Diffusion & Diffusion Coefficient
Understanding Fick's Law and the mathematics of concentration gradients.
While Graham's Law tells us how fast different gases diffuse relative to their mass, it doesn't give us the full mathematical picture of how particles move through space. To fully understand diffusion, we must look at the concentration gradient and the Diffusion Coefficient ($D$).
1. The Mechanism of Diffusion
Diffusion is the net movement of particles (atoms, ions, or molecules) from a region of higher concentration to a region of lower concentration.
- Driving Force: The movement is driven by the random thermal motion of molecules (Brownian motion). Particles naturally spread out to maximize entropy.
- Concentration Gradient: The difference in concentration over a distance. Diffusion continues until this gradient is eliminated (reaching dynamic equilibrium).
- No External Force: Diffusion occurs spontaneously without any external mechanical stirring or pressure differences (unlike bulk flow).
2. Fick's First Law of Diffusion
In 1855, Adolf Fick formulated the laws of diffusion. Fick's First Law relates the diffusive flux to the concentration under steady-state conditions.
Let's break down the variables:
- $J$ (Diffusion Flux): The amount of substance per unit area per unit time ($mol \cdot m^{-2} \cdot s^{-1}$). It tells us how much gas is flowing through a specific surface area.
- $D$ (Diffusion Coefficient): The constant of proportionality, measured in $m^2/s$ or $cm^2/s$.
- $\frac{dc}{dx}$ (Concentration Gradient): The change in concentration ($dc$) over a change in distance ($dx$).
- The Negative Sign ($-$): Crucial! It indicates that diffusion occurs in the direction of decreasing concentration (down the gradient). Since $\frac{dc}{dx}$ is negative, the minus sign makes the overall flux ($J$) positive.
3. The Diffusion Coefficient ($D$)
The Diffusion Coefficient ($D$) (also called Diffusivity) is a physical constant that measures how easily a specific substance diffuses through a specific medium.
A higher value of $D$ means the gas particles move and intermix very rapidly. A lower value means the process is sluggish. For gases, $D$ is typically in the range of $10^{-5}$ to $10^{-4} \text{ } m^2/s$ (much faster than liquids).
4. Factors Affecting the Diffusion Coefficient
The value of $D$ is not completely constant; it depends heavily on the physical conditions of the gas and its molecular properties.
| Factor | Effect on Diffusion Coefficient ($D$) | Reason |
|---|---|---|
| Temperature ($T$) | $D$ increases as $T$ increases. ($D \propto T^{3/2}$) |
Higher temperature increases the average kinetic energy and velocity of the molecules, causing them to spread faster. |
| Pressure ($P$) | $D$ decreases as $P$ increases. ($D \propto 1/P$) |
Higher pressure packs molecules closer together. This increases the frequency of collisions, shortening the mean free path (distance traveled between collisions), slowing down net forward movement. |
| Molecular Mass ($M$) | $D$ decreases as $M$ increases. ($D \propto 1/\sqrt{M}$) |
Heavier molecules move slower than lighter molecules at the same temperature (This directly aligns with Graham's Law). |
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