The binomial formula

is used in chemistry for: (a) calculating the intensities of NMR (nuclear magnetic resonance) lines due to spin-spin splitting (Pascal’s triangle);

(b) in mass spectrometry for calculating the ion abundances (intensities) for elements with several isotopes;

(c) to model the biochemical transition between the helix and coiled structures of proteins; and

(d) in statistical mechanics to calculate the probabilities of a given distribution of molecules over a set of quantum states.

You will meet all of these techniques later on in your Chemistry degree.

**Propane H = gray, C = dark gray**

The natural abundance of the stable carbon isotopes are ^{13}C ≈ 1% and ^{12}C ≈ 99%. Calculate the isotopic abundances arising from just the three carbon atoms in propane (ignoring the H-atom isotopes) by expanding the binomial formula (1 + 99)^{3} into the first 4 terms. These 4 terms correspond to the relative amounts of the following four isotopic molecules Bcgis, ^{13}C_{3} H_{8}, ^{13}C_{2}^{12} CH_{8} and ^{12}C_{3} H_{8} (these isotopic variants are called isotopomers).

**Solution**

The shorthand notation used above is called the “binomial bracket”. There is no dividing line and it does not mean n divided by k. It is pronounced “n choose k” and by the binomial theorem it is equal to

Where a fractional is defined as:

**n! = n (n-1) + (n-2) + (n-3) … … …1; and by definition, 0! = 1**

The carbon atom isotopic abundances in propane are found by expanding the binomial bracket below for the four different isotopomers which “choose” 0, 1, 2, or 3 carbons out of three carbons to be ^{12}C.

Expanding the binomial formulae gives

Cancelling out terms top and bottom within any single term, gives

**(1+99) ^{3 }= 1+(3X99) + (3X99^{2}) + (1X99^{3}) = 1+ (297) + (29403) + ( 970299)**

In mass spectrometry we normally express the ratios as a percentage of the largest abundance to give the following predicted mass spectrum. Notice how rarely we would find in our mass spectrometer a propane molecule with three ^{13}C atoms (about one in a million propane molecules). On the other hand, just over 3% of the propane molecules contain one ^{13}C atom.