2674
R. L. VOLDAND ADOLFO CORREA
Nuclear Magnetic Resonance Measurements of Proton Exchange in Aqueous Thiourea
by R. L. Vold and Adolfo Correa Department of Chemistry, University of California, La Jolta, California (Received December 17, 1969)
The semiclassical line shape expression of Hahn, Maxwell, and McConnell for chemical exchange between two unequally populated sites is extended to the case where spin l / Z nuclei in one site are coupled to a relaxing nucleus of spin I . The line shape expression is written in closed algebraic form for spin f = 1 and I = a/2. Experimental results are reported for acid-catalyzed protolysis of aqueous thiourea. Possible exchange reactions are discussed. Water line widths are used to derive rate parameters. The rate law for acid-catalyzed protolysis of aqueous thiourea is found to be R = (1.0 =!= 0.2) X 106 [thiourea] [Ha+O]0~98*0~02 h i ' sec-l. Protolysis of thiourea N-H protons is concluded to be catalyzed about 60 times more efficiently than N-methyl acetamide N-H protons.
Introduction
Theory Consider the case of two sites, A and B, distinguished by their chemical shifts w, and cob, with equilibrium populations of spin nuclei pa and pb, respectively. Assume that nuclei in site B are spin coupled (a radians/sec) to a nucleus of spin I , resulting in (21 1) subsites of €3, with chemical shifts W b -k la, W b (I l)a,.. . w b - la. Assume also that spin nuclei transfer from A to B at rate kab and from B to A at rate
+
The Journal of Physical Chemistry, Vol. 74, No. 13, 1970
Then kab and detailed balancing
kba
kba.
Nmr line shape methods have been widely used to measure proton transfer reaction rates in aqueous solutions of amines,I amides,2 and amino acidsS3 In most of these studies, less than the maximum possible information was obtained from both the water resonance and the N-H resonance, due to the complication of line broadening by scalar coupling to relaxing nitrogen nuclei. I n this paper the semiclassical line shape expression of Hahn, Maxwell, and McConnell4 for chemical exchange between two unequally populated sites is extended to the case where spin nuclei in one site are coupled to a relaxing nucleus of spin I. The line shape expression is written in closed algebraic form for spin I = 1and I = 3/z. Experimental results are reported for aqueous thiourea. A large number of proton transfer reactions are a priori possible for this molecule. Also, it is a rather effective protein denaturing agent, by mechanisms which may involve alterations of water structure by thiourea.5 The possibility exists that effects of water structure may be revealed in detailed studies of proton transfer kinetics of urea and substituted ureas. Thiourea is particularly suitable for initial investigations because the thiourea-water chemical shift is large, which allows reaction rates to be measured over a wide range of concentrations.
+
are related by the principle of
pahb
(1)
= pbkbs,
The rate of transfer of magnetization from A to any 1) and the rate of particular subsite of B is kab/(21 transfer from any subsite of B into A is kba. Assume that the spin I nucleus relaxes at rate R1 (due to the quadrupole mechanism), and that spin nuclei can be scrambled with equal probability among the ( 2 I + 1) subsites of B alone by a reaction with specific rate IC,. It is shown in the Appendix that the exact line shape expression for this model system is
+
I(0) = - R e { [ p a @ -k 7) -I- Pba 4zpapbl/[a(@
+ 7) - 4~2papbl ( 2 )
where I(w) is the absorption intensity at from the first moment of the spectrum. CY
=
-Rzo -
kab
p
=
-R2" -
kba
2 7 =
=
'/2(kab
w
radians/sec
+ - a) + i(wb - w )
(3)
i(wa
(4)
+ kba)
(5)
2/3a2(p - O.6R1 - k J / [ ( p - O.6R1 - IC,)
(a - R I - k d
+
X az/31
(6)
Except for the addition of 7 , eq 2 is identical with the Hahn-Maxwell-RilcConnell expression. Limiting expressions for slow or fast chemical exchange ( E ) or quadrupole relaxation (R1) can readily be obtained from eq 2. (1) E. Grunwald, C. F. Jumper, and 8. Meiboom, J . Amer. Chem. Soc., 84,4664 (1962). (2) C. Y.S.Chen and C. A.Swenson, {bid., 91,234(1969). (3) M. Sheinblatt, J . Chem. Phys., 39, 2005 (1963). (4) E. 1,. Hahn and D. E. Maxwell, Phys. Rev., 88, 1070 (1952); H. M. McConnell, J . Chem. Phys., 28,430 (1958). (5) M.Abu-Hamdiyyah, J . Phys. Chem., 69, 2720 (1965).
NMRMEASUREMENTS OF PROTON EXCHANGE IN AQUEOUS THIOUREA The fast quadrupole relaxation limit is obtained by expanding eq 6 in powers of a/(p - R1 - IC,) to second order and neglecting small imaginary terms. It is found that 7
-'//,a'/(Rl f
kba
f
ICs)
(7)
Equation 7 has sometimes been employed in (2) to correct for quadrupole relaxation effects on an ad hoc basis without including kba or k , in the denominator of q. This procedure can lead to serious errors, particularly in the fast chemical exchange limit where the exchange broadened, single peak has width Rz'/T Hz, where
Rz'
=
Rzo -I- PaPb(wa
- Wb)'/22
+
2/3Pbaz/(Rif
kba
f k ) (8)
As expected, the slow chemical exchange-slow quadrupole relaxation limit of eq 2 yields four lines, broadened by the appropriate uncertainty in the lifetime of the spin states. The slow chemical exchange/fast quadrupole relaxation limit of eq 2 yields two lines, centered at wa and W b , respectively. In this limit the A resonance is broadened by k a b / T Ha, independent of quadrupole effects but the B resonance is broadened by (ha- q ) , with 7 given by eq 7. In this limit, by contrast to that of eq 8, (kba -I- k,) may be ignored relative to kl in the denominator of q. The limiting expressions are presented here not because they are useful in evaluating experimental data (errors in such procedures are well documented6) but because they provide physical insight into effects of the various rate processes on I ( @ )which is lacking from numerical computation.
Experimental Procedures Thiourea (Matheson Coleman and Bell) was recrystallized from methanol, mp 180.0-181.4". Kinetic studies were done on 0.5 M thiourea (this concentration gives the minimum accurately measurable exchange broadening), 0.75, 1.0, and 1.4 thiourea (solubility limit) in the pH range 1 to 5 at 35". All solutions were made up no more than 2 hr prior to use. Acid concentration in the samples was adjusted using small quantities of strong hydrochloric acid, while monitoring the pH with a Corning Model 10 p H meter equipped with a Corning Riodel 476050 combination electrode. The measured pH values were found to be within rt0.02 unit of -log [HCl] by calibration for each thiourea concentration with known volumes of standard acid, prepared from constant boiling HCI. Solution pH was maintained constant by use of R4cIlvaine's buffer' above pH 2 and by Clark and Lubs's buffers between pH 1 and 2. Preliminary experiments demonstrated that the pH remained constant within =tO.01 unit for at least 2 hr for each solu-
2675
tion, and that line widths were reproducible within rt0.l Hz independent of buffer concentration. Therefore catalysis by buffer components of those reactions which influence nmr lineshape is negligibly slow. As further evidence for this fact, all the data presented here for 1.4 M thiourea were obtained in unbuffered solutions. Although the precision of these data is less than that for the other thiourea concentrations, the rate constant derived from it does not differ significantly from that for the other thiourea concentrations. Xmr spectra were obtained using a Varian T-60 spectrometer operating at 60 MHz with ambient probe temperature of 35". For all solutions studied, the resonance consisted either of a single averaged resonance or two lorentzian resonances, the weaker being due to thiourea N-H protons. The internal chemical shift was found to be 150 He f 2 He, independent of thiourea concentration. Precalibrated frequency scales were used, checking the calibration periodically by the sideband method. The precision of the shift meawrement is low because the S-H resonance was much wider and weaker than the water proton resonance. The width of the water resonance above pH 5 was found to be 2.0 rt 0.1 Hz for all thiourea concentrations, and equal to the width of the single peak observed at very low pH values. This value was taken to be the line width in absence of exchange. It is larger than the line width (-0.5 Hz) due to field inhomogeneity because of radiation damping. The T-60 employs a small, single coil in its probe circuitry. Recording accurate complete line shapes is the method of choice for accurate nmr kinetic studies, but it could not be done in this case because of the great disparity in relative populations (pw = 0.96 in 1.4 M thiourea). Minor misalignment of the rf phase control introduces significant distortions in the wings of the very intense water line which interfere with the wing of the weaker and wider in;-H resonance. Therefore the lines were recorded separately, with a sweep rate of 0.2 Hz/sec for peaks less than 15-Hz wide and 2.0 Hz/sec for wider peaks. At least four measurements of each peak were made, at rf levels well below saturation. The intense (water) line was found to be lorentzian (Al/* = d 3 A ~ l Jin all cases; only full widths at half-maximum intensity (A,,,) are reported here. The urea peak was found to be about 50 Hz wide in solutions with pH > 5 , but the precision of the measurement is low, particularly for the lower urea concentrations, due to phase adjustment problems. Therefore the N-H line widths were not extensively used in deriving rate parameters (see below). (6) A. Allerhand, H. S. Gutowsky, J. Jones, and R. A. Meinzer, J . Amer. Chem. SOC.,88,3185 (1966). (7) ':Handbook of Chemistry and Physics,'' 39th ed, Chemical Rubber Publishing Co., p 1615. (8) "Handbook of Chemistry and Physics," 47th ed, Chemical Rubber Publishing Co., p D-79. The Journal of Physa'cal Chemistry, Vol. 7 4 , No. 13, 1970
2676
R. L. VOLDAND ADOLFOCORREA
Figure 1. Water line widths (full width/half height) for aqueous thiourea. measured with buffered solutions.
Results and Discussion The following proton transfer reactions were considered
S
lyzed proton transfers from thiourea to water. Its forward rate R is related to the pseudo-zero-order rate kba of eq 2 by the relation kba
S
II HzS-C-NHz
S
+ Hz+OH,
(9)
S
II + HZN-C-NHz
II
H2N-C-NHaHb
S
S
I1
HZN-C-NH,H
S
/I
+ HbHNC-YHz
(10)
S
I/
/I
I_ HbHNCNHH,
H,HJS-C-NHHb
Hb
(11)
Ha
Reaction 9 is a schematic representation of acid-cataThe Journal of Physical Chemistry, Val. 7 4 , No. IS, 1970
All line widths except those for 1.4 M thiourea were
=
1 d - [thiourea] = R/[thiourea] [thiourea] dt
(13)
Reactions 10 and 11 together give the total rate of transfer of protons among different N-H groups not involving water as an intermediate. Reaction 12 represents internal rotation, which in principle renders the representation of (9)-(11), as well as the line shape expression (eq 2) , incomplete. We have ignored internal rotation in the data analysis described below for two reasons. Firstly, at most a single N-H resonance was observed. Thus if hindered internal rotation is present, the rate of reaction 12 is sufficiently fast to average the resonance parameters of H, and Hb. Some of the observed N-H line width could, however, be due to incomplete averaging of these differences. Secondly, theoretical calculations were made by numerical solution of Alexander's density matrix equationsJgallowing H, and Hb in eq 12 to be chemically shifted by up to 20 Ha. For values of the N-H coupling constants less than 100 Hz, any value of the internal rotation rate, and all values of the thiourea-water exchange rates, the computed water line widths were found to be (9) 8 . Alexander,
J. Chem. Phys., 37, 967 (1962).
2677
NMRMEASUREMENTS OF PROTON EXCHANGE IN AQUEOUS THIOUREA
3.0
1 10
1000
100
-
k , sec-I
Figure 2. Exact computed water line widths as a function of average exchange rate 5. Curves similar to this were calculated using eq 2 of the text for other thiourea concentrations and for values of RI between 200 sec-1 and 1000 sec-1.
independent of the parameters used to describe internal rotation, as long as the quadrupole relaxation rate R1 was chosen large enough to partially collapse the N-H resonance. Water line widths measured as a function of p H are shown in Figure 1. Equation 2 was used to compute water line widths as a function of the average exchange rate 1 and the results are shown in Figure 2 for thiourea concentration of 1.0 M . Similar curves were calculated for the other thiourea concentrations. I n these calculations, the relative populations were chosen to correspond to the known sample composition and the chemical shift was taken as 150 Hz. Since the scalar " is not known for thiourea, the coupling constant a value of 60 Hz (compared with 63.4 Hz for urealo) was used, and R1 was chosen to yield a computed thiourea line width of about 50 He. The direct exchange rate k, was arbitrarily set equal to zero, because eq 7 is a t least approximately valid for aqueous thiourea, and so only the sum (R1 k,) is relevant. As can be seen from Figure 2, the "quadrupole parameters" R1 and a" have very little effect on the water line widths. Nevertheless, those curves in Figure 2 which predict less than the maximum observed line width in Figure 1 cannot correspond to reality. We used the curve in Figure 2 which does predict the correct maximum line width together with the line widths of Figure 1 to estimate & (and hence k b a ) via eq 1 and 5 as a function of p H for each thiourea concentration. The results are shown in Figure 3. Values of ~ I \ . - H and R1 used in
this computation may be significantly in error, but their ratio predicts the correct thiourea line width in absence of proton exchange, and possible errors in the individual parameters do not affect the accuracy of the thiourea-water proton exchange rate measurement. I n the calculations leading to Figure 3, the value of Rzowas varied slightly from 2.0 Hz found by direct measurement. The linearity and slope of Figure 3 was found to be dependent on the value used for Rzo;typically the slope varied by 10% for a 7% change in Rzo. Such behavior is expected when the natural line width
Table I : Rate Law of Protolysis of Aqueous Thiourea [CHaNzSl, M
0 50 0 75 1 00
+
1.4OC a
Slope of Figure 3"
-0 -1 -0 -0
95 00 99 97
Y intercept of Figure 3*
4 95
10-6,
seo-1
0 9 1 2 11 0 7
5 07 6 04 4 85
The general form of the rate law is Rate
x
IM-1
k
=
~[CH~NZS]".
[€II]", from which it follows that, log k b a = log k[CHaN2Sjm-1npH. The slope of a plot of log k b a vs. p H (Figure 3) indicates that the order of the rate law with rerpect t o hydrogen ion concentration is 0.98 i 0.02. b The Y intercept indicates that the rate law is first order with respect to thiourea concentration. KO buffer was used for this thiourea concentration.
(10) J. B. Lambert, B. W. Roberts, G. Binsch, and J. D. Roberts, J . Amer. Chem. Soc., 86, 5564 (1964).
The Journal of Physical Chemistry, VoZ. 7 4 , N o . IS, 1970
2678
R. L. VOLDAND ADOLFO CORREA 4.00
0.50
3.00
onA. 0
a
A 0.75
&.
n 1.00
0
&
0
-
1.40
bo
0
2.00
1.00 I
1.00
I
I
2.00
I
I
3.00
I 4.00
I
! 30
Figure 3. Logarithm of kba, the inverse lifetime of a proton on thiourea nitrogen, plotted against pH, for various thiourea concentrations.
contributes at least 20-25% of the maximum observed line width. The slope and intercept for each thiourea concentration shown in Figure 3 are listed in Table I, from which the rate law for protolysis is found to be
R
= (1.0 f 0.2) X
105[thiourea][Hg+O]o~g8*0~02 (14)
Errors of the type just discussed for RZ0lead to larger errors in estimating the order of the reaction as well as its rate constant; analogous errors in enthalpies and entropies of activation have been pointed out for variable temperature studies.6 For the present case the slope of the curves in Figure 3 is most certainly either integral or half integral, and probably should be 1.00. Use of this fact has allowed us to estimate Rzo properly, and also to conclude that the thiourea-water chemical shift does not vary significantly with pH, for if it did, the slopes in Figure 3 would not have been close to unity. The use of water line widths to compute lifetimes of protons on thiourea (l/kb, from measured values of kab) depends crucially on the validity of eq 1, and one situation where this relation could fail is if protonation of sulfur as opposed to nitrogen were slow enough to produce extra broadening of the water peak. From the slow exchange limit of eq 2, if eq 1 is valid, the ratio of exchange broadening of the thiourea and water resonance should be p,/pb. The water peak exchange The Journal of Physical Chemistry, Vol. 74, No. IS, 1970
broadening can be estimated by subtracting the natural line width from the observed line width. The thiourea peak exchange broadening can be estimated by subtracting the thiourea line width at neutral pH from that observed at lower pH. Although eq 7 shows that this procedure is not exact, in the present case if the slow exchange condition is met, R1 >> k b a and little error is made. For 1 M thiourea, p,/pb is 26, and the ratio of thiourea to water exchange broadening was found to vary randomly between 22 and 32 at pH between 3.6 and 4.0. For lower pH values (larger rates), this ratio is significantly smaller, but calculations using eq 2 indicate that the slow exchange approximation, rather than eq 1, is breaking down. Therefore we conclude that the rate law of eq 14 is an accurate representation of acid-catalyzed protolysis in aqueous thiourea. For X-methylacetamide,ll the protolysis rate constant for acid catalysis is 400 M-' sec-', and for base catalysis, 2 X 106. The acid-catalyzed reaction is supposed to be slow because protonation occurs preferentially on oxygen rather than nitrogen,l' and the latter process only is observed by nmr. The mechanism for base catalysis is thought to be a bimolecular collision. Correcting the rate constant for acid-cata(11) A. Berger, A. Loewenstein, and 8. Meiboom, J . Amer. Chem. Soc., 81, 62 (1959).
2679
NMRMEASUREMENTS OF PROTON EXCHANGE IN AQUEOUS THIOUREA lyzed protolysis of thiourea by a statistical factor of 4 (for one of the four equivalent N-H protons the rate constant is 2.5 x lo4M-l sec-l), its value is much closer to the base-catalyzed rate constant of N-methylacetamide than to the acid rate constant. However, thiourea is known to protonate preferentially on sulfurl2 and not on nitrogen. We therefore conclude that the protolysis of thiourea N-H protons is catalyzed by acid about 60 times more efficiently than N-methylacetamide N-H protons.
Conclusions Proton lifetimes on aqueous thiourea have been determined from measurements of water and N-H nmr line widths. Effects of quadrupole relaxation of nitrogen may be separated experimentally from chemical exchange effects. Acid-catalyzed protolysis of thiourea N-H groups is about 60 times more efficient than acid-catalyzed protolysis of N-methylacetamide N-H groups. Factors responsible for this difference could perhaps be determined by studies of substituted ureas and thioureas.
Acknowledgment. We gratefully acknowledge partial support of this work by the National Science Foundation and the Office of Naval Research. Mr. Thomas DiGennaro provided many useful suggestions leading to the closed form of the line shape expression.
can readily be obtained from the discussion following eq 1 of the text. K, describes the scrambling of thiourea protons among themsleves. R1 describes quadrupole relaxation of spin s. It consists of the quadrupolar matrices obtained previou~ly,'~ bordered by a row and column of zeroes. SL is a diagonal matrix whose first entry is ua,the resonant frequency of spins in site A, and all other diagonal elements are Wb, the site B resonant frequency. J describes scalar coupling of spins in site B to the quadrupolar nucleus. Its diagonal elements are 0, I @ ,(I - 1) a,. . . -I@. Off-diagonal elements of J are due to the nonsecular part of the heteronuclear scalar coupling. They are neglected in this work, limiting validity of eq 2 to situations where the rates are less than the difference in resonant frequency of protons from nitrogen. I n order to explicitly calculate the inverse required by eq A l , it is useful to transform (Al) by a similarity transformation. For any nonsingular matrix P
1
I ( w=) -Re{lP{P-lAP - ~ w I I - ~ P - ~ W(444)
Equation A4 may be used successively choosing P each time to accomplish any desired simplification. R1 and K, commute, and P may be chosen to diagonalize them simultaneously. For spin1 = 1
0
rl 0
1
0
Appendix A general expression for the nmr line shape I(.) given by
I ( w ) = -Re{lA - iwI)-lW
1
is (Al)
where I and I are the unit matrix and the unit vector, respectively, W is a vector of relative weights, 0 is the frequency offset in radians/sec from the first moment of the spectrum, and A is a complex, nonhermitian matrix. For the case discussed in the text, all the matrices and vectors in (Al) are of dimension 21 2 where I is the spin of the nucleus to which spin l/z nuclei are coupled in subsite B. The vector W is given by
and for spin I
=
1 0
0
0
0 '/z
3/24E
-1/2&
'/z
0 '/z
-1/24E
-3/22/'5
-'/z
+
+ 1))
PI
= col(l,O,O,. . .O)
(A2) An analytical formula for I ( w ) can be obtained by obtaining an explicit expression for the inverse (A iuI)-l in terms of elements of A. A consists of submatrices as follows
Pz
= c01(0,1,1,1,** I 1 ) 4 2 I
PB
= COl(O,I,(I - 1). . . .
w
= (Pa,Pb/(21
+ 11, Pb/(21 + 1 ) ~
For arbitrary spin I , the first three eigenvectors (columns of P) are
' ''
'Pb/(21
+ + + + +
A = RzO K K, R1 iJa iJ (A3) Rz0 refers to the width of each transition in absence of the exchange processes included in the other matrices. We assume for simplicity that RzO is diagonal and proportional to the unit matrix, the proportionality constant being Rzo of eq 3 and 4 of the text. K refers to the thiourea-water proton exchange; its elements
+1
- 1)/41/J(I
+ 1)(2I + 1)
(A7) Equation A7 and the facts that P is orthonormal, and RI itself is symmetric about the diagonal and the antidiagonal, allow additional relations t o be obtained among the elements Pij. For example, since any column Pj of P is orthogonal to Pz (12) T. Birchall and R . J. Gillespie, Can. J . Chem., 41,2642 (1963). (13) H. L. Gutowsky, R. L. Vold, and E. J. Wells, J. Chem. Phys., 43, 4107 (1965). The Journal of Physical Chemistry, Vol. 7 4 , N o . 13, 1970
2680
E(. M. CRECELY, R. W. CRECELY, AND J. H. GOLDSTEIN 21+2
j=1
PI, = 0
( j = 1,3,4,.. . 2 I
+ 2)
(A8)
After transforming (Al) to a form in which R1 and k, are diagonal, it is convenient to further transform the result by the diagonal matrix d i a g ( d ( 2 I l)Pa, dEl,l,. . . 1). This procedure symmetrizes the upper 2 X 2 block of A, and yields the result
ro 1
+
where
The elements of Az, cannot be explicitly determined without knowledge of all the eigenvectors ofP. Nevertheless eq A9-Al3 suffice to yield eq 2 of the text, where 7 is the only nonvanishing element of thc matrix
and
Q = XA2r-lY (A14) For the case of spin I = 1, we obtain eq 6 of the text while for I =: 3/2
I(w) = -Re{L,(B - iw1)-lLr]
(A9)
rl = 5 / 4 w
The matrix Az is precisely the matrix form of the HMM equations13 obtained previously. The rectangular matrix X is given by
and Y is given by
- R1)@
- 2Rl)/[(P Ri)'(P - 2R1) + '/&')I
(A15)
This derivation could be extended to cases of I > 3/2 by using a computer t o calculate the eigenvalues and eigenvectors of R1, but the resulting expression for would be very cumbersome. The derivation can also be applied to cases of line broadening caused by scalar coupling to nuclei relaxing by other mechanisms than quadrupole coupling to permanent field gradients.
Carbon-13 Nuclear Magnetic Resonance Spectra of Monosubstituted Cyclopropanes by K. M. Crecely, R. W. Crecely, and J. H. Goldstein Chemistry Department, Emory University, Atlanta, Georgia $0829 (Received February 9, 1970)
High-resolution '*C spectra have been obtained and analyzed for cyclopropane and its monosubstituted 4 1 , -Br, -I, -NH2, and -COCl derivatives. The long-range W-H coupling constant in cyclopropane has been found to be -2.55 Ha. Relationships between the long-range W-H couplings and the H-H couplings in these cyclopropanes are discussed.
Introduction There have been numerous nmr studies of cyclopropane and its monosubstituted derivatives.l-13 The proton chemical shifts and H-H coupling constants have been reported for many of these compounds.1-8 I n several cases the 13C-H satellite spectra have been the l a G H Such bonded couplings and information about the differences The Journal of Physical Chemistry, Vol. 74, N o . 18, 1970
between long-range 1%-H coupling constants. The l3C spectra of cyclopropane and several monosubstituted derivatives have appeared in the literature, (1) H . M. Hutton and T.Sohaefer, Can. J. Chem., 41, 2774 (1963). (2) K . B. Wiberg and B. J. Nist, J. Amer. Chem. Soc., 85, 2788 (1963). (3) T. Sohaefer, F. Hruaka, and G . Kotowycz, Can. J. Chem., 43, 75 (1965).
