9.5Counting Principles

Counting the microstates available to N particles in n boxes as nᴺ (2⁴=16 for four particles in two boxes), then grouping them into distributions (six of the sixteen put two particles in each box), is the multiplication and combination counting the algebra course names — chemistry is where ‘how many arrangements are there’ turns into a physical entropy.

Enumerating the geometric, optical, linkage, and ionization isomers of one complex — e.g. the three isomeric forms of [Co(en)₂Cl₂]⁺: trans, and a cis enantiomer pair — is a small, hand-drawn counting problem; systematically listing arrangements without double-counting a mirror-image pair is exactly what counting principles formalizes for cases too large to sketch.

The alkane table lists structural-isomer counts exploding from 1 (methane) to 75 (decane) to 60,523 (octadecane) as chain length grows — a visible, memorable instance of combinatorial growth that motivates why counting principles, rather than drawing every isomer by hand, becomes necessary past about ten carbons.