Question

The unit of diffusion coefficient is

• A. mol m$^{-2}$ s$^{-1}$
• B. mol m$^{-3}$
• C. m$^{2}$ s$^{-1}$
• D. kJ mol$^{-1}$

Hint:

The diffusion coefficient always has the dimension of {area per unit time}, so its SI unit is {m$^{2}$ s$^{-1}$}.

Answer 1

The Correct Option is C

Step 1: Understanding diffusion coefficient.
The diffusion coefficient ($D$) represents how fast particles spread out due to random motion from a region of higher concentration to a region of lower concentration. It depends on factors such as temperature, medium, and size of particles.
Step 2: Using Fick’s First Law.
According to Fick’s first law of diffusion: \[ J = -D \frac{dC}{dx} \] where $J$ is the flux (mol m$^{-2}$ s$^{-1}$) and $\frac{dC}{dx}$ is the concentration gradient (mol m$^{-4}$).
Step 3: Deriving the unit of diffusion coefficient.
Rearranging the equation: \[ D = \frac{J}{\frac{dC}{dx}} \] Substituting units: \[ D = \frac{\text{mol m}^{-2} \text{ s}^{-1}}{\text{mol m}^{-4}} = \text{m}^2 \text{ s}^{-1} \]
Step 4: Final conclusion.
Thus, the SI unit of diffusion coefficient is m$^{2$ s$^{-1}$}.

Answer 2

Instead of starting from Fick's first law, this question can be worked out from the random-walk picture of diffusion, where a diffusing particle's mean-squared displacement grows in proportion to time, and checking each option's units against that relationship.

1. mol m\(^{-2}\) s\(^{-1}\): These are the units of molar flux, describing how much substance crosses a unit area per unit time, which is a different physical quantity from the diffusion coefficient itself.
2. mol m\(^{-3}\): These are the units of molar concentration, the amount of substance per unit volume, not a measure of how fast that substance spreads.
3. m\(^{2}\) s\(^{-1}\): The random-walk relation for diffusion states that the mean-squared displacement of a particle, \( \langle x^2 \rangle \), is proportional to \( D t \), so rearranging gives \( D = \langle x^2 \rangle / t \); since \( \langle x^2 \rangle \) has units of area (m\(^{2}\)) and \( t \) has units of time (s), the coefficient \( D \) must carry units of m\(^{2}\) s\(^{-1}\).
4. kJ mol\(^{-1}\): These are units of molar energy, relevant to quantities like activation energy, not to a rate of spatial spreading.

The random-walk relationship between mean-squared displacement and time independently confirms the same unit that Fick's law gives for the diffusion coefficient.

Therefore, the correct answer is m\(^{2}\) s\(^{-1}\).