946). (18) G. Briegleb, Z. physik. Chem., 51B, 9 (1941); G. Briegleh anti W. Strohmeier, Z. Elektrochem., 57, 668 (1953). (19) E. G. Hoffmann, i b i d . , 53B, 179 (1943). (20) N. D. Coggeshall and E. L. Saier, J . Am. Chem. Soc.. 73, sill& (1951). 121) M. Davies and D. K. Thomas, J . P h y s . Chem., 60, 763 (1956).
