The preexponential and activation energy for the diffusion of chromium in nickel are \(1.1 \times 10^{-4} \mathrm{m}^{2} / \mathrm{s}\) and \(272,000 \mathrm{J} / \mathrm{mol},\) respec- tively. At what temperature will the diffusion coefficient have a value of \(1.2 \times 10^{-14} \mathrm{m}^{2} / \mathrm{s} ?\)

The Arrhenius equation is a formula that relates the rate of a chemical reaction to temperature. It's represented by the following expression:
\[ k = A \times \text{exp}\bigg(-\frac{E_a}{RT}\bigg) \]
where:

• \( k \) is the rate constant of the reaction.
• \( A \) is the pre-exponential factor or frequency factor, which is a constant that includes factors like the frequency of collisions and their orientation.
• \( E_a \) is the activation energy required for the reaction.
• \( R \) is the gas constant.
• \( T \) is the temperature in Kelvin.

In the context of diffusion, the Arrhenius equation helps us understand how diffusion coefficients change with temperature, which is particularly useful in materials science for processes like alloying where diffusion is key. When we use the equation to calculate the diffusion coefficient \( D \), the pre-exponential factor \( D_0 \) corresponds to the diffusion coefficient at an infinite temperature where the impact of activation energy is considered negligible.

Activation energy, denoted as \( E_a \), plays a crucial role in chemical reactions and diffusion processes. It is the minimum amount of energy required for reactants to transform into products in a chemical reaction, or for diffusing species to move from one site to another in a material.

• A higher \( E_a \) indicates that the particles need more energy to overcome the energy barrier for the reaction or diffusion to occur.
• At a given temperature, a reaction or diffusion with lower activation energy happens at a faster rate than one with higher activation energy.

In the exercise provided, the activation energy for the diffusion of chromium in nickel is given as 272,000 J/mol – a substantial energy requirement that dictates the temperature at which diffusion occurs at a specified rate. Understanding \( E_a \) is vital for estimating how fast a substance can diffuse through another, influencing the design of materials and the conditions for industrial processes.

Diffusion is inherently dependent on temperature, a relationship that is explained by the Arrhenius equation. As the temperature increases, the diffusion coefficient also increases, leading to a faster diffusion rate. This temperature-dependence can be understood by considering that higher temperatures provide more energy to the particles, allowing them to overcome the activation energy barrier more readily.
In the context of the problem given concerning chromium in nickel, we are asked to calculate the temperature for a certain diffusion coefficient value. The rate at which chromium atoms can mix with the nickel matrix is mostly determined by the temperature, which directly affects the mobility of these atoms. For any material scientists or engineers, understanding this temperature-dependence is essential to control the properties of materials and to ensure the desired outcomes in processes such as hardening, alloying, or annealing metals.