The steady-state diffusion flux through a metal plate is \(7.8 \times 10^{-8} \mathrm{kg} / \mathrm{m}^{2}-\mathrm{s}\) at a tem- perature of \(1200^{\circ} \mathrm{C}(1473 \mathrm{K})\) and when the concentration gradient is \(-500 \mathrm{kg} / \mathrm{m}^{4}\). Calculate the diffusion flux at \(1000^{\circ} \mathrm{C}(1273 \mathrm{K})\) for the same concentration gradient and assuming an activation energy for diffusion of \(145,000 \mathrm{J} / \mathrm{mol}\).

In the realm of materials science, Fick's first law is an essential principle that connects the idea of diffusion flux with how substances move in response to concentration differences. It states that the flux of material through an area is directly proportional to the concentration gradient across that area. In simpler terms, it explains how particles flow from regions of high concentration to regions of low concentration. This behavior is mathematically represented as

\[ J = -D \frac{dC}{dx} \]

where \(J\) stands for the diffusion flux, \(D\) is the diffusion coefficient indicative of how easily particles diffuse within the material, and \(\frac{dC}{dx}\) represents the concentration gradient, or how the concentration changes with distance. The negative sign signifies that diffusion occurs in the direction of decreasing concentration.

The Arrhenius equation is a vital tool that describes how the diffusion coefficient \(D\) varies with temperature. It is particularly useful in materials science for predicting how temperature changes affect the rate at which atoms diffuse. The Arrhenius equation is given by:

\[ D = D_0 e^{-\frac{Q}{RT}} \]

where \(D_0\) is known as the pre-exponential factor (a constant that involves factors like vibration frequency), \(Q\) is the activation energy for diffusion, \(R\) is the universal gas constant (8.314 J/mol·K), and \(T\) is the absolute temperature in Kelvins. According to this equation, as the temperature increases, the exponential term decreases, allowing \(D\) to increase, which means the diffusion process accelerates.

The term concentration gradient plays a pivotal role in the study of diffusion. Simply put, it is the measure of how the concentration of a substance changes from one location to another. Mathematically, it's the change in concentration \(dC\) over a distance \(dx\), expressed as \(\frac{dC}{dx}\). A high concentration gradient implies a steep change in concentration over a short distance and typically results in a higher diffusion flux according to Fick's first law. Imagine a hill—the steeper the hill (or the higher the concentration gradient), the faster a ball (or diffusing particles) would roll down (or move across the gradient).

In the processes of diffusion and chemical reactions, the concept of activation energy, represented by \(Q\), is crucial. It is the minimum energy barrier that must be overcome for atoms or molecules to move and rearrange themselves. In the context of diffusion, it is the energy required for atoms to break away from their positions and jump to new ones, enabling the transport of matter. A high activation energy means that the substance will diffuse more slowly at a given temperature, and vice versa. The Arrhenius equation incorporates this activation energy into the exponential term, linking it to temperature and highlighting its influence on the rate of diffusion.