It is known that isotope patterns of molecules and ions form according to the binomial expansion (2). The perennial Pascal’s triangle is a graphical form of such an expansion and it is easy to see that the coefficients in the standard Pascal’s triangle resemble the isotope patterns of bromine. Although the use of Pascal’s triangle is widely advocated in textbook explanations of multiplets in ¹H NMR spectra, considerably less attention is paid to the fact that isotopic distributions of molecules can be explained in a similar fashion. As a slight modification of the standard Pascal’s triangle, isotopic distributions can be rendered in the form of cellular automata, which generate complex isotopic patterns in an educationally friendly manner (3).

As an example, the isotope pattern of PbCl₂ can be considered. Lead has three isotopes in a ratio of ∼1:1:2 (²⁰⁶Pb, ²⁰⁷Pb, and ²⁰⁸Pb) and chlorine has two isotopes in a ratio of ∼3:1 (³⁵Cl and ³⁷Cl). The rather complex isotopic distribution of PbCl₂ can be easily obtained as shown in Figure 1. In this process, the isotopic distribution of PbCl₂ is obtained in two consecutive steps: first, the isotopic distribution of Cl₂ is obtained and then the isotopic distribution of Pb is added leading to the PbCl₂. Such construction of complex isotope patterns captures the very essence of the problem—complex isotope patterns can be generated using very simple rules. As can be seen from the above example, no calculators are needed to obtain the isotopic distribution of PbCl₂.

Figure 1. Complex isotopic distribution of PbCl₂.

Literature Cited

1. Sein, Lawrence T., Jr. J. Chem. Educ. 2006, 83, 228–232.
2. Margrave, J. L.; Polansky, R. B. J. Chem. Educ. 1962, 39, 335–337.
3. Meija, J. Anal. Bioanal. Chem. 2006, 385, 486–499.