In the Laboratory
Conformation Interchange in Nuclear Magnetic Resonance Spectroscopy Keith C. Brown, Randy L. Tyson, and John A. Weil* Department of Chemistry, University of Saskatchewan, 110 Science Place, Saskatoon, SK S7N 5C9, Canada
Introductory courses in chemistry usually contain a spectroscopy component that includes a discussion of nuclear magnetic resonance (NMR) spectroscopy. Generally, however, the effect of chemical exchange processes is not included in this discussion but is left until later. Herein we present a simple experimental example introducing this topic. This experiment has been performed in our department for several years as part of an undergraduate course in spectroscopy. The term exchange, in our case, denotes the interchange of magnetic environments of two spin-bearing nuclei of the same type. When this happens there will be a change in the shielding factor σ and hence of the resonant (Larmor) frequency ν of each nucleus, according to the equation ν = ᎑(γ /2π )Bo(1 – σ), where γ is the gyromagnetic ratio for the magnetic nucleus and Bo is the magnitude of the externally applied magnetic field Bo. There is a mean environmental lifetime τ for each nucleus, in each environment, which is set by the kinetics of the exchange process and is a function of temperature. As will be shown, the rate constant kT = τ᎑1 of the exchange process may be extracted from the NMR line shape so that, using a series of experiments done at different temperatures, one may determine the desired kinetic parameters of the process. One type of exchange process that has been studied extensively is hindered intramolecular rotation, the first reported example being that of N,N-dimethylformamide (1). In our case, we will be studying the occasional interchange of conformation about the hydrazinic-nitrogen picrylring-carbon bond, observable in liquid solutions of 2,2-diphenyl-1-picrylhydrazine (2): O
H (A)
N
O
N
O
N
H
O
O N
H (B) O
N
N
H (A)
N
N
O
N
H
O N
O
O O
H (B)
O
where “picryl” (= Pic) denotes the 2,4,6-trinitrophenyl group. The same phenomena will be observed in general for molecules of type RNHPic, when group R ≠ H. The 1H and 13 C NMR characteristics of the hydrazine have been reported in some detail (3). For our purposes we will consider two nuclei, A and B, that are interchanging magnetic environments, labeled 1 and 2. Such exchange occurs when a given nucleus experiences a change from one magnetic environment to a different one, and therefore changes resonance frequency from ν1 to ν2 while the second nucleus experiences the opposite change. Each such exchange (in our case, rotations) occurs at a random time (perhaps stimulated by an intermolecular collision) and lasts a brief interval. Of course, a whole huge ensemble of molecules bearing such nuclear pairs are present and are active in this way. *Corresponding author.
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Figure 1. The 300-MHz pulsed NMR spectrum of the picryl protons of 2,2-diphenyl-1-picrylhydrazine in CDCl3 in the slow-exchange limit (T = 230 K). The scale is in ppm from tetramethylsilane.
In our experiment, we shall take the two picryl protons as convenient probes (A and B, having nuclear spins I = 1/2) for the process. They see different environments because the two inner nitro groups (on carbons 2 and 6, respectively) are at different orientations relative to the picryl ring, owing to hydrogen bonding to the hydrazinic hydrogen and steric effects from the diphenyl-amino group. Since the picryl protons thus are inequivalent (chemical shift difference ∆ H–H = |σ1 – σ 2| νo = | ν1 – ν 2| = 219 Hz [s ᎑1] for 300-MHz NMR, for the hydrazine in CDCl3), the magnetic dipolar interaction between the protons shows up (Fig. 1) as a small line splitting, described by spin–spin coupling parameter JH–H measured herein in frequency units (2.5 Hz). Here νo = (ν1 + ν2)/2. The dynamic process brings the hydrazinic hydrogen from near one of these nitro groups to near the other, while these groups interchange orientations. Thus the picryl protons interchange environments. Clearly the latter have the same mean lifetime τ. Chemical exchange processes have been studied extensively via the use of NMR spectroscopy (4, 5). In many cases the experiments are designed to gain information on molecular dynamics and involve the comparison of computer-synthesized line shapes with NMR spectra taken over a range of temperatures. The objective is to extract the rate parameter kT for the process at each of various temperatures T. For convenience, three distinct thermal situations can be envisioned: 1. “Very slow” exchange (τ long). This occurs when the lifetime τ at each exchange site is very long compared to the inverse of the NMR transmitter frequency (ca. ν o). In other words, visualized classically, each nuclear spin magnet precesses many times at its Larmor frequency about B o, before an interchange event is likely to occur. The NMR spectrum of a substance in which such a very slow exchange process is occurring shows this very clearly, since two doublets at frequencies ν 1 and ν 2 are observed (Fig. 1; at somewhat higher temperatures, these spin–spin doublets are no longer resolved, as is evident in Fig. 2).
Journal of Chemical Education • Vol. 75 No. 12 December 1998 • JChemEd.chem.wisc.edu
In the Laboratory 2. Intermediate rate of exchange. Here the exchange rate is roughly equivalent to the chemical-shift difference, and the result on warming is a broadening and eventual averaging (coalescence) of the two lines. Analysis of the line shapes under these conditions requires numerical methods, as we shall see. Since this is where the NMR spectrum is most strongly affected by the exchange, this is also the region in which the most reliable rate information can be obtained. 3. “Very fast” exchange ( τ short). Here, each nucleus can be visualized as having very little time to precess at its Larmor frequency in one state before it is likely to undergo exchange to the other one. In this case, as τ approaches 0, the spectrum will appear as a single peak at frequency ν o, when one has equal spin populations in sites 1 and 2 as is the case in our example.
In the present type of experiment, one wishes to compare experimental line shapes with calculated line shapes. A set of equations exist which describe such NMR line shapes as a function of temperature T via the single temperature-dependent parameter τ. The relevant peak intensity function F(ν) is given (6 ) compactly by
F=C
r +b + – sa + 2
a +2 + b +
+
r ᎑b ᎑ – sa ᎑ a ᎑2 + b ᎑
2
(1)
where C is an adjustable amplitude factor and a± = 4π2(νo – ν ± J/2)2 – (τ᎑1 + τ2᎑1)2 – π2( ν1 – ν2)2 – π2J 2 + τ᎑2 (2a) b± = 4π(νo – ν ⫾ J/2)(τ᎑1 + τ2᎑1) ⫾ 2π J τ᎑1
(2b)
r± = 2π(ν o – ν ⫾ J )
(2c)
s = 2τ᎑1 + τ2᎑1
(2d)
In equations 2, JH–H is abbreviated to J. As τ approaches infinity, the slow-exchange limit, F converges to the well-known AB spectrum (Fig. 1) consisting of four lines located at the four frequencies ν o ± J/2 ± [(ν 1 – ν2 )2 + J 2/4]1/2
where there are four sets of sign choices. As τ approaches zero, the rapid-exchange limit, F converges to a single Lorentzian absorption line located at νo. The transverse relaxation time τ2 is simply related to the individual line width W1/2 at half height via the relation τ2 = (πW1/2) ᎑1. Symbol W1/2 denotes the width in the absence of exchange broadening. The observed line width of an NMR resonance is generally due to magnetic field inhomogeneity and spin–spin relaxation (for which τ2 is the time constant). If τ2 is not too large, it can be measured directly from W1/2. Since field inhomogeneity generally contributes at least 0.1 Hz to the line width, to do so, τ2᎑1 must be larger than about 2 Hz. To ensure, in measuring τ2 from W1/2, that the observed width is not affected by τ, we make the measurement at low temperature, where there is no contribution due to exchange broadening. In reality, to obtain τ2 from the observed line width of the picryl proton resonances at 230 K, where splitting due to scalar J coupling is observed (Fig. 1), one is confronted with the line width of a doublet (i.e., of the two lines partially merged). Thus this measurement does not yield an accurate
Figure 2. Experimental and synthesized 300-MHz pulsed NMR spectra of the picryl 1H resonances of 2,2-diphenyl-1-picrylhydrazine in CDCl 3 at various temperatures. The scale is in ppm from tetramethylsilane. Temperatures T and rate parameters kT are listed.
value of τ 2 but rather, depending on the magnetic-field inhomogeneity, approximately twice the true value. Fortunately, the fitting of the synthesized line shapes to the observed lines is not very sensitive to the exact value of τ2. In our case, a rough measurement of W1/2 at 230 K gives a value of 2.5 Hz, from which we calculate 0.13 s for τ2. Various computer programs have been written to perform line-shape calculations (eq 1). In our case, we use the Quantum Chemistry Program Exchange program 450, DNMR3H (7 ). One could, equally well, use one of the symbolic computation programs such as Maple, Mathematica, or WinDNMR (8) to achieve the same results. The program DNMR3H requires input containing information about the chemical shifts of the resonances, the spin–spin coupling constant, line width, and a rate constant. The chemical shifts, coupling constants, and line widths of the resonances can best be measured at low temperatures, before dynamic broadening and line merging begin. Taking spectra at several low temperatures is recommended because temperature dependences of the line positions may exist. Appropriate extrapolations to the higher temperatures of interest can then be made. After the initial measurement of the shift, spin–spin coupling, and width parameters, a series of spectra at higher temperatures is acquired. These are obtained in the intermediate-rate-exchange region and the fast-exchange region. Subsequently, these spectra are to be simulated, using the computer program. Comparison of the experimental and simulated spectra by eye then allows the determination of the appropriate rate parameter kT by trial and error. The Arrhenius equation k T = k ∞ exp(᎑ ∆ E/RT)
(3)
where k ∞ is the temperature-insensitive preexponential factor,
JChemEd.chem.wisc.edu • Vol. 75 No. 12 December 1998 • Journal of Chemical Education
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In the Laboratory
Figure 3. Experimental rate-constant data plotted as ln(k T) vs T ᎑1 (k T in s᎑1, T in K ).
R is the ideal gas constant, and T is the absolute temperature, may then be used to estimate the molar activation energy ∆ E for the process. To do so, a plot of ln(k T) versus 1/T is made (Fig. 3). Here the slope is ᎑ ∆ E/R and the intercept at 1/T = 0 is ln(k ∞). Finally, activation pseudothermodynamic parameters ∆ H ‡ and ∆ S ‡ , respectively the molar activation enthalpy and entropy changes at any particular temperature T, may be determined from ∆ E and ln(k ∞) using the equations ∆ H ‡ = ∆ E – RT
(4)
∆ S = R(ln [k ∞hN A /(eRT )])
(5)
‡
where NA is Avogadro’s number, h is Planck’s constant, and e is the base of the natural logarithm (9). The Gibbs free energy of activation ∆ G ‡ = ∆ H ‡ – T∆ S ‡ (6) for the process may be determined from these, or from ∆ G ‡ = RT (ln [RT /(N A hk T)])
(7)
where kT is the rate constant calculated from the Arrhenius plot, at the temperature T of interest. It turns out that, of the above kinetic and activation pseudothermodynamic parameters, ∆ G ‡ has the least statistical error associated with it (6 ), so that it is of the most interest. It can be correlated with hydrogenbonding power of the solvent used (9). Experimental Procedure A 10-mg sample of purified 2,2-diphenyl-1-picrylhydrazine (Aldrich Chemical Company, Handbook of Fine Chemicals 1996–97: 28,168-9) is dissolved in 0.6 mL of chloroform-d, transferred to a 5-mm NMR tube and degassed, and the glass tube is sealed. The high vapor pressure of the solvent necessitates the last step. Our experiment can be performed on any of a variety of spectrometers with different field strengths and continuouswave (cw) or pulsed transmitters. We have observed coalescence of the picryl proton resonances on 80, 200, 300, and 500 MHz NMR spectrometers. In fact, the original work done on diphenylpicrylhydrazine was performed at 60 MHz 1634
(2). The major difference in the spectra obtained is in the shift ∆H–H between the picryl proton resonances. Thus, since resonant frequency is proportional to field strength, doubling the applied field will double ∆H–H. Note that ∆H–H is solvent dependent and slightly temperature dependent. The sample is then used to obtain 1H NMR spectra (herein at 300 MHz) over a range of temperatures from 230 to 340 K. The variable-temperature probe is calibrated by studying the temperature dependence in the chemical shifts of the methylene and hydroxyl resonances in an 80% ethylene glycol–20% dimethylsulfoxide sample (6 ). For each temperature, a spectrum containing only the picryl resonances (with “peak picking” output) is plotted. From the spectra obtained at two or three sufficiently low temperatures, determine for the picryl protons the value of ∆H–H, JH–H (from the lowest-temperature spectrum), and W1/2. Using appropriate computer software, the line shapes of the picryl protons at each of the experimental temperatures are calculated. Input to the program consists of ∆H–H, JH–H, W1/2, and a guess as to the appropriate value of the rate constant kT = τ᎑1, all in Hz. (It may be helpful to remember that the value of the rate constant will increase by approximately a factor of two as the temperature is raised by 10 K.) The plot of F(ν ) produced is used for comparison with the experimental spectrum at hand. It is worth noting that it is also possible in principle to measure the same rate parameters using 13C and 14,15N NMR spectra. Results and Discussion Once a set of best-fit spectral plots is produced for each temperature, the student will have a set of temperatures and corresponding rate-constant values. These are used to create an Arrhenius plot for the rate of reorientation of the picryl ring relative to the diphenylamino group. Parameters ∆ E and ln(k∞) are calculated from the slope and intercept, respectively, and ∆ H ‡ and ∆ S ‡ are calculated from the above equations. The Gibbs energy ∆ G ‡ at 298.1 K, calculated by use of eq 6, can be compared to the value obtained from eq 7. A series of the 1H NMR spectra of the picryl protons at various temperatures is displayed in Figure 2, to demonstrate the temperature dependence of the line shapes observed with the hydrazine. Also shown in Figure 2 are the synthesized spectra, obtained using best-fit values of kT. Figure 3 shows an Arrhenius plot obtained from the rate constants used to best simulate the spectra in Figure 2 and from their corresponding temperatures. From this plot, we can extract a value for ∆ E of 68.63 kJ/mol᎑1, using eq 3. The resulting parameters ∆ H ‡, ∆S ‡, and ∆ G ‡ are 66.15 kJ mol᎑1, 1.67 J K᎑1 mol᎑1, and 65.65 kJ mol᎑1, respectively, at T = 298.1 K. Consideration of how one may visualize the actual molecular motions involved leads to several ideas. The hydrazinic hydrogen is almost certainly involved in hydrogen bonding with one of the ortho-nitro groups on the picryl ring. In order for rotation of the picryl ring about the hydrazinic N–C bond to occur, this hydrogen bond must be broken. In other words, there is a minimum energy requirement for this process to occur. Other likely relevant factors are the partial double-bond character in the hydrazine-N–picryl-C bond and the steric hindrance to rotation within the solvent due to the presence
Journal of Chemical Education • Vol. 75 No. 12 December 1998 • JChemEd.chem.wisc.edu
In the Laboratory
of the phenyl rings. The diagram below shows one of the various resonance structures exhibiting the double-bond character of the hydrazinic-nitrogen–picryl-carbon linkage. (–)
O
O N
N (+)
O
(–)
O
(+)
N
N H O
N O
The student might consider extending the experiment so that several different solvents are used, a set of ∆ E, ∆ H ‡, ∆ S ‡, and ∆ G ‡ values being determined for each solvent. Such measurements have been published (9) and interpreted in terms of the H-bonding capabilities of the solvents (i.e., how these bond to the picrylhydrazine). The student may also wish to visualize and compare the several characteristic times encountered, at least implicitly, in this experiment. These include: 1. The average environment lifetime, that is, how long the resonance frequency ν1 or ν2 is likely to be appropriate for each probe nucleus (1 HA and 1HB). 2. The average transition time, that is, how long the transfer of environment (to go from ν1 to ν 2, or reverse) is likely to take for any probe nucleus. 3. The average spin-orientation lifetime, that is, how long a given spin state (quantum number m 1) of a probe nucleus is likely to last. This is of course related to the radiofrequency power level of the excitation magnetic field applied as well as to the transverse and longitudinal magnetic relaxation times of the nucleus.
Finally, we note that dynamic averaging of conformations occurs universally and thus can affect any magnetic resonance spectrum (NMR and EPR) taken above cryogenic temperatures. The decisive factor is the “NMR or EPR time scale” (10, Chapter 5), which is set by (the inverse of ) the difference in line positions (frequencies) to be averaged, as compared to the mean probability per unit time of molecules changing their conformation. Acknowledgment This work was supported by funding from the Natural Sciences and Engineering Research Council of Canada. Literature Cited 1. Gutowsky, H. S.; Holm, D. H. J. Chem. Phys. 1957, 25, 1288. 2. Heidberg, J.; Weil, J. A.; Janusonis, G. A.; Anderson, J. K. J. Chem. Phys. 1964, 41, 1033. 3. Currie, P. F.; Quail, J. W.; Rusk, A. C. M.; Weil, J. A. Can. J. Chem. 1983, 61, 1760. 4. Binsch, G.; Kessler, H. Angew. Chem. 1980, 19, 411. 5. Lamber, J. B.; Nienhuis, R. J.; Keepers, J. W. Angew. Chem. 1981, 20, 487. 6. Sandström, J. Dynamic NMR Spectroscopy; Academic: London, 1990. 7. Stempfle, W.; Klein, J.; Hoffman, E. G. DNMR3H; QCPE Program No. 450; Department of Chemistry, Indiana University, Bloomington, IN. 8. WinDNMR; JCE Software 1995, 3D(2). 9. Tyson, R. L.; Weil, J. A. J. Phys. Chem. 1990, 94, 3951. 10. Harris, R. K. Nuclear Magnetic Resonance Spectroscopy; Pitman: London, 1983.
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