Polymerization Evaluation bv Spectrophotometric Measurements Jaume Dufiach Departament de Quimica Analitica, ~niversitatAut6noma de Barcelona, Bellaterra, Spain Many solutions of associable, (UV-visible) light-absorbent monomers do not follow Beer's law with increasing concentrations. Often dimers and oligomers are formed. The lesser number of absorbent particles per unit volume produces an "absorbance contraction." Examples range from surfactants, like sodium alkylbenzenesulfonates ( I ) creating micelles of numerous associates, to dies that originate dimers, like Basic Violet 10 (2). The monomers can be biopolymers, like the tobacco mosaic virus (3), complex molecules, like chlorophyll a in pyridine ( 4 ) , medium-sized ones, like 2,2'-bipyridine in cyclohexane (51,or smaller ones, like ethanol in carbon tetrachloride (6). After setting up some underlying hypotheses and definitions, the differences in absorbance will be calculated using association degrees. Later, association equilibria constants will be introduced and the absorbance differences w~llbr expressed with rhe~n.In pursuing these paths, some intrrcsting relations and features will he cncuuntered.
in the ith associate, divided by the total numher of moles of monomer, evaluated by dividing the weight of dissolved substance by the molecular weight of the monomer. The degree of ouerall or global association a of a solute in a solution is the number of moles of monomer that are associated, regardless of the species they form, divided by the total number of moles of monomer dissolved, evaluated as above. so, a = a z + a 3 +...+ a , + . . . (1)
System and Definitions The monomer and its aaareaates -- - are homoaeneouslv dispersed, in water or in another convenient solvent. " ~ a c h species, considered alone, follows Beer's law, in the concentiation interval taken. This is not always admissible (7). No species is decomposed by the radiation used. The association number i is the number of associated monomers in a species. In a solution i can take several values. It is an indexing letter; i = 1is called the monomer, i = 2, the dimer, etc. The degree of association aj of the ith associate in solution is the fraction of the number of moles of monomer involved
Expression of Absorbance The absorbance of a monomer and its associates. measured in a spectrophotometer using a monochromatic radiation not absorbed bv the cuvette or the solvent. can be ex~ressedas the sum A = A 1 + A 2 + ... + A i + ... (2)
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Journal of Chemical Education
The nonassociated fraction, or free monomer fraction, is 1 a = al. Only i - 1 values of aj are independent. If c is the overall concentration of the dissolved substance, in moll-' of monomer, the concentration of the i t h associate is cajli. Association Degrees and Molar Absorptivlties
A1 being the absorbance of the monomer and A; the one of the i t h associate, which molai absorptivity is rj. Applying Beer's law to each term, aiti
A=IcEi
L
(3)
Figure I.Absorbance versus concentration in a solution of a monomer with oligomers, showing extrapolation of absorbance from lower to higher m c e n nations and the difference "A - A', or absorbance contraction.
where 1 is the optical length of the cuvette. This equation can take another useful form, avoiding the inconvenience that when a1 increases, the other aj's decrease. Replacing 1- a2 - as - . . .- a; - . . . for a1 and grouping the terms with the same a;, gives
Suppose now that the o v e r d concentration increases n times, becoming c' = nc. According to the laws of equilibrium, the association increases with the concentration, hut it does not mean that the concentration of each species will increase n times. There will be new association degrees aj', greater than the respective former ones, a;. The new absorbance A' will be A' = &A;' or
centrated solutions, even if associates existed in solution, provided that the monomers that integrate them would keep their proper and same E value, so that rl = €212 = . . . = rjli, which normally does not occur, or provided that the degrees of association would not vary with concentration and a;' = a;, which is also hypothetical. In the polymer solution, Beer's law can be understood as a limiting law, holding for the entire solution only when the variation of all the association degrees with concentration is sufficiently small or when all the differences between the molar absorptivity of the monomer r1 and the molar ahsorptivities of the associated species divided by their association number, rili, are experimentally unappreciable. In conclusion, for Beer's law to be experimentally established for the whole solution, it is not required that the associates not exist, but only that the added products (ai' - aj)(rl - ~ i l ibe ) negligible. The following example refers to a solution containing monomer and dimer with exclusion of other associates. If for M , 1 = 1cm, = 1500, EZ = 2000, and the instance, c' = difference nA -A' is a t the lowest detectable values, say nA -A' = 0.005, the difference az' - a z between the association degrees of dimers is found to be equal to 1. 0.005 absorbance =1 (12' - a2 = 2000 mol absorb. an.-Ilmol 1.10-5. 1500 - 2 1 em This is the maximum possible value, meaning aa' = 1,that is, a t c' all the dissolved substance is in the dimer form. At the concentration c'ln, the monomer exists. Still Beer's law for the entire solution, can be said to hold from c'ln to c'. Association Degrees and Equilibrium Constants
Formation Products of Association Degrees For the successive association equilibria 2MeD M+DeT M (i - 1)th ith
(8)
+
If Beer's law would be applicable to the whole solution, the absorbance would be nA, the extrapolated value of A,
(where M, D, and T are solvated monomers, dimers, and trimers, respectively) the thermodynamic formation constants can be condensed in the form where Q; represents the formation concentration product (also called quotient) of species i. For example
In comparing nA with A' only the aj"s differ from the respective aj's, and as they are greater, but in negative terms, the whole A' shall he less than nA: A' < nA. So, on concentrating, the absorbance will not increase so much as the value nA (see Fig. 1). I t is possible that all the differences €1- rili may he positive, i.e., the association number i times the molar absorptivity of the monomer may he greater than the molar absorptivity of any of the other species. Absorbance Contraction and Beer's Law The difference
takes into account the possibility that the association degrees he differently modified one from another. This is equivalent to admitting a series of successive i - 1 equilibrium constants. Beer's law for the whole solution can hold, i.e., nA = A', either when all a;' = a; or all = rili. Beer's law could be extended from diluted to more con-
F; is the corresponding product of activity coefficients
Note $1 = 1and K I = 1. If the association degrees are introduced in Q; at a concentration c, it can he written:
-.-
0:"
1
Qi =
-
Qi-l c.alc
i-1 L
a;
1 i-18;
oi-lo, c
L
c
(12)
i-1
The Q;representing the formation product of the association degrees (Q-FPAD). Volume 62
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45 1
Figure 2. Proportionatity between % ard n. Straight lines are shown for i = 2 and i= 5, with arbitrary K2 and KSvalues. The concentration ratios between n = 1 and n = 0 indicate dilutions.
Replacing Q; in K; gives:
3f, heing 8;.Fi or (K-FPAD). When passing from the concentration c to another one c' = cn, a t the same temperature, thermodynamic constants keep their values and
Bi
Figure 3. Log as a function of log nfw i= 2 wd i=5. Arbitrary values are taken for & and The values of the intercepts with the axes are indicated, as well as the intersection of the two straight lines, log &' and log &'.
&.
Global Formation Plwucts of Association Degrees If global /3; = K&3. . .K; constants are used, R;, g;, and fi; expressions are similarly defined as Qi, P f , , and Q;, respectively.
as well as
where 8;'represents the association degrees product a t the new concentration c' = nc, and F;', the activity coefficient product a t this concentration. Equalizing and simplifying gives:
The products ff, are proportional to the concentration ratio n, increasing n times when the concentration of the total dissolved substance increases n times (see Fig. 2). If the activity coefficients products F; would not vary with the change in concentration, or if the activity coefficients f ; could he taken as unity, as it is approximately the case when the monomer are not charged or when they are sufficiently diluted, the above statement could he made with the Q; only
with
In varying the concentration from c to c' new pi's, called &' originate. The relation between them is obtained from
and is
B.' - 8...;-I L -
I
(25)
a'
or
-- a; a;-,
.a,
-1
a;'
n ai-I
,
01'
(16)
The association degrees product increases proportionally t o the concentration ratio. Dissociation Products When dissociation constants are used instead of formation ones, the conclusion is that the products expression of the association degrees is n times lower when concentration increases n times. 452
Journal of Chemical Education
The or (8-FPAD) vary following the (i - 1)th power of the concentration ratio. The log Pi', vary linearly with the log of the concentration ratio, log n, the angular coefficient being ( i - 1) and the intercept log Ti: log Bi' = log Bi + (i - 1) log n
(26)
(see Fig. 3). Negativr values of log n mean that c' is leis than c and that there is a dilution. The intersection with the ahscissa axis, log 75,' = 0, are found at values of
i.e.. when D' = n'-1. if the activitv coefficients can be cancelled. A dimer accomplishing Beer's law realizes a;' = ail&. As the total concentration increases n times, the free monomer becomes × lower. In the case of a high association number i, the firs, memher d e a n . (91) will awroximate sufficirntlv to 1. for more thm one d u e of i; ~ e e &law will hold, then, hpradice for several species. Observe from eqn. (30) that, ordinarily, this first member in eqn. (31) is >1, as a;' > ai, and the association degrees increase with concentration, except for the monomer. I t follows that n > p and the relative reduction in monomer concentration cannot be larger than the increase in relative overall concentration.
Figure 4. Variation of with concernlion. An arbitrary value is given to &. The graph shows the pronounced change of &o. Examples. From eqn. (25) it follows that with only a monomer and a dimer present, and taking t h e n values equal to 1/Pz or less, the dimer molar fraction a2/2 is less than 20% (Fp = 1). For i = 3, n 5 (1/&)1/2 and only trimer as oligomer, the trimer is less than 11%. Provided the n ratios be realizable, any two lines, i and j , intersect, showing a same = Oj' value. At a certain concentration ratio n the species i and j will have the same D-PFAD, when
If i and j are successive, the value of n equals, then, the inverse of the expression (K-FPAD) of the superior species n = 1R;.
Association k g r e e s a t Different Concentrations From eqns. (25) and (23) come
Effect of Concentration on Global Formation Association Products The value of i3;. or fii is a certain measure of the progress of association degrees. From eqn. (19), 15, = &ic'-1 or I& = Riici-' i t can he seen that the influence of concentration, especially with great values of i, is considerable in & or Ci, (see Fig. 4). An equilibrated value of Fj = 1, or fi, = 1is obtained with a concentration of c = l/(@i.i)'li-l, respectively c = 11 (Ri.i)'li-'. Higher values of concentration than this one tend to increase & of fi, rapidly and hence the association. Below it, association remains reduced. Absorbance and Equlllbriurn Constants
The equilibrium constant 0;can he introduced in the absorbance expression (3), taking the value of aili from eqn. (17):
The use of Di and of thermodynamic constants implies the use of activity coefficients, which can be avoided with concentration products like R; = Qz-Q3 . . . Q;, as shown, or with global formation products of association degrees, Ri:
Absorbance Contraction andEquilibrium Constants The difference nA - A' is expressed by
and
where p; = allal'is the monomer fraction quotient, >l.So the new association degrees a;', at a total concentration c', are expressed as a function of the' association degrees ai, a t the total concentration c, and at the same temperature. I t is reasonable to use p , which is equivalent to the monomer concentration ratio, instead of another possible quotient, as the monomer are often the most abundant and determinable forms. Beer's Law of an Associated Species Generally an associated mixed species does not follow exactly Beer's law between concentrations c and c', due to modifications suffered.
This result indicates that Beer's law for the entire solution would he satisfied if n(al)'lFg, = ni(al)'lFpi', for every i. This condition is the same as that found earlier in eqn. (31); for a given n and its p = allalr Beer's law cannot he satisfied for every i. Instead, it can only he verified for one determined value of i, entire and positive. So Beer's law is not possible for the whole solution in media containing monomers and associated monomers, a t least theoretically. Presumably this is the case of most of the molecules which are not very insoluble. However, approximations may be found in current use, particularly in regions of low concentration.
Ai' f nAi
Literature Cited
because a;' Z ai Beer's law will hold for this species when a;' = a; or, From eqn. (30). when
(1) Berthwl, A., J Chlm Phya., 80,407 (1983). (21 Smirl, A.L.,Clark,J. B.,V8nStryland. E. W.,andRwdl,B.R..J. Chem.Phya. 77.631
11982).
(3) Kam, L a n d Rider, R., Biaphys, J., 39.7 (1982). (4) Do Wilton, A,, Haley, L. V., and Koninpstein, J. A,, Con. J Chem.. 60,2189 (1982). ( 5 ) Agresfi, A,, Bscci, M., Casfellueci, E., and Salvi, P. R., Chem. Phya. Loft.,89, 324
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