The diffusion coefficients for silver in copper are given at two temperatures: $$ \begin{array}{cc} \hline T\left({ }^{\circ} \mathrm{C}\right) & D\left(\mathrm{~m}^{2} / \mathrm{s}\right) \\ \hline 650 & 5.5 \times 10^{-16} \\ 900 & 1.3 \times 10^{-13} \\ \hline \end{array} $$ (a) Determine the values of \(D_{0}\) and \(Q_{d}\). (b) What is the magnitude of \(D\) at \(875^{\circ} \mathrm{C}\) ?

The diffusion coefficient, often symbolized as D, is a vital parameter in the study of diffusion processes. It quantifies the rate at which particles, such as atoms or molecules, spread out due to random motion from an area of higher concentration to an area of lower concentration. In materials science, understanding the diffusion coefficient is crucial for predicting how fast a substance will diffuse through another substance.

For instance, when considering how silver atoms move within a copper matrix, the diffusion coefficient tells us not only the speed at which the silver atoms will distribute themselves but also provides insights into the material's behavior at different temperatures. The Arrhenius equation, mentioned in the original exercise, connects the diffusion coefficient to the temperature and activation energy for diffusion. By understanding the values of the coefficient at different temperatures, one can predict the behavior of diffusion at a new temperature.

Activation energy for diffusion, represented by Q_d, is the minimum energy required for atoms or molecules to overcome barriers and move from one location to another within a material. This concept is fundamentally related to the rate of diffusion—which changes with temperature and reflects the inherent resistance to the flow of particles within a substance.

The higher the activation energy, the more energy is needed for the atoms to diffuse. In practice, the activation energy is a critical factor in controlling processes like the hardening of alloys, the sintering of ceramics, and the lifetime of electronic components. The calculated activation energy of 84990 J/mol from the exercise indicates the energy barrier for silver atoms migrating through copper. Recognizing the activation energy provides insight into how robust or fast these diffusion-driven processes can be, especially when fine-tuning materials for specific applications.

Temperature plays a substantial role in diffusion and is typically measured in Kelvin (K) when dealing with scientific equations like the Arrhenius equation. The Kelvin scale is preferred in scientific work because it is an absolute temperature scale with a starting point of zero, which corresponds to absolute zero – the temperature at which molecular motion stops.

To convert Celsius to Kelvin, which is crucial for accurate inputs into the Arrhenius equation, you simply add 273.15 to the Celsius temperature. For example, in the exercise given, the temperatures are converted from degrees Celsius to Kelvin before being substituted into the equation: 650°C becomes 923.15K and 900°C becomes 1173.15K. Without this conversion, the calculations for the diffusion coefficient and activation energy wouldn't be correct, highlighting the importance of temperature units in scientific endeavors.