Check the dimensional consistency of the equation J = D x A x ( L)/( x)

• $J$ (Drug diffusion rate): Mass of drugs crossing membrane per second. $$[J] = [M T^{-1}]$$
• $A$ (Area): Length squared. $$[A] = [L^2]$$
• $\frac{\Delta L}{\Delta x}$ (Concentration gradient): Concentration ($\Delta L$) is typically mass per unit volume. $$[\text{Concentration}] = [M L^{-3}]$$ $\Delta x$ is a change in position, so it has dimensions of length. $$[\Delta x] = [L]$$ Therefore, the dimensions of the concentration gradient are: $$\left[\frac{\Delta L}{\Delta x}\right] = \frac{[M L^{-3}]}{[L]} = [M L^{-4}]$$
• $D$ (Diffusion coefficient, denoted as $\Delta$ in the image): The standard dimensions for a diffusion coefficient are length squared per unit time. $$[D] = [L^2 T^{-1}]$$

The equation is $J = D \times A \times \frac{\Delta L}{\Delta x}$.
We will compare the dimensions of the Left Hand Side (LHS) with the dimensions of the Right Hand Side (RHS).

LHS dimensions:

RHS dimensions:

Substitute the dimensions found in Step 1:

Combine the terms:

The dimensions of the LHS are $[M T^{-1}]$.
The dimensions of the RHS are $[M T^{-1}]$.

Since the dimensions on both sides of the equation match, the equation is dimensionally consistent.