Consider a diffusion couple composed of two cobalt-iron alloys; one has a composition of \(75 \mathrm{wt} \%\) Co-25 wt\% Fe; the other alloy composition is \(50 \mathrm{wt} \%\) Co-50 wt \(\% \mathrm{Fe}\). If this couple is heated to a temperature of \(800^{\circ} \mathrm{C}(1073 \mathrm{~K})\) for \(20,000 \mathrm{~s}\) determine how far from the original interface into the \(50 \mathrm{wt} \%\) Co-50 wt \% Fe alloy the composition has increased to \(52 \mathrm{wt} \%\) Co-48 wt Fe. For the diffusion coefficient, assume values of \(6.6 \times 10^{-t}\) \(\mathrm{m}^{2} / \mathrm{s}\) and \(247,000 \mathrm{~J} / \mathrm{mol}\), respectively, for the pre exponential and activation energy.

The Arrhenius equation is a formula that provides the temperature dependence of reaction rates, and it is also used in the study of diffusion processes in materials. The equation is expressed as:
\begin{align*}D = D_0 \times e^{-\frac{Q}{RT}}\begin{align*},
where:\begin{itemize}\item \(D\) is the diffusion coefficient, which can vary with temperature.\item \(D_0\) is the pre-exponential (frequency) factor, reflecting the inherent diffusivity of the material.\item \(Q\) is the activation energy for diffusion, representing the energy barrier for an atom to move from one site to another.\item \(R\) is the gas constant, typically \(8.314 \frac{J}{mol \cdot K}\).\item \(T\) is the absolute temperature in Kelvin.\end{itemize}
In the context of alloys, the diffusion coefficient calculated using the Arrhenius equation allows us to understand how quickly atoms or molecules can move through the metal at a given temperature. For example, in our exercise, the given activation energy and pre-exponential factor are used to calculate the diffusion coefficient at \(800^\circ C\), which is crucial for predicting how far the alloying elements have diffused over time.

Fick's Second Law is a fundamental equation describing the diffusion process when the concentration varies with both position and time. The general form of the law can be written as:
\(\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}\),
where:\begin{itemize}\item \(C\) represents the concentration of the diffusing species.\item \(t\) stands for time.\item \(D\) is the diffusion coefficient, indicating how fast diffusion occurs.\item \(x\) is the position in the material.\end{itemize}
For a given initial and boundary condition, this partial differential equation can be solved to give a concentration profile as a function of time and space. In the step-by-step exercise solution, the error function solution to Fick's Second Law is used:
\(C(x, t) = C_0 + (C_s - C_0) \cdot erf\left(\frac{x}{2\sqrt{Dt}}\right)\),
illustrating how the concentration gradients due to diffusion diminish over time and how one can determine the position \(x\) where the concentration reaches a specific value after a set period.

A concentration gradient is the change in the concentration of a substance across a distance. It is a crucial factor in diffusion, as it dictates the direction and rate at which atoms or molecules diffuse. In a general context, particles tend to move from regions of higher concentration to lower concentration, down the concentration gradient, until equilibrium is reached.

In the exercise provided, we have two alloys of cobalt and iron with different compositions, which form a concentration gradient at the interface. Over time, due to the concentration gradient, atoms from regions of high concentration (75 wt% Co in one alloy) will diffuse towards regions of lower concentration (50 wt% Co in the other alloy) when the couple is heated. The diffusion coefficients and Fick's Second Law are then employed to determine how far the composition changes in the alloy with lower cobalt concentration.

The error function, denoted as \(erf\), is a special mathematical function crucial in the solution of the diffusion equations for certain boundary conditions. It arises when solving differential equations associated with normal distributions and random diffusion processes.

In the context of diffusion in alloys, the error function is used to model the smooth transition of concentration from one value to another over a distance. Specifically, it helps find the position where the concentration reaches a certain value after a given time, as shown in the exercise solution:
\(x = 2\sqrt{Dt} \cdot erf^{-1}\left(\frac{C(x, t) - C_0}{C_s - C_0}\right)\),
where \(erf^{-1}\) is the inverse error function. This mathematical tool allows us to calculate how far from the initial interface the concentration of cobalt has changed to the desired percentage in the alloy, thereby providing a clear and quantitative understanding of the diffusion process. It is practical for scientists and engineers in predicting the behavior of diffusing species at different temperatures and timescales.