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(Solved): A substance has a diffusion constant \( D \) and its concentration is distributed in space accordi ...
A substance has a diffusion constant \( D \) and its concentration is distributed in space according to \( C(x, t)=A(t) \sin (2 \pi x / L) \) where \( L \) is the wavelength and \( A(t) \) is the amplitude of the distribution. (a) Use the one dimensional diffusion equation to show that the concentration decays exponentially with time as \( A(t) \sim \exp (-t / \tau) \). (b) Determine an expression for the time constant \( \tau \) in terms of \( L \) and \( D \). Which decays faster, a long-wavelength (diffuse) distribution or a shortwavelength (localized) distribution?
Expert Answer
A) C(x,t) = a(t) sin(2??/L) one dimensional diffusion equation:- ?c/?t = D ?2c/??2 ?A/?t sin (2??/L) = -A(t)D(2?/L)2 sin(2??/L) ?dA/A = ?-D(2?/L)2 dt