A rod of length carries a diffusing species with

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Numerical and computational mathematics

Computational Physics, Ch Rules on a lattice: diffusion, percolation and growth

Problem

A rod of length L=0.50 mL = 0.50\,\mathrm{m} carries a diffusing species with D=1.0×10−9 m2 s−1D = 1.0\times10^{-9}\,\mathrm{m^2\,s^{-1}}. Both ends of the rod are held fixed at zero concentration, representing perfectly absorbing walls, and at t=0t=0 the concentration already has the shape of the slowest surviving mode of the rod, c(x,0)=c0sin⁡(πx/L)c(x,0) = c_0\sin(\pi x/L) with c0=4.0 mol m−3c_0 = 4.0\,\mathrm{mol\,m^{-3}}. That shape decays without changing shape, so the total amount in the rod falls as e−π2Dt/L2e^{-\pi^2 D t/L^2}. (a) Find the time τ\tau at which the total amount has fallen to 1/e1/e of its initial value. (b) Predict the average concentration in the rod at t=τt=\tau.

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