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

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

• A. The center of the sphere
• B. The surface of the sphere
• C. The equator of the sphere's cross-section
• D. Uniform concentration throughout the sphere

Answer: (A)

In steady-state diffusion into a sphere, diffusion is spherically symmetric, so the concentration depends only on radius. The governing equation in that case leads to dC/dr proportional to 1/r^2, which comes from ensuring the net flux through any spherical shell is the same regardless of the radius. Since dC/dr scales as 1/r^2, its magnitude grows as r gets smaller, reaching its largest value toward the center. Physically, the same amount of diffusive flux must pass through every concentric spherical surface, so as the surface area decreases with smaller radius, the local concentration gradient must increase to carry that flux. Therefore the location where the gradient is steepest is the center of the sphere, making that the best description. The surface would have a smaller gradient in this steady-state, the equator isn’t singled out in this radial symmetry, and uniform concentration would imply no gradient at all.

In steady-state diffusion into a sphere, diffusion is spherically symmetric, so the concentration depends only on radius. The governing equation in that case leads to dC/dr proportional to 1/r^2, which comes from ensuring the net flux through any spherical shell is the same regardless of the radius. Since dC/dr scales as 1/r^2, its magnitude grows as r gets smaller, reaching its largest value toward the center. Physically, the same amount of diffusive flux must pass through every concentric spherical surface, so as the surface area decreases with smaller radius, the local concentration gradient must increase to carry that flux. Therefore the location where the gradient is steepest is the center of the sphere, making that the best description. The surface would have a smaller gradient in this steady-state, the equator isn’t singled out in this radial symmetry, and uniform concentration would imply no gradient at all.