(Solved): Please show all work, Thank You!! Consider a diffusion couple composed of two semi-infinite solid ...

Please show all work, Thank You!!

Consider a diffusion couple composed of two semi-infinite solids of the same metal, and that each side of the diffusion couple has a different concentration of the same elemental impurity; assume each impurity level is constant throughout its side of the diffusion couple. For this situation, the solution to Fick's second law (assuming that the diffusion coefficient for the impurity is independent of concentration), is as follows: \( C_{x}=C_{2}+\left(\frac{C_{1}-C_{2}}{2}\right)\left[1-\operatorname{erf}\left(\frac{x}{2 \sqrt{\mathrm{Dt}}}\right)\right] \) The schematic diffusion profile in the figure shows these concentration parameters as well as concentration \( t=0 \) and \( >0 \). Please note that at \( \mathrm{t}=0 \), the \( \mathrm{x}=0 \) position is taken as the initial diffusion couple interface, whereas \( \mathrm{C} 1 \) is the impurity concentration for \( x<0 \), and \( \mathrm{C} 2 \) is the impurity content for \( x>0 \). Schematic concentration profiles in the vicinity of the interface (located at \( x=0 \) ) between two semi-infinite metal alloys before (i.e., \( t=0 \) ) and after a heat treatment (i.e., \( t>0 \) ). The base metal for each alloy is the same; concentrations of some elemental impurity are different \( -C_{1} \) and \( C_{2} \) denote these concentration values at \( t=0 \). A diffusion couple composed of two silver- gold alloys is formed; these alloys have compositions of 98 wt\% Ag-2 wt\% Au and 95 wt\% \( \mathrm{Ag}-5 \mathrm{wt} \% \mathrm{Au} \). Determine the time this diffusion couple must be heated at \( 750^{\circ} \mathrm{C}(1023 \mathrm{~K}) \) in order for the composition to be \( 2.5 \mathrm{wt} \% \) Au at the \( 50 \mu \mathrm{m} \) position into the \( 2 \mathrm{wt} \% \) Au side of the diffusion couple. Preexponential and activation energy values for Au diffusion in \( \mathrm{Ag} \) are \( 8.5 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s} \) and \( 202,100 \mathrm{~J} / \mathrm{mol} \), respectively. hours