Directional symmetry of the time lag for downstream absorptive permeation studied by the matrix method

The directional symmetry of the downstream time lag for absorptive permeation, accompanying a first-order reaction, across a heterogeneous membrane has been proven by the matrix method based on the theory of Siegel. Owing to the heterogeneity the partition coefficient (K), diffusivity (D) and rate constant (k) are all dependent on position. The directional symmetry was first shown for multiple laminates with each laminate having a distinct K, D and k. The transmission matrix of the system is then a sequential product of those of the individual laminates. The proof is for the unit value of the determinant of the transmission matrix for each laminate and, hence, for the whole system. Directional symmetry is thus proven for the heterogeneous membrane, since it can be visualized to be an assembly of an infinite number of infinitely thin, homogeneous laminates.