In steady-state diffusion into a sphere, the center is defined as the location where the concentration gradient is steepest. Which option best represents the center?

In steady-state diffusion into a sphere, the center is defined as the location where the concentration gradient is steepest. Which option best represents the center?

• A. The center of the sphere
• B. The surface of the sphere
• C. The midpoint along the radius
• D. Uniform throughout the sphere

Answer: (A)

In a spherically symmetric steady-state diffusion problem, the gradient is radial and becomes stronger as you move toward the center because the radial derivative scales as dC/dr ∝ 1/r^2. This means the smallest radius (the center) exhibits the largest magnitude of the concentration gradient. The center is therefore the location where the gradient is steepest. The surface is where the boundary condition is applied, not where the interior gradient is maximized. The midpoint along the radius isn’t singled out by the math, and a uniform concentration throughout would imply no gradient at all. Hence the center best represents the point of steepest gradient.

In a spherically symmetric steady-state diffusion problem, the gradient is radial and becomes stronger as you move toward the center because the radial derivative scales as dC/dr ∝ 1/r^2. This means the smallest radius (the center) exhibits the largest magnitude of the concentration gradient. The center is therefore the location where the gradient is steepest.

The surface is where the boundary condition is applied, not where the interior gradient is maximized. The midpoint along the radius isn’t singled out by the math, and a uniform concentration throughout would imply no gradient at all. Hence the center best represents the point of steepest gradient.