A sheet of steel 2.5 \(\mathrm{mm}\) thick has nitrogen atmospheres on both sides at \(900^{\circ} \mathrm{C}\) and is permitted to achieve a steady-state diffusion condition. The diffusion coefficient for nitrogen in steel at this temperature is \(1.2 \times 10^{-10} \mathrm{m}^{2} / \mathrm{s},\) and the diffusion flux is found to be \(1.0 \times 10^{-7} \mathrm{kg} / \mathrm{m}^{2}\) -s. Also, it is known that the concentration of nitrogen in the steel at the high-pressure surface is \(2 \mathrm{kg} / \mathrm{m}^{3} .\) How far into the sheet from this highpressure side will the concentration be 0.5 \(\mathrm{kg} / \mathrm{m}^{3} ?\) Assume a linear concentration profile.

Diffusion is a fundamental material's behavior that explains how particles, such as atoms, ions, or molecules, move through a medium, like a solid, liquid, or gas. This movement is driven by the kinetic energy of particles at temperatures above absolute zero, allowing them to randomly vibrate and migrate from regions of higher concentration to regions of lower concentration, a phenomenon known as the concentration gradient.

In the context of materials science, diffusion is particularly important for processes like the hardening of steel, where elements like carbon and nitrogen penetrate the metal's surface. The rate of this process can depend on factors such as the temperature, the nature of the diffusing substance, and the medium it's diffusing through. In the given exercise, nitrogen atoms are diffusing into the steel sheet, and understanding this mechanism is crucial for tailoring materials with desired properties.

Steady-state diffusion refers to a condition in which the diffusion flux does not change with time. This means that the amount of substance entering a volume is equal to the amount leaving, resulting in a constant concentration profile over time. Such a state is typically achieved after sufficient time has passed since the beginning of the diffusion process, allowing it to reach equilibrium.

In the exercise provided, the steel sheet has reached a steady-state diffusion condition, implying that the amount of nitrogen entering and leaving at any point within the steel is consistent. This allows for the simplification of Fick's First Law since the flux (J) and the concentration gradient (dC/dx) remain constant across the medium. Calculating when the steady-state condition is met can be vital for controlled material fabrication processes where the consistent distribution of diffusing particles is crucial.

The concentration gradient is a term used to describe the rate of change of concentration of a substance within a region. It is essentially the driving force of diffusion; particles will naturally move down the concentration gradient, from higher to lower concentrations. Mathematically, the gradient can be represented as a derivative, indicating how much the concentration changes per unit distance.

In the exercise, a linear concentration profile is assumed, which simplifies the computation of the concentration gradient. This linearity implies a direct relationship between the distance from the source and the concentration of nitrogen within the steel sheet. The concept of concentration gradient is invaluable for engineers and scientists as it informs the design of systems and materials for targeted substance distribution, catalysis, and sensor operations, ensuring optimal performance.