In the chemical concentration equation for non-steady state diffusion from a constant source, A(x,t) equals which expression?

In the chemical concentration equation for non-steady state diffusion from a constant source, A(x,t) equals which expression?

• A. exp(-x^2/4Dt)
• B. ln(x)/sqrt(Dt)
• C. x/2sqrt(Dt)
• D. 1/(4 pi D t) exp(-x^2/4Dt)

Answer: (C)

Non-steady diffusion from a constant source often leads to a self-similar solution where the profile depends on the combination x divided by the diffusion length √(D t). In other words, the concentration (or the related quantity A) scales with x/(2√(D t)) because the governing diffusion equation ∂A/∂t = D ∂^2A/∂x^2 enforces this √t scaling and a linear dependence on distance near the source in the self-similar form. The expression x/(2√(D t)) directly reflects that self-similar behavior, making it the appropriate representation for A(x,t) in this scenario. The other forms correspond to different physical setups—for example, Gaussian or exponential factors typical of instantaneous point sources in infinite space, or logarithmic forms that don’t arise from the standard diffusion boundary conditions—so they don’t match the non-steady diffusion with a constant source.

Non-steady diffusion from a constant source often leads to a self-similar solution where the profile depends on the combination x divided by the diffusion length √(D t). In other words, the concentration (or the related quantity A) scales with x/(2√(D t)) because the governing diffusion equation ∂A/∂t = D ∂^2A/∂x^2 enforces this √t scaling and a linear dependence on distance near the source in the self-similar form. The expression x/(2√(D t)) directly reflects that self-similar behavior, making it the appropriate representation for A(x,t) in this scenario. The other forms correspond to different physical setups—for example, Gaussian or exponential factors typical of instantaneous point sources in infinite space, or logarithmic forms that don’t arise from the standard diffusion boundary conditions—so they don’t match the non-steady diffusion with a constant source.