Question

The dimensions of diffusion coefficient is given by:

• A. $M\ L\ T^{-2}$
• B. $L^2\ T^{-1}$
• C. $L\ T^{-1}$
• D. $M\ L^{-2}\ T$

Hint:

Diffusivity always has the dimension of area per unit time, i.e., $L^2/T$, whether it's mass, heat, or momentum diffusion.

Answer 1

The Correct Option is B

The diffusion coefficient, also called diffusivity, represents the rate at which particles (mass, heat, or momentum) spread out over space.
From Fick’s first law for mass diffusion:
\[ J = -D \frac{dc}{dx} \]
Where:
- $J$ is the diffusion flux (amount per unit area per unit time),
- $D$ is the diffusion coefficient,
- $\frac{dc}{dx}$ is the concentration gradient.
To derive the dimensional formula for $D$, rearrange:
\[ D = \frac{J}{\frac{dc}{dx}} \]
Let’s consider the dimensional units of each term:
- $J$: [mass]/[area]/[time] = $M\ L^{-2}\ T^{-1}$
- $dc/dx$: [mass]/[volume]/[length] = $M\ L^{-3}\ L^{-1} = M\ L^{-4}$
Therefore, dimensions of $D$ are:
\[ D = \frac{M\ L^{-2}\ T^{-1}}{M\ L^{-4}} = L^2\ T^{-1} \]
This is consistent with the dimensional form of diffusivity in all contexts — mass diffusivity, thermal diffusivity, or momentum diffusivity (kinematic viscosity).
So, the correct dimension of diffusion coefficient is:
\[ \boxed{L^2\ T^{-1}} \]