At temperatures between \(775^{\circ} \mathrm{C}(1048 \mathrm{~K})\) and \(1100^{\circ} \mathrm{C}(1373 \mathrm{~K})\), the activation energy and preexponential for the diffusion coefficient of \(\mathrm{Fe}^{2+}\) in \(\mathrm{FeO}\) are \(102,000 \mathrm{~J} / \mathrm{mol}\) and \(7.3 \times 10^{-8} \mathrm{~m}^{2} / \mathrm{s}\), respectively. Compute the mobility for an \(\mathrm{Fe}^{2+}\) ion at \(1000^{\circ} \mathrm{C}(1273 \mathrm{~K})\).

The Arrhenius equation is fundamental for understanding how temperature influences reaction rates, including diffusion processes. It's expressed as \[D = D_0 \cdot e^{\frac{-E_a}{RT}}\] where

• \(D\) is the diffusion coefficient,
• \(D_0\) is the pre-exponential factor indicating the frequency and orientation of molecular collisions,
• \(E_a\) represents the activation energy required to initiate the diffusion,
• \(R\) is the ideal gas constant, and
• \(T\) is the absolute temperature in Kelvin.

The exponent \( -E_a/(RT) \) illustrates the exponential decrease in diffusion rate with the increase of activation energy or the decrease of temperature, showing how crucial these variables are in the diffusion process. By plugging in the relevant values, students can calculate the diffusion coefficient at any given temperature, providing insights into the mobility of particles under thermal influence.

Activation energy (\(E_a\)) is the required energy barrier that reactants must overcome to transform into products. In the context of diffusion, it's the minimum energy needed for particles to move from one location to another. This concept is central because it determines how easily particles can migrate through materials, like \(Fe^{2+}\) ions in \(FeO\).

The higher the activation energy, the lower the diffusion coefficient at a given temperature, as described by the Arrhenius equation. Knowing the activation energy, which in the case of \(Fe^{2+}\) ions is 102,000 J/mol, allows us to predict how temperature changes will affect their diffusion. The activation energy is a constant for a given material and is significant in estimating reaction rates and understanding the characteristics of a material's structure at an atomic level.

Ion mobility (\(\mu\)) refers to the movement of an ion under the influence of an electric field, which is directly related to the diffusion coefficient (\(D\)) through the equation \[\mu = \frac{qD}{RT}\] where

• \(\mu\) is the mobility,
• \(q\) is the ion charge,
• \(D\) is the diffusion coefficient,
• \(R\) is the gas constant, and
• \(T\) is the temperature in Kelvin.

Ion mobility is a measure of how fast an ion can travel in a certain medium when an electric field is applied. The mobility of ions, such as \(Fe^{2+}\) at 1273 K, tells us how efficient the ions are at conducting current in that material. Higher mobility indicates faster ion movement, which is critical for applications in battery technology and semiconductor devices. The derived mobility, in this case, demonstrates the feasibility of using \(FeO\) in different electrochemical processes.