Usually we think of NMR as a structural tool, but it is equally powerful as a means to measure rates.

The most obvious way NMR can be used to measure a reaction rate is to monitor the relative sizes of peaks representing starting material and product, which in general would come at different chemical shifts. As the reaction proceeds the product peaks grow and the starting material peaks shrink in a way that reveals the rate of reaction.

But there is a more subtle approach that allows measuring rates that would be too rapid to measure by most techniques. It involves averaging.

We know that the anisotropy of proton-proton interactions (and less dramatically of chemical shift) would give very different spectra for differently oriented molecules. But for molecules tumbling rapidly in solution these anisotropies average, so that the peaks for protons in given molecular positions are sharp and appear at an average position, rather than being a broad "envelope" spanning perhaps hundreds of thousands of Hz for molecules in many different orientations.

Similarly, when the spectrum of ethanol is recorded with a trace of acid present, the CH₂ group shows splitting only by the CH₃ protons on one side of it, not by the OH proton on the other side. The reason is that there is not a single proton in the OH position. Rather, in the presence of acid, protons rapidly add to, and then disappear from, the oxygen atom, so that the CH₂ "sees" only the average of many different protons which have essentially equal probabilities (within 1 part in tens of thousands) of having the two different magnetic orientations that would give rise to an additional splitting of the CH₂ peaks in a sample without acid catalyst. (see Text p. 362, figures 13.44,45).

Another averaging example is for the signal of the sole proton in cyclohexane-d₁₁. There are essentially equal numbers of chair cyclohexanes with the lone proton in the axial and equatorial positions (no big difference in axial and equatorial vibrational frequencies implies no basis for an isotopic bias). One would expect peaks representing two different chemical shifts, unless the chairs interconvert rapidly, in which case one would expect a single peak at an average position. As shown in Figure 13.42 (page 360) the chair-chair interconversion is slow at -89°C (two sharp peaks), but fast at -49° (single peak). At higher temperature the single peak would be even sharper.

Such observations raise the crucial question:

How fast does interconversion have to be in order to see an averaged peak rather than the separate peaks that would be characteristic of a static sample?

The Text (p. 359) answers this question by invoking the Heisenberg Uncertainty Principle. There is nothing really wrong with this, unless, like me, you prefer intuitive understanding rather than appeals to the authority of famous physicists. Here is an alternative way of understanding what is going on:

Consider how long you would have to "observe" a sample in order to measure the frequency of its lines precisely. You measure the frequency by counting cycles (e.g. of nuclear precession) and dividing by the time interval. You're not allowed to count fractional cycles, just the nearest integer.

(The reason for this rule is that you are counting by matching the magnetic vector of a light wave with the precession, if during the interaction the light wave stays more-or-less in-phase with the precession, it can be absorbed. But if it gets half a cycle out-of-phase during the interaction, it will have no net interaction and will not be absorbed. Thus a really short pulse of light can interact with a wide range of adjacent frequencies, because it's not around long enought to get out-of-phase with any of them, while a long pulse can interact only with a very narrow range.)

If you want to distinguish between two nearby peaks, you must count at least long enough that they would differ by about one cycle. Thus if one peak were at 100,000,000 Hz (100 MHz, a typical nmr frequency) and the other at 100,000,100 Hz (1 ppm higher) one would have to count for about 1/100 sec to tell them apart (the first would make 1,000,000 cycles, the second 1,000,001 cycles).

If during this 1/100 sec the protons giving rise to these two peaks exchanged environments and frequencies several times (e.g. by the axial-equatorial interconversion of cyclohexane-d₁₁), a wave that would stay in phase overall would need to have the average frequency of 100,000,050 Hz, more or less.

Thus the "NMR time-scale" for averaging of two peaks is the reciprocal of the difference in frequency of the peaks. This is how long the frequencies would have to remain distinct in order to be distinguishable.

Thus in pure ethanol at room temperature (figure 13.44), where the separation among the sharp triplet lines of the OH signal is about 6 Hz, protons must be jumping from one ethanol to another much more slowly that once every 1/6 seconds, while with 1% HCOOH added they much be jumping much more rapidly than this.

Or in the case of cyclohexane-d₁₁ the rate of chair-chair interconversion averaging changes from "slow" to "fast" on the "NMR time-scale" at about -60°C. The book tells us that the difference in chemical shift for these peaks is 0.5 ppm, but it does not tell us what the frequency difference is! The frequency difference will depend on how large the magnetic field of the spectrometer is. If the field used in measuring the spectra of Figure 13.42 were such as to give proton resonance at about 40 MHz, the peak difference (0.5 ppm) would be 20 Hz, meaning that the rate of chair-chair interconversion at -60°C is about 20 times per second, but if it gave proton resonance at about 800 MHz, the rate would be 40 times faster at -60°C.

In fact these spectra were measured in the olden days (mid-1960s) with a spectrometer operating at 60 MHz, meaning that the rate of interconversion must have been about 30 times per second.

In fact the Text is correct that the "coalescence" into a single broad peak occurs when the rate of interconversion is 2p times the difference in frequency, rather than the difference in frequency itself, but this is a rather minor correction to our qualitative picture of how averaging works.