In non-steady state diffusion from a constant source, the concentration's dependence on distance is the same as its dependence on what power of time?

In non-steady state diffusion from a constant source, the concentration's dependence on distance is the same as its dependence on what power of time?

• A. -1/2
• B. 1/2
• C. 0
• D. -1

Answer: (A)

The essential idea is a diffusion similarity scaling. For a constant surface source in a semi-infinite medium, the solution has the form C(x,t) = C_s erfc(x/(2√(Dt))). This shows that distance and time enter only through the combination x/√t, so the profile looks the same when distance is scaled with √t. If you look at how concentration changes with time at a fixed distance, the time dependence comes in through t^(-1/2) in the argument of the erfc function. Therefore, the distance dependence mirrors the concentration’s dependence on time to the power of -1/2.

The essential idea is a diffusion similarity scaling. For a constant surface source in a semi-infinite medium, the solution has the form C(x,t) = C_s erfc(x/(2√(Dt))). This shows that distance and time enter only through the combination x/√t, so the profile looks the same when distance is scaled with √t. If you look at how concentration changes with time at a fixed distance, the time dependence comes in through t^(-1/2) in the argument of the erfc function. Therefore, the distance dependence mirrors the concentration’s dependence on time to the power of -1/2.