An FCC iron-carbon alloy initially containing \(0.20 \mathrm{wt} \% \mathrm{C}\) is carburized at an elevated temperature and in an atmosphere wherein the surface carbon concentration is maintained at \(1.0 \mathrm{wt} \%\). If after \(49.5 \mathrm{~h}\) the concentration of carbon is \(0.35 \mathrm{wt} \%\) at a position \(4.0 \mathrm{~mm}\) below the surface, determine the temperature at which the treatment was carried out.

Understanding the basics of diffusion is essential in fields such as material science and metallurgy. Fick's Laws of Diffusion help us quantify the process. The first law states that the diffusion flux is proportional to the concentration gradient. In simple terms, atoms or molecules will move from an area of high concentration to an area of low concentration at a rate dependent on this gradient. The second law, which is used in the provided exercise, takes this a step further by describing how the concentration changes with time.

It assumes an initially uneven distribution of concentrations which levels out over time due to the motion of atoms or molecules. In carburizing of iron-carbon alloys, Fick's second law can be utilized to predict the change in carbon concentration at a given depth over time, which is a critical aspect of strengthening the material's surface.

In the context of carburizing, the carbon concentration profile is a graph that plots carbon concentration as a function of distance from the alloy's surface. Carburizing enhances the surface properties of the alloy by introducing additional carbon into the iron matrix.

The profile is not linear due to the nature of diffusion—it is steeper near the surface where carbon is introduced and flattens out with increasing depth. In our textbook exercise, an error function (erf) is used to represent this profile mathematically. The profile is crucial for understanding the distribution of carbon and how it affects the properties of the material at different depths.

The diffusion coefficient, denoted as D, is a critical parameter describing how quickly atoms or molecules diffuse within a material. It essentially represents the ease with which particles can move through a medium.

This value varies with temperature, which is why Fick's second law includes it in the form of a temperature-dependent variable. The diffusion coefficient is integral in calculating the amount of time required for the diffusion process to reach a certain level, which can be directly plugged into the mathematical model provided by Fick's law to predict the carbon concentration profile.

The concept of activation energy, noted as Q in diffusion calculations, is a fundamental concept in chemistry and materials science. It is the energy barrier that must be overcome for atoms to move from one position to another within a solid.

In the carburizing process, this energy barrier affects how rapidly carbon atoms can diffuse into the iron-carbon matrix. The relationship between the diffusion coefficient and activation energy is described by an Arrhenius-type equation, as shown in the exercise. A higher activation energy implies that higher temperatures are required for the diffusion process to proceed at a significant rate. Calculating the activation energy is necessary to determine the temperature at which a carburizing treatment should be performed to achieve the desired material properties.