# How to determine a boundary condition at a thin membrane for diffusion from experimental data

## TL;DR

This paper introduces a novel method to derive boundary conditions at thin membranes for diffusion processes directly from experimental data, revealing a fractional derivative term that indicates long memory effects in particle transfer.

## Contribution

It presents a new approach to determine boundary conditions from experimental data, including fractional derivatives, enhancing the modeling of diffusion through membranes.

## Key findings

- Boundary condition contains a fractional time derivative of order 1/2.
- Transfer through the membrane exhibits long memory effects.
- Method demonstrated on ethanol-water diffusion experiments.

## Abstract

We present a new method of deriving a boundary condition at a thin membrane for diffusion from experimental data. Based on experimental results obtained for normal diffusion of ethanol in water, we show that the derived boundary condition at a membrane contains a term with the Riemann--Liouville fractional time derivative of the $1/2$ order. Such a form of the boundary condition shows that a transfer of particles through a thin membrane is a `long memory process'. Presented method is an example that an important part of mathematical model of physical process may be derived directly from experimental data.