In a diffusion treatment at a fixed temperature, by what factor must the treatment time be increased to double the depth at which a given concentration is reached?

Answer: C. 4 times

In the error-function solution the concentration depends only on x/2√(Dt), so the depth x at which a given concentration is reached is proportional to √(Dt). At the same temperature D is constant, so doubling the depth requires doubling √t, that is, four times the time. Two times assumes the depth is proportional to time, and √2 times applies the proportionality the wrong way round. The diffusion-distance estimate x ≈ √(Dt) is used in many situations, including carburizing and dopant diffusion. When a deep layer is needed, raising the temperature to increase D is more effective than extending the time.