14 Jul 2014

Application of binomial distribution to interpret 31P NMR for aqueous solution of alpha-dodecatungstophosphoric acid, H3[PW12O40]

In-Class Activity

Submitted by Mariusz Kozik, Canisius College

31-P NMR spectrum of alpha-dodecatungstophosphoric acid is a combination of thirteen spectra, each spectrum representing the compound with a different number of 183W isotopes per molecule. In order to fully interpret the spectrum one needs to apply binomial distribution to calculate the mole fractions of the molecules with various numbers of 183W isotopes.

This LO requires an understanding of the satellites concept in NMR spectroscopy, originating from coupling with nuclei whose NMR active isotopes are not 100% abundant.

Learning Goals: 

In answering these questions, a student will

  • apply binomial distribution to calculate probabilities for presence of various numbers of NMR active isotopes of tungsten in the species containing multiple tungsten atoms.
  • use the calculated probabilities to interpret experimentally observed 31P NMR spectrum for alpha-dodecatungstophosphoric acid.
Equipment needs: 


The spectrum was never published in the literature. Only 183W NMR of this compound, 1:1 doublet, was published.

Implementation Notes: 

I do this in-class exercise in Modern Synthetic Method course after one class on multinuclear NMR. Students are given this problem with seven leading questions. I walk around the class and make sure that the teams (4-5 teams of 4 students per team) are on track. The class ends with teams reporting on the blackboard their predicted intensities and comparing their answers with other teams’ answers. Completing this asctivity requires around 50 minutes.


The power point listed as the web resource has some information on the NMR satellites when there is more than one NMR active nucleus with less than 100% abundance. This information is near the very end of the ppt. It discusses coupling with two Pt atoms.


I found no web resource on using binomial distribution in NMR spectroscopy for interpreting satellites.

Time Required: 
one class, 50 minutes
Evaluation Methods: 

This in class activity isn’t formally assessed but the problems that groups encounter are discussed in the "Evaluation Results".

Evaluation Results: 

I found that even after having the concept of satellites explained in class, when not given the leading questions, students try to use n+1 rule. They simply assign the heptet to 6 different W atoms, even though the problem specifically states that all W atoms in this compound are equivalent.

Only after introduction of P(183W)k(184W)12-k symbolism they begin to think in terms of different species exisiting in solution.

I also discovered that most students heard about binomial distribution, but they never thought it could be used in NMR spectroscopy!

Creative Commons License: 
Creative Commons Licence


n+1? I hope you teach them 2nI+1. If they are in an inorganic course they can handle the truth.

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