When \(\alpha\)-iron is subjected to an atmosphere of ( nitrogen gas, the concentration of nitrogen in the iron, \(C_{\mathrm{N}}\) (in weight percent), is a function of hydrogen pressure, \(p_{\mathrm{N}_{2}}\) (in \(\left.\mathrm{MPa}\right)\), and absolute temperature \((T)\) according to $$ C_{\mathrm{N}}=4.90 \times 10^{-3} \sqrt{p_{\mathrm{N}_{2}}} \exp \left(-\frac{37,600 \mathrm{~J} / \mathrm{mol}}{R T}\right) $$ Furthermore, the values of \(D_{0}\) and \(Q_{d}\) for this diffusion system are \(5.0 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\) and \(77,000 \mathrm{~J} / \mathrm{mol}\), respectively. Consider a thin iron membrane 1.5-mm thick at \(300^{\circ} \mathrm{C}\). Compute the diffusion flux through this membrane if the nitrogen pressure on one side of the membrane is \(0.10 \mathrm{MPa}(0.99 \mathrm{~atm})\) and on the

Understanding multiple principles governing the movement of atoms or molecules through materials is critical. One central concept is Fick's laws of diffusion, which provide a mathematical description of this process in solids. The first law states that the diffusion flux, which measures the amount of substance that will flow through a unit area over a unit time, is proportional to the concentration gradient. In simpler terms, diffusion will occur from regions of high concentration to regions of low concentration, and the rate of this movement depends on how steep the concentration difference is.

The second law of diffusion extends this concept. It suggests that the concentration of particles within a material will change over time as diffusion occurs. Mathematically, it is described as a partial differential equation which gives us insight into how substances spread out or diffuse over time.

The term activation energy for diffusion refers to the minimum amount of energy required for atoms or molecules to move from one location to another within a solid. This movement is essential for the diffusion process. A helpful analogy might be to think of activation energy as the effort needed to push a heavy ball over a hill before it can freely roll down; without enough initial push, the ball won't move. Similar is true for atoms - without the necessary energy, atoms will not jump to different positions within the material. The higher the activation energy, the slower the rate of diffusion, as fewer particles will have the energy to move at a given temperature.

• Activation Energy: Energy needed to initiate diffusion.
• Temperature's Role: Higher temperatures can reduce the effective barrier posed by activation energy, thus enhancing the diffusion rate.

A key driver of diffusion is the concentration gradient. The phrase describes the rate of change in concentration over a certain distance within a material. Where the concentration of particles is not uniform, they tend to move from an area of higher concentration to an area of lower concentration, creating what we call a concentration gradient.

This imbalance triggers the diffusion, aiming to achieve equilibrium, or equal concentration, throughout the material. The gradient can be depicted graphically with concentration on the y-axis and position on the x-axis, where a steeper slope represents a larger gradient and typically a faster rate of diffusion. A high concentration gradient indicates a significant difference over a small area, which often leads to a higher diffusion flux.

The diffusion coefficient, often represented with the symbol D, is a fundamental parameter quantifying the ease with which atoms or molecules diffuse through a material. This coefficient depends on several factors, including the temperature, the medium through which diffusion occurs, and the size and charge of the diffusing particles.

In mathematical terms, the diffusion coefficient is a factor in Fick's first law and it directly influences the diffusion flux. A higher diffusion coefficient means that particles will spread through the material more rapidly. In the case of the given exercise, calculating the diffusion coefficient involves using both the pre-exponential factor and the activation energy, taking into account the temperature of the system. This understanding helps us predict how fast nitrogen will diffuse into iron at a given temperature.