Linear Properties of Solutions Absorption of Light TERRELL L. HILL, Western Reserve University, Cleveland, Ohio, and S . ARONOFF, University o f California, Berkeley, California
R
ECENTLY, spectrophotometric measurements have been found to be of considerable value in studies concerned with colored ions in solution and with the equilibrium relationships between such ions.' In connection with some aspects of this work (e. g., the determination of the absorption spectra of intermediate ions and of the equilibrium constants involved), however, the methods of analysis have not been systematized or generalized. For this reason and because of the fact that any other linear2 property can be subjected to the same treatment, we have felt that it might be worth while to outline a general procedure.
we may call G a linear property of the solution. It is sometimes convenient to distinguish one component of the solution and refer to it as the solvent. Thus, if the nth component is denoted as solvent, equation (3) or equation (5) may be rewritten in various forms making use of equation (2). If a. = 0, that is, if the solvent makes no contribution4to the property G, If, further, nl =
.. . =
=
an-1
= n
LINEAR PROPERTIES OF SOLUTIONS
and we may call G a colligutive property of the solution. The total vapor pressure over a solution serves to fistrate equation (3), while the total vapor pressure over ideal is- an examole of eauatiou (5). -- a.n .~-~ ~ - solution - ~ ~, Amonp other orooerties which follow eauation (5) mav " G = G(N.. (.1.) be listed absor~tionof lirht, ~ .. N,.. . . . N J... - . conductkty a t 'iifinite dilution, refractive index, and rotation i f the plane where N, is the mole fraction of the ith of the n com- of polarized light, ti^^ (6) could be used to be considered a mnms POnents (each absorntion of lieht bv a solution for . ..=. . . .t h e total ...-..- .- - r component, when convenient). The additional rela- a wave length a t which the solvent does not absorb. tionship The usual colligative properties of dilute solutions obey N ., +. N 2.~ + . . . +N . = 1 ( 2 ) equation (8). keturning to linear properties and equation (5), would, of course, allow the elimination of one variable let GI be an exp&mentally measured value of G when from the function G. . the mole fractions are NIL,Nz,, . . N.,. By varying the If G is such a function that it may be written as a one can obtain other values of G. ~f n sum of contributions of the various components of the such values are thus known, one has solution (one term for each component that conGI = N,, at NII nr . . . + N n l a . ) tributes to the property), that is, Let be the measure Of a Property of a which is afunction of composition alone (temperature, Pressure, and other possible variables being held constant). That is, ~
~
~~
~
~~
~
-
~
a
~~
~
~~
~
d
.
+
we may call G a qumi-additivea property of the solution. If the functions G, are constants, G; = w (i = 1, 2 , . . . n) so that
-
G =N m
+ Nmr + . .. + N.u.
(4)
(5) See, for example, FXBINOWITCH AND STOCKMAYBR, 1.Am. Chem. Soc., 64, 335 (1942);.La~aomAND KIEHL, ibid., 64,291 (1942); McCumonoH. rbwl., 64, 2672 (1942); HARING AND HELLER, ibid., 63, 1024 (1941). ' Defined below., , T h e term "add~twe"is usually reserved for extensive properties, that is, homogeneous functions of the first degree in the mole numbers. Equation 3 cannot, in general, be homogeneous in the mole fractions because of the restriction imposed by Equation ( 2 ) . See footnote" however. 1
G.
= Nwx,
+
+ Nma* + . . . f N..n.
The solution of this set of nonhomogeneous linear equations is then immediate for m,, as,. . . an. Of more interest than this trivial case is the one in which the composition is not completely known but certain equilibrium relationships between the various molecular species present can be made use of. Thus the unknown molefractionscan be writtenasfunctionsof known mole fractions and either of thermodynamic equilibrium constants (if the activity coefficients may be taken as unity) or of concentration equilibrium constants (at
--
4 In this case, G is an additive property over a certain range of mole fractions. That is, N,, . . N,-I may all be multiplied by a constant k at the expense of N., hut the value of k is limited by equation ( 2 ) .
615
.
homogeneous linear equation in the (2p - 1) unknowns Y, = a,, y, = arKz.y, = arK,K1,.. .y, = a , K , K n . . .K,_, y.1 = K I , YD+, = KLKI,. . . YP.-I = K l K t . . . Kp-1 If now G is measured experimentally a t (2p - 1) diferent values of x (the ionic strength must, of course, be the same for all values of x), we have (2p - 1) linear equations which may be solved for the above unknowns. If other values of x are available and another choice of the (2p - 1) values of x is made, the results should not differ. This may be used as a check. In any case the values of x used should be well disG = u l a ~ ulan . . . u,a, (10) tributed. Fmally al, . . ,a*, KI, . . ,K*I may be where u, is the mold concentration of the ith com- found from the relationships: ponent, the a's are the new constants replacing the at = YI. a. = ydy,+,, ar = YI/Y~+~, . . ..ar = Y~/Y*~-I m's, and m = n - 1. The validity of equation (10) . . K,-1 = for the concentration range of interest should be con- K I = YP+~,K I = y,+Jy,+~ Ks = y,+t/y,+., Yld~2P-z sidered in every case. Stepwise Dissociation. Perhaps the most important Clearly, if any of the a's or K's are previously known type of equilibrium relationship between the various or can be found by other means, the number of unmolecular species is that of stepwise dissociation, knowns and equations may be reduced.' It is possible that by successive lowerings of the represented by the system ionic strength and determination of the equilibrium constants a t each ionic strength, the thermodynamic B* are the only equilibrium constants may be approximated by in which the substances BI, B s ones contributing to the property of interest, and the extrapolation. It is not difficult to see from the nature of the function mold concentration of A may be measured experiG in equation (15) [a quotient of two polynomials, mentally. We have, then, each of degree (p - I)] that if the experimental values of G are plotted against x over a sufficiently wide range in x, the maximum number of points of inflection in the resulting curve is (p-1). This fact may be and of use in determining the number of species of B when it is not otherwise known.' We have assumed above that the substances B1, where ul is the mold concentration of B,, the K's are B, are all present. However, the method is unequilibrium constants (see above), the molal con- Be, centration of A is x, and c is the gross mold con- changed if any of them are absent.' Thus if the sub= B&I 2A, in centration of B in the solution. Froma equations stance B, does not exist [i. e., B,I the system (ll)], equation (16) is altered formally by (12) and (13), omitting the terms containing xu-' and setting KrlK, = K-1, , as can easily be verified. There are, then, two fewer unknowns in equation (16). A similar chanre is made in more complfcated cases. For example, if Putting equation (14) in equation (lo), we obtain both B, and B&I do not exist, equation (16) is altered c(a,rs-' + a2K,xP-' + a&K,zP-s + ., .+ a p K I K , ...Kp-,) by omitting the terms containing xc-I and fl and G= setting K,IK&I = K,I, , &I, etc. For the system X P - 1 + K , z * - ~+ K , K , x ~ -+ ~ . . . + K , K , . . . Kn_, (15) BI = B, 3A, Bk = Bs A: which can be written in the form K m = u,ea -, K , = u s and u, ua us = c the ionic strength of the solution being studied). Then the use of additional linear equations (i. e., additional experimental measurements) may allow the evaluation not only of ml, m2, . . . a, but also of the equilibrium constants. It is sometimes more convenient to make use of concentrations rather than mole fra~tions,~ and since in sutficiently dilute solutions mole fractions are proportional to concentrations, we may in this case rewrite the above equations. Furthermore, a solvent is often chosen which does not contribute to the property. Then
+
+
+
.
.
.
...
.. .
+
+
G
=
@)at
U,
+ (E) arKx + ( s ) a a ~ & + . . . +
(s,). . Q,KIK,
( ),
. K,-1-
. (
. .-
. .,
-.u,
+ +
Equation (16) becomes
(E)K, ),
+
G 1 )
x may be determined experiThe quantities G, c, mentally and consequently equation (16) is a non6 Indeed, far some properties, the molal concentrations may havemore physical significancethan the mole fractions. HILL,J. Phys. Chew., 46,417 (1942).
= ca,
+ -c a&= + 2r a&dG 28
G G -2 KW - 2 KmKa
Equation (16) applies to stepwise reactions only, but if other complications are introduced, an analogous equation may be derived and used, so that the method is not restricted to the particular case we have discussed.
-
'
This question will be considered again in connection with absorption of light.
x.
ABSORPTION OF LIGHT
In studying this particular linear property a new variable, the wave length of the light absorbed, must be introduced. Thus, equation (10) may be written -if we are still dealing with stepwise dissociation-
This is easily seen from equation (15): lim G = cat and lim G = cap x-
x-0
rn
This, of course, simplifies the analysis by reducing the number of unknowns (equation 17). D(h, X ) = u ~ ( ~ ) s l ( X unb)rr(X) ) . .. u l l b ) 4 ) (19) It was pointed out above that plotting G against x and noting the number of points of inflection in the where D is the total absorption by the solution, X is resulting curve may give some information about tde the wave length, and et is the extinction coefficient of value of p. I n any case, the maximum number of BIa t X. For a particular wave length points of idection is (P- 1). Here we plot D against D ( x ) = U,(X)., ~ X ) O . . up(x)ep (20) r, for a given wave length. By examining such curves If, for any suitable though arbitrary wave length, D is a t several wave lengths we increase the probability measured a t 2p-1 different values of x (see above), that the largest number of points of inflection observed the preceding method of analysis gives the values of (in any one curve) is indeed the maximum number posn, 4, %, K I ,Kz, K,I. If a different wave length sible, namely (p - I). That is, the maximum number of is chosen, the same equilibrium constants should result inflections may be obtained a t one wave length but not (thus serving as a check) but the r's will in general be a t others. The number (P-1) mentioned above assumes that different. That is, the equilibrium constants are inp substances BI, Bz, . . B, are present. If only r of dependent of the wave lengths but the extinction them actually exist (see above), the result is the same. coefficients are not. In fact, if the analysis is perThat is, the maximum number of points is (r - 1). formed for an adequate number of wave lengths Examfile. It will not be necessary to include an the spectra of all the substances B,, Bz, . . 4% experimental example here since an opportunity arose may be obtained, since for each X we obtain a elsewheres to make use of this method. different set of e's, and by plotting e,(X) against Swmmary and Relation to Earlier Methods. In the X we have the absorption spectrum of the subcase of stepwise dissociation (whether or not all interstance B,. It should be mentioned that after the mediate species exist), using this general method we equilibrium constants are checked a t several wave are thus able to determine the equilibrium constants lengths they need not be considered unknowns a t the K J , Ka, . . . K,J, the absorption spectra of BI, Bz, remaining wave lengths. Instead, the values of B,, and, in general, also the number of diierent varirl, . . e9, a t a wave length for which the equilibrium constants are assumed known, may be obtained [see eties of B. If the system is more complicated than stepwise, the method may still be used after suitable equation (16)] by the use of modification of the eauations derived here. However, the present method does not allow the &Ka +) *..I i-I = cq + "+ ( 7 determination of which of the species of B actually cKLK,. .. occur. Vosburgh and his coworkersghave developed . ..+ 1 * a procedure to accomplish this. Job,Io in addition to originating the method extended which is a nonhomogeneous linear equation in P un- by Vosburgh and his coworkers, calculated equilibrium h o r n s (€1, d. Therefore, a t such a wave length, constants for certain particular cases. However, his measurements of at 9 method, as he pointed out, is notgeneral, values of x are necessary. 'ARON~RT AND CALVIN,J . O*g. Chm. 8,205 (1943). It is often possible to obtain one or both of the ex0 Vossmn AND COOPER, 3. Am. C h m . Soc.,63, 437 (1941); treme spectra (i. e., spectra of BI and B,) by separate GO~,,D AND V O S B ~ ~ H ;bid., . 61,1630 (1942). experiments, using very small and very large values of 10 JOB. Ann. chim. (lo), 9,113 (1928); (11) 6 , 9 7 (1936).
.. .
+
+ +
+
+ .+
.. .
.
.
. ..
.
~ ' g ' ( " " ~ ' ~ ' - ~ (??) )
(
...
)
