Carbon is allowed to diffuse through a steel plate \(15 \mathrm{~mm}\) thick. The concentrations of carbon at the two faces are \(0.65\) and \(0.30 \mathrm{~kg} \mathrm{C} / \mathrm{m}^{3}\) Fe, which are maintained constant. If the preexponential and activation energy are \(6.2 \times\) \(10^{-7} \mathrm{~m}^{2} / \mathrm{s}\) and \(80,000 \mathrm{~J} / \mathrm{mol}\), respectively, compute the temperature at which the diffusion flux is \(1.43 \times 10^{-9} \mathrm{~kg} / \mathrm{m}^{2} \cdot \mathrm{s}\).

At the heart of understanding how substances move within solids lies Fick's First Law of Diffusion. This fundamental principle expresses how diffusion flux, the rate at which particles spread out, is driven by differences in concentration. Mathematically, it is given by the equation:
\[\begin{equation}J = -D\frac{dC}{dx}\end{equation}\]
Here, 'J' signifies the diffusion flux, indicating the amount of substance that flows through a unit area per unit time. A critical factor in this equation is 'D', the diffusion coefficient. It quantifies the ease with which particles can move within the solid and typically relies on factors like temperature. The negative sign in the equation indicates that diffusion occurs from regions of higher concentration to regions of lower concentration. Breaking down the terms, 'dC' represents the concentration difference, and 'dx' is the distance over which this concentration change occurs. To understand how this law applies to real-world situations, consider carbon atoms diffusing through steel, where they move from areas of high carbon concentration towards areas where it is lower.When handling problems related to diffusion in solids, assessing the concentration gradient is step one, as this gradient is the driving force for diffusion. Emphasizing the role of the concentration gradient and the diffusion coefficient can significantly improve the comprehension of actual diffusion processes.

To delve deeper into how temperature affects diffusion in solids, we turn to the Arrhenius equation. This equation is key for understanding the temperature dependence of the diffusion coefficient, and it is given by:\[\begin{equation}D = D_0\,e^{\frac{-Q}{RT}}\end{equation}\]
The 'pre-exponential factor' (\[\begin{equation}D_0\end{equation}\]) and the 'activation energy' (Q) are intrinsic properties of the material and provide clarity on how temperature influences diffusion. The constant 'R' represents the ideal gas constant, and 'T' is the absolute temperature in kelvins.A crucial aspect of the Arrhenius equation is the exponential relationship, indicating that even small changes in temperature can result in significant variations in the diffusion coefficient. This is because the activation energy, typically expressed in joules per mole, denotes the energy barrier that the diffusing particles must overcome.Practically, this means that raising the temperature will increase the diffusion coefficient, enabling atoms or molecules to move more quickly through the solid. Students often benefit from visualizing this concept: picture a landscape with hills where temperature provides the energy for atoms to 'climb' over the hills, thus moving from one side to the other.

The concentration gradient is the cornerstone of diffusion processes. It determines the direction and rate of diffusion. Imagine a slope: just as a ball rolls from the higher end to the lower, particles move from an area of higher concentration to one of lower concentration. This motion persists until a uniform distribution is achieved, characterized by the absence of a concentration gradient.Mathematically, the concentration gradient is found by taking the difference in concentration between two points and dividing by the distance between those points. A steeper gradient means a faster rate of diffusion. In the context of the example from our exercise, carbon atoms distribute through steel according to the concentration gradient:\[\begin{equation}\frac{dC}{dx} = \frac{C_1 - C_2}{dx}\end{equation}\]Here, \[\begin{equation}dC\end{equation}\] is the change in concentration of the carbon and \[\begin{equation}dx\end{equation}\] is the distance over which this change occurs—in this case, the thickness of the steel plate. It's pivotal for students to grasp this concept as it not only applies to diffusion in solids but also to liquids and gases. In fact, gradients are ubiquitous in nature and understanding their role plays a fundamental part in fields ranging from physics and chemistry to biology and environmental science.