A sheet of BCC iron \(1 \mathrm{~mm}\) thick was exposed to a carburizing gas atmosphere on one side and a decarburizing atmosphere on the other side at \(725^{\circ} \mathrm{C}\). After reaching steady state, the iron was quickly cooled to room temperature. The carbon concentrations at the two surfaces of the sheet were determined to be \(0.012\) and \(0.0075 \mathrm{wt} \%\). Compute the diffusion coefficient if the diffusion flux is \(1.4 \times 10^{-8} \mathrm{~kg} / \mathrm{m}^{2} \cdot \mathrm{s}\). Hint: Use Equation \(4.9\) to convert the concentrations from weight percent to kilograms of carbon per cubic meter of iron.

Understanding the fundamental principles of how atoms or molecules move within materials is essential in materials science, and this is where Fick's First Law of Diffusion plays a crucial role. This law gives us a quantifiable relationship between the rate of diffusion and the concentration gradient. Specifically, it's expressed as

\[ J = -D \frac{dC}{dx} \]
In this equation, \(J\) represents the diffusion flux, which is the amount of substance that flows through a unit area per unit time, \(D\) is the diffusion coefficient, which is a property that describes the ease with which atoms or molecules diffuse through a material, \(dC\) is the change in concentration over \(dx\), the change in distance. The negative sign indicates that diffusion occurs in the direction of decreasing concentration.

In the context of the exercise, the concentrations given are on the two surfaces of a sheet after it's reached steady-state diffusion, implying that the diffusion flux does not change with time at any point in the material. Furthermore, we can see that Fick's First Law was used to relate the given concentration gradient to the flux in order to calculate the diffusion coefficient.

Delving deeper into the realm of diffusion, we encounter the concept of steady-state diffusion. This is a scenario where the diffusion flux does not change with time. In more practical terms, the amount of diffusing species entering a volume is equal to the amount leaving that volume over any time period. In such cases, the concentration gradient remains constant over time, which simplifies the calculations.

This constant state is precisely what was achieved in the exercise when the iron sheet was exposed to carburizing and decarburizing atmospheres until reaching steady-state before the cooling process. The constant concentration gradient across the sheet allows us to use Fick's First Law to find the unknown diffusion coefficient by simply plugging in the values for the diffusion flux and the change in concentration across the sheet thickness.

When dealing with diffusion problems, particularly involving solutes in metals like iron, we often need to convert concentration measurements into different units to apply equations properly. In our exercise, we're dealing with carbon concentration conversion from weight percent to kilograms per cubic meter. This conversion is pivotal because our diffusion equations require consistency in units.

The given formula for this conversion is

\[ C = k(\text{wt}\%) = k\frac{100g_{\text{carbon}}}{100g_{\text{iron}} + g_{\text{carbon}}} \]
where \(C\) is the desired concentration in \(kg/m^3\) and \(k\) is a proportionality constant that factors in the density of iron and the specifics of the weight percent to mass-per-unit-volume conversion. This step in the solution process is crucial because it ensures that we are working with the correct units. Once we have the correct units, we can use them to find the diffusion coefficient using Fick's Law, as demonstrated in the problem. In the exercise, the calculation of \(C_1\) and \(C_2\) using the respective weight percentages allows us to define the concentration gradient, a necessary step in solving such diffusion-related problems.