How To Calculate J Coupling Constants

Organic chemists use a technique called nuclear magnetic resonance spectroscopy, or NMR for short, to analyze organic molecules based on hydrogen and carbon. The test results in a deceptively simple graph show a peak for each atom in the molecule. Defining the relationship between them – the J coupling constant – enables researchers to determine the sample's makeup.

The NMR graph measures the location of each ion by how it resonates within the spectroscope's magnetic field. The resonance shows as a series of peaks. Each peak in the graph corresponds to an element in the molecule, so a molecule containing one carbon atom and three hydrogen atoms shows four peaks. Each grouping of peaks is referred to generally as a multiplet, but they also have specific names determined by the number of peaks. Those with two peaks are called duplets, those with three peaks are triplets and so on. Some are trickier: Four peaks might be either a quadruplet, or it might be a duplet of duplets. The difference is that all peaks within a quadruplet have the same spacing, while a duplet of duplets would show two pairs of peaks with a different spacing between the second and third peaks. The same holds true for quadruplets and other multiplets: The peaks within a given multiplet have the same relative spacing. If the spacing varies between them, you have a grouping of smaller multiplets rather than one large one.

Peaks are measured in parts per million, which – in this context – means millionths of the spectrograph's operating frequency, but J constants are expressed in hertz, so you'll need to convert the peaks before determining the value of J. To do this, multiply the ppm by the spectrograph's frequency in hertz and then divide by a million. If your value was 1.262 ppm, for example, and your spectrograph operated at 400 MHz or 400 million hertz, this gives a value of 504.84 for the first peak.

Repeat that calculation for each peak in the multiplet, and write down the corresponding values. There are online calculators to speed that process, or you can use a spreadsheet or physical calculator if you prefer. To calculate J for a duplet, simply subtract the lower value from the higher. If the second peak results in a value of 502.68, for example, the value for J would be 2.02 Hz. The peaks within a triplet or quadruplet all have the same spacing, so you'll only need to calculate this value once.

In more complex multiplets, such as a duplet of duplets, you need to calculate a small coupling constant within each pair of peaks and a larger one between the pairs of peaks. There are a couple of ways to arrive at the larger constant, but the simplest is to subtract the third peak from the first, and the fourth peak from the second. The spectrograph usually has a margin of error that's roughly plus or minus 0.1 Hz, so don't worry if the numbers vary slightly. Average the two to arrive at the larger constant for this specific example.

In a duplex of triplets, the same reasoning applies. The smaller constant among the three peaks is identical, within the spectrograph's margin of error, so you can calculate J by choosing any peak in the first triplet and subtracting the value for the corresponding peak in the second triplet. In other words, you can subtract the value of peak 4 from the value of peak 1, or the value of peak 5 from the value of peak 2, to arrive at the larger constant. Repeat as needed for larger multiplets, until you've calculated J for each set of peaks.