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In this paper we investigate the linear stability behavior of a flow within an adsorption/diffusion model for porous materials summarized in Section 2. We consider a 1D stationary flow through a poroelastic medium which is perturbed by transversal (2D) disturbances with mass exchange. The eigenvalue problem for the first step field equations is solved using a finite-difference scheme. We present the instability regions in dependence on the three most important model parameters, namely the bulk and surface permeability coefficients, and the mass density of the adsorbate on the inner surface.