Matrix representation of solution mixing by aliquot exchange

We present a mathematical description of mixing two solutions by exchanging aliquots back and forth between them. We propose that this method of aliquot exchange can be used to automate calibration curve preparation in a way that produces less solvent waste than conventional serial dilution methods. We also show its use in quickly mixing solutions. The process of aliquot exchange is represented mathematically by a 2 × 2 symmetric matrix, A, that is a function of the volume or percentage of liquid, p, that is exchanged. Each cycle of aliquot exchange is represented by operating the matrix, A, on the previous concentrations. That is, after n mixing cycles, the final concentrations (C(i)(n)()) are given by A(p(n))A(p(n)(-1))...A(p(1)) operating on the initial concentrations (C(i)(0)), or (A(p))(n)()C(i)(0) if the same amount of liquid is exchanged in each step. We observe close agreement between theory and experiment. For solutions that have equal initial volumes, both the matrix A(p) and the product of any number of such matrices that may have the same or different values of p have equal diagonal and equal off-diagonal elements (they are symmetric), the sum of the elements in any row or column sums to unity, and the operation of any of these matrices on a set of concentrations produces two new concentrations that sum to the same value as the sum of the initial concentrations. We follow the mixing process for two solutions of equal volume by plotting the matrix element A(12)(n)() of A(n)(), which approaches 0.5 as n increases. As expected, the larger the exchanged aliquot, the more quickly the solutions mix. By varying the fraction, p, that is exchanged, we show that it should be possible to produce a calibration curve with values that vary in concentration over at least 3 orders of magnitude from just two solutions.