Fick's first law
/ FIKS /
Fick's first law answers a simple, practical question: if atoms are diffusing steadily through a slab, how fast do they cross it? The everyday intuition is that things flow faster when the difference driving them is steeper. Heat flows faster across a wall when it is much hotter on one side than the other; water runs faster down a steeper hill. Diffusion obeys the same common sense: atoms flow faster when the concentration is much higher on one side than the other. Fick's first law just writes that idea as an equation.
In symbols, J = -D times (dC/dx). Here J is the flux — how many atoms cross one square meter of area per second (units like kg/m^2-s or atoms/m^2-s). The term dC/dx is the concentration gradient — how steeply the concentration C changes with position x (concentration per meter). D is the diffusion coefficient, a number measuring how mobile the atoms are in this particular material at this temperature. The minus sign simply says atoms flow downhill, from high concentration toward low, opposite to the direction the concentration increases. Double the gradient and you double the flux; nothing on the left is changing with time.
The crucial fine print is the phrase steady state: Fick's first law applies only when the concentration profile is not changing with time — atoms enter one face at exactly the rate they leave the other, so the gradient stays fixed. Think of a pressurized hydrogen tank whose steel wall has a constant high hydrogen concentration on the inside and near-zero outside: once settled, hydrogen leaks through at a steady rate you can compute directly from J = -D dC/dx. When the profile is still building up and changing with time, you need Fick's second law instead.
A steel sheet 2 mm thick separates a gas holding 1.2 kg/m^3 of carbon on one face from 0.8 on the other, with D = 3 times 10^-11 m^2/s. The gradient is (1.2 - 0.8)/0.002 = 200 kg/m^4, so the steady flux is J = -(3 times 10^-11)(−200) = 6 times 10^-9 kg per square meter per second. Steeper gradient or higher temperature (bigger D) would push more carbon through.
Flux equals diffusivity times gradient — the steady-state workhorse for leakage and membrane problems.
Fick's first law holds only at steady state (a fixed, unchanging profile). For a case that is still building — like carbon soaking deeper into a gear over time — you must use Fick's second law.