Learning physics by experiment: IX. Entropy

The creation of a molecule requires the constituent atoms to be in the right place at the right time. The factors involved in this become increasingly more challenging as one progresses from simple molecules to organic molecules, which are necessary precursors to cells. Some insight into the temporospatial aspects involved in even the simplest of building blocks of living matter is provided in this study by examining an intentionally primitive example with equal numbers k of only two types of particle. The statistics for arranging an initially random state of these particles into a highly-ordered configuration are determined, and the number of trials n required to produce at least one instance of this at a given threshold of observation p is also computed. Values obtained from such calculations begin with the case of 2k = 40 (cf. atoms in nucleotides), which has n = 10⁹ to 10¹¹ for p = 0.01 to 0.99, respectively. For more complex molecules where 2k = 100 (and 200) might apply, n = 10²⁷ to 10²⁹ (and 10⁵⁶ to 10⁵⁹) for the same p. Results of physical experiments and virtual simulations conducted for small k agree closely with analytical values. An account of the instructional approach for presenting the above material to students is included, with an emphasis on ab initio methods.