The activation energy for the diffusion of copper in silver is \(193,000 \mathrm{J} /\) mol. Calculate the diffusion coefficient at \(1200 \mathrm{K}\left(927^{\circ} \mathrm{C}\right),\) given that \(D\) at \(1000 \mathrm{K}\left(727^{\circ} \mathrm{C}\right)\) is \(1.0 \times 10^{-14} \mathrm{m}^{2} / \mathrm{s}\).

The Arrhenius equation plays a fundamental role in understanding how temperature affects the rate of a chemical process, such as diffusion. It is a mathematical expression that describes the temperature dependence of reaction rates or diffusion coefficients. The equation is given by:
\[\begin{equation}D = D_0 e^{-\frac{Q}{RT}}\end{equation}\]
In the context of diffusion, this equation reveals how the diffusion coefficient () changes as a function of temperature (). The ) is a measure of how quickly atoms or molecules diffuse through a material, and it increases as temperature goes up. ) is the pre-exponential factor, representing the diffusion coefficient at an infinite temperature where the movement of particles is free from any barriers. The exponential factor reveals how the diffusion rate decreases exponentially with an increase in activation energy () or decrease in temperature (). The activation energy is a critical threshold energy that must be overcome for diffusion to occur. is the universal gas constant, and is the absolute temperature in Kelvin. This fundamental relationship is widely used in material science, chemistry, and physics to predict how diffusion behavior will change with temperature.

Activation energy () is a pivotal concept in chemistry and material science, serving as the minimum amount of energy required to initiate a chemical reaction or a diffusion process. In the realm of diffusion, it is the energy barrier that atoms or molecules must overcome to move from one position to another within a substance.
A higher indicates that the particles need to acquire more energy to overcome the barrier, making the diffusion process slower. This energy barrier is influenced by the type of material and the nature of the diffusing species. For instance, the activation energy for the diffusion of copper in silver is relatively high, thus requiring significant thermal energy for the atoms to diffuse effectively.
The activation energy can be determined experimentally by measuring the diffusion coefficient at different temperatures and applying the Arrhenius equation. The calculated value of reveals information about the diffusion mechanism and is indispensable in design and prediction of material behavior at varying temperatures. It is important for students to comprehend activation energy not only as an abstract number but also as an indicator of how challenging it is for a reaction or process, such as diffusion, to occur under different temperature conditions.

Diffusion is inherently a temperature-dependent process, with its rate significantly influenced by temperature variations. As temperature increases, particles within a material gain kinetic energy and are more likely to overcome the activation energy barrier, resulting in higher diffusion rates. This increase in particle motion and diffusion coefficient with temperature is quantifiable by the Arrhenius equation, which defines an exponential relationship between temperature and diffusion.
In practice, as seen in the textbook exercise with copper diffusing in silver, a rise in temperature from (727 degrees Celsius) to (927 degrees Celsius) led to an increase in the diffusion coefficient. The underlying reason is that at higher temperatures, particles vibrate more vigorously, allowing them to break free from their initial positions more readily and migrate through the material.
Understanding the temperature dependence of diffusion is crucial for students, as it is directly applicable to various industrial processes, such as the annealing of metals, the firing of ceramics, and the manufacturing of semiconductor devices. With this knowledge, students can predict how changes in temperature affect the diffusion rate and thereby control and optimize industrial processes to achieve desired material properties.