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Figure 10.5

Low temperature I » 15 ms Slow exchange

Intermediate temperature

T ~ 15 ms Coalescence

High temperature T « 15 ms Fast exchange

The DMF bond rotation can be considered as a dynamic equilibrium with equilibrium constant of 1 (k1 = k_i):

The average time spent in the Ha state (¿(CH3) = 2.94 ppm)) before jumping to Hb is 1/ki, as the k1 rate defines the end of its lifetime in the Ha state. The average time spent in the Hb state is likewise equal to 1/k-1, as the k-1 process defines the end of its life. Figure 10.6 shows a simulation of the DMF bond rotation starting with slow exchange (top) and raising the temperature to increase the average time spent in each state (rex = 1/k1 = 1/k-1 = 1.5 s, 0.15 s, 15 ms, 1.5 ms, and 0.15 ms). In the simulation, an equal amount of noise has been added to each spectrum and the spectra are scaled vertically to the tallest peak. Comparing the average lifetime rex to the coalescence lifetime (shutter time) rc, we see sharp lines at the two chemical shift positions when rex (1.5 s) is 100 times longer than rc (Fig. 10.6, top).

The exchange process is much slower than the "shutter time," which means that the nucleus spends much more time on average than the NMR timescale in each environment. In this slow exchange limit, the intensity (integral area) of each peak will reflect the fraction of time that the nucleus spends in that environment (1/(Keq + 1) for Ha, Keq/(Keq + 1) for Hb), in this case a 1:1 ratio (Keq = 1). When the temperature is increased to the point where tex (0.15 s) is only 10 times longer than tc, the peaks begin to broaden and the peak maxima begin to "creep" inwards each other. This "exchange broadening" can be used to measure the rate constants k1 and k-1 (see below). When the temperature is raised further until tex is equal to the coalescence time (Fig. 10.6, center), there is no longer a "dip" between the two peaks and we see one single, very broad peak. Because the same peak area (two CH3 groups) is now spread over a very broad peak, the peak height is much lower and the noise appears larger in comparison. It is not uncommon for peaks to be broadened out of existence (into the noise) by exchange!

Continuing to raise the temperature, we arrive at a point where tex (1.5 ms) is 10 times shorter than tc. The single peak is now much sharper but still broadened relative to the natural linewidth in the absence of exchange. This is a case where we might see one peak in the spectrum broader and shorter than the others and start thinking about the possibility of an exchange process. Finally, at the high temperature limit where tex is 100 times or more shorter than tc, we see a single sharp peak at the average chemical shift position (Fig. 10.6, bottom), with no broadening and a peak area representing the total of both environments (in this case 6H).

Note that on a 200-MHz instrument, Av is 30 Hz (0.15 ppm x 200 Hz/ppm) and the NMR timescale tc is 15 ms (1/(2.22 x 30)), but the "shutter speed" is faster as we go to higher field instruments because the chemical shift difference Av is measured in hertz, not ppm. Thus, moving to higher field shortens the shutter time tc in a way that is inversely proportional to Bo. Figure 10.7 shows simulated spectra of the DMF sample at the same temperature that gives tex = 15 ms, analyzed on three different spectrometers with YHBo/2n = 60, 200, and 600 MHz. At 60 MHz (top), we have an exchange-broadened fast

ppm

exchange spectrum (rex = Tc/3.33); at 200 MHz (middle), we are at the coalescence point (tex = tc); and at 600 MHz (bottom), we have an exchange-broadened slow exchange spectrum (tex = 3 tc). In terms of its effect on the spectrum, going to higher field is like cooling down the sample and going to lower field is like heating it up. If you simply want to verify if a broadened peak is due to exchange it might be much simpler to try a different field strength instead of doing a variable temperature study.

If the nucleus is farther away in the molecule from the site of chemical change, its chemical shift may be affected less so that Av is small or even zero. We can have as many NMR timescales as there are distinct nuclei within a molecule. Thus, the NMR timescale is not an absolute time, but rather it depends on field strength and on the significance of the chemical exchange in terms of its effects on the chemical shift of a particular nucleus.

10.2.1 Slow Exchange

Let's consider in more detail what happens in slow exchange as we increase the exchange rate and the single NMR peak begins to broaden. A simple 1D NMR spectrum is recorded by rotating the sample magnetization into the x-y plane with a pulse and then observing the precession of this sample magnetization in the x-y plane as it induces a sinusoidal signal in the probe coil. As the sample magnetization rotates during the recording of the FID, individual Ha spins become Hb spins, keeping the same spin state they had before, but changing their resonant frequency (chemical shift) from va to vb. Each individual spin undergoes the jump from one chemical environment to another at random intervals, so that the phase coherence is rapidly lost as it switches its precession frequency from va to vb and back again (Fig. 10.8). The process is random ("stochastic") from the point of view of any one nucleus, but we can say that on average the nucleus will spend half its time in the Ha state and half its time in the Hb state (if Keq = 1). We can also say that on average a nucleus will remain in the Ha state for ta = 1/k 1 s and it will remain in the Hb state for tb = 1/k -1 s. Phase coherence is lost because after one spin changes to a different frequency for a brief period, it cannot jump back into the original frequency at the same position in the x-y plane (the same phase) that it would have occupied if it had stayed at the original frequency for the whole time. So now it has lost phase coherence with other spins that did not make the "jump." This randomization of individual phases leads to a "fanning out" of the individual vectors that make up the Ha net magnetization, and the net magnetization vector for the population of Ha spins rapidly decays to zero magnitude as it rotates. This is similar to the T2 relaxation process, and it adds to the loss of coherence resulting from T2 relaxation:

Ha slow exchange linewidth = Av1/2 = 1/(nT2) + 1/(nT2) + 1/(nTa)

Spin 1:

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