Nitrogen from a gaseous phase is to be dif? fused into pure iron at \(700^{\circ} \mathrm{C}\). If the surface concentration is maintained at \(0.1 \mathrm{wt} \% \mathrm{~N}\) what will be the concentration \(1 \mathrm{~mm}\) from the surface after \(10 \mathrm{~h}\) ? The diffusion coefficient for nitrogen in iron at \(700^{\circ} \mathrm{C}\) is \(2.5 \times 10^{-11} \mathrm{~m}^{2} / \mathrm{s}\).

Understanding diffusion in solids is crucial for grasping how substances like nitrogen can penetrate materials such as iron. Diffusion, a thermal process, involves the movement of atoms or molecules from regions of higher concentration to areas of lower concentration. In solids, this motion is generally slower compared to liquids and gases due to the tightly packed structures of solid materials.

In the context of the exercise, nitrogen atoms are diffusing into the iron's lattice. This process is significant in many industrial applications, such as hardening surfaces and altering the chemical composition of metals to improve their properties. Factors like temperature, which is high in the given example, play a crucial role in increasing the diffusion rate as the atoms gain kinetic energy and move more freely through the lattice voids.

The concentration gradient is a term used to describe the variation in the concentration of a substance within a region. It acts as a driving force behind diffusion, where the substance moves from a place of higher concentration to one of lower concentration trying to reach equilibrium.

In the exercise, the gradient is established between the high nitrogen concentration on the iron surface and the lower concentration within the iron. Over time, the nitrogen atoms will spread into the iron, decreasing the gradient, and hence, reducing the diffusion rate. This concept is fundamental because the rate at which nitrogen diffuses into the iron is directly proportional to the steepness of this gradient.

The error function, denoted as 'erf', appears in diffusion problems to address the complex behavior of diffusion in error function solutions to Fick's second law. It's a special mathematical function that cannot be expressed as a simple equation but is tabulated or computed using software.

In our problem, the error function quantifies the amount of nitrogen that has diffused into the iron at a certain distance from the surface after a specific time. Calculating it, as we've seen in the solution steps, is essential to finding the concentration at 1 mm from the surface. Through 'erf', we can precisely define the relationship of concentration with both distance and time in a diffusion process.

Mass transport is a broad term in materials science, including the movement of atoms and molecules through diffusion as well as other mechanisms like convection and migration under an electric field. It is integral to understanding material behavior and engineering material properties.

Diffusion, a type of mass transport, is exemplified by the nitrogen atoms moving into iron in our example. Knowledge of mass transport phenomena is essential for devising materials with desired character, such as strength, ductility, or corrosion resistance. By controlling variables like concentration, temperature, and time, material scientists engineer the mass transport to modify material at atomic levels, optimizing their usage in various applications.