Equilibrium Constants from Richard W. Ramettel Corleton College Northfield, Minn. 55057
Spectrophotometric Data practice, a n d programming
For decades chemists working toward go:lls or great variety have shown interest in the determination of equilibrium constants for reactions of the typ: D+X%DX
(1)
I'amiliar examples include weak association between douor aud acceptor molecules to form charge-transfer complexes; coordination of ligands to metal ions; proinn-transfer equilibria, including metal ion "hydrolysis;" ion pair formation; and even dimerization (mhere D = X ) . Of course, many techniques have been applied to the st,udy of these equilibria. Measurements of conduct,ance, voltage, solubility, nmr, distribution, etc., have proved useful and, in certain cases, indispensable. But in terms of general applicability and simplicity, spect~rophotometricmethods stand supreme. In addit,ion, absorbance measurements do not disturb the system under study, permit data to be obtained over a very wide concentration range, and can be of high accuracy. However, while the literature shows an impressive variety of spectrophotometric equilibrium shdies involving widely different media and types of association, only recently has there been recognition t,llat data processing for these diverse systems can be neatly generalized, particularly with the aid of the digit,al computer (1, 8). The desire to avoid iterative cnlculationshas led to intentional compromises in experiment,al design, and inattention to other factors has led t,o poor data of little value. Indeed, we note several publications where authors have "dropped their lumpy I)nt,t,eron the griddle, only to have it slapped and tossed by the greasy spatulas of some of the best fry cooks in the bu~iness."~ There is no need here for an historical review, though Ihc references cited made a useful initial bibliography. I t suffices to say that much of the molecular, nonaqueous work has been patterned after the approachused by Benesi and Hildebrand (S), and we see frequent reference to the "B-H equation." A more rigorous algebraic approach, requiring graphical solution of simult,ancous equations, has been proposed by Rose and -
' Wavk performed while the author was on leave as Visiting Srie~ltistat Argonne National Laboratory under a joint program with i,he Associated Colleges of the Midwest, 1966-67. hIetaphor created by Wayne Carver, Professor of English, Carletan College, for use in 8. short article deploring the sad wcord of college convociltior speeches. His text (The Carlefoaian, Nov. 7, 1959) repays the closest possible scrutiny, and EI.PP ?"pies vill he enclosed with reprints.
Drago (4), who criticize the careless application of the simple B-H equation. Research in ionic equilibria has been more individualized, with tendencies to derive data processing methods specifically tailored to the system under study. We can unify the treatment of spectrophotometric equilibrium data by starting only with eqn. (I), Beer's Law, and simple stoichiometry. The computation approach proposed can be handled manually for simple cases, and lends itself easily to computer programming, which is highly preferable. Further, the equations stimulate awareness of the factors which are important for good experimental design and accurate results. Finally, the overall presentation fills a gap in the chemical education literature. Deflning the System
If we are to make measurements leading to the equilibrium constant for eqn. (I), it is necessary to have one (preferably several) solutions containing the three species at equilibrium. Such solutions can be prepared in a variety of ways, but typically a small concentration of substancex will bemixed with asomewhat higher concentration of substance D. We can think of the concentration of D as a variable parameter which dictates (in conjunction with the equilibrium constant) the extent of conversion of X to the complex, DX. In the derivations which follow we shall used brackets, [ 1, to indicate equilibrium concentrations. The initial conceqtrations used to prepare the mixture will be called CD and CX. Thus, CD = [Dl
CX
=
+ [DXI
[XI + [DX]
When we refer to the equilibrium constant, K, for eqn. (1) we mean K
=
activity of DX aetivit,y of D activity of X
The equilibrium quotient, Q, is identical in form, but uses concentrations instead of activities:
The relationship between K and Q involves the activity coefficients, f;, as follows:
where F is the composite value of the collection of activity coefficients. Thus, the determination of K reVolume 44, Number 1 1 , November 1967
/
647
quires knowledge of three quantities: the equilib~ium concentration of D, the equilibrium ratio of concentrations of DX and X, and the value of F. Each of these deserves discussion. The Equilibrium Cuncmtration of D. Substance D participates in the reaction, and its equilibrium concelltration will differ from the initial value, CD. This change can be ignored when CX is very much sn~aller than CD, or when some sort of buffer system maintains a nearly constant [Dl in spite of the reaction. If no buffer control of D is used, the equilibrium concentration can be calculated. If CD is much larger than C X , no correction is necessary and we say that ID1 = CD. - 1fthe initial solution is a mixture of D and X, the concentration of D is diminished by whatever concentration of complex is formed, so, [Dl = CD - [DX]. If the initial solution is a mixture of D and DX, the concentration of D is raised by whatever amount of DX dissociates in reaching equilibrium. In this case, [Dl = CD [XI = CD CX - [DX]. Before these corrections can be made, it is necessary to know [DXJ,and we will see that this followsfrom the optical absorbance measurements, but in units of n i n larity. If CD is also expressed as molarity there is no problem. But it is often desired, for nouaqueous niolecular systems, to use mole fraction units for [Dl. II corrections are to be made, the mole fraction conccntrations must be converted to molarities (requiring the density of the solution), corrected, and then changed back for use in the expression for K. Therefore, experiments using mole fraction units for D usually are dcsigned so that CD is much greater than C X , eliminating the need for correction. The Equilibrium Ratio. The concentration ratio. [DX]/ [XI, is a simple function of the optical absorption data as shown below. Because the absorbance values are directly related (Beer's Law) to molar concentrations, we will use molarities in the derivations. However, for the usual low concentrations of X and DX the concentration ratio will be independent of the concentration scale, because the proportionality constalits relating mole fraction, molality, and molarity are the same for both specie8. The Value of F. The activity coefficient term, I.', can he handled in various ways. First, if the work is in a nonionic systen~,and if the concentrations are not, high, we often may assume that the activity coefficients of X, D, and DX are unity, so that F = 1. But if it is necessary to use high concentrations of D in order to form a sufficient quantity of the complex, the resulting change in the medium (sometimes extending to the limit of using pure D as the solvent) makes this assumption unrealistic. Then we are up against it, hecausc there is usually no convenient way to assign values to the molecular activity coefficients. Even if the activity coefficients of X and DX are equal, and therefore cancel, we are still faced with the problem off^. Even for an ideal system (obeying Henry's Law) only the mole fraction activity coefficient is independent of concentration. Therefore, one prefers to use mole fraction units for [Dl. For ionic systems, the situation is quite different, because of the great effect of interionic forces and the dependence off, on the charge carried by thc spccies.
+
+
648 / Journol of Chemical Education
One common approach is to carry out all measurenients on solutions of constant ionic strength, on the assumption that this establishes constant values of the activity coefficients,even though they may he far removed from unity. Then F is incorporated with K, and the calculations establish Q-values. The second approach is to work a t ionic strengths low enough so that it is reasonable to estimate the individual activity coefficient values by an extended form of the Debye-Hiickel equat,ion. A useful semi-empirical form is:
where Z is the charge on the ion, A is a constant depending on the temperature (0.51 for water solutions at 25"C), fi is ionic strength, and wland w2are adjustable parameters. If there is no empirical reason for assigning particular values, then it is common practice (6) to let wl= 1and wz= 0.1. If a similar form of equation is used for the collection of activity coefficients which we have called F, then
(4
Again, it is useful to let W , = 1 and Wz = 0.1 in the :~bsenceof better empirical information. The Basic Algebra
Suppose the absorption spectra of the three species : r e as shown in Figure 1, where molar absorptivities are plotted versus wavelength. At different wavelengths, the absorptivities bear quite different relationships to each other. Typically, we might decide to make measurements at X2,where D does not absorb but X and DX have finite molar absorptivities, EX and E D X , respectively. When the equilibrium mixture is placed in an absorption cell having a path of B em, the absorh:tnce will be A
A1
=
EX . B . [XI
+ EDX . B . [DX]
A3
A2
WAVELENGTH Figure 1.
Hypothetical absorption spectra for the system D+X*DX.
(5)
where the concentrations are, of course, the equilibrium values. We may also define an apparent absorptivity, E, by t he equation The value of E will necessarily lie between t,he values of E X and E D X , as indicated in Figure 1, where t,ha line for E has arbitrarily been drawn just midway bctween the other curves. By equating eqns. (5) and (6) and letting B cancel, we obtain E .CX = E X . [XI + E D X . [DXI (7) By letting [XI = C X - [ D X ] ,eqn. (7) can hc rrarranged to give
and similarly, letting [DX] = CX yields
-
1x1, rqn.
(7)
The desired concentration ratio now follows direct,ly from dividing eqn. (8) by eqn. (9) : [DX] - E - EX [XI EDX - E Referring hack to eqn. (2), and subst,it.ut.ingfrom eqn. (lo), we can discuss the equilibrium qnot,ient,, Q
=
1
[Dl
.
E-EX EDX - E
or the equilibrium constant,
K = - 1. E - E X , p [Dl EDX - E As for the concentration of D, note that cqn. ( 8 ) gives, in terms of measurable data, the expression for [DX I needed for correction of the D concentrations. It ii. interesting t o note that eqn. (8) shows that. t,hc fraction converted to the complex is a simple funct,ion: [DX1 = fraction converted =
CX
E-EX EDX - E S
The Four Experimental Situations
Suppose one has a number of equilibrium rnixturrs, each with a different concentration of D, and each with some known (not necessarily all the same) value of CA-. Absorbance measurements at Xz have yielded (eqn. (6)) the corresponding values of apparent absorptivity. E . The activity coefficient term, F , is being handled by the procedure listed earlier. According to eqn. (11) or (12), all that is left is the determination of thc absorptivity values, E X and E D X . We can rlistinguish four possible situations: Case I.
Bath EX and EDX Can Be Directly Determined
This is the well-behaved system, exemplified by thi: acid-base equilibrium of the indicat,or hromcrmol green (fi) : colorless
blue
yellow
I n dilute sodium hydroxide solution this indica1,or is fully dissociated to the blue form, and an absorbance measurement on a known concentration readily gives
1.h~ v:aluc of E X . When bromcrcsol green is placed in n di1ut.e hydrochloric acid solut,ion, it in quantitatively prot,onat,ed to the yellow form, and another direct absorbance measurement gives t.he value of E D X . At int,ermediate concentrations of hydronium ion, convcnient.ly controlled by acet.at,ebulrers of const,ant ionic strengt,h, the green color is due tr) n mixture of the blue :md yellow forms, and t,he app:rrcnf, :~hsorpt,ivit,y,E, can vary over a range of values. I n t,his desirable situation eqn. ( I 1) can he used for t,he st,raight,forward calculat,ion of a ()-value for eaeh of t,hc green solutions measured. A nice example in this cat,{:gory is (.lie st.udy of t,riiodidc ion format,ion,
rcpostod lry Daniele (7). His spcedlr~pl~ol~~~lnetric mc:~snrements were carried nnt. at, t.wo wavelengths (t,he absorption maxima for I:,- at 29s and 350 nm), using a wide variation of very low iodine and iodide conccnt~rat.ions,with closely agrecing results for about, SO mca,mrements at each of fivr t,rmperatures. In scpawt,e experiments he dct,ermined t,he rnnlar absorpt,ivit,y, E X , of iodine t,o be IS 1 mole-'em-' (at 350 nm) while that of the complex, ICDX, was 26,400 1 mole-' cm-'. The data at, 25°C for t,his wavelengt,h yieldcd apparent absorpt,ivit,y v:dncs, I$', ranging from 2.500 ~ , I I21,700. -4few of Daniele's results are shown in 1 : l c 1 Tlir averago rcsr~ltfor :3O such meawreFormation Constant for 13- -1nilinl coocentrationsII2 oh. A
Table I.
O.000146.~ 0.00001996 O.001367 0.00003500 O.OOR509 0.00003467
0.050 04.X 0.735
at 25'C -.
-
(Daniele) -
K (eqn. 12)
2505 1308.5 21774
720 727 726
nicnt,s at 350 nm is K = 723, wit,h an werage deviation of only 0.5v0. At 288 nm (EX = 95 I mole-' cm-' M I X = 39,900 1 mole-' em-') t,he average for 28 measnrcments was K = 729 wit,h the same average ilcviat,ion. It should be rrot,t:d t,liat we assume the :ict,ivit,ycoefficient term t.1) he nnity for t h i ~case, bee:u~sc at t,he low ionic st,rengt,hs wed t,he activity cncfFicin~tof molecular I2 shor~ld hi: nnity, and the :~ct,ivit.y coefficients of I and 1%should be nearly equal. Ot,lirr Case I examples in t,ho lit,erat,nre are dinitrophenol (a),substituted anili~~es (Q), sulist,it,ut,edphenols ( 1 0 , l I ) , : r i d hydrogen chrom:tt,eion (12). Case 11.
EDX Can Be Measured Directly; EX is Unavailable
Consider a very weak mnnoprotic acid, HA, with a, diss~~ciati~m constant of abont 10-'? I t is easy to I)rcp:lre a solution with a known concentration of H A . As long as the pH is below 9-10, there will be negligible ~lissociat,ionto form A-. The :lhsorptivity of HA can l.lrus be determined. But even at, pH = 14 t,he dissocial.ion would be only about 90%, :~ndso the absorptivity of A- cannot be reliably det,c:rmincd wil.l~ontdrast,ically eh:rnging t,hemedium. At. first,one might want. 1.ogive up, because a value of ICx is needed for eqn. (12). Rut fortunately, that wlu:i,t.ion is easily rearranged 1r1t.he following form: Volume 44, Number 1 1 , November 1967
/
649
E
=
EX
+ FK-
- [Dl - (EDX - E )
(13)
This has the useful property that the unknown quantity, EX, is fully separated from the experimental quantities. We infer that one should be able to make a series of determinations of E at varying concentrations of D, and that a plot of E versus the quantity [Dl. (EDX - E ) / F should be linear, with an intercept equal to the otherwise unavailable EX, and a slope equal to the desired K. Of course, if the values of F are unknown, they cannot be used in plotting. If E is plotted versus [Dl. (EDX - E ) , the plot will be curved or will have the wrong slope if the values of F vary, and it will be straight if F is somehow a constant. In the latter case, the slope will be K/F, or Q. The application of eqn. (13) is appropriate in the study of the hydrolysis of ferric ion, written here in reverse to correspond in form to eqn. (1) :
Measurements in strongly acidic solutions (15) show that a t 355 nm the molar absorptivity of ferric ion (EDX) is small, about 1.0 1 mole-' cm-'. At this wavelength, the absorptivity of FeOH2+ is much higher, but E X cannot be directly determined because when the acidity is lowered there is appreciable formation of Fe(OH),+ and even precipitation of Fe(OH)3. Siddall and Vosburgh (14) made a series of absorbance measurements on 1 X lO-*M iron solutions in the acidity range of [H+] = 0.001-0.015, and it appears that in this range the only significant iron(II1) species are FeOH2+ and FeS+. The ionic strength was maintained constant at 0.0166, so the activity coefficient factor, F, can be presumed constant. Figure 2 shows a plot of the data, E versus [H+].(1.0 - E ) according to eqn. (13). The linearity supports the assumptions, and from the slope the value of Q is 227 1 mole-'. I n terms of the ionization quotient for the Bronsted acid, Fe(aq)3+, Q. = 1/& = 4.4 X mole I-'. The intercept of the plot yields 499.5 for the molar absorptivity of FeOH2+. Siddall and Vosburgh used a different approach for their computations, but the results are in good agreement.
650
/
Journal of Chemical Education
Equations nearly identical to eqn. (13) have hew used for the BiCls2- complex (16), for studies of ionization of certain phenol derivatives (l7), and for ceriun(IV) hydrolysis (18). Chemical systems meeting tho criteria for Case I1 are not particularly common, but eqn. (13) can probably find additional applicatiou in studies of the highest coordination step of stable metal complexes. As for application to very weak acids (19), as mentioned above, one might prefer the alternative view of the system as OHD
+ IIA % AX DX
where the ur~linown absorptivity is EDX (see Casc 111), because it is experimentally more accurate to cmtrol moderate values of [OH-] than to establish known, very small values of [H+]. Further, the above reaction will not be as sensitive to ionic strength (thc activity coefficients tend to cancel), and the value of Q = K = l/Kb, where Kb is the Bronsted base constant for A-. If K . for the very weak acid is desired, it can he calculated by the use of K, = K,/K,. Case 111.
EX Con Be Directly Determined, but EDX Cannot
In many cases the complex is so weak that eve11 high concentrations of D are insufficient to drive reaction (1) quantitatively to the right. In other cases, the complex is indeed stable, but the use of higher concentrations of D leads to the formation of additional (successive) complexes, so that the desired species, DX, cannot be formed by itself in known concentration. When it is thus impossible to make a direct determination of EDX, eqn. (12) cannot be used for straightforward calculation of individual K values. However, if both sides of eqn. (12) are divided by K and multiplied by (EDX - E) it follows that F (E - E X ) E=EDX+K[Dl
Again the unknown quantity is separated from t.hc experimental variables. A plot of E versus the quantity F . (E - EX)/ [Dl should give a straight line with h'L1.Y as intercept, and with K = - l/slope. As with Case 11, if no account is taken of the values of F, only the quantity (E - EX)/[D] is plotted. If F is a const,ant,, the line will be straight, but the quantity (-l/slope) is equal to Q, not K. If F is not constant, the plot either will be curved or will yield a misleading value of the slope. A familiar example of the Case I11 situatiou is t,he weak charge-transfer complex between benzene and iodine in carbon tetrachloride solutions. When elemental iodine is dissolved in pure CC14the purple color indicates that the molecules are free It>as in the vapor of iodine. However, as benzene is added to the snlut~i~n, the color changes gradually to reddish-brown. But even in pure benzene the complexationis not complete; about 40% still remains "purple." It is something of a tradition to use the data of Benesi and Hildebrand (3) to illustrate new methods of photometric data processing. They made measurements at 297 nm, where the complex absorbs strongly, and where the free, uncomplexed iodine has an absorptivity, flX, of 42 1 mole-' em-'. Figure 3 shows t,hc plot of the data according to eqn. (14), using nwlc fract,ion units for the benzene concentration. Any
I 15858
I
I
I
I
I
I
I
I
I
I
'
- 100 - 90 * -80 % E
C.He
+I, s complex ICCI, solution1
-70
Ip in pure benzene-
-60
E
- 5 0
8
e c
.-0
g
-40
$
-30
:
U D
- 2 0
-
XtsHs = 0 . 0 4 3 -
g
n
10 0
42
Figure 3.
Care Ill plot for charge-transfer s m p l e r .
if neither EX nor EDX can be determined? At first,, this might be dismissed as a hopeless system for which the equilibrium constant cannot be determined by spectrophotometric means. Admittedly, it is preferable (for best accuracy) to know either or both of the two absorptivities, but even without them the determination of K may be possible. Hope lies in the expectation that i f we did know the value of E X then the plot discussed under Case III would be linear. Now, suppose that for EX we simply guess some value which is not quite correct. Because eqn. (14) depends on the quantity ( E - E X ) , any error in EX becomes more serious as the values of [Dl become lower and E approaches E X . This will lead to curvature in the plot. If the error in EX is positive, the curvature will be in one direction, while negative error in EX will cause curvature in the other direction. Only that value of EX which is correct will yield a plot which is straight. So "all" we (that is, the computer) must do is try values of E X over a range which includes the correct value, and "zero in" on the one which gives a linear plot. You may object that any set of real data suffers from experimental errors which might contribute a misleading curvature even when the correct value of EX is used. But if the errors are random, and if the data cover a fair range of D concentration with a number of points, there is promise in this approach. Anyway, no one is proposing that this method is preferred over direct and accurate knowledge of E X or EDX. But it may be all we have. A Case IV approach is required for the data of Offner and Skoog (18) on the hydrolysis of cerium(1V). i.e., the reverse of the reaction.
deviation from linearity is approximately within experimental error, but it comes as a bit of a surprise to see such a straight line all the way to the solution which is merely iodine dissolved in pure benzene. Apparently t,he activity coefficient term, F, remains nearly constant as t,he solvent is changed from pure CCll to pure CsHs.s The extrapolated intercept gives 15,857 1 mole-' cin-' for EDX, and the slope yields 1.60 for Q. These values differ somewhat from those reported by Benesi and Hildebrand, because they assumed the absorptivity of free iodine, E X , to be zero. Equations nearly identical to eqn. (14) have been applied to the formation of the CeSO,+ complex (20), 1.0 the anion-zwitter ion step of methyl red (16), and to This is chemically similar to the iron(II1) hydrolysis suhst.ituted salicylaldehyde complexes with metal ions discussed under Case 11, but Ce4+ is a stronger acid (21). Many other studies which have used other than Fe3+, and even in 1M perchloric acid the cerium met.llods of data processing can be nicely interpreted t.l~roughapplication of eqn. (14). These include the mnny charge-transfer complex systems studied by Drago, et al. (ZZ), the formation of NdNOaZ+(M), the association of phenol with triethylamine (Z), and the ion pair association of C U ~ + - S O (29) ~ ~ -and C ~ ( e n ) ~ ~ + H+ CeOH3+ t Ce4+ S2012--(30). A point worth making is that the plots for Case I1 and Case 111 determinations are bounded at both ends. This is summarized in Table 2. -
+
Table 2.
Limiting Values of Plot Variables
Chemical View Case 11
No formation of D X
EX EDX
No farmhtion of DX
EX ED EDX
complete oonversian
Cnso 111
Core IV.
Ordinate
eomolete conversion
+
E
Abseima 0
(EDX - EX)/Q (EDX - E X - 8DI.Q 0
1200
-
Neither EX nor EDX Con Be Determined
We have seen how eqn. (12) can be used directly, and how it can be pushed around to permit calculations \\-hellone of the absorptivities is not known. But what 1000 But similar studiea on other systems yield curved plots, implying changes in F or formation of other complexes. Then it is probably better to estimate the limiting slope (as the mole fraet i w of D approaches zero) (88).
2000
3000
E-EX -
4000
5000
[H+I
Figure 4.
Care IV onolyrix of coriumllV) hydrolysis.
Volume 44, Number 1 1 , November 1967
/
651
is only 80% converted to t,he Ce4+ion. Therefore, it is not possible to make a direct determination of EDx. If one attempts, using lower acidities, to prepare a solution of CeOHa+the difficulty is in the formation of Cd(OH)22+and other more complex species, so it is also impossible to make a direct determination of EX. However, in the range of 0.1-1.0 M acid it appears t,h:ll only CeOH3+ and Cd4+ are present. E values wero obtained by absorbance measurements at 305 urn. Figure 4 shows three curves, plotted according to eqn. (14). When E X is assigned the value 600, there is obvious upward curvature a t the low-E end. When E X is taken as 900, the curvature is clearly in t,he opposite direction. Somewhere in between there must, be a value of E X which would give a graph with "no" curvature. The computer program does not really look for curvature as such. But with each trial value of E X the intercept of the plot (EDX) is calculated using least squares, and then a Q value is calculated for each point. The "best" value of E X i s defined as that which gives the lowest aveyage deviation of the Q values. This is equivalent to looking for the "straightest" line which can be constructed wit,h the available data, deviatiolis and all. Figure 5 shows how the average deviation varies with assumed vdues of EX. For the cerium
is uegligibly small. The equations developed to this point have been based on this idea. However, we should provide for the circumstance that D will have an appreciable absorptivity, as at XI. Because the concentration of D will often he relatively large, its contribution to the total absorbance can be important. Then, the absorbance is the sum of three parts: A
= =
+
A*". ta D ED .B. [Dl
Aduo
to
X
+
Adua ta DX
+ IJX - B. [XI + EDX .B. [DXI
A derivation analogous to that given earlier leads to l.he more comprehensive expression: & , = -1 .
[Dl
E - EX - EDCDICX - K EDX - E - ED (1 - CDICX) - F
(15)
This replaces eqns. (11) and (12), and rearrangements or eqn. (15) can be used as replacements for eqns. (13) and (14) for Case I1 arid Case I11 situations. In the derivation of eqn. (15) we would find the expression for [DX],which is needed for the correction of t,he concentrations of D: [UX]
=
- EX - ED -CD/CX cx.E EDX - EX - ED
Of course, t,his becomes identical with eqn. (8) when ED = 0. Other Equilibria Coexist
0
600
700
800
906
Assumed volue of EX kCIOH~+) Figure 5. Effect of choice of EX on overage deviation for cerium(lV1 ~atul~tions.
case, the results are: ICX = 741, EDX = 1739, a d Q = 4.8, in close agreement with the results calculated by Offner and Skoog, who deduced EDX by solving simultaneous equations, and then used a Case 11 plot to find Q. Another example of a Case IV system is the "red-red" equilibrium of methyl red, which w : ~ analyzed by detcrmiuing tho "best" value of t,hc c;~t,io~l absorptivity (16). Some Common Complications Substance D Absorbs at the Wavelength Used
I t is ideal to work a t wavelength such as X:, (I'ig. 1) where D is transparent and either E X o r E D S is zero while the other one passes through a maximum. RIore frequently otle must use a wavelength like A,, when! both X aud DX have linit,eabsorptivities, hut whcre fill 652
/
Journal of Chernicol Education
In the cases presented above it was possible to use data obtained under chemical conditions such that the equilibrium system was described fairly by the reaction of eqn. (1). However, it is common to find appreciable effects due to associated equilibria, such as dimerization of X (and/or DX), or the "overlapping" of successive complexes. Dimerization can he minimized by using concentrations as low as permissible without sacrificing accuracy in absorbance, or by applying separate corrections if the dimerization constant is known. For calculations made with the B-H equation, the theoretical errors caused by unsuspected dimerizatiou have been fully discussed (94). The prescnce of successive equilibria may he difficult to detect (%), aud work at multiple wavelengths may he necessary. The effects of stepwise complexation are serious when successive equilibrium constants do not diier greatly. It may he possible to use successive approximations to analyze the system. For example, suppose where Q, aud Qz are only a factor of ten apart. All solutions contain appreciable concentrations of all three species, except at extreme concentrations of D, where E X and EDeX can be measured. Using the lower concentrations of D, with a Case I11 plot, one obtains an approximate value of Q,. Data for higher D concentrations are then approximately corrected for the presence of X , before being used in a Case I1 plot which gives an approximate value for Q,. This, in turn, is used in more accurate corrections of the low-D data, and now a better value can be obtained for Q,. This back-and-forth process is continued until there are no further changes in the calculated values of Q1 and Q2. The author has used this approach successfully for the two-step dissociation of isophthalic acid as reported by Thtlmer and Voigt (26).
However, for a comprehensive and definitive treatment of the spectrophotometric analysis of successive complexation, including mixed ligand species, the reader should consult the paper by Newman and Hume (16). Calculation by Computer
A researcher armed only with a desk calculator would look with trepidation at the equations developed above. Given a series of absorbance, composition, and ionic strength values, he faces a discouraging program of successive approximations, least squares fits, minor corrections, etc. Especially for the situation where the concentration of D is changed by the equilibrium reaction, the labor of calculations is considerable. It is not surprising, therefore, that many workers have resorted to compromises in experimental design, merely to simplify the calculations. For example, measurements have been made a t other than the optimum wavelengths, merely to avoid the complications due to absorption by more than one species. Often, the concentrations of D have been maintained at undesirably high levels, merely to minimize the errors due to changing [Dl. Or, when [Dl was known to be affected by reaction, approximate rather than exact equations have been used to process the data. But with the digital computer readily available for routine use as a laboratory tool, the only concern is the effort of writing and debugging a program which can carry out the desired logic and sequence of operations. This task has been completed by the present author, for use with the approach advocated above, and by others who favor different computation strategies (1, 8). The FORTRAN program used here handles all calculations described in this paper, including ionic activity coefficient calculations. It has been used successfully on a large number of varied cases in the
literature, and in one version can be used to drivc 11 Calcomp plotter to make graphs of the type describcd for Case I1 and Case I11 situations. The basic strncture of the program (called PHODEC, for photometric determination of equilibrium constants) is shown in Figure 6. The input data are the initial concentratious of D and X (or DX), the corresponding absorbanccs and (if desired) ionic strengths. Absorptivities arc entered with code digits (0 or 1) to show whether they are fixed or merely first approximations, thus defining the "Case." A FORTRAN listing is available on request. Table 3.
--
0.013 -0.015 0.007 -0,022 0.004 0.031 -0.020
COMPUTE VARIABLES FOR LINEAR PLOT
COMPUTE VARIABLES FOR LINEAR PLOT
Y -EX+Q.X
Y =EDX-Wa
*
Use intercept for W A LEAST SQUARES unknown abrorplivily. FIT0FYvr.X repeat least w a r e s l i l . ~ , h o u l d becorrrcted?
NOW THAT ABSORPTlVlTlES .HAVE BEEN CALCULATED. COMPUTE INDIVIDUALVALUES OF Q, and AvE.i"
I S THIS A "CASE i v " 7
i
no
PUNCH FINAL RESULTS
1
MAKE PLOT F i g u r e 6.
F l o w sheet for b a s i c PHODEC c o m p u t e r p r o g r a m .
CHANGE ElNC TO -ElNCIZ
yes
Line 1. The FeaC ahsorptivity, ED, was taken as aem
s t 450 nm. The SCNshsorptivity, E X , was tsken as zero, while the absorptivity of the complex was roughly guessed to he 2000 1 mole-' em-'. Line 8. After computation, EX is still fixed as zem (of course), but iteration has shown the vdue of EDX to he 3552.5. Column 1. The initial values of [Fe8+]were varied from 0.001 to 0.008 M. Column 8. The initid value ol [SCN-] was alweys 0.0003 M. I CHANGE THE Column 4 . The measured ah--+ VALUE M EX sorbencea, using a 1.305 om cell. Column 4. Once EDX had been established, s. QLOWER value was calculated for THAN BEFORE? each experimental paint. The average is 134.6. Column 6. Individual relst,ivc deviations of Q-values from their average. The merage deviation is .OlG,
EX is guesed
\
\/
0.000965 O.OOl!J:iX 0.002914 0.002916 0.004880 0.007X43 0.007847
A typical PHODEC output is shown in Table 3. This is part of a study of the Fe3+-SCN- associatiou reported by Frank and Oswalt (31). This system was studied at constant, high ionic strength, using perchloric acid to repress the hydrolysis of ferric ion. The concentration of Fea+ (substance D) was not very high compared to that of SCN- (substance X), so it is necessary to account for the loss of Fe3+due to complex formation. Frank and Oswalt used a truncated series expansion to process the data, and a variation of their equation was proposed for a student experiment (27). PHODEC is su~eriorto both of these approaches. The output shows:
C X V
I
0.116 0.205 0.283 0.278 0.398 0.521 0.509
Averaee Values:
EX is known EDX is guerred
EX is guessed EDX is known
--
Initial Values ED = 0; EX = 0, @xed); EDX = 2000, (guessed). ED = 0: EX = 0: EDX = 3552.5 (from o l d ) Final Values ' Dev. DXICX ' [D)
READ DATA linitial concentiationr, abiarbancer. abrorptivitier, ionic strengths, code numberrl
X i s known DX is known
PHODEC Output (notation modified) for Few-SCN- Association
I S AVE. OW. ATMINIMUM?
or 1.6%.
Volume 44, Number 7 I , November 1967
/
653
used in the equilibrium solution. This is s. differentialabsorbance measurement and is more accurate than taking two separate measurements, one for E and one for EX. Eqn. (14) can then be put into a useful form by subtracting E X from both sides: ( E - IFX) = (EDX - EX) - F - . (E - E X ) K ID1. . and the directly-determined difference can thus be used on the left. The only change in interpretation is that the intercept will no longer be merely EDX, hut rather (EDX - EX). Similar considerations can be applied to eqn. (13), where EDX e m be suhtracted from each side, and a solution of the pure complex is used as the reference solution. The measured absorptivities, E, should cover a range as broad as possible. This is another way of saying that the composition of the equilibrium solutions should be varied to permit broad variation in the percentage of conversion of X to DX. In still otherwords, ID] should vary as far as possible on either side of the value of the equilibrium constant. This will require the minimum extrapolation in the plots to obtain the unknown absbsorptivity, which therefore will be estimatcd with greater accuracy. Repeat the whole experiment sever-1 times before publishing. E -
[~e'+l
Figure 7.
Case Ill plot for Fetf-SCN-
ossociotion.
Column 6 . The fraction of SCN- which was converted to the complex, ranging from 11 to 52y0. Column 7. Equilibrium values of [Feat], appreciably smaller than the initial values shown in Column 1.
For aid in making a graph (in this case of the Case I11 type, as shown in Figure 7) the values of Y and X are also tabulated. Experimental Design
I n the development and testing of the computer program, PHODEC, it was useful to examine a large number of published spectrophotometric studies. Some of the best, have been used as examples in this paper, and other good studies are referenced. But deliberately omitted are some "horrible examples" of spectrophotometric equilibrium studies. A few even gave negative equilibrium constant values when processed, while in others the bad scattering of points made them virtually worthless. Because a good data-processing method can do nothing to improve the raw data, it seems worthwhile to conclude with a few comments on getting the best, most useful data a given system has to offer. Choose a wavelength where the absorptivity of X or DX goes through s. maximum, or is on a. "shoulder." Then, small errairs in resetting the wavelength will he less important.. Choose a wrtvelennth where the absorotivitv of X is as laree
ity values. Measure absorbance values at more than one wilvelength. This helps minimize the chance that ~rnsuspectedequilibria will go undetected. Use temperature control in the cell compartment, or at least keep the solutions in a. controlled temperature bath nntil just before measurement. Equilibrium constants and ahsorplivit,ies often are temperaturedependent. Don't feel t,hat CX hhas to be the same for all solutions. The important thing is to obtain accurate values for E X , E, and EDX for use io the equations, and the concentration of absorbing species should be in the range where the absorbance is about 0.3-0.6. This leads to minimum photometric error. Consider eqn. (14), and notice that the variable on the right involves s. difference in absorptivities, ( E - EX). This difference can be determined directly if the reference solution contains substance X (uncomplexed) in a concentration equal to C X , as
654 / Journal of Chemical Educafion
Literalure Cited ( 1 ) CONROW, K., JOHNSON, G. D., AND BOWEN, R. E., J . Am. Chem. Sac., 86, 1025 (1964). W. E., HIRSCH,W., AND CHEN,E., J . Phy8. (2) WENTWORTH, Chem.. 71. 218 (1967). . . (3) B E N E S I ,A., ~ AND HILDEBRAND, J. H., J. Am. Chem. Soc., 71, 2703 (1949). R. S., J. Am. Chem. Soe.. 81,6138 (4) ROSE,N. J., AND DRAGO, (1959); DRAW,R. S., AND ROSE,N. J., J . Am. Chem. SOC.,81, 6141 (1959). (5) KING,E. J., "Acid-Base Equilibria," The MaeMillan Co., New York, 1965, p. 20. (6) RAMETTE,R. W., J. CHEM.EDUC.,40,252 (1963). G., D m .ehim. ital., 90, 1068 (1960). (7) DANIELE, (8) See reference (6j, p. 93. R. A., J . Res. N. B. S., 68A, 159 (1964). (9) ROBINEON, (10) ALLEN,G. F., ROBINSON, R. A,, AND BOWER,V. E., J . Phys. Chem., 66, 171 (1962). (11) ROBINSON, R. A., ANDPEIPERL, A,, J . Phys. Chem.. 67,1723, 2860 (1963). Y., Acla Chem. Scand., 16,719 (1962). (12) SASAKI, W. C., J . Am. Chem. (13) MI-URN, R. M., AND VOSBURGH, Soe.. 77., 13.52 lg.5.5). - f,--, (14) SIDDALL. T. H., 111, AND VOSBURGH, W. C., J. Am. Chem. Sac., 73, 4270 (1951). L. N., AND HUME,D. N., J. Am. Chem. Soc., 79, (15) NEWMAN, 4571 (1957). (16) RMETTE, R. W., DRATZ,E. A,, AND KELLY,P. W., J. Phys. Chem., 66, 527 (1962). L. B., P o s m s , C., JR., AND CRAIG,C. A,, (17) MAGNUSX~N, J . Am. Chem. Soe., 85, 1711 (1963). (18) OFFNER,H. G., AND SK000, D. A,, A d . Chem., 38, 1520 (IQfifiI - - --,. (19) STEARNS, R. S., AND WAELAND, G. W., J. Am. Chem. Soe., 69, 2025 (1947). T. W., AND ARCAND, G. M., J. Am. Chem. Soe., (20) NEWTON, 75, 2449 (1953). L. B.. A N D CRAIG.C. A.. (211 . . POST^. c... JR:. , MAONUSSON. Znorg. k'hem., 5, 1154 (1966)' (22) DRAGO, et al., J . Am. Chem. Soe., 83,3572 (1961); 84,2320 (1962); 85, 505 (1963); 86, 1694 (1964); 88, 3921 (1966). N. A., h n r ~KISER,R. W., J . P h w Chem., 70,213 (23) COWARD, (1966). D. W., AND BRUICE,T. C., J . Phys. Chem., 70, (24) TANNER, 3816 (1966). G. D., AND BOWEN,R. E., J . Am. Chem. Soe., 87, (25) JOHNSON, 165.5 (1965). B. J., AND VOIGT,A. F., J. Phys. Chem., 56, 225 (26) THAMER, (1952). R. W., J. CHEM.EDUC.,40, 71 (1963). (27) RAMETTE, (28) M a o ~ u s s o ~ L.. B., AND KIRCHHOFF. W. H. (hberpub-lished). (29) MATHESON, R. A., J . Phys. Chem., 69, 1537 (1965). R. A., J. Phys. Chem., 71, 1302 (1967). (30) MATHESON, (31) FRANK,H. S. A N D OSWALT, R. L., J. Am. Chem. Soe., 69, 1321 (1947). \
