296
JORULFBRYNESTAD AND G. PEDRO SMITH
Isosbestic Points and Internally Linear Spectra Generated by Changes in Solvent Composition or Temperature'"
by Jorulf Brynestad and G. Pedro Smith'b Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 87880 (Received August 7 , 1967)
We show that isosbestic points generated by changes in temperature or solvent composition may be formed under a wider variety of conditions than previously recognized, so that the rules previously proposed for interpreting these points in terms of the number of absorbing species or the number of independent reaction parameters are unreliable. Examples of experimental cases in which these rules break down are cited. We also point out that in most applications it is more useful t o show that spectra are linearly related than to show that they have isosbestic points.
Introduction The presence or absence of isosbestic points in a set of spectra generated by changing the composition or temperature of a liquid system is often used as evidence regarding the number of absorbing species or the number of independent reaction parameters. The most complete isosbestic point theories are those by Cohen and Fischer,2 and Morrey.a We shall show these theories to be incomplete and partly wrong, and shall cite experimental results that illustrate some of the deficiencies. I n so doing, we do not single out these authors for special criticism. On the contrary, we cite them because their treatments are the most complete thus far. Cohen and Fischer2 showed that iscsbestic points may occur in a system containing several absorbing species when the variation in concentration of all these species is linearly related by a single reaction parameter. They then generalized on this result and concluded that the existence or absence of isosbestic points provides a basis for distinguishing between systems in which the absorbing species are interrelated by a single reaction parameter and those requiring more than one such parameter. They also considered the effect which changes in temperature or solvent composition have on the spectra of such systems. They pointed out that these changes will, in general, not only shift the chemical equilibrium but also alter the shapes of the absorption bands, so that one would expect to find isosbestic points only for limited ranges of variations in temperature or solvent composition. Recently, Angell and Gruen4v6 applied the CohenFischer type of model to large temperature variations under the assumption that the effects of temperature on band shapes could be ignored. They proposed that the presence of temperature-generated isosbestic points implies the presence of two absorbing species in equiThe Journal of Phyeical Chemistry
librium, while their absence implies the presence of a single such species which is continuously distorted by changing temperature. The Cohen-Fischer and Angell-Gruen treatments overlooked the possibility that the effect of temperature (or solvent composition) on the spectra of species that are not in chemical equilibrium might be of a unique type that generates isosbestic points. Morreya experimentally demonstrated that just this behavior sometimes occurs in the case of temperature variations. He measured the effect of temperature on the spectra of several systems that plausibly contained a single absorbing species and showed, first, that isosbestic points are formed on both the absorbance and absorptivity scales over wide temperature ranges, and, second, that the absorbance and absorptivity a t any wavelength are linear functions of temperature. He proposed a phenomenological theory of this behavior and concluded, among other things, that only one species contributes to the absorbance a t a given temperature-generated isosbestic point, and that any equilibrium involving this species is not appreciably affected by temperature changes. I n all of these theories it is taken for granted that isosbestic points occur only if the spectra are linearly interrelated. That this is not a generally valid assumption is illustrated by the measurements of Angell and Gruen,4 who published several sets of spectra that have well-defined isosbestic points but that are not (1) (a) Research sponsored by the U. S. Atomic Energy Commission under contract with the Union Carbide Corp. (b) Oak Ridge National Laboratory and Department of Chemistry, University of Tennessee, Knoxville, Tenn. (2) D. M. Cohen and E. Fischer, J . Chem. SOC.,3044 (1962). (3) J. R. Morrey, J . Phys. Chem., 66, 2169 (1962); 67, 1569 (1963). (4) C. A. Angell and D. M. Gruen, ibid., 70, 1601 (1966). (5) C. A. Angell and D. M. Gruen, J . A m . Chem. SOC.,88, 6192 (1966).
GENERATED ISOSBESTIC POINTS AND INTERNALLY LINEARSPECTRA even approximately linearly interrelated. We shall treat this problem, but we shall also show that even when the spectra are known to be linearly interrelated, still the Cohen-Fischer and Morrey theories are inadequate and misleading. Furthermore, we shall contend, in opposition to Angell and Gruen, that the absence of isosbestic points provides no evidence one way or the other concerning the number of species.
Definitions The absorbance, A = log (lo/l), of a liquid system is a function of the wavelength A, the path length, and a set of N macroscopic parameters that uniquely fix the state of the system (temperature, pressure, external fields, composition, etc.). We denote these N parameters { x ~ } ~ . It is convenient to factor A as follows A = ACA,{XkJN) = E(~,IXkJN)/V({XkJN) (1) where V is the volume of the system. The quantity E is related to the formal absorptivity ef via the expression ef = E / M , where M is the mass parameter for a component that gives rise to the absorbing entities under consideration. We designate the external parameters in a reference state as {XkO} and the spectrum of the reference state as A0 = AO(X,{X,OJ,) E o = Eo@,{XkOJN) When one of the external parameters, say X,, is changed so that X, = Xp AX,, the resultant change in the absorbance can be written
+
A(A,Xj) = A’
+ AA(A,AXj)
with similar expressions for E and V , namely
+ +
E(A,Xj) = E o AE(A,AXj) V(Xj) = Vo AV(AXj) We say that a set of spectra has an X,-generated isosbestic point on the E (or A ) scale at a wavelength X I if AE (or AA) is zero at A1 for all values of AX, over a specified range. There are a number of well established cases6-10 in which a set of spectra may be described by the relation
+
E = EO[l h(A)g(AXj)I
(2)
where g is a monotonous function of AX,. We refer to such sets as internally linear because any spectrum that conforms to eq 2 can be expressed as a linear combination of any two other members of the set. That is
EO)
= =
+
(1 - Pr)Ei(Q PiEz(A) Eo{1 h(A)[(I - Pi)g(AX,(’))
+
+
PdQ(Ax1(2)) 11 (3) where Pc is a number, and AX, (’) and AX, (2) are the values of X , at which E1 (A) and Ez (A), respectively, were measured.
297
Role of Internal Linearity in Isosbestic Point Theories I n the Cohen-Fischer2 and Morrey3 theories, and in all previous theories as well, isosbestic points appear as a mathematical consequence of internal linearity, so that these theories are, in fact, theories of internal linearity. According to the Cohen-Fischer model, isosbestic points occur only if the spectra are of the type specified by eq 2. Morrey3 concluded that temperature-generated isosbestic points occur only if a special form of eq 2 is obeyed (his eq 11). The possibility that isosbestic points can be caused by functional forms other than those in accord with eq 2 has been overlooked or else ignored as physically improbable. As a result, the observation of isosbestic points is commonly considered a sufficient proof that the system behaves in accordance with some theory of internal linearity. The fallacy in such an approach is easily recognized by realizing that although internal linearity is a sufficient condition for isosbestic points, it is by no means a necessary condition. For example, if E (or A ) is any monotonous function of X , a t all wavelengths, isosbestic points will occur at wavelengths where bE/bX, = 0. (Equation 2 is a particularly simple class of functions monotonous in X,.) There may also be cases where isosbestic points occur “accidentally” without underlying monotony at all wavelengths. Not only are there no a priori physical reasons for excluding these possibilities, but in the recent literature there are a number of examples in which isosbestic points are observed without internal linearity. Angell and Gruen4 present several such sets. The danger in this situation is illustrated by the fact that these authors proceed to use these isosbestic points as evidence for a Cohen-Fischer2 type of situation, i.e., as if the spectra were internally linear. Since isosbestic points are not reliable indicators of internal linearity, other methods must be employed to test for this property. Several methods are possible and two have been published7pQthat can be used whether isosbestic points are present or not. We conclude that isosbestic points are not sound evidence for anything, although they may serve as eyecatching indicators of possible internal linearity. The question of whether or not internal linearity is reliable evidence of something ill be treated in the remainder of this paper. Existence Conditions for Internally Linear Sets of Spectra We consider first systems which obey the relation of additive absorbances, that is (6) Examples are given in ref 3 and 7-10. (7) J. Brynestad, C. R. Boston, and G. P. Smith, J. Chem. Phys., 47,3179 (1967). (8) J. Brynestad and G. P. Smith, ibid., 47,3190 (1967). (9) C.R.Boston and G. P.Smith, J. Phys. Chem., 66, 1178 (1962) (10) G. P. Smith, C. R. Boston, and J. Brynestad, J. Chem. Phys., 45,829 (1966). Volume 72,Number 1 January 1968
JORULF BRYNESTAD AND G. PEDRO SMITH
298
where E $ and n, are the absorptivity and number of moles, respectively, of the ith species in the system. Both e t and n, are, in the general case, functions of the external parameters {X,},. Equation 4 can be expected to hold for dilute solutions or absorbing species that do not interact. This equation can be recast in terms of a reference state designated by EO, e?, and n,O as "
l i
EO =
etOn,o i-1
C
E
(eto
I=
i=l
=
Eo
+
+ Ae,)(n? + Ant) c
+
+
C
( n t A ~ eioAni) i-1
AeiAnt (5) i=l
I n the following, we discuss the more important conditions under which systems that obey eq 4 will give rise to internal linearity. A . Case I : All Ae, = 0, A11 Ani # 0. I n thismodel case, the absorptivities of the individual species are invariant with respect to the external parameter in question, X,. Equation 5 reduces to
mental tolerance. If there are no good reasons for believing that these conditions are fulfilled, it is not valid to assume that observed internal linearity proves that a Cohen-Fischer type of mechanism is operative. There are many valid examples of internal linearity of the Case I type generated by composition changes, and these are discussed by Cohen and Fischer.2 Also, an example of the Case I type generated by temperature changes is presented by Angel1 and G r ~ e n . These ~ authors report a set of spectra with one temperaturegenerated isosbestic point for nickel complexes in concentrated aqueous solutions of magnesium chloride (their Figure 3a), and they use the isosbestic point as evidence of a temperature dependent two-species equilibrium. Their spectra also appear to be internally linear, in accordance with the requirements of the Cohen-Fischer model. Note, however, that in Rforrey's3 theory, the same experimental behavior is used as evidence that only one absorbing nonreacting species is present. We look more closely at this discrepancy in the section that follows. B. Case I I : All An, = 0, All Ae, # 0. In this case the absorptivities E ( are not invariant with respect to X,, whereas the amounts of the absorbing species are constant, i.e., these species do not participate in chemical equilibria. Equation 5 reduces to c
E
=
Eo
+ CntOAel i-1
Suppose that in a specific instance all the Ant are linearly related, i e .
An, = k,A[ = k,g(AXj)
(7) where the ki are constants (stoichiometric coefficients), finite or zero. Then eq 6 gives
(8)
where h(X) = (l/EO)Z(i)Qk,. Hence, we obtain an equation of the same type as eq 2, the internal linearity relation. Isosbestic points occur when Z(i)e?k, = 0. This case is the same as that discussed by Cohen and Fischer.2 It applies when there is only one chemical equilibrium in the system, in which case the n, are linearly interrelated. It automatically includes the situation in which there is an arbitrary number of species that do not take part in the equilibrium. For these species, An, = 0 and their spectra are included in EO. A basic requirement for this model to be applicable is that the absorptivities e c be invariant with respect to the variable parameter, X,. If X, is a composition parameter, the species must obey the Bouguer-Beer law t o within experimental tolerance. If X, is the temperature (or pressure), the e, must be insensitive to changes in temperature (or pressure) to within experiThe Journal of Physical Chemistry
(9)
Obviously, if the spectrum of a single species behaves according to eq 2, then the absorptivity must be of the same functional type, i.e. 6
=
eo[l
+ h(X)g(AXj)I
( 10)
Now there is no known physical reason to expect that the spectrum of a pure species should in general have this particularly simple form with separable X and X, dependence. Therefore, one might expect internal linearity of the Case I1 type to be rather rare except, possibly, for special types of optical transitions. Furthermore, if there are several species, the only way of converting eq 9 into a relation with internal linearity (eq 2 ) is for the e l of all of the species to have the form of eq 10 and also the same X, dependence, that is Et
= ea0[1
+ h,(X)g(X,)l
(11)
This would lead to
+ h(X)g(AXj)I
= EO[l
(12)
where h(X) = 1 /EO[Z(i)h,(X)n?]. We do not expect eq 12 to be fulfilled very often except in cases where all of the absorbing species are "similar." There are a substantial number of examples of internal linearity generated by changes in temperatures that cannot be rationalized in a satisfactory way as in
GENERATED ISOSBESTIC POINTS AND INTERNALLY LINEARSPECTRA terms of Case I, so that one is forced to conclude that Case I1 applies. I n the known instances where a Case I1 situation is suspected, the spectra are internally linear, or almost so, on both the A and E scales over most of the wavelength range and A and E at a fixed wavelength are linear functions of temperature. This behavior was first pointed out by Morrey.3 He presumed that this behavior would hold a t all wavelengths, but we have recently found that in X regions where the relative change AE/Eo (or A A / A o ) is not small compared to unity, linearity tends to break down. An empirical explanation for this behavior is as follows. A condition for observing temperature-generated isosbestic points on the A scale is that E must have the same temperature dependence as the volume V a t given wavelengths, i.e., AE/EO = AV/Vo, and in order to have isosbestic points on the E scale, AE must be zero at given wavelengths. This can be accounted for by assuming that in these cases E has the same, or almost the same, type of temperature dependence as V at all wavelengths. Since it is commonly true that the volume expansion coefficient a = l/V(bV/bT), is approxiinately constant for liquids over substantial temperature ranges, and hence V = Vo exp(cyAT) this would require that
E = Eo exp[h(X)AT] (13) with isosbestic points on the E scale wherever h(X) = 0, and on the A scale wherever h(X) = a. Moreover, if aAT and h(X)AT
