How to Find the Coupling Constant

Written by john brennan
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Chemists often use nuclear magnetic resonance (NMR) spectroscopy to determine the structure of organic compounds. The data from proton-NMR will be a series of "peaks" on a graph. You can use this data to deduce a surprising amount of information about the molecule's structure. Sometimes it can be helpful to measure the coupling constants, the distance between the individual peaks in a group of peaks. If you have a simple group of peaks like a doublet, a triplet or a quartet, this will usually be fairly straightforward. It may not be possible, however, to find a coupling constant for more complex multiplets.

Skill level:
Moderately Challenging

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  • NMR data

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  1. 1

    Start by thinking about how NMR works. Just like electrons, neutrons and protons have a property called spin. A hydrogen nucleus has only one lone proton, so it has two possible spin states, either spin = +1/2 or spin = -1/2. Ordinarily these two states have equal energy. When you place the sample in a magnetic field, however, the hydrogen nucleus behaves somewhat like a tiny magnet in the sense that it can either line up with the field, in which case it will have one spin state, or line up against it, in which case it will have the other. The two spin states now have unequal energy. At any one given time, the majority of the hydrogen nuclei in a given population will be in the lower-energy spin state, but a small fraction will be in the higher state. When the sample is irradiated with radio waves, the radio photons will be absorbed by hydrogen nuclei if and only if their energy is equal to the difference in energy between the two spin states.

  2. 2

    Notice that the electron cloud around a hydrogen atom will move in a way that creates a small magnetic field countering the much larger external magnetic field. Consequently, the more electron-rich the hydrogen atom, the less the difference in energy between the spin states will be. Depending on their environment in the molecule, some hydrogen atoms will be more electron-rich than others. A hydrogen atom attached to an oxygen, for example, would be much more electron-poor than one attached to a -CH3 methyl group.

  3. 3

    Recall that NMR data is in the form of a graph that uses tetramethylsilane as a reference. In other words, rather than displaying the frequency of the radio waves that were absorbed, the graph displays the difference in Hz between the frequency absorbed by tetramethylsilane and the frequency absorbed by your sample divided by the spectrometer frequency in MHz. The resulting units are ppm or parts per million. The number of ppm increases as you go down the graph towards the left, and decreases as you go towards the right. This means that the frequency of radio waves absorbed by the sample increases as you go towards the left.

  4. 4

    Notice that for hydrogen atoms situated such that other hydrogens are attached to neighbouring carbons, there are several different possibilities. The neighbouring hydrogens could be spin +1/2 or spin -1/2, and both of these situations will be found in different molecules in the solution at any given time. If the neighbour is aligned with the magnetic field, it will strengthen the field felt by the first hydrogen atom, and thereby increase the frequency of radiation it absorbs. If the neighbour is aligned against the magnetic field, it will decrease the field felt by the first hydrogen atom, and hence the frequency of radiation it absorbs as well. This effect is called spin-spin coupling, and the difference in Hz between the peaks created by spin-spin coupling is called the coupling constant. It's important to note that equivalent nuclei do not couple with each other.

  5. 5

    Start with the simplest situation: a coupling constant for a doublet (two peaks). Convert the ppm value (the chemical shift) for each peak into a value in hertz (Hz) by multiplying it by the spectrometer frequency in MHz. If the two peaks have chemical shifts of 0.89 ppm and 0.91 ppm, for example, and the spectrometer frequency is 90MHz, the chemical shifts as expressed in Hertz are 80.1 and 81.9, and the coupling constant is just the different between them: 81.9 - 80.1 = 1.8 Hz.

  6. 6

    Tackle a slightly more difficult situation -- two chemically equivalent neighbours (i.e. neighbours that are in the same chemical environment, like two hydrogens on a -CH2 group). These will turn the first hydrogen into a triplet. In this case, the coupling constant will be equal to the difference in Hz between either of the side peaks and the central peak. Just as with the doublet, you need to convert from ppm to Hz before you calculate the coupling constant. If the triplet has peaks with 80 Hz, 88 Hz and 96 Hz, for example, the coupling constant would be 8 Hz, because both the 80 Hz peak and the 96 Hz peak are 8 Hz from the central peak.

  7. 7

    Consider a slightly more interesting situation, where a peak is split by three chemically equivalent neighbours. We might have a hydrogen on a carbon atom, for example, which is attached to a methyl group (-CH3); the three hydrogens of the methyl group will split our first hydrogen's peak into a quartet of four peaks. The distance between any two neighbouring peaks in the quartet will be the coupling constant. If the four peaks are 80 Hz, 88 Hz, 96 Hz and 104 Hz, for example, the coupling constant would again be 8 Hz, since the difference between any two neighbouring peaks is 8 Hz.

Tips and warnings

  • It's very difficult to calculate the coupling constant for complex multiplets and situations where the neighbouring hydrogens are not chemically equivalent, in which case there may be different coupling constants for each. Generally you won't need to calculate the coupling constant to solve basic NMR problems, however, because the number of peaks in a group and its location is actually far more useful than the coupling constants as a general rule.

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