# 1-D Fick's first law - partial derivative?

I've recently been reviewing some concepts, including diffusion. Fick's 1st law:

$$J = -D\frac{\partial C(x,t)}{\partial x}$$

as I understand it, applies to the steady state.

For 1-D diffusion and in the steady state, why is the expression not

$$J = -D\frac{\mathrm dc(x)}{\mathrm dx}$$

s there is only one spatial variable and no time dependence (steady state)?

• Fick's law also applies to non-steady states. The only requirement is that the $t$ and $x$ dependence is slowly varying compared to collisional time and length scales. Commented Feb 3, 2016 at 2:55
• I'm not sure I understand your comment. I thought Fick's second law applies to the unsteady state? Commented Feb 4, 2016 at 15:24

The distinction between Fick's first and second law is mostly historical (it is not a distinction you will find in any reasonably modern treatment of non-equilibrium physics). Fick's law is $$\vec{\jmath}= -D\vec{\nabla}c(\vec{x},t).$$ Then Fick's first law is the trivial special case $c(\vec{x})$, and Fick's second law is Fick's law combined with current conservation $$\partial_0 c+\vec{\nabla}\cdot\vec{\jmath}=0.$$ As to why we think Fick's law is correct (for general, but smooth, non-equilibrium states), see, for example, here: Rigorous derivation of Fick's first law .