The diffusion coefficients for carbon in nickel are given at two temperatures: $$\begin{array}{cc}\hline T\left(^{\circ} C\right) & D\left(m^{2} / s\right) \\\\\hline 600 & 5.5\times 10^{-14} \\\700 & 3.9 \times 10^{-13}\end{array}$$.(a) Determine the values of \(D_{0}\) and \(Q_{d}\) (b) What is the magnitude of \(D\) at \(850^{\circ} \mathrm{C} ?\)

Fick's Laws of Diffusion describe how particles spread from an area of high concentration to an area of low concentration. The first law, in particular, is an essential tool in materials science for quantifying the diffusion rate. It can be written as:
\[ J = -D \frac{\partial C}{\partial x} \]
where \( J \) represents the diffusion flux (amount of substance per unit area per unit time), \( D \) is the diffusion coefficient, \( C \) is the concentration, and \( x \) is the position. The negative sign indicates that diffusion occurs in the direction of decreasing concentration.

To further understand its significance, let's relate this to an exercise where we calculate the diffusion coefficient at different temperatures for carbon in nickel. Fick’s law allows us to establish a relationship between the diffusion flux and the concentration gradient, while the diffusion coefficient signifies the ease with which particles can move through a material. Different materials and different conditions, such as temperature, can significantly affect the diffusion coefficient, leading to practical implications in areas like metallurgy and semiconductor manufacturing.

The Arrhenius equation provides a quantitative basis for understanding temperature dependence in diffusion processes and chemical reactions. This critical relationship is expressed as:
\[ D = D_0 e^{-\frac{Q_d}{RT}} \]
where \( D \) is the diffusion coefficient, \( D_0 \) is the pre-exponential factor (a constant), \( Q_d \) is the activation energy for diffusion, \( R \) is the universal gas constant, and \( T \) is the absolute temperature (in Kelvin).

In practical terms, this equation predicts how the diffusion coefficient rises with increasing temperature due to thermal energy overcoming the activation energy barriers for diffusion. This concept has been utilized in our exercise to derive the values of \( D_0 \) and \( Q_d \), enabling us to predict the behavior of diffusion across temperatures. The Arrhenius equation illustrates the profound effect of temperature on material properties and is essential for engineers and scientists designing processes where temperature control is critical.

Activation energy, denoted as \( Q_d \), represents the energy barrier that atoms or molecules must overcome to move from one place to another within a material. In the context of diffusion, it is the energy needed for a particle to break away from its current position and move to a neighboring empty site.

This concept is crucial for understanding material behaviors at the atomic level, especially when we aim to calculate or predict diffusion rates like in our exercise. The higher the activation energy, the slower the diffusion will be, as only a few particles will possess enough thermal energy to move. Conversely, lower activation energy corresponds to more rapid diffusion rates. By calculating \( Q_d \), which we found to be 165.264 kJ/mol in our exercise, we can infer how easily carbon atoms migrate within a nickel matrix at different temperatures. This understanding helps in designing and optimizing industrial processes, including the manufacturing of alloys and the treatment of materials under thermal stress.