QualitativeAnalysis and the Binomial Expansion A question that often confronts an instructor in preparing qualitative unknown mixtures is "How many &-different solutions can he prepared from a limited number of ions or compounds?" The answer can he neatly supplied by the hinomid exnansion. -~~ Let un limit ourselves to four different amino acids that can be mired cogpther in all possible combinations. The cotal numher of different unknowns would be 2' or 16. These 16 would contain all four, any three, two, one, or none at all of the amino acids. The expansion of (a + hPgiven a cmnplete rundown of the poiarhilities: r
-
~~
~
~
~
~
The exponent of each a indicates the number of amino acids in the unknown, while the coefficient tells us how many combinations are possible with that exponent. Obviously there is only one way to have all four amino acids present in one sample. If three of the four are to be induded, we see that there are four ways to leave out just one ofthe four. This agrees perfectly with the term 4a3b;four ways to have three of the amino acids present. Incidentally, the exponent of the b tells how many amino acids are omitted. Thus the final term in the expansion, b4, which is equivalent to a%', says there is one way to leave out all four compounds. The sum of all the coefficients in the expansion is equal to 16.
momas McCullough, C.S.C. St. Edward's Universily Austin. TX 78704
490
Journal of Chemical Education
