Sequential Filling of a Line by Intervals Placed at Random and Its Application to Linear Adsorption

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Abstract
A line with integral length n is filled sequentially at random with nonoverlapping intervals of integral length a, their end points having integer coordinates. It is shown that, as n tends to infinity, the average value of the length left vacant tends asymptotically to (n+a) A1(a) while its variance tends to (n+a) × aA2(a), where A1(a) and A2(a) are constants whose values are given numerically for all a; these asymptotic results are accurate to four significant figures for n/a>8. Some results are also given for the (average) relative numbers of vacant spaces of length i=0, 1,..., a—1. With certain restrictions, the above results can be applied to the complete adsorption of linear molecules which when once adsorbed remain fixed in position in the line troughs of suitable crystal surfaces such as (110) on an fcc crystal or (112) on a bcc crystal. These restrictions concern the way in which the molecule sits in equilibrium on the underlying crystal sur face.