An Introduction to NMR Titration for Studying Rapid Reversible Complexation Roger S. Macomber University of Cincinnati, Cincinnati, OH 45221-0172

Nuclear magnetic resonance (NMR) spectroscopy is one of the most useful techniques available to chemists for the investieation of dvnamic molecular Drocesses. Most basic - treatments of NMR include at least a qualitative descrip- tion of the effect of "exchanre". that is. rcvcrsiblc dvnamlc - . processes, on the appearance of NMR spectra (I).

Fast and Slow Exchange The topic is often introduced with a discussion of a sim-

ple two-site exchanging system, for example, the methyl groups in dimethylformamide.

I t is stated that under "slow-exchange" conditions the spectrum consists of two signals: one for species A at chem- ical shift 6. and one for species B at 66. By contrast, under "fast-exchange" conditions there is just one signal at the population-averaged chemical shift.

S = n,6, + nb& (fast exchange) (1)

where n represents the mole fraction. In this context, "slow exchange" means that the differ-

ence in chemical shift between A and B, AS = 8, - f i b , ex- pressed in Hz, is much larger than the rate constant for interconversion k = kl + k-, in Hz. Conversely, fast ex- change means thatk >> A6 (1).

By measuring the spectrum under slow-exchange condi- tions (e.g., at sufficiently low temperature), it is usually possible to determine both 8, and tib directly. Furthermore,

the equilibrium constant K can be calculated from the sig- nal integrals (n of the two signals.

As the rate. of exchange is allowed to increase (e.g., by raising the temperature), a rather involved technique known as complete lineshape analysis (I) is required to ex- tract values for kl and k_l. Fortunately, it is somewhat sim- pler to determine the value of K in the fast-exchange re- gion, provided that 6, and 6b are known. Since nb = 1 - n., eq 1 can be written

6 = n,S, + (1 - n,)Sb = n,(A6) + 6b (2)

Thus,

and

nb So-6 K=-=- (fmt exchange) n. S - Sb (5)

Complexation One of the most active areas in modern chemical re-

search involves "molecular recognition", the formation of so-called host-guest (or H-G) complexes (C).

Volume 69 Number 5 May 1992 375

. - K = 0 . 1 / 0 . 5 / 1 / 5 / 1 0 / 5 0 / 1 0 0 AND CHI0 = 0 . 1 a

I YI ^

<

Figure 1. Plots of eq 13 relating S to R = [G]d[H],The values for,Kare (A) 0.10; (B) 0.50; (C) 1.0; (D) 5.0; (E) 10; (F) 50; (G) I00 M- . For each line, [HI,= 0.10 M, 6,= 0 Hz, and a,= 10 Hz.

Cram, Pedersen, and Lehn shared the 1987 Nobel Prize in chemistry for their pioneering work in this area. The re- cent chemical literature is replete with examples in which NMR has been used to study such complexation (2-7). Of particular interest is the strength of the complexation, as measured by equilibrium wnstant K.

where nhis the mole fraction of uncomplexed host and n, is the mole fraction of complexed host.

To use NMR methods to study such complexation phe- nomena, a t least one site (i.e., nucleus) in the uncomplexed host (or guest) molecule must give rise to a signal at chem- ical shift 6h that is significantly different from the same site in the complexed host (or guest) molecule (6,). The magnitude of this difference (A6 = Zh - 8J often gives infor- mation about the structure of the complex, and may be as large as several ppm (7).

DILUTION CURVE V/ 1-18 & I N I T I A L [GI0 = [HI0 = 1 . 0

= r

Figure 3. The effect of dilution on a solution initially with [HI, - [GI, = 1.0 M. Input parameters are K = 10 M', 8h = 0 Hz, 6,= 10 Hz; output from eq 14.

With +is information, we can determine the value of K under slow-exchange conditions by first rewriting eq 7 as

where [HI. and [GI. are the formal (i.e., initial) concentra- tions of host and guest ([C]. = 0). The relative signal inte- grals at 61, and 6, give

I, -- l ,+lh -% (slow exchange)

and

[CI = n,[Hlo (10)

Thus, eq 8 can be rewritten as

K = nJ% (1 - nJ(R - nJ (11)

where R is the ratio [GIJHI,.

Determining Kunder Conditions of Fast Exchange How do we determine the value ofKunder fast-exchange

conditions? We can rewrite eqs 1 4 , substituting h for a, and c forb. Then, due to the relationship

6,-s n -- ' - 6, - a, (12)

K = 0 . 1 / 0 . 5 / 1 . 0 AND LHl0 = 0 . 1

we can use the observed chemical shift 6 (with 6h and 6, from slow-exchange measurements and the known values of [HI, and [GIo) to calculate K from eq 11.

The observed chemical shift under fast-exchange condi- tions is a complicated function of [HI,, [GIo, ah, 6,. and K.

8 = 8. - [$](b - m) (fast exchange) (13)

where

Figure 2. Same as Figure 1, extended to larger values of R.

376 Journal of Chemical Education

The derivation of this expression is given in Appendix 1. Equation 13 describes the complexation-induced shift (CIS) of an NMR signal. The variation of 6 with changing R forms the basis of a general technique known as NMR titration (2-9) for the determjnation of K values.

Table 1. Nonlinear Curving-Fining of Equation 12 to Find Kand 6~

Initial estimate of K = 100 Initial estimate of K = 0.01

True K R ka &tc" best fit K best fit h &lcD best fit K best fit h 0.44 0.84

1.00 ( '" 1.21 2.00 1.56

:: 1.56 j\ 100 10.0 10.0 1.56

3.00 2.18 2.18 2.18

- --

'Generaled from eq 12 using the indicated "true" Kvalue and [HI. = 0.10 M. 6h = 0.00 Hz. & = 10.0 Hz (see Fig. 1).

'output from the pmgram in Table 2, with the following input: [HI. = 0.10,60 = 0.00 Hz, and the five &, VS. Rdata poinfs.

Figure 1 is a plot of eq 13 that shows graphically how 6 varies with R for several values of K.' As one might quali- tatively expect, values of K greater than 5 lead to rapid "saturation" of the host sites at relatively low ratios of [GI. to [HI, and consequent rapid changes in 6. By contrast, for K = 0.10 the observed chemical shift has changed by only 9% of A6 when R = 10 and by only 50% when R = 100 (See Fig. 2).

For convenience, it is often easiest to carry out an NMR- CIS experiment by preparing just one solution of known [HI. and [GI., then diluting it successively and measuring

'BASIC programs for plotting eqs 13 and 14 on a Sweet P Moael 100 Potter are avadab e from tne author.

6 as a function of the dilution. In this case, eq 13 can he rewritten.

where

Note: successively reduced values for [HI. and [GI. must be calculated for each dilution. In this case, the CIS titration curve is plotted with [GI. along the abscissa, as in Figure 3.' It is also possible to plot 6 against n, (see eq 4a in the Appendix), which yields a straight line with y-intercept Sh.

Table 2. Nonlinear Curve-Fitting Program

DIM H(100), G(100), D0(100), ~(100) NPUT "ENTER VALUES FOR DH, K. AND N DH, K, N TT=O:TS=O:CN=I F O R I = l T O N PRINT "FOR I = ";I;

INPUT "ENTER H(I), G(I) AND DO(I) "; H(I), G(I), DO(!) NEXT SE = 0 : BC(N) = H(N) + G(N) + I / (K) TC(N) = BC(N)-SQR(BC(N)"24*H(N) 'G(N) ) DC = 2'HINI' (DO(NI-DHI TTCINI + DH . . . . . . . . PRINT : PRINT : PRINT "CURRENT VALUES OF K AND DC ARE ": K. DC . . F O R I = I T O N e(l) = H(I) + G(I) + 11 (K) IF (B(1)"24'H(I) 'G(I) ) 0 GOT0 145 T(I) = B(I)-SQR(B(I)"24'H(I) 'G(I) ) D(I) = DH + (11 (2'H(I) ) ) 'T(I) ' (DGDH) PRINT H(I), G(I). DO(I). D(I) SE = SE + DO(I)-D(I)

NEXT T = T S

IF SE 0 GOT0 170 TS = + I : GOT0 175 TS = -1

IF TSTT 0 THEN CN = CN+1 K = K'(1 +TS.&CN) IF CN = 8 THEN 200 GOT0 60 INPUT "DO YOU WANT TO BEGIN AGAIN:AN$ IF AN$ = "YTHEN 10

STOP Sample prompted input: 0.0,100,5 0.10.0.05,0.44 0.10.0.10,0.84 0.10.0.15,1.21 0.10.0.20,1.56 0.10,0.30,2.18

Volume 69 Number 5 May 1992 377

Nonlinear Curve Fitting Alas, there is a fly in the ointment. Equations 12,13, and

14 require that both 6h and 6, be known. Although the for- mer is easily determined by direct measurement in the ab- sence of guest, the latter cannot usually be determined di- rectly for two reasons. First, the rate constants for com~lexation and decom~lexation (k, and k-,, ea 6) are often very large, making ii impossible reach the slow-ex- change limit. Sewnd, unless the value ofK is substantially greater than 10, it miy be difficult to approach the 6,li&t of 6 at any readily attainable ratio of [GI, to [HI, (see Figs. 1 and 2). So, we are left with one equation (eq 13 or 14) with two unknown parameters, 6, andK.

But all is not lost. It is possible to use nonlinear curve fitting to iteratively determine the values of K and 6, that best simulate an experimental set of 6 vs. R (or 6 vs. [GI,) data (10). A short program for doing this is given in Table 2. The input includes the experimental values of 6 vs. [GI, and [HI. (n data ooints), alonn with the value of Sh and an estimate of the value ofii. ~okever , as shown below, even when this estimate is orders of magnitude too large or too small, the program still converges on a best fit.2

Taking the data point corresponding to the largest [GI, value, the program uses the estimate of K and eq 14 to calculate a value of 6,. With these values of 6, and K, it calculates a value of 6 for each experimental value of [GI. and [HI. from eq 14. Then it computes the difference

for each point. If the sum of these differences is positive, the value of K is incremented geometrically; if the sum is negative, K 1s decremented geometrically. Then the entire pr6cess is repeated until convergence, which is arbitrarily assumed when the successive change in K reaches < 1%.

Table 1 lists five "experimental" vs. [GI, (at constant Ma) data ooints for each of three different "true" Kvalues. ( ~ h i s e data points were generated from eq 13, using the same parameters as in Figure 1.) These data were input to the curve-fitting program in Table 2, along with the indi- cated K estimates, which were chosen to be at least an order of magnitude larger or smaller than the "true" value. As can be seen, the program converges on the best-fit val- ues ofK and $, which accurately match the "true" values.

It is to be expected that a curve-fitting approach to the determination ofK values will be most accurate when the experimental data points cover a range of R values that is broad enough to allow a significant degree of host satura- tion. In fact. it can be shown (8) that the ootimum wndi- tions for data collection are [HI.. about equalto 0.10/K, with IG1, soannine the ranee O.1OiK to 10K. In such a case. the . ." . resulting titpation cu&e will resemble line C in ~ i g u r e 2.

Limitations It is important to be aware of certain assumptions that

are implicit in the derivation of eqs 1-14. Most im- portantly, it is assumed that 6h and 4 are themselves inde- pendent of concentration and temperature effects. While this can usually be experimentally verified for 6h, it is gen-

'A more sophisticated nonlinear NMR titration cuwe-fining pm- aram (HOSTEST I11 has been develooed bv Craio Wilmx at the Uni- tersity of pinsburgh (8). The authorwishesio thank Professor Wilcox for his helpful communication on the topic.

erally impossible to verify for 6, unless the slow-exchange limit can be attained.

Another potential error is the failure to include activitv coeRcien& in the equilibrium expressions, even thou& concentrations may exceed 1 M. Since nonlinear iterative curve fitting of& involves locating a relatively shallow minimum, effects such as these can lead to significant error in derived K values. Further discussion of these lim- itations are addressed elsewhere (8, 11).

Conclusions The bottom line is this: When an NMR spectrum exhibits

a shift in 6 as a function of the changing concentration of other ~otential interactine soecies. this mav well be due to cornpiexation-induced sG& under fast-exchange condi- tions. Note also that a wncentration-dependent chemical shift itself may be indicative of a shift induced by solvent complexation, that is, solvation.

Acknowledgment is made to the Donors of the Petroleum Research Fund, administered bv the American Chemical Society, for partial support of this work.

Appendix 6 = nhSh + nc&

= (1 -n36h + n,6, = 6, + n,(& - tih) = Sh - nJA6)

From eq 11 in the text we find

n, -- K[HI.

- (1 - nJ(R - n,) =R - (R + l)n, + (nJ2

or

(n& ( l+R+- K & ] ~ + R = ~

The real root of this quadratic equation is given by

n, = b - @ z i T

2

where 1 b=l+R+-

K[HI.

Substituting eq 4a into eq 2a gives eq 13.

Literature Cited 1. Msmrnber, R. S. -NMR speetmsmpy, Esaentkl Theory and P,aetices; Har-rt

Brace Jovanadch; San Diego, 1988. 2. Liu, R.; Sanderaon, P. E. J.; Still, W. C. J. Org. Chpm. 1890,55,6184. 3. Schneider, HJ.; Blatter, T.: Sirnova, S. J. Am Chpm. Soc 1991, 113, 1966 (and

refereneee therein). 4. Gads-TaUado, F; Doswami, S.: Chang. S. K.: Geib. 5. J.; Hamilton, k D. J Am.

chpm. Soc 1880,112, 7393. 5. Tanaka,Y.; Kato,Y.; A0yama.Y. J Am Chpm. Sae. 1630,112,2807. 6. Cowart,M.D.:Sueholeiki,I.;BuLowniL,R.R.;W~mr,C.S.J.Am.Chpm. Sac. lw,

110,6204Adrian, J.C., Jr; Wiimr,C. S . J A m . Chem Sae. 1991,113,678. 7. Siibesrna, R. P; Kentgens, A. P M.; No1te.R. J. M. J. Org. Chem 1991,56.3199. 8. For scornprehensive and time1yrevlew of the topie ofthe NMR titration method as

applied to rnolffularremgnition, see Wilmx. C. S. Iksiw, Synthesis endEvalue t i o n o r a n ~ ~ e a t i o u a ~ v n d i ~ ~ ~ l G ~ U ~ D Y ~ ~ . ~ ~ t h ~ d ~ and~unitationa in theuse of NMR for Mesalving Hosffiuest Interadiona" in Fmnfiers in Svpmmokulor Orgonic Chemistry and Phofoehpmiafry: Schneider, H.J.; Dux. H., Ed.: VCH: Weinheim, 1991.

9.. For two ar?iclea in thia Jaumnl dealing with NMR titration experiments in bidern- i c d systems, see Burt. C.T. J. Chpm. Edue. I m , 59,1056; Wder , E J.;Hariman, I. S.; Kwong, S. T. J. Chpm Ed-. 1917,54,447.

10. See, for example, Sprame, E. D.: Lamabee, C. E., Jr J. C h . Edue. IW,6J,23& 11. D o d , G.; Bernstein, H. J. J Chpm Php. 1987,47,2818.

378 Journal of Chemical Education